Proof. The exact valuation conditions are finite divisibility inclusion–exclusion: 1_{\{v_\ell(m)=e_\ell\ (\ell\in H)\}} =1_{h_{\bf e}\mid m} \sum_{a\mid M_H}\mu(a) 1_{a\mid m/h_{\bf e}}. After also dividing by d, each summand is a difference of two values of \Psi(\,\cdot\,,y). The fixed factors A,B,h_{\bf e},a alter the smoothness parameter by O_H(1/L), and \rho is Lipschitz on the resulting compact range. Theorem 2.1, with the exact formula below parameter one, therefore gives (2.4) after summing \frac1{h_{\bf e}} \sum_{a\mid M_H}\frac{\mu(a)}a =\delta_{\bf e}. The endpoint floor error is O(1). Since n/d\geq n/y^4=n^{1/9}, it is absorbed by O(n/(dL)). Stieltjes partial summation proves (2.5); division by the case d=1 proves (2.6). ◻

Proof. For r\le200, use the array in Lemma 4.1. For r\ge201, let p(r) be the least prime strictly larger than r, and put x_{r,p(r)}:=\alpha_r, \qquad x_{r,q}:=0\quad(q\ne p(r)). Bertrand’s postulate gives r<p(r)<2r, so p(r) is an allowed cofactor. The row identities (4.3) follow.

It remains to check the prime capacities. A tail row contributes only to the prime p(r). If p and p_- are consecutive primes, the indices for which the least-prime rule routes to p are precisely \begin{equation} p_-\le r\le p-1. \label{eq:least-prime-tail-block} \end{equation} The actual tail contribution is the intersection of this block with r\ge201. For p\le200, there is no tail contribution, while for 200<p\le401, the finite-plus-tail inequality is exactly the third check in Lemma 4.1.

Now let p>401. Every finite cofactor is at most 401, so its p-load is zero. Nagura’s theorem [Nag52], applied to p_->25, gives p<6p_-/5. Hence p-p_-<p_-/5, and \begin{equation} \begin{split} (p-1)\sum_{r=p_-}^{p-1}\alpha_r &< p\,\frac{p-p_-}{2p_-^2}\\ &<\frac{3}{25} <C_0. \end{split} \label{eq:nagura-tail-capacity} \tag{4.5} \end{equation} The first strict inequality uses (r+1)(2r+1)>2p_-^2 throughout the block, and the last one follows directly from the rational value of C_0. Dividing (4.5) by p-1 proves (4.4) for every remaining prime. ◻

Proof. Take R=R(c) as above, set F_{\mathrm{anc}}:=F_R, D:=D_R, and \mathcal H:=\mathcal H_R. Equations (4.19), (4.21), and (4.22) give all assertions. ◻

Proof. At an ordinary scale Q, first choose one donor for every eligible marker, as above, and split the resulting marker–donor pairs into the two orientation pools. Let m_Q^\pm be the numbers of component requests of the two orientations. A path has O(1) components at a given scale, by the literal descent estimate preceding (5.12), and (5.1) gives m_Q^\pm=O(y). On the other hand, each orientation pool has \asymp K_Q pairs. Uniformly for Q\le y, \frac{m_Q^\pm}{K_Q} \ll \frac{y\max(Q,L)}N \le \frac{y^2+yL}{N}=o(1) by (5.7). A greedy assignment therefore uses a fresh pair for every request. Successive ordinary marker intervals are disjoint by the calculation following (5.7), so assignments made at different scales cannot meet.

For a bottom pool, (5.10) supplies (c_i+o(1))N/L marker–donor occurrences, whereas the request count is O(y)=o(N/L). The same greedy construction works separately in each oriented subpool. The four displayed marker intervals are disjoint for large n, and the last ordinary marker interval lies below the first bottom interval, so these choices do not meet any ordinary choice.

Finally, at an ordinary scale the three cofactors q,b,u are y-smooth and P>y, P^2>M, by (5.8); the same statement is immediate from the four fixed bottom rows. Hence every factor in one component has complete signature P with exponent one. In the last two bottom rows the donor and one state coincide by definition, and the construction counts that single occurrence only once. This proves all assertions. ◻

Proof. For a 5\to4 marker P\in(n/3,M/6], one has, for all sufficiently large n, \left\lfloor n/P\right\rfloor=2,\qquad \left\lfloor2n/P\right\rfloor=5,\qquad v_P(C_n)=1. Thus it lies in the row-2 carry interval, and we force its anchor to be 3P. The relevant certificate coordinate is strictly positive: \begin{equation} x_{2,3}=\frac{3432239399}{51940076370}>0. \label{eq:bank-positive-row-two-cell} \end{equation} Likewise a terminal marker P\in(n/2,M/4] belongs to the row-1 carry interval and is assigned the anchor 3P, where \begin{equation} x_{1,3}=\frac{597400199}{51940076370}>0. \label{eq:bank-positive-row-one-cell} \end{equation} Each of these two certificate cells contains x_{r,3}N+o(N)\asymp N marker primes by (4.9). The number forced into either cell is O(y)=o(N), by (5.1); hence the integer partition can accommodate every forced marker. The terminal bank switch is still only 4P\leftrightarrow2P: the anchor 3P uses the central P-occurrence, while either bank state uses its separately backed tail occurrence.

We record explicitly why the fixed prefix meets only O(y) bank components, although the entire bank has O(yL) components. If an ordinary component at scale Q meets a fixed-prefix anchor, equality of the two integers forces equality of their complete rough signatures and hence forces the component marker P to be the anchor prime: for fixed R and large n, its prime is >n/(R+1)>y, whereas its cofactor is at most 2R+1<y. A fixed-prefix anchor has P>n/(R+1), while P\in I_Q\subset(4n/(3Q),3n/(2Q)]. Consequently Q<\frac{3(R+1)}2. There are only O_R(1) geometric scales satisfying this inequality. By the literal descent estimate preceding (5.12), each of the O(y) paths has O(1) components at any one scale. (The stronger assertion “at most one” is neither needed nor generally true for the finite descent table.) Hence ordinary components meeting the prefix number O_R(y). The four bottom moves contribute at most another O(y). Thus \begin{equation} \#\{\text{bank components meeting a fixed-prefix anchor}\} =O_R(y)=O(y). \label{eq:bank-prefix-component-count} \tag{5.15} \end{equation}

At any other fixed prefix row r\ge2, the legal cofactors r+1,\ldots,2r+1 contain at least three integers. At most two are the incident component cores, so choose a third legal cofactor. It need not have had positive limiting allocation weight: by (5.15), adding only O(y)=o(N) occurrences changes every fixed capacity by o(N). Beyond the fixed prefix, a promoted nonsmooth anchor has power-of-two smooth core, whereas the ordinary descent uses non-power cores. The only unavoidable core 4 is covered by the forced row-2 assignment; the 4\to3 and 3\to2 bottom pools have no competing central occurrence.

Distinctness is now literal. A prefix anchor has its unique marker prime P>n/(R+1)>2R+1; different components use different markers, and for a fixed marker we selected a cofactor different from both state cores. Promoted anchors were already mutually distinct, and the power-of-two-core observation separates them from ordinary states.

For each changed occurrence let a_i and b_i be its old and new cofactors. If D is written as the product of its individual cofactor occurrences, deletion of the selected occurrences and insertion of their replacements gives the integer \begin{equation} D':=D\prod_i\frac{b_i}{a_i}. \label{eq:bank-modified-divisor} \end{equation} This notation does not assert a_i\mid b_i: integrality follows because the displayed denominators cancel the corresponding factors in that occurrence-level product. Every b_i lies in a fixed-prefix legal range, so all its prime divisors belong to F_{\mathrm{anc}}. Moreover, by (5.15), for each f\in F_{\mathrm{anc}}, \begin{equation} |v_f(D')-v_f(D)|=O_{F_{\mathrm{anc}}}(y)=o(N). \label{eq:bank-anchor-change-cost} \tag{5.16} \end{equation} Replacing the corresponding anchors changes their product by exactly the same ratio, and hence \begin{equation} D'\mid T_n,\qquad \prod_{a\in\mathcal H'}a=C_nD'. \label{eq:bank-modified-anchor-identity} \tag{5.17} \end{equation} Here divisibility and the surviving positive linear reserve follow from (4.23) and the o(N) loss in (5.16). This proves the lemma. ◻

Proof. The Dickman function is Lipschitz on compact intervals. Extend \rho' by zero to the left of 1. On a fixed compact interval this extension has bounded variation (including its single jump at 1). For a compactly supported function q of bounded variation, \int_{\mathbb R}|q(t+s)-q(t)|\,dt \le \operatorname{Var}(q)|s|. Apply this to the translated copies of \rho' in (6.7), using 0\le\{y^v\}y^{-v}\le1. The moving endpoint is included by the zero extension. This proves the assertion also at the transition u=1. ◻

Proof. Put u=\log x/\log y and d_i=\log t_i/\log y. Theorem 2.1, (6.7), and the exact formula below y give, uniformly across every Dickman transition, \Psi(t_ix,y)=t_ixG_y(u+d_i) +O\!\left(1+x\exp\{-cL^{1/2}\}\right). Subtract t_ixG_y(u), sum, and use Lemma 6.1 and (6.8). This gives (6.9); the exponentially small term is absorbed because x\ge n^{\delta_*}/C. For the coprime version use the exact finite identity \mathbf 1_{(m,P_{\rm hd})=1} =\sum_{d\mid P_{\rm hd}}\mu(d)\mathbf 1_{d\mid m} and apply the first assertion at x/d. This proof retains every endpoint O(1), and differentiates neither \rho nor \Psi at a transition. ◻

Proof. Write b=1-K\kappa, u=\log x/\log y, e=\log d/\log y, and \Delta_e(s):=G_y(s-e)-G_y(s). The fixedness of d, Lemma 6.1, and the triangle inequality give, on the relevant compact parameter interval, \begin{equation} |\Delta_e(s)|\ll_W L^{-1},\qquad |\Delta_e(s)-\Delta_e(t)|\ll_W |s-t|. \label{eq:rough-head-shift-modulus} \tag{6.11} \end{equation} The transition normalization used in the proof of Lemma 6.2, with every endpoint floor retained, therefore gives \begin{align} B_{\rm ph}(x/d)-d^{-1}B_{\rm ph}(x) &=\frac{x}{d}\left[ \alpha\{\Delta_e(u)-b\Delta_e(u+\log b/\log y)\}\right.\notag\\ &\hspace{26mm}\left. +\frac\beta L\{b\Delta_e(u+\log b/\log y) -\tfrac12\Delta_e(u-\log2/\log y)\}\right]\notag\\ &\quad+O_W(xe^{-cL^{1/2}}+1). \label{eq:rough-head-shift-expansion}\tag{6.12}\end{align} Here 1-b=K\kappa=O(L^{-1}) and |\log b|/\log y=O(L^{-2}). Rewrite the first brace as (1-b)\Delta_e(u) +b\{\Delta_e(u)-\Delta_e(u+\log b/\log y)\}. Both of its terms are O_W(L^{-2}) by (6.11); the second brace in (6.12) is O_W(L^{-1}) and already has the prefactor L^{-1}. The exponential error is O(x/L^2) in the stated range. This proves (6.10), including the O(1) endpoint terms. ◻

Proof. Changing the inclusion convention at either endpoint of I changes \Phi(I;y) by at most two, which is absorbed by the claimed error. We may therefore write I=(A,B]. For 1\le j\le4, define the ordered count \begin{equation} \mathcal N_j(I;y):= \#\{(p_1,\ldots,p_j):p_i>y\ {\rm prime},\ p_1\cdots p_j\in I\}. \label{eq:rough-ordered-count} \end{equation}

We first record the reciprocal-prime discrepancy with all constants uniform on the compact range used below. On [1,4.8], put d\mu_y(s):=\sum_{p>y}\frac1p \delta_{\log p/\log y}(ds), \qquad d\lambda(s):=\frac{ds}{s}. For 1\le T\le4.8, partial summation gives \begin{equation} \mu_y([1,T]) =\frac{\pi(y^T)}{y^T}-\frac{\pi(y)}y +\int_y^{y^T}\frac{\pi(v)}{v^2}\,dv. \label{eq:rough-reciprocal-partial-summation} \tag{6.18} \end{equation} Write E(v)=\pi(v)-\operatorname{li}(v). The \operatorname{li}-part of (6.18) is exactly \int_y^{y^T}dv/(v\log v)=\log T. By (2.7), uniformly in T, \begin{align} &\left|\frac{E(y^T)}{y^T}\right| +\left|\frac{E(y)}y\right| +\int_y^{y^{4.8}}\frac{|E(v)|}{v^2}\,dv\notag\\ &\quad\ll e^{-c_0\sqrt{\log y}} +\int_y^{y^{4.8}}e^{-c_0\sqrt{\log v}}\frac{dv}{v} \ll L e^{-c_0\sqrt{(2/9)L}} \ll e^{-c_1\sqrt L}. \label{eq:rough-reciprocal-error-sum} \end{align} Consequently \begin{equation} D_y:=\sup_{1\le T\le4.8} \left|\mu_y([1,T])-\lambda([1,T])\right| \ll e^{-c_1\sqrt L}. \label{eq:exceptional-Stieltjes-PNT} \end{equation} The same bound holds for either convention at the moving endpoint: changing one atom costs at most 1/y\ll e^{-c_1\sqrt L}. In particular, if 1\le a\le4.8 and f has bounded variation on [1,a], Stieltjes integration by parts gives \begin{equation} \left|\int_{[1,a]}f\,d(\mu_y-\lambda)\right| \le2(D_y+y^{-1}) \{\|f\|_\infty+\operatorname{Var}_{[1,a]}f\}. \label{eq:rough-one-dimensional-BV} \tag{6.19} \end{equation}

We next expose every moving endpoint in the ordered count. Fix j, put m=j-1, and write q=p_1\cdots p_m, with the empty product interpreted as one. If B/q\le y, this tuple contributes nothing. Otherwise put a_q=\max(y,A/q). The two nonempty endpoints a_q,B/q lie between y=n^{2/9} and CZ\le C y^{4.6}. Subtracting (2.7) at these exact endpoints gives \begin{align} \#\{p_j>y:qp_j\in(A,B]\} &=\pi(B/q)-\pi(a_q)\notag\\ &=\int_{a_q}^{B/q}\frac{dv}{\log v} +O_C\!\left(\frac Zq e^{-c_2\sqrt L}\right)\notag\\ &=\int_A^B\frac{\mathbf 1_{qy<t}}{q\log(t/q)}\,dt +O_C\!\left(\frac Zq e^{-c_2\sqrt L}\right). \label{eq:exceptional-last-prime}\tag{6.20}\end{align} This is an absolute cumulative-PNT error, not a relative assertion for a short interval. Moreover, \begin{equation} \sum_{\substack{p_1,\ldots,p_m>y\\q<B/y}}\frac1q \le\left(\sum_{y<p\le CZ/y}\frac1p\right)^m \ll_{C,m}1 \label{eq:rough-reciprocal-tuple-mass} \end{equation} by (2.10). Thus the total error in (6.20) is O_{C,j}(Ze^{-c_2\sqrt L}), uniformly in both endpoints.

Summing the main term in (6.20) and interchanging the finite sum with the t-integral gives \begin{equation} \mathcal N_j(I;y) =\int_I S_{j,y}(t)\,dt +O_{C,j}(Ze^{-c_2\sqrt L}). \label{eq:rough-ordered-after-last-prime} \tag{6.21} \end{equation} If u=\log t/\log y, then \begin{equation} S_{j,y}(t)=\frac1{\log y} \int_{\substack{s_1,\ldots,s_m\ge1\\ s_1+\cdots+s_m<u-1}} \frac{d\mu_y(s_1)\cdots d\mu_y(s_m)} {u-s_1-\cdots-s_m}; \label{eq:rough-exact-product-Stieltjes} \tag{6.22} \end{equation} for m=0, the last integral means 1/u. The strict face causes no outer-integral error: for each fixed prime tuple it is a single value of t.

For t\in[Z/C,CZ], the hypotheses on Z imply, for all sufficiently large n depending only on C, \begin{equation} 4.1\le u\le4.7. \label{eq:rough-padded-u-range} \end{equation} Replace the m\le3 copies of d\mu_y in (6.22) one at a time by d\lambda. Conditioned on the other variables, the current variable ranges over [1,a], where a=u-1-\sum_{\ell\ne i}s_\ell, or over the empty set. On a nonempty interval the conditional kernel f(s)=\frac1{u-s-\sum_{\ell\ne i}s_\ell} has denominator at least one, and hence \begin{equation} \|f\|_\infty\le1,\qquad \operatorname{Var}f\le1. \label{eq:rough-conditional-BV} \end{equation} For large n, both \mu_y and \lambda have mass at most M_0:=1+\log4.8 on the relevant compact interval. Therefore the literal m-term telescope and (6.19) give \begin{align} &\left| \int_{\sum s_i<u-1} \frac{d\mu_y(s_1)\cdots d\mu_y(s_m)}{u-\sum s_i} -\int_{\sum s_i\le u-1} \frac{ds_1\cdots ds_m} {s_1\cdots s_m(u-\sum s_i)} \right|\notag\\ &\hspace{35mm} \le4mM_0^{m-1}(D_y+y^{-1}) \ll_j e^{-c_1\sqrt L}. \label{eq:rough-product-measure-telescope}\tag{6.23}\end{align} This includes a moving or degenerate simplex face with one constant.

Define \begin{align} \mathcal K_1(u)&=\frac1u,\notag\\ \mathcal K_j(u)&= \int_{\substack{s_1,\ldots,s_{j-1}\ge1\\ s_1+\cdots+s_{j-1}\le u-1}} \frac{ds_1\cdots ds_{j-1}} {s_1\cdots s_{j-1}(u-s_1-\cdots-s_{j-1})} \quad(2\le j\le4). \label{eq:exceptional-Kj} \end{align} Equations (6.21) and (6.23) prove \begin{equation} \mathcal N_j(I;y) =\frac1{\log y}\int_I \mathcal K_j(\log t/\log y)\,dt +O_{C,j}(Ze^{-c\sqrt L}). \label{eq:rough-ordered-asymptotic} \tag{6.24} \end{equation}

For completeness, the regularity of the kernels is also uniform. For j\ge2, put m=j-1, T=u-j, and \Delta_m=\{z_i\ge0:\sum z_i\le1\}. The substitutions s_i=1+Tz_i give \begin{equation} \mathcal K_j(u)=T^m\int_{\Delta_m} \frac{dz_1\cdots dz_m} {\prod_{i=1}^m(1+Tz_i) \{1+T(1-\sum_i z_i)\}}. \label{eq:rough-kernel-fixed-simplex} \end{equation} On 4.1\le u\le4.7 and 2\le j\le4, one has T\ge0.1 and T\ll1. The integrand and its first T-derivative are bounded on the fixed simplex, so differentiation under the integral proves \begin{equation} \sup_{4.1\le u\le4.7} \{|\mathcal K_j(u)|+|\mathcal K_j'(u)|\}\ll_j1. \label{eq:rough-kernel-C1} \end{equation} The same assertion is immediate for \mathcal K_1.

It remains to pass from ordered tuples to rough integers. Put \Omega(r)=\sum_pv_p(r), and let A_j(I;y) count the r\in I with P^-(r)>y and \Omega(r)=j. Since y^5=n^{10/9}>3n, every integer counted by \Phi(I;y) has 1\le\Omega(r)\le4. If r=\prod_{\nu=1}^kq_\nu^{e_\nu}, \qquad \sum_{\nu=1}^ke_\nu=j, then its exact multiplicity in \mathcal N_j is \begin{equation} \frac{j!}{e_1!\cdots e_k!}. \label{eq:rough-exact-ordered-multiplicity} \end{equation} Thus division by j! gives weight one for a squarefree r, and weight 1/(e_1!\cdots e_k!)\in(0,1] otherwise. Consequently \begin{equation} 0\le\Phi(I;y)-\sum_{j=1}^4\frac{\mathcal N_j(I;y)}{j!} \le\#\{r\in I:q^2\mid r\text{ for some prime }q>y\}. \label{eq:rough-ordered-unordered-discrepancy} \tag{6.25} \end{equation} The last quantity is at most \begin{align} \sum_{\substack{y<q\le\sqrt{CZ}\\q\ {\rm prime}}} \left(\frac{|I|}{q^2}+1\right) &\ll_C\frac{|I|}{y\log y}+\frac{\sqrt Z}{\log Z}\notag\\ &\ll_CZe^{-c\sqrt L}+\sqrt Z. \label{eq:rough-repeated-prime-total} \end{align} Here the first bound follows by partial summation from \pi(t)\ll t/\log t, and the last uses y^{-1}\ll e^{-c\sqrt L}. Combining (6.24) and (6.25) proves (6.17) with \mathcal K(u)=\sum_{j=1}^4\frac{\mathcal K_j(u)}{j!}. ◻

Proof. The first global bounds follow from the construction of the central anchors, |E|=h=O(N), and (5.14). Anchor modification affects O(y) occurrences by the fixed-prefix scale count (5.15). A promoted anchor has trivial rough signature only when its base prime is at most y, which gives O(\pi(y)) possibilities.

A nonsmooth prefix or row-zero anchor has one unique rough marker P>y; a promoted nonsmooth anchor has one complete signature p^{e_p}>1. There is at most one such anchor in a given nontrivial row. Distinct bank components have distinct marker primes by Lemma 5.1, so a nontrivial marker row contains at most one component, its two possible endpoint occurrences, and its one donor occurrence. The set G_{\rm fix} is supported only on exceptional rows. This proves the rowwise assertion.

Finally, a prefix anchor has smooth cofactor supported on F_{\rm anc}\subset[2,W], a row-zero anchor is prime, and a promoted nonsmooth anchor has a power of two as smooth cofactor. None is divisible by a label in (W,y]. Only promoted smooth anchors remain possible in a lower common-multiplier list. The fixed exceptional factors and all withheld donors lie above 2n; bank endpoints contribute O(yL) numbers. This proves the last assertion. ◻

Proof. Before guards, (6.16) supplies at least c_WX_R broad candidates. By Lemma 6.5, at most a fixed number are deleted in an active nontrivial row: at most one anchor, and the endpoints and donor role of at most one bank component. Fixed exceptional factors lie above 2n and occur only in exceptional rows. Since X_R\ge X_0\to\infty, at least c_WX_R/2 clean candidates remain for all sufficiently large n, proving the second assertion in (6.43).

For the first assertion, let \mathcal D_R be exactly the lower coordinates removed from the raw row. Subtracting the definitions of q_R and \mathcal A_R gives the identity \begin{align} &\left(q_R-\sum_{a\in\mathcal A_R}x_a^{\rm raw}\right) -\left(t_R-\sum_{R_y(a)=R}x_a^{\rm raw}\right)\notag\\ &\qquad =-m_R(G_{\rm fix})-m_R(G_{\rm bank}^0) +\sum_{a\in\mathcal D_R}x_a^{\rm raw}. \label{eq:guard-exact-local-ledger} \end{align} Here m_R(G_{\rm fix})=0. If a bank component occurs in the row, its contribution to the right side is -1+\sum_{\substack{a\in\mathcal D_R\\ a\ {\rm is\ one\ of\ its\ lower\ endpoints}}} x_a^{\rm raw}=O_W(1); the expression is not claimed to vanish. The at-most-one anchor contributes another O_W(1). There are no further classes in the census. Therefore \begin{equation} \left|q_R-\sum_{a\in\mathcal A_R}x_a^{\rm raw}\right| \le |\varepsilon_R|+O_W(1), \label{eq:guard-local-ledger-bound} \end{equation} and (6.15) proves the first assertion. Dividing by \#\mathcal P_R\gg_WX_R gives (6.44). Since X_R\ge n^{\delta_*}, this is o(L^{-1}), so a fixed fraction of both the lower and upper broad margin remains. ◻

Proof. We verify the valuation inequality in three disjoint prime ranges.

If p\in F_{\rm anc}, the definition of \gamma above (6.1) gives v_p(T)-v_p(D')\ge\gamma N/p. Equations (6.47) and (6.48), together with A_{\rm exc}\delta_*/\theta<\gamma/4, show that v_p(B_0)<\gamma N/(2p) for all sufficiently large n. Thus the required inequality has a fixed positive margin.

If p\le y but p\notin F_{\rm anc}, then v_p(D')=0, while (6.49) and the definition of \gamma give v_p(T)\ge\gamma N/p. The same two charge estimates again give v_p(B_0)<\gamma N/(2p). This includes p\in F_{\rm hd}\setminus F_{\rm anc}: no anchor valuation is charged at such a prime.

Finally let p>y. Then v_p(D')=0, because F_{\rm anc}\subset[2,W]. The injection (6.41) maps every factor of G_{\rm fix}\cup G_{\rm bank}^0 to a distinct actual factor of E, preserving the entire complete rough signature and hence every p-adic valuation. A fixed exceptional factor is mapped to itself; a bank-base factor is mapped to its designated donor. These two image classes are disjoint by construction. Therefore v_p(B_0)\le v_p(T) for every p>y.

The three ranges cover every prime and prove v_p(D'B_0)\le v_p(T) coordinatewise. This proves both assertions in (6.50). In particular, the conclusion is not obtained by multiplying two separately known divisors of T; it is a simultaneous verification for the product D'B_0. ◻

Proof. The exact valuation pattern, unlike a general residue class, has the finite divisibility identity \begin{equation} \mathbf 1_{\{v_\ell(m)=e_\ell\ (\ell\in H)\}} =\mathbf 1_{h_{\bf e}\mid m} \sum_{a\mid M_H}\mu(a)\mathbf 1_{a\mid m/h_{\bf e}}. \label{eq:marked-head-inclusion-exclusion} \end{equation} Since (d,M_H)=1, it follows exactly that \begin{align} A_{{\bf e},A,B}(d) =\sum_{a\mid M_H}\mu(a) \left\{\Psi\!\left(\frac{Bn}{h_{\bf e}ad},y\right) -\Psi\!\left(\frac{An}{h_{\bf e}ad},y\right)\right\}. \label{eq:marked-cell-Psi-identity}\tag{7.6}\end{align} The smallest unscaled cofactor is \begin{equation} \frac n{y^4}=n^{1/9}=y^{1/2}\longrightarrow\infty. \label{eq:marked-four-cofactor-margin} \tag{7.7} \end{equation} Thus every smooth parameter in (7.6) lies in the fixed compact interval [1/2+o(1),9/2+o(1)]. When it is at least one, apply Theorem 2.1 and (2.3) [HT93, Sai89]; when it is at most one, use the exact identity \Psi(x,y)=\lfloor x\rfloor. The fixed factors A,B,h_{\bf e},a shift the parameter by O_H(1/L), and \rho is Lipschitz on this compact range. Hence, uniformly for C=A,B, \begin{equation} \Psi\!\left(\frac{Cn}{h_{\bf e}ad},y\right) =\frac{Cn}{h_{\bf e}ad} \left\{\rho(U-v(d))+O_H(L^{-1})\right\}+O(1). \label{eq:marked-Psi-specialization} \tag{7.8} \end{equation} By (7.7), the endpoint O(1) is O(n/(dL)). Substitute (7.8) in (7.6) and use \frac1{h_{\bf e}}\sum_{a\mid M_H}\frac{\mu(a)}a =\delta_{\bf e}. This proves (7.2). With d=1, it also gives \begin{equation} \#\mathcal C_{\bf e}(A,B) =\delta_{\bf e}(B-A)n\{\rho(U)+O_W(L^{-1})\} \asymp_W n, \label{eq:marked-cell-size} \tag{7.9} \end{equation} so division proves (7.3).

Apply (7.2) uniformly to every cumulative subinterval (An,tn], A\le t\le B. Its error is O_W(n/(dL)), independent of t. Stieltjes partial summation against \phi proves (7.4). Every cell has the same normalized main divisibility profile, so finite disjoint unions and compact convex mixtures preserve it; this also proves (7.5). ◻

Proof. Put \Omega(m)=\sum_{W<p\le y}v_p(m). Since \Omega(m)\log W\le\log(Bn), the factor e^{B\Omega/L} is uniformly bounded. Moreover, for j=1,2, elementary counting gives \begin{equation} \mathbb E_C\{\mathbf 1_{d\mid m}\Omega(m)^j\} \ll_{W,j}\frac{(1+\log L)^j}{d}. \label{eq:marked-Omega-moments} \tag{7.16} \end{equation} Indeed write m=dk, discard smoothness and the head restrictions, and enlarge to k\le Bn/d. Since d has at most two prime factors, \Omega(dk)\le2+\Omega(k). Expanding the first two powers into prime-power divisibility indicators and using \lfloor X/e\rfloor\le X/e gives \frac1X\sum_{k\le X}\Omega(k)^j \ll_j\left(1+ \sum_{W<r\le y}\sum_{a\ge1}\frac1{r^a}\right)^j \ll_j(1+\log L)^j. Division by the cell size (7.9) proves (7.16).

Taylor’s formula, with the uniform exponential bound just noted, gives \begin{align} \mathbb E_Ce^{S/L} &=1+L^{-1}\mathbb E_CS+O_B((\log L)^2/L^2),\notag\\ \mathbb E_C(\mathbf 1_{d\mid m}e^{S/L}) &=\mathbb P_C(d\mid m)+L^{-1}\mathbb E_C(\mathbf 1_{d\mid m}S) +O_B((\log L)^2/(dL^2)). \label{eq:marked-tilt-Taylor}\tag{7.17}\end{align} It remains to use cancellation in the centered first-order term. Expand S into prime-power indicators. If the score prime r is not a factor of d, Proposition 7.1 applies to dr^a for r^a\le y; the product is then at most y^3. Its main covariance is bounded by \frac1{dr^a} |h_U(v(d)+at_r)-h_U(v(d))h_U(at_r)| \ll_U\frac{at_r}{dr^a}, because the expression vanishes at at_r=0 and h_U is Lipschitz on the relevant compact range. The at most two score primes dividing d contribute O_B(1/d) directly. The marked errors and the tail r^a>y are summable by elementary common-multiple bounds. Since \begin{equation} \sum_{W<r\le y}\sum_{r^a\le y}\frac{at_r}{r^a}=O_W(1), \label{eq:marked-score-summability} \end{equation} we obtain \begin{equation} |\operatorname{Cov}_C(\mathbf 1_{d\mid m},S)|\ll_{B,W}\frac1d. \label{eq:marked-first-order-cancellation} \end{equation} Divide the two lines of (7.17) and use (\log L)^2/L^2=o(1/L). This proves (7.14).

Insert \phi(m/n) into the same calculation. In every leading marked term, (7.4) factors the same Lebesgue average of \phi. More explicitly, the first derivative at zero of the tilted covariance is the third centered combination \begin{align} \kappa_C(\mathbf 1_d,\phi,S) &:={\mathbb E}_C(\mathbf 1_d\phi S) -{\mathbb E}_C(\mathbf 1_d\phi){\mathbb E}_CS -{\mathbb E}_C(\mathbf 1_dS){\mathbb E}_C\phi\notag\\ &\quad-{\mathbb E}_C(\phi S){\mathbb P}_C(d\mid m) +2{\mathbb P}_C(d\mid m){\mathbb E}_C\phi\,{\mathbb E}_CS. \label{eq:marked-physical-three-centering} \end{align} For each prime-power term of S, the common Stieltjes main term in these five summands cancels exactly. Summing the remaining Stieltjes errors gives \frac1L|\kappa_C(\mathbf 1_d,\phi,S)| \ll_{B,W}\frac{\|\phi\|_{BV}\log L}{dL^2}. The three-factor Taylor remainder is O_{B,W}(\|\phi\|_{BV}(\log L)^2/(dL^2)), hence is o(\|\phi\|_{BV}/(dL)). Together with the un-tilted covariance (7.12), this proves (7.15). ◻

Proof. Put \Omega_E(m)=\sum_{\substack{W<r\leq y\\r\notin E}}v_r(m). Writing m=dk, discarding smoothness and the fixed cell restrictions, and expanding one or two powers of \Omega_E into divisibility indicators gives, exactly as an elementary multiple-counting estimate, \begin{equation} \mathbb E_C\{\mathbf 1_{d\mid m}\Omega_E(m)^j\} \ll_{W,j}\frac{(1+\log L)^j}{d}\qquad(j=1,2). \label{eq:marked-omitted-local-moments} \tag{7.19} \end{equation} The endpoint contribution is absorbed because d\leq y^3 and n/y^3=n^{1/3}. Since |S_E|\leq B\Omega_E and e^{B\Omega_E/L}\ll_{B,W}1, Taylor’s formula reduces the desired ratio to bounding the centered first-order term.

Expand S_E as \sum_{r\notin E}\eta_r\sum_{a\geq1}\mathbf 1_{r^a\mid m}. When r^a\leq y, the product dr^a\leq y^4, and Proposition 7.1 gives the main covariance \frac{h_U(v(d)+at_r)-h_U(v(d))h_U(at_r)}{dr^a}. The numerator vanishes at at_r=0 and is O_U(at_r) by Lipschitz continuity. Hence these main terms sum to O_{B,W}(1/d), because \sum_{W<r\leq y}\sum_{r^a\leq y}\frac{at_r}{r^a}=O_W(1). Their marked errors sum to O_{B,W}((1+\log L)/(dL)). For r^a>y, elementary common-multiple counting gives \mathbb P_C(dr^a\mid m)\ll_W\frac1{dr^a}+\frac1n. The reciprocal tails are O_W(1/d); the literal endpoints total O_W(yL/n)=O_W(1/d), since d\leq y^3 and y^4L=o(n). We have therefore proved \begin{equation} |\operatorname{Cov}_C(\mathbf 1_{d\mid m},S_E)|\ll_{B,W}\frac1d. \label{eq:marked-omitted-local-first-order} \end{equation} Using (7.19), the second-order Taylor remainder is O_{B,W}((1+\log L)^2/(dL^2))=o(1/(dL)). Division by the analogous denominator expansion, followed by (7.3), proves (7.18). A product of two divisibility indicators is the indicator of their lcm. Finally, inserting \phi and using (7.4) makes the same Lebesgue average appear in every leading centered term; it cancels, while the Stieltjes errors are bounded by O_{B,W}(\|\phi\|_{BV}/(dL)). This proves all assertions. ◻

Proof. Let \nu be the probability law obtained by tilting the allowed base law \nu_C by e^{S_{\{p,q\}}/L}. If C is an actual cell, \nu_C is its normalized counting measure; if C denotes a finite mixture, \nu_C is the probability mixture defined in (7.1). The density of \nu relative to \nu_C is bounded above and below by constants depending only on B,W. Elementary multiple counting and (7.9) give the following estimate on every actual cell. For a mixture, apply that estimate cellwise to the tilted numerator and use the uniform two-sided bounds for the (cellwise and mixture) normalizing constants. Thus, for every integer D, \begin{equation} \nu(D\mid m)\ll_{B,W}\frac1D+\frac1n. \label{eq:marked-omitted-elementary-bound} \tag{7.26} \end{equation} For D\le y^3 supported on p,q, the stronger formula (7.18) applies.

Proof. Four-mark chamber transfer. We first transfer the structured-cell formula through the complete four-mark range while omitting the forced local primes from the tilt. For a set E of one or two primes in (W,y], put \Omega_E(m)=\sum_{\substack{W<r\le y\\r\notin E}}v_r(m), \qquad S_E(m)=\sum_{\substack{W<r\le y\\r\notin E}}\eta_rv_r(m), and let \nu_E be the probability law obtained by tilting the allowed base law \nu_C by e^{S_E/L}. If D is supported on E, then \Omega_E(Da)=\Omega_E(a). For each actual cell C_i, one has \#C_i\asymp_W n, and elementary multiple counting consequently gives the following bound on C_i, uniformly for D\le y^4. Since the number of cells is fixed, averaging the cellwise expectations with the weights in (7.1) gives exactly the same bound for an allowed finite mixture: \begin{equation} \mathbb E_C\!\left(\mathbf 1_{D\mid m}\Omega_E(m)^j\right) \ll_{W,j}\frac{(1+\log L)^j}{D}\qquad(j=1,2). \label{eq:marked-omitted-four-moments} \tag{7.38} \end{equation} Here are the details needed for the second moment. On writing m=Da and discarding smoothness and the cell restrictions, expand \Omega_E(a)=\sum_{\substack{W<r\le y\\r\notin E}} \sum_{b\ge1}\mathbf 1_{r^b\mid a}. The distinct-prime terms in its square have mean at most A_y^2, where A_y:=\sum_{W<r\le y}\sum_{b\ge1}\frac1{r^b} =\sum_{W<r\le y}\frac1{r-1}\ll_W1+\log L. The same-prime terms have mean at most \sum_{W<r\le y}\sum_{a,b\ge1}\frac1{r^{\max(a,b)}} =\sum_{W<r\le y}\sum_{c\ge1}\frac{2c-1}{r^c} \ll_W1+\log L. This proves (7.38) literally.

We have |S_E|\le B\Omega_E, while \Omega_E(m)\log W\le\log(C_*n) for a fixed physical endpoint C_*. Thus e^{B\Omega_E/L}\ll_{B,W}1 pointwise. Taylor’s formula and (7.38) show that \begin{align*} \mathbb E_C(\mathbf 1_{D\mid m}e^{S_E/L}) &=\mathbb P_C(D\mid m) +O_{B,W}\!\left(\frac{1+\log L}{DL} +\frac{(1+\log L)^2}{DL^2}\right),\\ \mathbb E_Ce^{S_E/L} &=1+O_{B,W}\!\left(\frac{1+\log L}{L} +\frac{(1+\log L)^2}{L^2}\right). \end{align*} The marked formula (7.3), valid uniformly through D\le y^4, may now be inserted before division. We obtain \begin{equation} \nu_E(D\mid m)=\frac{h_U(v(D))}{D} +O_{B,W}\!\left(\frac{1+\log L}{DL}\right) \qquad(D\le y^4, \operatorname{supp}D\subset E). \label{eq:marked-omitted-four-transfer} \tag{7.39} \end{equation} There is no separate endpoint here: n/D\ge n^{1/9}, so the endpoint in the structured-cell count is absorbed by 1/(DL). For arbitrary D, the bounded density ratio and elementary multiple counting give the fallback \begin{equation} \nu_E(D\mid m)\ll_{B,W}D^{-1}+n^{-1}. \label{eq:marked-omitted-four-fallback} \tag{7.40} \end{equation}

We next record the product estimate for the Dickman profile using only the regularity available across its kink. Put H=h_U|_{[0,4]}. The function H is bounded and Lipschitz, H(0)=1, and it is C^{1,1} on a fixed neighborhood of zero. Hence \begin{equation} |H(x+z)-H(x)H(z)|\le C_Kxz \qquad(x,z\ge0,\ x+z\le4), \label{eq:marked-four-product-kernel} \tag{7.41} \end{equation} where C_K depends only on U. Indeed, in a sufficiently small square at the origin the kernel vanishes on both axes and its mixed derivative is bounded. If x is in that small interval and z is outside it, Lipschitz continuity and H(0)=1 give |H(x+z)-H(x)H(z)| \le |H(x+z)-H(z)|+|H(z)|\,|H(x)-1|\ll x\ll xz; the transposed case is identical. When both variables are outside the small interval, boundedness gives O(1)\ll xz. This proves (7.41); in particular, no C^{1,1} assertion on all of [0,4] is being used.

It remains to restore the primes of E exactly. Use the truncated coefficients c_{p,a} from the proof of Lemma 7.4, so that (7.20) is exact on every allowed cell. Fix B,W and take n large enough that \lambda_p\le3/2. For r\ge0, define D_p(r):=\sum_{a\ge1}\frac{|c_{p,a}|}{p^{\max(r,a)}}, \qquad C_p(r):=p^{-r}+D_p(r). Splitting at a=r, using |c_{p,a}|\le(2B/L)\lambda_p^{a-1} and \lambda_p^{A_p}\ll_{B,W}1, and summing the geometric tail gives \begin{align} \sum_{a\ge1}|c_{p,a}|&\ll_{B,W}1, &D_p(0)&\ll_B\frac1{Lp},\notag\\ D_p(r)&\ll_{B,W}\frac{r+1}{Lp^r}\quad(r\ge1), &C_p(r)&\ll_{B,W}p^{-r}\quad(r\ge1), \label{eq:marked-four-local-coefficients} \end{align} and therefore \begin{align} \sum_{r\ge2}D_p(r)&\ll_{B,W}\frac1{Lp^2}, &\sum_{r\ge2}C_p(r)&\ll_{B,W}\frac1{p^2},\notag\\ \sum_{r\ge2}(2r-3)D_p(r)&\ll_{B,W}\frac1{Lp^2}, &\sum_{r\ge2}\frac{2r-3}{p^r}&\ll\frac1{p^2}. \label{eq:marked-four-local-aggregate}\tag{7.42}\end{align} For example, in D_p(r) the two ranges satisfy p^{-r}\sum_{1\le a\le r}|c_{p,a}| \ll_{B,W}\frac r{Lp^r},\qquad \sum_{a>r}\frac{|c_{p,a}|}{p^a} \ll_{B,W}\frac1{Lp^{r+1}}. These estimates also prove the aggregate displays.

Take first E=\{p,q\}. If N_{r,s} is the numerator for p^rq^s\mid m after restoring the two local fugacities, exact expansion gives \begin{equation} N_{r,s}=\sum_{a,b\ge0}c_{p,a}c_{q,b}\, \nu_E\!\left(p^{\max(r,a)}q^{\max(s,b)}\mid m\right), \qquad \mathbb P_{C,S}(p^rq^s\mid m)=\frac{N_{r,s}}{N_{0,0}}. \label{eq:marked-four-exact-restoration} \end{equation} Apply (7.40) to every term other than (a,b)=(0,0). The coefficient sums just proved yield \begin{equation} |N_{r,s}-\nu_E(p^rq^s\mid m)| \ll_{B,W}D_p(r)C_q(s)+p^{-r}D_q(s)+n^{-1}. \label{eq:marked-four-restoration-error} \tag{7.43} \end{equation} The endpoint is really O_{B,W}(n^{-1}), because the unweighted absolute coefficient sums are bounded. Taking r=s=0 shows \begin{equation} N_{0,0}=1+O_{B,W}\!\left(\frac1{Lp}+\frac1{Lq}+\frac1n\right). \label{eq:marked-four-restoration-denominator} \tag{7.44} \end{equation} Thus the local denominator is 1+o_{B,W}(1), not merely bounded by a box-dependent constant.

For clarity, we make the summation implicit in the next displays literal. Put a_n=C_{B,W}\frac{1+\log L}{L},\qquad z_{p,q}=C_{B,W}\left(\frac1{Lp}+\frac1{Lq}+\frac1n\right), where the constants are enlarged once below. Division by (7.44), together with (7.39) and (7.43), gives, whenever rt_p+st_q\le4, \begin{align} &\left|\mathbb P_{C,S}(p^rq^s\mid m) -\frac{H(rt_p+st_q)}{p^rq^s}\right|\notag\\ &\quad\ll_{B,W} \frac{a_n+z_{p,q}}{p^rq^s} +D_p(r)C_q(s)+p^{-r}D_q(s)+\frac1n. \label{eq:marked-four-pointwise-error-ledger} \end{align} The one-prime restoration gives symmetrically \begin{equation} \left|\mathbb P_{C,S}(p^r\mid m)-\frac{H(rt_p)}{p^r}\right| \ll_{B,W}\frac{a_n+z_p}{p^r}+D_p(r)+\frac1n, \qquad z_p\ll_{B,W}\frac1{Lp}+\frac1n. \label{eq:marked-four-marginal-pointwise-ledger} \end{equation} The required geometric sums are \begin{equation} \sum_{r\ge2}\frac1{p^r}=\frac1{p(p-1)},\qquad \sum_{r\ge2}\frac r{p^r}=\frac{2p-1}{p(p-1)^2},\qquad \sum_{r\ge2}\frac{2r-3}{p^r}=\frac{p+1}{p(p-1)^2}; \label{eq:marked-four-geometric-ledger} \tag{7.45} \end{equation} in particular, all three are O(p^{-2}).

Combine (7.39) and (7.42)– (7.45), and sum these pointwise error majorants at fixed p,q before passing to a prime-row norm. There is a quantity \begin{equation} \alpha_{B,W}(n)\ll_{B,W}\frac{1+\log L}{L}, \qquad \alpha_{B,W}(n)\log L\longrightarrow0, \label{eq:marked-four-pair-error-rate} \end{equation} such that, throughout the indicated four-mark ranges, \begin{align} \sum_{\substack{r\ge2\\rt_p+t_q\le4}} \left|\mathbb P_{C,S}(p^rq\mid m) -\frac{H(rt_p+t_q)}{p^rq}\right| &\le\frac{\alpha_{B,W}(n)}{p^2q}+O_{B,W}(L/n),\notag\\ \sum_{\substack{s\ge2\\t_p+st_q\le4}} \left|\mathbb P_{C,S}(pq^s\mid m) -\frac{H(t_p+st_q)}{pq^s}\right| &\le\frac{\alpha_{B,W}(n)}{pq^2}+O_{B,W}(L/n),\notag\\ \sum_{\substack{r,s\ge2\\rt_p+st_q\le4}} \left|\mathbb P_{C,S}(p^rq^s\mid m) -\frac{H(rt_p+st_q)}{p^rq^s}\right| &\le\frac{\alpha_{B,W}(n)}{p^2q^2}+O_{B,W}(L^2/n). \label{eq:marked-four-joint-aggregate}\tag{7.46}\end{align} The one-prime restoration gives in exactly the same way \begin{align} \sum_{\substack{r\ge2\\rt_p\le4}} \left|\mathbb P_{C,S}(p^r\mid m)-\frac{H(rt_p)}{p^r}\right| &\le\frac{\alpha_{B,W}(n)}{p^2}+O_{B,W}(L/n),\notag\\ \left|\mathbb P_{C,S}(q\mid m)-\frac{H(t_q)}q\right| &\le\frac{\alpha_{B,W}(n)}q+O_{B,W}(1/n). \label{eq:marked-four-marginal-aggregate}\tag{7.47}\end{align} By symmetry, the high-power marginal needed in the IJ and JJ products is \begin{equation} \sum_{\substack{s\ge2\\st_q\le4}} \left|\mathbb P_{C,S}(q^s\mid m)-\frac{H(st_q)}{q^s}\right| \le\frac{\alpha_{B,W}(n)}{q^2}+O_{B,W}(L/n); \label{eq:marked-four-symmetric-marginal-aggregate} \tag{7.48} \end{equation} the analogous first-power estimate at p follows by exchanging p,q in the second line of (7.47). All errors in these displays have been accounted for explicitly: (7.39) contributes O_{B,W}((1+\log L)/L) times the reciprocal series, (7.42) contributes O_{B,W}(1/L), and there are only O_W(L), respectively O_W(L^2), endpoint terms.

Let \Gamma_{r,s}:=\operatorname{Cov}_{C,S}(I_{p^r},I_{q^s}),\qquad \mathcal K_{r,s}:= \frac{H(rt_p+st_q)-H(rt_p)H(st_q)}{p^rq^s}. Use |ab-a_0b_0|\le |a-a_0||b|+|a_0||b-b_0| with (7.46)– (7.48). For the JJ marginal error only, enlarge the coupled chamber to the containing Cartesian product. The geometric sums in (7.45) then give the literal covariance error ledger \begin{align} \sum_{\substack{r\ge2\\rt_p+t_q\le4}} |\Gamma_{r,1}-\mathcal K_{r,1}| &\le\frac{\alpha_{B,W}(n)}{p^2q}+O_{B,W}(L/n),\notag\\ \sum_{\substack{s\ge2\\t_p+st_q\le4}} |\Gamma_{1,s}-\mathcal K_{1,s}| &\le\frac{\alpha_{B,W}(n)}{pq^2}+O_{B,W}(L/n),\notag\\ \sum_{\substack{r,s\ge2\\rt_p+st_q\le4}} |\Gamma_{r,s}-\mathcal K_{r,s}| &\le\frac{\alpha_{B,W}(n)}{p^2q^2}+O_{B,W}(L^2/n). \label{eq:marked-four-covariance-error-ledger} \end{align} Finally (7.41) and \sum_{r\ge2}r/p^r\ll p^{-2} show that the complete four-mark chamber contributes at most \begin{align} C\frac{t_pt_q}{p^2q} +\frac{\alpha_{B,W}(n)}{p^2q}+O_{B,W}(L/n),\qquad C\frac{t_pt_q}{pq^2} +\frac{\alpha_{B,W}(n)}{pq^2}+O_{B,W}(L/n),\notag\\ C\frac{t_pt_q}{p^2q^2} +\frac{\alpha_{B,W}(n)}{p^2q^2}+O_{B,W}(L^2/n), \label{eq:marked-four-covariance-aggregate}\tag{7.49}\end{align} in the JI, IJ, and JJ orientations. The constant C depends only on U, and in particular is independent of B,W.

Beyond-four-mark reciprocal tails. We now treat every forced product beyond y^4. The full tilted measure also has bounded density ratio, so \begin{equation} \mathbb P_{C,S}(D\mid m)\ll_{B,W}D^{-1}+n^{-1}. \label{eq:marked-full-tilt-tail-count} \tag{7.50} \end{equation} For JI, if rt_p+t_q>4, then (r-2)t_p>4-t_q-2t_p\ge1, so the reciprocal tail is at most O_{B,W}(Ly^{-1}/(p^2q)). The transpose is identical. For JJ, put E_{r,s}=(r-2)t_p+(s-2)t_q. The case r=s=2 cannot occur. If exactly one exponent exceeds two, say r\ge3,s=2, then rt_p>2 and E_{r,2}=(r-2)t_p>\frac{2(r-2)}r\ge\frac23. If both exceed two, then rt_p+st_q=2(t_p+t_q)+E_{r,s}\le3E_{r,s}, so E_{r,s}>4/3. The JJ reciprocal tail is consequently O_{B,W}(L^2y^{-2/3}/(p^2q^2)). The elementary inequality |\operatorname{Cov}(\mathbf 1_A,\mathbf 1_B)| \le\mathbb P(A\cap B)+\mathbb P(A)\mathbb P(B) and (7.50) also show that all literal endpoints are O_{B,W}(L/n) in JI,IJ and O_{B,W}(L^2/n) in JJ. Thus the complete tail and endpoint ledger is \begin{equation} \begin{array}{c|c|c} &\text{reciprocal tail}&\text{literal endpoint}\\ \hline JI&O_{B,W}(Ly^{-1}/(p^2q))&O_{B,W}(L/n)\\ IJ&O_{B,W}(Ly^{-1}/(pq^2))&O_{B,W}(L/n)\\ JJ&O_{B,W}(L^2y^{-2/3}/(p^2q^2))&O_{B,W}(L^2/n). \end{array} \label{eq:marked-four-tail-endpoint-ledger} \tag{7.51} \end{equation} Together with (7.49), this proves (7.34)–(7.35) with an intermediate pair error \begin{equation} \epsilon^{(0)}_{B,W}(n) \ll_{B,W}\frac{1+\log L}{L}+\frac Ly+\frac{L^2}{y^{2/3}}, \qquad \epsilon^{(0)}_{B,W}(n)\log L\longrightarrow0. \label{eq:marked-pair-error-explicit-rate} \tag{7.52} \end{equation}

For the diagonal, repeat the omitted-score and exact-restoration argument with E=\{p\}. Since J_p^2=\sum_{r\ge2}(2r-3)\mathbf 1_{p^r\mid m}, (7.42) and the one-prime version of (7.39) give, for rt_p\le4, an absolute leading bound C/p^2, an \epsilon^{(0)}_{B,W}(n)/p^2 error, and an O_{B,W}(L^2/n) endpoint. If rt_p>4, then (r-2)t_p>4-2t_p\ge2, so the remaining reciprocal tail is O_{B,W}(L^2y^{-2}/p^2). This proves (7.36) with a leading constant independent of B,W.

Aggregate row norm. Finally, sum the three off-diagonal bounds in the row norm. For fixed p, the contraction uses the following literal finite-sum factorizations (with q=p omitted throughout): p\sum_{q\ne p} \left(\frac{C_{\rm pow}t_pt_q}{p^2q}, \frac{C_{\rm pow}t_pt_q}{pq^2}, \frac{C_{\rm pow}t_pt_q}{p^2q^2}\right) =C_{\rm pow}\left(\frac{t_p}{p}\sum_{q\ne p}\frac{t_q}{q}, t_p\sum_{q\ne p}\frac{t_q}{q^2}, \frac{t_p}{p}\sum_{q\ne p}\frac{t_q}{q^2}\right). The same identities with the t’s removed factor the three error terms. Thus no interchange of an unscaled o(1) with the prime sum is being used. These exact identities give the explicit orientation ledger \begin{align} \mathcal R_{JI}^{\rm main} &\ll \frac{t_p}{p}\sum_q\frac{t_q}{q}, &\mathcal R_{JI}^{\rm err} &\ll \frac{\epsilon^{(0)}_{B,W}(n)}p\sum_q\frac1q,\notag\\ \mathcal R_{IJ}^{\rm main} &\ll t_p\sum_q\frac{t_q}{q^2}, &\mathcal R_{IJ}^{\rm err} &\ll \epsilon^{(0)}_{B,W}(n)\sum_q\frac1{q^2},\notag\\ \mathcal R_{JJ}^{\rm main} &\ll \frac{t_p}{p}\sum_q\frac{t_q}{q^2}, &\mathcal R_{JJ}^{\rm err} &\ll \frac{\epsilon^{(0)}_{B,W}(n)}p\sum_q\frac1{q^2}. \label{eq:marked-row-orientation-ledger} \end{align} The leading terms use \sum_{W<q\le y}\frac{t_q}{q}=O(1),\qquad \sum_{q>W}\frac1{q^2}\ll\frac1W. The only harmonic loss is in the unweighted JI pair error, and (7.52) gives \epsilon^{(0)}_{B,W}(n)\sum_{q\le y}\frac1q \ll\epsilon^{(0)}_{B,W}(n)\log L=o(1). For the endpoints, p\le y, \pi(y)\le y, and the worst JJ term in (7.51) give explicitly p\,\pi(y)\frac{L^2}{n}\le\frac{y^2L^2}{n}=o(1). On the diagonal, I_pJ_p=J_p, 0\le J_p\le J_p^2, and hence |\operatorname{Var}(V_p)-\operatorname{Var}(I_p)| \le2|\operatorname{Cov}(I_p,J_p)|+\operatorname{Var}(J_p) \le3\mathbb E J_p^2. Thus its weighted row contribution is O(1/p)\le O(1/W), with the same box-independent leading constant. With a sufficiently large constant C_{B,W}, define \epsilon_{B,W}(n):=C_{B,W}\left\{ \epsilon^{(0)}_{B,W}(n)\log L+\frac{y^2L^2}{n}\right\}. This single function simultaneously dominates the pair and row remainders and tends to zero. This proves (7.37) and completes the proof. ◻

Proof. The elementary inequality \Omega(m)\log W\leq\log(C_*n) gives |S(m)|/L\leq B/\log W+O_{B,W}(1/L). The physical and head scores in (7.62) are O_{B,W}(1/L). Normalization therefore proves (i). Taylor expansion of the latter two factors has a uniform O_{B,W}(1/L) remainder. The Stieltjes formula makes their leading physical average cancel in centered marked moments, while the common marked profile of the head cells makes the between-cell term cancel. Combining this with Lemma 7.2 and Lemma 7.5, together with (7.60) proves (ii). We shall also use the following component-level consequence of the same calculation. Let \nu_{i,\xi} be the conditional law on the i-th exact component cell before the numerical guards are deleted. Then, uniformly over the finitely many nonzero components and the fixed box, \begin{equation} \sup_{i,i'}\left| \mathbb E_{\nu_{i,\xi}}V_p-\mathbb E_{\nu_{i',\xi}}V_p \right|\ll_{B,W}\frac1{pL} \qquad(W<p\leq y). \label{eq:marked-full-tilt-component-valuation-comparison} \tag{7.64} \end{equation} Indeed (7.11) proves this before tilting. For p^k\leq y^2, the one-local specialization of the omitted-local-score expansion and exact fugacity restoration used in Lemma 7.5 writes every component mean as the same restored local model, with error O_{B,W}\!\left(\frac1{p^kL}+\frac1n\right). Summing k gives O_{B,W}(1/(pL)+L/n)=O_{B,W}(1/(pL)), since pL^2/n\leq yL^2/n=o(1). The omitted powers are o(1/(pL)) by the tail calculation immediately preceding (7.61). A physical tilt is 1+O_B(1/L), and a head tilt is constant on each exact component, so neither changes this comparison. Thus (7.64) is a displayed consequence of (ii), rather than an implicit appeal to a common profile.

We first make the component-weight bookkeeping literal. If the later construction prescribes weights \lambda_i for the individually guarded component laws, put s_i:=\nu_i((\mathcal R_n^{\rm guard})^c),\qquad \widetilde\lambda_i:= \frac{\lambda_i/s_i}{\sum_j\lambda_j/s_j}. Every component has \Theta_W(n) candidates, so 1-s_i\ll_W\frac{yL^A}{n}=o(1) uniformly in the fixed finite component family. Hence all s_i are positive and the weights \widetilde\lambda_i remain in a fixed compact subset of the relative interior whenever the prescribed \lambda_i’s do. The exact measure identity \begin{equation} \left(\sum_i\widetilde\lambda_i\nu_i\right) \left(\,\cdot\mid(\mathcal R_n^{\rm guard})^c\right) =\sum_i\lambda_i \nu_i\left(\,\cdot\mid(\mathcal R_n^{\rm guard})^c\right) \label{eq:marked-guard-component-weight-identity} \end{equation} follows because the conditioned weight of component i on the left is \widetilde\lambda_is_i/\sum_j\widetilde\lambda_js_j=\lambda_i. The identity remains exact after applying and normalizing the same positive exponential tilt on both sides. We may therefore apply the pre-deletion estimates to the compact mixture with weights \widetilde\lambda_i, and recover precisely the prescribed guarded mixture after conditioning. This also covers the simpler case in which the guards are deleted directly from a globally weighted mixture.

Let \nu_\xi denote the resulting pre-deletion tilted law, \widehat\nu_\xi the actual guarded tilted law, and put \delta_\xi:=\nu_\xi(\mathcal R_n^{\rm guard}). Every component cell has \Theta_W(n) candidates, and (i) makes the probability of a single candidate O_{B,W}(1/n). Hence, uniformly on the fixed box, \begin{equation} \delta_\xi\ll_{B,W}\frac{yL^A}{n}=o(1). \label{eq:marked-guard-deleted-mass} \tag{7.65} \end{equation} We use the following elementary conditional-deletion identities. If \mu is a probability law, G has \mu(G)=\delta<1, and \widehat\mu=\mu(\,\cdot\mid G^c), then, for every bounded F, \begin{equation} \mathbb E_{\widehat\mu}F-\mathbb E_\mu F =\frac{\delta\mathbb E_\mu F-\mathbb E_\mu(F\mathbf 1_G)}{1-\delta}, \qquad \left|\mathbb E_{\widehat\mu}F-\mathbb E_\mu F\right| \leq\frac{2\delta}{1-\delta}\|F\|_\infty. \label{eq:marked-conditional-deletion-expectation} \tag{7.66} \end{equation} Writing the covariance difference as \begin{align*} \operatorname{Cov}_{\widehat\mu}(F,H)-\operatorname{Cov}_\mu(F,H) &=\{\mathbb E_{\widehat\mu}(FH)-\mathbb E_\mu(FH)\}\notag\\ &\quad-\{\mathbb E_{\widehat\mu}F-\mathbb E_\mu F\} \mathbb E_{\widehat\mu}H\notag\\ &\quad-\mathbb E_\mu F \{\mathbb E_{\widehat\mu}H-\mathbb E_\mu H\} \end{align*} and applying (7.66) three times gives the explicit recentered bound \begin{equation} \left|\operatorname{Cov}_{\widehat\mu}(F,H) -\operatorname{Cov}_\mu(F,H)\right| \leq\frac{6\delta}{1-\delta}\|F\|_\infty\|H\|_\infty. \label{eq:marked-conditional-deletion-covariance} \tag{7.67} \end{equation} Thus both normalization and the two changes of means are included.

Apply this with \mu=\nu_\xi, G=\mathcal R_n^{\rm guard}, and V_p=v_p(m). On every candidate, for all sufficiently large n, \begin{equation} pV_p(m)\ll yL, \qquad \sum_{W<r\leq y}V_r(m)\leq\frac{\log m}{\log W}\ll_W L. \label{eq:marked-guard-score-envelopes} \tag{7.68} \end{equation} Let E^{\rm guard}_{pr}:= \operatorname{Cov}_{\widehat\nu_\xi}(V_p,V_r) -\operatorname{Cov}_{\nu_\xi}(V_p,V_r). For fixed p, duality with H_b=\sum_r b_rV_r, \|b\|_\infty\leq1, followed by (7.67)– (7.68), gives \begin{align} p\sum_{W<r\leq y}|E^{\rm guard}_{pr}| &=\sup_{\|b\|_\infty\leq1} p\left|\operatorname{Cov}_{\widehat\nu_\xi}(V_p,H_b) -\operatorname{Cov}_{\nu_\xi}(V_p,H_b)\right|\notag\\ &\ll_{B,W}\delta_\xi yL^2 \ll_{B,W}\frac{y^2L^{A+2}}n \leq C_{B,W}\frac{y^2L^{A+3}}n. \label{eq:marked-guard-explicit-prime-row}\tag{7.69}\end{align} The expectation bound in (7.66) proves the analogous one-mark estimate; replacing either V by I_p=\mathbf 1_{p\mid m}, or the second score by a bounded physical or finite-head score, only makes the envelope smaller.

It remains to identify the norm after band aggregation. For \|b\|_\infty\leq1, the covariance-error contribution on band i is bounded exactly by \begin{align*} \frac1{H_i}\left|\sum_{p\in\mathcal P_i}\sum_r E^{\rm guard}_{pr}b_{j(r)}\right| &\leq\frac1{H_i}\sum_{p\in\mathcal P_i}\frac1p \left(p\sum_r|E^{\rm guard}_{pr}|\right)\\ &\leq\sup_u u\sum_r|E^{\rm guard}_{ur}|. \end{align*} Denote this raw normalized band covariance-error operator by E^{\rm guard}_{\rm band}. Thus harmonic averaging is a contraction of the normalized prime-row norm (7.33); the uniformly bounded arithmetic gauge projections do not alter its order. For completeness, the sharp conjugation used later also follows from the displayed relative rate. Put S_n=\operatorname{diag}(\alpha_0,\ldots,\alpha_s). If \|q\|_\infty\leq1, apply the raw row estimate to b_j=\alpha_jq_j, for which \|b\|_\infty\leq1. Dividing output row i by \alpha_i gives \left|\bigl(S_n^{-1}E^{\rm guard}_{\rm band}S_nq\bigr)_i\right| \ll_{B,W}\frac{y^2L^{A+3}}{n\alpha_i} \leq C_{B,W}\frac{y^2L^{A+3}}{n\alpha_0}=o(1). Indeed, once the cells are nonempty, \alpha_0<\delta<\alpha_j for every j\geq1: the low cell is contained in [t_0,\delta], every positive cell lies in (\delta,1], and nonemptiness follows for the fixed finite permitted mesh from the prime number theorem. Thus \alpha_0 is the smallest cell center. The arithmetic quantities H_i,\alpha_i are prime geometry and hence are unchanged by deleting candidates. The gauge projections are uniformly bounded in this sharp norm by (8.41). This proves (7.63). Finally y^2L^{A+3}/n=n^{-5/9}L^{A+3}=o(1/\log L)=o(\alpha_0). ◻

Proof. For a fixed head pattern and sign, let C_{e,\sigma,n} be the smooth members of the corresponding scaled interval with that exact valuation and coprimality pattern, after disjointification and guard deletion. The marked cell estimate gives \Theta_W(n) members before deletion. By Lemma 6.5, the whole excluded set has size O_c(N)+O(yL)=o(n) (and its intersection with the active smooth row is smaller). Hence the same \Theta_W(n) estimate remains after deletion. The strict barycentric margin in (8.7) and the strict two-sided physical split put every normalized assigned mass in a fixed compact subset of the open simplex. This proves (i), (ii), and (8.9).

Since q_n\asymp n/L, formula (8.10) gives the upper bound in (iii); the compact lower mass and the upper candidate-count bound give the asserted zero-head lower bound. These cells lie below 2n-Kh, hence are disjoint from the top component, and all other unavailable coordinates were deleted as guards. The only allowed overlap is with the protected layer, whose weight is itself O_W(1/L). This proves (iv). Finally the barycentric equations are exactly the head equations, the two physical split parameters were chosen to solve the ordinary logarithm, and their weights sum to q_n. These are finite linear identities, proving (v). ◻

Proof. Let \mathcal D:=\{f\in L^\infty(0,1]: f=0\ \hbox{a.e. on }(0,\varepsilon) \hbox{ for some }\varepsilon>0\}. This is dense in \mathcal H=L^2((0,1],dt/t). For f\in\mathcal D, the additive sum S_f=\sum_{x\in\Pi}f(x) has only finitely many nonzero terms. The one- and two-point Palm formulas therefore give, first for f,g\in\mathcal D, \begin{align} \operatorname{Cov}_U(S_f,S_g) &=\int_0^1h(t)f(t)g(t)\,\frac{dt}{t}\notag\\ &\quad+\int_0^1\!\int_0^1K(s,t)f(s)g(t) \,\frac{ds}{s}\frac{dt}{t} =\langle f,Ag\rangle_{\mathcal H}. \label{eq:PD-core-covariance}\tag{8.15}\end{align} Multiplication by h is boundedly invertible on \mathcal H. Also, by (8.11), \int_0^1\!\int_0^1|K(s,t)|^2 \,\frac{ds}{s}\frac{dt}{t}<\infty. Thus the integral part is Hilbert–Schmidt, the bilinear form (8.15) extends continuously from \mathcal D to \mathcal H, and A is the resulting bounded self-adjoint operator. Nonnegativity on the dense core passes to the extension. Since A is an invertible multiplication operator plus a compact operator, it is Fredholm of index zero.

If Af=0, the integral equation and (8.11) give the explicit Cauchy–Schwarz bound \begin{equation} |f(s)|\leq \frac1{\min h} \left(\int_0^1|K(s,t)|^2\frac{dt}{t}\right)^{1/2} \|f\|_{\mathcal H} \ll s\|f\|_{\mathcal H} \label{eq:PD-kernel-pointwise-control} \end{equation} for almost every s. Here \min_{[0,1]}h>0, since U-s\in[7/2,9/2]. Change the representative on a null set so that this holds for every s\in(0,1]. Then \sum_{x\in\Pi}|f(x)|\ll\sum_{x\in\Pi}x=U \qquad\mathbb P_U\hbox{-almost surely}. The same bound makes all one- and two-point second-moment integrals finite. Truncating at x\geq\varepsilon and using dominated convergence extends (8.15) to this particular f. Consequently \operatorname{Var}_U(S_f)=\langle f,Af\rangle_{\mathcal H}=0. Thus S_f is a genuine almost-sure constant: for some c, Z_c(\pi):=\sum_{x\in\pi}f(x)-c=0 \quad\text{for }\mathbb P_U\text{-almost every }\pi. We spell out the Palm disintegration which turns this almost-sure statement into an additive identity. Campbell–Mecke, disintegrated with respect to the total mass, says for every nonnegative measurable \Phi \begin{align} &\mathbb E_U\sum_{r\in\Pi} \Phi(r,\Pi\setminus\{r\}) =\frac1{\rho(U)}\int_0^1 \rho(U-r)\mathbb E_{U-r}\Phi(r,\Pi)\,\frac{dr}{r}, \label{eq:smooth-one-palm-disintegration}\tag{8.16}\\ &\mathbb E_U\sum_{\substack{r,s\in\Pi\\r\ne s}} \Phi(r,s,\Pi\setminus\{r,s\}) =\frac1{\rho(U)}\int_0^1\!\int_0^1 \rho(U-r-s)\mathbb E_{U-r-s}\Phi(r,s,\Pi) \frac{dr}{r}\frac{ds}{s}. \label{eq:smooth-two-palm-disintegration}\tag{8.17}\end{align} Terms with a negative residual mass are interpreted as zero. These formulas follow first with all atoms at least \varepsilon, where the sums are finite, and then by monotone convergence. Apply (8.17) to (Z_c(\pi)+f(r)+f(s))^2 and (8.16) to (Z_c(\pi)+f(u))^2. Their left sides vanish. Since \rho(U-r-s)>0 for r+s\leq1, we obtain, for almost every such pair (r,s), under the same residual law \mathbb P_{U-r-s}, \begin{align*} Z_c(\Pi)+f(r)+f(s)&=0,\\ Z_c(\Pi)+f(r+s)&=0. \end{align*} For the second equality we used the one-point formula at u=r+s; the pullback of its null set under (r,s)\mapsto r+s is null by Fubini. Subtraction gives f(r)+f(s)=f(r+s) for almost every r,s>0 with r+s\leq1. For completeness, the measurable local Cauchy step can be seen by convolution. On each compact interval I\Subset(0,1), choose \psi\in C_c^\infty(0,1-\sup I) with \int\psi=1, and put F_0(x):=\int\{f(x+y)-f(y)\}\psi(y)\,dy. The bound f(t)=O(t) gives local integrability, so translation continuity in L^1_{\rm loc} makes F_0 continuous. Fubini and the almost-everywhere Cauchy identity give F_0=f almost everywhere. If two compact intervals overlap, their continuous representatives agree almost everywhere on the overlap and hence, by continuity, agree everywhere there. They therefore glue to a continuous representative on (0,1). The identity F_0(x+z)=F_0(x)+F_0(z) holds first almost everywhere and then everywhere on each compact subtriangle by continuity. A continuous local additive function is \lambda x; the glued representative shows that the same \lambda applies on all overlapping intervals. Thus f(t)=\lambda t almost everywhere. Conversely this function is in the kernel because the conditioned sum is constant. Since zero is now an isolated eigenvalue of a Fredholm self-adjoint operator, (8.12) follows.

For completeness, put \widetilde K(s,t):=\frac{K(s,t)}t, using at t=0 the continuous value F'(s)-F(s)F'(0). On L^\infty(0,1], Af=M_hf+\mathcal K_\infty f,\qquad (\mathcal K_\infty f)(s) =\int_0^1\widetilde K(s,t)f(t)\,dt. The image of the L^\infty unit ball under \mathcal K_\infty is uniformly bounded and equicontinuous in C[0,1]. Arzelà–Ascoli therefore makes \mathcal K_\infty compact as an operator into L^\infty. Since M_h is invertible, A is Fredholm on L^\infty. If a bounded f lies in its kernel, the integral equation first gives f(t)=O(t), so the preceding L^2 kernel calculation applies and \ker_{L^\infty}A =\operatorname{span}\{t\}. The induced map \overline A:L^\infty/\operatorname{span}\{t\} \longrightarrow\operatorname{Ran}A,\qquad [f]\longmapsto Af, is a bounded bijection between Banach spaces, because a Fredholm range is closed. The quotient norm here is \|[f]\|=\inf_{\lambda\in\mathbb R}\|f-\lambda t\|_\infty. The open mapping theorem gives (8.13).

For the weighted assertion write f(t)=tQ(t). Then \begin{equation} \frac{A(tQ)(s)}s =h(s)Q(s)+\int_0^1\frac{K(s,t)}sQ(t)\,dt. \label{eq:PD-Xt-operator} \tag{8.18} \end{equation} The quotient kernel in (8.18) is even more explicit: \begin{equation} \frac{K(s,t)}s\longrightarrow F'(t)-F'(0)F(t) \quad(s\downarrow0), \label{eq:smooth-weighted-kernel-axis} \tag{8.19} \end{equation} uniformly in t\in[0,1]. Since F\in C^2[0,2], this defines a continuous kernel on the whole compact square. The corresponding conjugated operator (\mathcal BQ)(s):=\frac{A(tQ)(s)}s =h(s)Q(s)+\int_0^1\frac{K(s,t)}sQ(t)\,dt maps the L^\infty unit ball in its integral part into a uniformly bounded, equicontinuous subset of C[0,1], and that part is compact by Arzelà–Ascoli. Thus \mathcal B is an invertible multiplication operator plus a compact operator. Its kernel consists of constants, because Q\mapsto tQ is an isometry from L^\infty onto X_t and tQ\in\ker A exactly when Q is constant. The induced map from L^\infty/\mathbb R, equipped with the quotient norm \inf_{\lambda\in\mathbb R}\|Q-\lambda\|_\infty, to the closed range of \mathcal B is a bounded bijection, and the open mapping theorem gives (8.14). ◻

Proof. The quadratic assertion follows by subtracting the first bound in (8.24). On \mathcal G, factor the projected operator as PD^{-1}(B-E)P =PD^{-1}BP\left\{I- (PD^{-1}BP|_{\mathcal G})^{-1}PD^{-1}EP\right\}. The second factor differs from the identity by an operator of \ell^\infty-norm at most C\varepsilon<1/2, so its Neumann series has norm at most two. This proves the inverse assertion. ◻

Proof. Step 1: the continuum inverse and form gap, uniformly in the moving low cell. This step contains no primes and no tilt parameter. In particular its constants are independent of W,B,n, and of the locations of the positive-cell endpoints subject to (8.20). Write f=J_{\mathcal P}b. Since h(s)=1+O(s) and |K(s,t)|\leq Cst, direct harmonic averaging on I_0=[t_0,\delta] gives \begin{equation} (B_{t_0,\mathcal P}b)_0 =\{1+O(\delta)\}b_0+ O\!\left(\alpha_0^{(c)}\right)\|b\|_\infty; \label{eq:smooth-low-row} \tag{8.27} \end{equation} indeed \frac1{H_0^{(c)}}\int_{I_0}s\,\frac{ds}{s} =\alpha_0^{(c)},\qquad \int_{t_0}^1t|f(t)|\,\frac{dt}{t}\leq\|b\|_\infty. The C^2-regularity of F also gives |h'(s)|\leq C,\qquad |\partial_sK(s,t)|\leq Ct. Hence, on a positive relative cell I_j, the oscillation of A^{(t_0)}f is at most C\eta\|b\|_\infty. These are ordinary supremum-norm estimates and their constants do not involve the growing harmonic mass H_0^{(c)}.

Suppose the projected inverse were not uniform. There would be t_{0,\nu}\downarrow0, permitted partitions, and gauge-fixed vectors b_\nu such that \|b_\nu\|_\infty=1,\qquad \|P_{\alpha,c}B_{t_{0,\nu},\mathcal P_\nu}b_\nu\|_\infty=o(1). By (8.22), for bounded scalars \beta_\nu=O(1), \begin{equation} B_{t_{0,\nu},\mathcal P_\nu}b_\nu =\beta_\nu\alpha^{(c)}+o_{\ell^\infty}(1). \label{eq:smooth-unprojected-compressed-equation} \tag{8.28} \end{equation} More explicitly, applying I-P_{\alpha,c} gives \beta_\nu= \frac{(\alpha^{(c)})^TD_c B_{t_{0,\nu},\mathcal P_\nu}b_\nu} {(\alpha^{(c)})^TD_c\alpha^{(c)}}+o(1). The denominator is bounded below by the cells meeting [1/2,1], while \sum_jH_j^{(c)}\alpha_j^{(c)}=1-t_0 and the compressed operators are uniformly bounded. Hence \beta_\nu=O(1). Formula (8.27) now gives \begin{equation} |b_{\nu,0}|=O(\delta)+o(1). \label{eq:smooth-low-value-control} \tag{8.29} \end{equation} On each positive cell, the oscillation estimate turns (8.28) into A^{(t_{0,\nu})}J_{\mathcal P_\nu}b_\nu(s) =\beta_\nu s+O(\delta+\eta)+o(1). On the low cell both sides are O(\delta)+o(1), by (8.29) and (8.11), so the same estimate holds there. Extend the low value to (0,t_{0,\nu}) by b_{\nu,0}t/t_{0,\nu}. For s\geq t_{0,\nu}, the newly inserted integral term is bounded by \int_0^{t_{0,\nu}}|K(s,t)|\,|b_{\nu,0}| \frac{t}{t_{0,\nu}}\frac{dt}{t} \ll |b_{\nu,0}|s t_{0,\nu}=O(t_{0,\nu}). For 0<s<t_{0,\nu}, the diagonal term is O(|b_{\nu,0}|)=O(\delta)+o(1), while the integral term is O(s); this is already within the error allowed below. Thus, for the resulting bounded function \widetilde f_\nu, \begin{equation} A\widetilde f_\nu=\beta_\nu t+O(\delta+\eta)+o(1) \quad\hbox{in }L^\infty. \label{eq:smooth-full-reconstructed-equation} \tag{8.30} \end{equation} Pairing with the kernel vector t shows \beta_\nu=O(\delta+\eta)+o(1). Then (8.13) puts \widetilde f_\nu within O(\delta+\eta)+o(1) of \lambda_\nu t. Finally, (\alpha^{(c)})^TD_cb_\nu=0 is precisely \int_{t_0}^1tJ_{\mathcal P}b_\nu\,dt/t=0; the linear extension changes this pairing by O(t_0). Hence \lambda_\nu=O(\delta+\eta)+o(1), contradicting \|b_\nu\|_\infty=1 once \delta+\eta is below a fixed threshold. For the quadratic estimate normalize \|b\|_{D_c}=1 and use the same linear extension on (0,t_0). Its extra squared norm is \int_0^{t_0}\left|b_0\frac{t}{t_0}\right|^2\frac{dt}{t} =\frac{|b_0|^2}{2} \leq\frac1{2H_0^{(c)}}=o(1), and its change in the t-pairing is at most |b_0|t_0/2=o(1). If the compressed quadratic gap failed, the quadratic forms, and not only the norms, compare as follows. With f=J_{\mathcal P}b on [t_0,1], harmonic averaging gives the exact identity \begin{equation} b^TD_cB_{t_0,\mathcal P}b =\int_{t_0}^1 f(s)(A^{(t_0)}f)(s)\,\frac{ds}{s}. \label{eq:smooth-compressed-form-identity} \tag{8.31} \end{equation} If e(s)=b_0s/t_0 on (0,t_0), then |K(s,t)|\ll st, Cauchy–Schwarz, and the preceding norm bound give \begin{align} \left|\int_0^{t_0}h(s)e(s)^2\,\frac{ds}{s}\right| &\ll |b_0|^2=o(1),\notag\\ \left|\int_0^{t_0}\!\int_{t_0}^1 K(s,t)e(s)f(t)\,\frac{ds}{s}\frac{dt}{t}\right| &\ll |b_0|t_0\|f\|_{\mathcal H}=o(1),\notag\\ \left|\int_0^{t_0}\!\int_0^{t_0} K(s,t)e(s)e(t)\,\frac{ds}{s}\frac{dt}{t}\right| &\ll |b_0|^2t_0^2=o(1). \label{eq:smooth-low-extension-form-error}\tag{8.32}\end{align} Consequently \begin{equation} \langle\widetilde f,A\widetilde f\rangle_{\mathcal H} -b^TD_cB_{t_0,\mathcal P}b=o(1) \label{eq:smooth-full-compressed-form-comparison} \tag{8.33} \end{equation} under unit D_c-normalization. If the compressed quadratic gap failed, the extended functions would therefore have L^2(dt/t)-norm 1+o(1), asymptotically zero pairing with t, and \langle\widetilde f,A\widetilde f\rangle=o(1). The Poisson–Dickman Poincaré inequality (8.12) would put them o(1)-close to \operatorname{span}\{t\}, while the gauge would force the scalar to be o(1), a contradiction. This proves the continuum quadratic gap uniformly as the low endpoint moves to zero. We have therefore proved, with structural constants C_c<\infty and \kappa_c>0, \begin{align} \left\|\left(P_{\alpha,c}B_{t_0,\mathcal P}P_{\alpha,c} \bigm|_{\mathcal G_c}\right)^{-1}\right\|_{\infty\to\infty} &\leq C_c,\label{eq:smooth-continuum-band-inverse}\tag{8.34}\\ b^TD_cB_{t_0,\mathcal P}b &\geq\kappa_c\,b^TD_cb\qquad(b\in\mathcal G_c). \label{eq:smooth-continuum-band-gap} \end{align} Both constants are uniform for all sufficiently small t_0>0, hence uniform as t_0\downarrow0, and for every permitted regular mesh once w\leq w_0. This is precisely the range used because t_0=(\log W)/(\log y)\to0. These are the continuum results; no arithmetic object has yet been used.

Step 2: arithmetic cell data and normalized operator convergence. We now give the discrete transfer rather than hiding it in a convergence phrase. Put I_p=1_{p\mid m} and \Omega_i^{\rm sf}=\sum_{p\in\mathcal P_i}I_p. For a prime coefficient vector a=(a_p), define the squarefree and full normalized prime operators and band averaging by \begin{align} (\mathsf A^{\rm sf}_{n,\xi}a)_p &:=p\operatorname{Cov}_{\nu_{n,\xi}} \left(I_p,\sum_q a_qI_q\right),\notag\\ (\mathsf A_{n,\xi}a)_p &:=p\operatorname{Cov}_{\nu_{n,\xi}} \left(v_p,\sum_q a_qv_q\right), & (\mathsf E_na)_i&:=\frac1{H_i} \sum_{p\in\mathcal P_i}\frac{a_p}{p}. \label{eq:smooth-prime-band-operators} \end{align} Thus D^{-1}\operatorname{Cov}_\xi(\Omega^{\rm sf},\Omega^{\rm sf}) =\mathsf E_n\mathsf A^{\rm sf}_{n,\xi}J_n, where (J_nb)_p=b_{j(p)}. The squarefree marked formulas give a diagonal multiplier plus a compact kernel and an error E_{pq}: \begin{align} \operatorname{Var}(1_{p\mid m}) &=\frac{h(t_p)}p+E_{pp},\notag\\ \operatorname{Cov}(1_{p\mid m},1_{q\mid m}) &=\frac{K(t_p,t_q)}{pq}+E_{pq}\quad(p\ne q), \label{eq:smooth-arithmetic-kernel}\tag{8.35}\\ \sup_{W<p\le y}\sum_{W<q\le y}p|E_{pq}| &\le \frac{C_{\rm sf}}W+\epsilon^{\rm sf}_{B,W}(n). \label{eq:smooth-normalized-prime-row-error}\tag{8.36}\end{align} Here C_{\rm sf} is absolute and \epsilon^{\rm sf}_{B,W}(n)\to0 for each fixed compact tilt box. The 1/W term contains the squarefree diagonal correction -h(t_p)^2/p^2; equivalently one may include it in the exact diagonal multiplier. The leading functions are the same h,K throughout a compact effective-tilt box: Lemma 7.2 puts the entire tilt dependence in E_{pq}. Thus the continuum kernel is independent of the limiting fugacity. The definition of the box above and \sum_{p>W}v_p/L\leq1/\log W+O(1/L) are exactly what make the constants in this assertion uniform.

The passage from the prime row norm to the band norm is the exact calculation \begin{equation} \left|\frac1{H_i}\sum_{p\in\mathcal P_i}\frac1p \sum_qpE_{pq}b_{j(q)}\right| \leq\frac1{H_i}\sum_{p\in\mathcal P_i}\frac1p \sum_qp|E_{pq}|\,|b_{j(q)}|. \label{eq:smooth-band-normalized-row-transfer} \tag{8.37} \end{equation} Thus the large value H_0\asymp\log L cancels rather than multiplying the error. Mertens’ theorem gives, uniformly cell by cell, \begin{equation} H_0=\log L+O_{W,\delta}(1),\qquad \alpha_0=\frac{\delta+o(1)}{H_0}. \label{eq:smooth-discrete-low-asymptotics} \end{equation}

We record precisely which discrete cell data converge to their continuum counterparts. On every positive cell, quantitative Mertens summation gives \begin{equation} H_j-H_j^{(c)}=o_n(1),\qquad H_j\alpha_j-H_j^{(c)}\alpha_j^{(c)}=o_n(1)\quad(j\geq1). \label{eq:smooth-positive-cell-data-convergence} \tag{8.38} \end{equation} On the moving low cell the first absolute convergence need not hold and is not used: the fixed lower cutoff can leave a bounded Mertens constant. The correct statements are \begin{equation} H_0=H_0^{(c)}+O_W(1),\quad \frac{H_0}{H_0^{(c)}}=1+o_n(1),\quad H_0\alpha_0=\delta-t_0+o_n(1) =H_0^{(c)}\alpha_0^{(c)}+o_n(1). \label{eq:smooth-low-cell-data-convergence} \tag{8.39} \end{equation} Indeed the last relation is partial summation applied to (\log y)^{-1}\sum_{W<p\leq y^\delta}(\log p)/p. It follows, for the fixed finite mesh, that \begin{align} \max_j|\alpha_j-\alpha_j^{(c)}|&=o_n(1),& \max_j\left|\frac{\alpha_j}{\alpha_j^{(c)}}-1\right|&=o_n(1),\notag\\ \alpha^TD\alpha-(\alpha^{(c)})^TD_c\alpha^{(c)}&=o_n(1),& \|P_{\alpha,n}-P_{\alpha,c}\|_{\infty\to\infty}&=o_n(1). \label{eq:smooth-two-projection-convergence}\tag{8.40}\end{align} For the relative assertion, use (8.39) on the low cell; on every positive cell, \alpha_j^{(c)}\gg\delta, so the absolute convergence is already relative. This is exactly the additional information needed in the sharp norm. Put S_n:=\operatorname{diag}(\alpha_0,\ldots,\alpha_s),\qquad S_c:=\operatorname{diag}(\alpha_0^{(c)},\ldots,\alpha_s^{(c)}). Then \begin{equation} \|S_nS_c^{-1}-I\|_{\infty\to\infty}=o_n(1),\qquad \|S_n^{-1}P_{\alpha,n}S_n -S_c^{-1}P_{\alpha,c}S_c\|_{\infty\to\infty}=o_n(1). \label{eq:smooth-sharp-projection-convergence} \tag{8.41} \end{equation} Indeed, if \omega_{n,j}=H_j\alpha_j^2 and \omega_{c,j}=H_j^{(c)}(\alpha_j^{(c)})^2, then S_n^{-1}P_{\alpha,n}S_n =I-\mathbf1\frac{\omega_n^T}{\mathbf1^T\omega_n},\qquad S_c^{-1}P_{\alpha,c}S_c =I-\mathbf1\frac{\omega_c^T}{\mathbf1^T\omega_c}. The positive-cell coordinates of \omega_n-\omega_c tend to zero by (8.38) and the relative center convergence. On the low cell, both coordinates are (\delta-t_0)^2/H_0^{(c)}+o(1)=o(1). Their total masses converge to the same positive limit by (8.40), proving the second operator convergence. Thus the continuum and arithmetic gauges are compared only through quantities which genuinely converge; no assertion \|D-D_c\|\to0 is made at the moving low cell.

For clarity, here is the double prime-band quadrature as a normalized operator calculation. Put (TJ_{\mathcal P}b)(s):=\int_{t_0}^1 K(s,t)(J_{\mathcal P}b)(t)\,\frac{dt}{t}. Let e_k be the step vector supported on one cell. Quantitative Mertens summation and partial summation give, for every positive cell, \sup_{0\leq s\leq1}\left| \sum_{t_q\in I_k}\frac{K(s,t_q)}q -\int_{I_k}K(s,t)\,\frac{dt}{t}\right|=o_n(1). For the moving low cell, split first at a fixed \tau>0. The same estimate applies on [\tau,\delta], whereas |K(s,t)|\ll st makes both the prime sum and the integral over [t_0,\tau] at most Cs\tau. Letting first n\to\infty and then \tau\downarrow0 proves the same conclusion for I_0, uniformly in s. Since the fixed positive-cell mesh has finite dimension, summing over the vectors e_k gives input operator-norm convergence for \|b\|_\infty\leq1.

The output averaging is a second application of the same argument. Define the actual doubly averaged kernel band operator by (\widehat B_n^K b)_i:=\frac1{H_i} \sum_{p\in\mathcal P_i}\frac1p \sum_{W<q\leq y}\frac{K(t_p,t_q)}q b_{j(q)}. Equicontinuity in the first variable and Mertens summation now give \begin{equation} \sup_{\|b\|_\infty\leq1} \left\|\widehat B_n^K b -E_{\mathcal P}TJ_{\mathcal P}b\right\|_\infty=o_n(1), \label{eq:smooth-collective-compact-quadrature} \tag{8.42} \end{equation} for each fixed permitted mesh. On the low output row the unnormalized kernel numerator is O(\delta), while H_0\asymp H_0^{(c)}\asymp\log L; hence the fixed-cutoff Mertens constant is absorbed after normalization. The diagonal multiplier is handled at the same time: \frac1{H_i}\sum_{p\in\mathcal P_i}\frac{h(t_p)}p -\frac1{H_i^{(c)}}\int_{I_i}h(t)\,\frac{dt}{t}=o_n(1). This includes I_0, since h(t)=1+O(t) there. Thus both prime indices and the diagonal, rather than only the input prime index, converge to the compressed continuum operator. The operator constants are mesh-uniform, but the convergence threshold may depend on the one fixed mesh. Equation (8.37) handles the remaining error E.

Fix one permitted positive-cell mesh; only its low endpoint now moves with n. Define the leading arithmetic band operator, before any marked, Bernoulli, prime-power, guard, or nuisance error is added, by \begin{equation} (\widehat B_n^{(0)}b)_i:=\frac1{H_i} \sum_{p\in\mathcal P_i}\frac1p\left\{ h(t_p)b_i+\sum_{\substack{W<q\leq y\\q\ne p}} \frac{K(t_p,t_q)}q\,b_{j(q)}\right\}. \label{eq:smooth-leading-arithmetic-band-operator} \end{equation} The omitted p=q kernel diagonal has operator norm o_n(1): \max_i\frac1{H_i}\sum_{p\in\mathcal P_i} \frac{|K(t_p,t_p)|}{p^2}=o_n(1). Indeed K(t,t)=O(t^2); on I_0 the numerator is O_W((\log y)^{-2}), and on a positive cell every prime tends to infinity. Consequently the preceding double quadrature, the multiplier comparison, and (8.40) imply \begin{equation} \left\|P_{\alpha,n}\widehat B_n^{(0)}P_{\alpha,n} -P_{\alpha,c}B_{t_0,\mathcal P}P_{\alpha,c} \right\|_{\infty\to\infty}=o_{n;W,\mathcal P}(1). \label{eq:smooth-leading-projected-operator-convergence} \tag{8.43} \end{equation} All operators in this display act on the common coordinate space; the two ranges are still different and are identified next.

Put \Phi_n:=P_{\alpha,n}|_{\mathcal G_c}: \mathcal G_c\longrightarrow\mathcal G_n,\qquad \varepsilon_n:=\|P_{\alpha,n}-P_{\alpha,c}\|_{\infty\to\infty}. For x\in\mathcal G_c, \|\Phi_nx-x\|_\infty\leq\varepsilon_n\|x\|_\infty. Hence, if \varepsilon_n<1, \Phi_n is injective; both gauge spaces have codimension one, so it is onto, and \begin{equation} \|\Phi_n\|\leq1+\varepsilon_n,\qquad \|\Phi_n^{-1}\|\leq(1-\varepsilon_n)^{-1}. \label{eq:smooth-ordinary-gauge-isomorphism} \tag{8.44} \end{equation} If y=\Phi_nx\in\mathcal G_n, the same calculation gives the ambient-space estimate \|\Phi_n^{-1}y-y\|_\infty =\|x-P_{\alpha,n}x\|_\infty \leq\frac{\varepsilon_n}{1-\varepsilon_n}\|y\|_\infty. Pulling the left side of (8.43) back by \Phi_n therefore gives the following comparison. Indeed, if \widehat T_n=P_{\alpha,n}\widehat B_n^{(0)}P_{\alpha,n} and T_c=P_{\alpha,c}B_{t_0,\mathcal P}P_{\alpha,c}, then for x\in\mathcal G_c \widehat T_n\Phi_nx-\Phi_nT_cx =(\widehat T_n-T_c)\Phi_nx +T_c(\Phi_n-I)x+(I-P_{\alpha,n})T_cx. The last two terms are O(\varepsilon_n)\|x\|_\infty, because T_cx\in\mathcal G_c and the leading operators are uniformly bounded. Together with (8.44), this gives \begin{equation} \left\|\Phi_n^{-1} \left(P_{\alpha,n}\widehat B_n^{(0)}P_{\alpha,n} \bigm|_{\mathcal G_n}\right)\Phi_n -P_{\alpha,c}B_{t_0,\mathcal P}P_{\alpha,c} \bigm|_{\mathcal G_c}\right\|_{\infty\to\infty}=o_n(1). \label{eq:smooth-pulled-ordinary-operator-convergence} \tag{8.45} \end{equation} The bounds in (8.34) and a Neumann series now establish the leading arithmetic inverse, with norm at most 2C_c, for all sufficiently large n. Only after this inverse has been obtained do we use (8.36), (8.37), and (8.47): choose W_0 so that their box-independent O(1/W) operator norm, after multiplication by the uniformly bounded projections and gauge maps, is at most (8C_c)^{-1}, and then, for the fixed box and mesh, increase n_0 until their vanishing remainders have the same bound. A second Neumann series gives the projected inverse for the actual un-Schur covariance. This order removes any possibility of using an inverse to justify its own leading approximation.

Step 3: quadratic-form convergence on the moving unit sphere. Let b^TDb=1. Then |b_0|\leq H_0^{-1/2},\qquad \sum_{j\geq1}H_j|b_j|^2\leq1. The norm comparison is now literal. On positive cells the dimension is fixed, H_j\gg_{\delta,\eta,c_{\rm mesh}}1, and H_j-H_j^{(c)}=o_n(1). On the low cell, |H_0-H_0^{(c)}|\,|b_0|^2 \ll_W H_0^{-1}=o_n(1). Thus \|J_{\mathcal P}b\|_{L^2([t_0,1],dt/t)}^2=1+o_n(1). The extension on (0,t_0) has squared norm |b_0|^2/2=o_n(1), so the full reconstructed norm is 1+o_n(1). The same decomposition, with one factor t, shows that an arithmetic gauge vector has continuum t-pairing o_n(1).

We next compare the leading forms, splitting both prime indices into I_0 and the positive cells. The positive–positive block converges uniformly on the present unit sphere by the fixed finite-dimensional two-index quadrature. For a block with one low index, |K(s,t)|\leq Cst, Cauchy–Schwarz on the positive cells, and \sum_{p\in\mathcal P_0}t_p/p=O(\delta) give |\mathscr K_{0,+}(b)| \ll |b_0|\left(\sum_{p\in\mathcal P_0}\frac{t_p}{p}\right) \sum_{j\geq1}|b_j| \sum_{q\in\mathcal P_j}\frac{t_q}{q} \ll H_0^{-1/2}=o_n(1). Here the last sum is uniformly bounded by Cauchy–Schwarz: \sum_{j\geq1}H_j\alpha_j|b_j| \leq\left(\sum_{j\geq1}H_j|b_j|^2\right)^{1/2} \left(\sum_{j\geq1}H_j\alpha_j^2\right)^{1/2} \ll1. The identical bound holds for the continuum low–positive block. The two low–low blocks are each O\!\left(|b_0|^2 \left(\sum_{p\in\mathcal P_0}\frac{t_p}{p}\right)^2\right) +O\!\left(|b_0|^2 \left(\int_{I_0}dt\right)^2\right) =O(H_0^{-1})=o_n(1). For the multiplier part, write h=1+(h-1). The constant term leaves only (H_0-H_0^{(c)})b_0^2=o(1) on the low row, and the remainder is handled by weighted Mertens summation because (h(t)-1)/t extends continuously to t=0. The positive multiplier rows converge by ordinary cellwise quadrature. Finally, the removed discrete p=q kernel diagonal is o_n(1) on the unit sphere by the estimate preceding (8.43). Combining these four estimates with (8.31)– (8.33) proves, uniformly for b^TDb=1, \begin{equation} b^TD\widehat B_n^{(0)}b- \langle\widetilde f,A\widetilde f\rangle_{\mathcal H}=o_n(1), \label{eq:smooth-discrete-full-form-comparison} \tag{8.46} \end{equation} where \widetilde f is the linear low extension of J_{\mathcal P}b. This explicitly proves both the norm and quadratic-form convergence needed for the Poincaré gap; no equivalence constant depending on H_0 has been used. Indeed, if in addition b\in\mathcal G_n, then the norm and gauge comparisons above give \|\widetilde f\|_{\mathcal H}^2=1+o_n(1),\qquad \langle\widetilde f,t\rangle_{\mathcal H}=o_n(1). Since \|t\|_{\mathcal H}^2=1/2, the exact Hilbert-space projection formula yields \inf_{\lambda\in\mathbb R} \|\widetilde f-\lambda t\|_{\mathcal H}^2 =\|\widetilde f\|_{\mathcal H}^2 -\frac{|\langle\widetilde f,t\rangle_{\mathcal H}|^2} {\|t\|_{\mathcal H}^2} =1+o_n(1). Thus (8.12) and (8.46) give, for all sufficiently large n, \begin{equation} b^TD\widehat B_n^{(0)}b\geq\frac{\kappa_0}{2}\,b^TDb \qquad(b\in\mathcal G_n). \label{eq:smooth-leading-arithmetic-form-gap} \end{equation} To transfer every remaining arithmetic error literally, let R=(R_{ij}) be any symmetric band covariance error satisfying \sup_i\frac1{H_i}\sum_j|R_{ij}|\leq\varepsilon. The normalized operator D^{-1}R is self-adjoint for the D-inner product, and \begin{align*} |b^TRb| &\leq\frac12\sum_{i,j}|R_{ij}|(|b_i|^2+|b_j|^2)\\ &=\sum_i|b_i|^2\sum_j|R_{ij}|\\ &\leq\varepsilon\,b^TDb. \end{align*} Thus its D-Hilbert operator norm is at most \varepsilon. Applying this estimate to the squarefree, prime-power, and quadrature errors transfers the continuum quadratic gap to the prime-band covariance form.

Replacing 1_{p\mid m} by v_p(m) adds \begin{equation} \sup_p\sum_qp|E^{\rm pow}_{pq}| \le \frac{C_{\rm pow}}W+\epsilon_{B,W}(n) \label{eq:smooth-power-row-transfer} \tag{8.47} \end{equation} by (7.56). The constants C_{\rm sf},C_{\rm pow} are known before the compact ODE box is chosen. This is the estimate invoked only after the leading inverse in (8.45) was established. The already chosen W\geq W_0 makes its 1/W contribution smaller than the reserved continuum inverse and form gaps; after a fixed box is specified, only the threshold for n is increased to absorb the vanishing remainder.

Step 4: the finite nuisance block and its Schur perturbation. It remains to justify the joint Schur complement in (8.23). Every head pattern retains a fixed mass in each of two separated positive-length physical subpools. If a centered linear combination aR+\sum_\nu b_\nu H_\nu^\circ vanished, varying R inside either physical subpool forces a=0, and the affinely independent simplex head patterns then force every b_\nu=0. The fixed mass and separation margins make this quantitative: \begin{equation} \Gamma_\xi\geq\gamma_W I \label{eq:smooth-joint-head-physical-gap} \tag{8.48} \end{equation} throughout every subsequently fixed compact tilt box, with \gamma_W chosen before the radius of that box. We prove this uniformity directly; no convergence of the baseline mixture weights is required. By Lemma 8.1, there is a number \lambda_W>0 such that \lambda_{e,\sigma}(n)\geq\lambda_W for every head pattern e, both signs \sigma, and all sufficiently large n. Moreover \frac{\inf I_+(n)}n-\frac{\sup I_-(n)}n=a_+-b_->0, so the corresponding ranges of R=\log(m/n) have a fixed positive separation.

We first prove a finite-n, uniform baseline bound. For F=aR+\sum_\nu b_\nu H_\nu^\circ, compare, within each fixed head pattern, its conditional means on the two physical cells. The head term is identical on those cells, whereas their R-means are separated by at least \log a_+-\log b_->0. The elementary two-point variance bound and the lower bound \lambda_W therefore give \operatorname{Var}_{\nu_n}(F)\geq c_{R,W}a^2. On the other hand the finitely many head-pattern vectors are affinely independent after the constant indicator has been deleted. Their masses are all at least 2\lambda_W, so, uniformly in n, \operatorname{Var}_{\nu_n}\!\left(\sum_\nu b_\nu H_\nu^\circ\right) \geq c_{H,W}\sum_\nu b_\nu^2. Since R is uniformly bounded, \operatorname{Var}_{\nu_n}\!\left(\sum_\nu b_\nu H_\nu^\circ\right) \leq2\operatorname{Var}_{\nu_n}(F) +2a^2\operatorname{Var}_{\nu_n}(R), and the preceding a^2-bound absorbs the last term, giving \operatorname{Var}_{\nu_n}\!\left(\sum_\nu b_\nu H_\nu^\circ\right) \leq C_W\operatorname{Var}_{\nu_n}(F). The head-pattern lower bound now yields \operatorname{Var}_{\nu_n}(F)\geq(c_{H,W}/C_W)\sum_\nu b_\nu^2. Combining this with the a^2-bound, one may choose \gamma_W>0, independently of n and of every later tilt box, such that \Gamma_{0,n}\geq2\gamma_W I \qquad(n\geq n_0(W)).

It remains only to compare each tilted covariance with its own finite-n baseline. If S_\xi=\sum_j\eta_j\Omega_j is the medium-prime part of the score, then |S_\xi|\leq B\Omega, e^{|S_\xi|/L}\ll_{B,W}1, and the one-mark estimates give \mathbb E_{\nu_n}\Omega\ll_W\log L. Consequently \sup_{\xi\ \text{ in the fixed }B\text{-box}} \mathbb E_{\nu_n}\left|e^{S_\xi/L}-1\right| \ll_{B,W}\frac{\log L}{L}=o(1). The physical and finite-head factors are 1+O_{B,W}(L^{-1}), and normalization changes the same estimate by only a constant factor. Thus \sup_{\xi\ \text{ in the fixed }B\text{-box}} \|\nu_{n,\xi}-\nu_n\|_{\rm TV}=o_{n;B,W}(1). Every coordinate of Z is uniformly bounded and its dimension depends only on W. If d_W=\dim Z and \max_\nu\|Z_\nu\|_\infty\leq M_W, then for every Euclidean unit vector a, \left|\operatorname{Var}_{\nu_{n,\xi}}(a^TZ) -\operatorname{Var}_{\nu_n}(a^TZ)\right| \leq 6d_WM_W^2\|\nu_{n,\xi}-\nu_n\|_{\rm TV}. Taking the supremum over a gives \sup_{\xi\ \text{ in the fixed }B\text{-box}} \|\Gamma_{\xi,n}-\Gamma_{0,n}\|_{\mathrm{op}} =o_{n;B,W}(1). For each fixed B, increasing only n_0(B,W) now proves (8.48). Its constant \gamma_W was chosen from the uniform finite-n geometry before B, so no unproved limiting mixture and no box-dependent main constant is involved. The constant score is absent: normalization has removed it, and one centered head indicator was deleted before \Gamma_\xi was formed. The head cross rows are O_{B,W}(1/L) throughout the fixed box, and the physical cross row is given by the Stieltjes estimate (7.61). Write U_\xi:=\operatorname{Cov}_\xi(\Omega,Z),\qquad C_\xi:=\operatorname{Cov}_\xi(\Omega,\Omega). The actual block elimination is \begin{equation} q_n^{-1}\mathcal C^Z_\xi =C_\xi-U_\xi\Gamma_\xi^{-1}U_\xi^T. \label{eq:smooth-displayed-joint-block-elimination} \end{equation} The marked head/physical bounds give, for \|b\|_\infty\le1, |(U_\xi^Tb)_\nu| \leq\sum_j|(U_\xi)_{j\nu}|\,|b_j| \ll_{B,W}\frac1L\sum_jH_j \ll_{B,W}\frac{\log L}{L}. Together with \|\Gamma_\xi^{-1}\|_{\rm op}\leq\gamma_W^{-1} this gives, row by row, \begin{equation} \left\|D^{-1}U_\xi\Gamma_\xi^{-1}U_\xi^Tb\right\|_\infty \ll_{B,W}\frac{\log L}{L^2}, \label{eq:smooth-joint-Schur-row-error} \tag{8.49} \end{equation} because |(U_\xi)_{i\nu}|\ll_{B,W}H_i/L and \sum_jH_j\ll\log L. This is the promised normalized row estimate, with an explicit rate.

Step 5: the exact physical direction and the full quotient form. The physical column is nevertheless responsible for the exact null relation, and we record it rather than discarding it as a small error. Let T_y:=\sum_{W<p\le y}t_pv_p,\qquad S_\alpha:=\sum_j\alpha_j\Omega_j,\qquad S_g:=\sum_{W<p\le y}(\alpha_{j(p)}-t_p)v_p. For a statistic X, put X^{\circ,\xi}:=X-\mathbb E_{\nu_{n,\xi}}X. This notation is distinct from the already defined baseline-centered head scores H_e^\circ; changing their centering subtracts only a constant and therefore leaves the centered covariance span unchanged. Every member of every active structured cell is y-smooth, so its logarithm is the sum of the p\leq y valuation terms. Moreover, for each p\leq W, the function v_p is constant on every exact head-pattern cell and hence is a linear combination of the retained head indicators and the deleted constant indicator. After centering, the constant disappears and the whole p\leq W term lies in the centered head span. Therefore, identically at finite n, \begin{equation} T_y^{\circ,\xi}= \frac{R^{\circ,\xi} -\sum_{p\le W}(\log p)v_p^{\circ,\xi}}{\log y} \in\operatorname{span}Z^{\circ,\xi},\qquad S_\alpha^{\circ,\xi}=T_y^{\circ,\xi}+S_g^{\circ,\xi}. \label{eq:smooth-physical-alpha-block-relation} \tag{8.50} \end{equation} Let P_{Z^\perp} denote orthogonal projection in L^2(\nu_{n,\xi}) onto the orthogonal complement of the centered coordinates of Z. Thus P_{Z^\perp} kills T_y^{\circ,\xi} exactly. More generally, for b=q+\lambda\alpha, with q\in\mathcal G_n, and for every \mu\in\mathbb R, \begin{equation} P_{Z^\perp}\{(b\cdot\Omega)^{\circ,\xi}\} =P_{Z^\perp}\left\{ \left(\sum_{W<p\leq y} (b_{j(p)}-\mu t_p)v_p\right)^{\circ,\xi}\right\}. \label{eq:smooth-exact-arithmetic-null-relation} \tag{8.51} \end{equation} This is the precise finite-n post-Schur relation used below. We next record its arithmetic geometry. Define \begin{equation} V_n:=\sum_{W<p\leq y}\frac{(\alpha_{j(p)}-t_p)^2}{p},\qquad \mathfrak d_n(b)^2:=\inf_{\mu\in\mathbb R} \sum_{W<p\leq y}\frac{|b_{j(p)}-\mu t_p|^2}{p}, \qquad A_{\alpha,n}:=\alpha^TD\alpha. \label{eq:smooth-arithmetic-physical-distance} \end{equation} Since \sum_{p\in\mathcal P_j}(\alpha_j-t_p)/p=0 in every band, and since q^TD\alpha=0, the following identities are exact. For all sufficiently large n, every permitted band has positive harmonic mass and \alpha_j>0; hence A_{\alpha,n}>0, while V_n\geq0. In particular the denominator in the third line below is nonzero: \begin{align} \mathfrak d_n(q+\lambda\alpha)^2 &=\inf_{\mu\in\mathbb R}\left\{ \|q+(\lambda-\mu)\alpha\|_D^2+\mu^2V_n\right\}\notag\\ &=\|q\|_D^2+\inf_{\mu\in\mathbb R}\left\{ (\lambda-\mu)^2A_{\alpha,n}+\mu^2V_n\right\}\notag\\ &=\|q\|_D^2+\lambda^2 \frac{A_{\alpha,n}V_n}{A_{\alpha,n}+V_n} \geq\|q\|_D^2. \label{eq:smooth-quotient-after-physical-geometry}\tag{8.52}\end{align} We also record here the size of V_n, independently of the later regression argument. Weighted Mertens summation and (8.40) give V_n=\sum_j\int_{I_j}|t-\alpha_j^{(c)}|^2\,\frac{dt}{t} +o_n\bigl((\delta+\eta)^2\bigr). On I_0=[t_0,\delta], the displayed integral is exactly \frac{\delta^2-t_0^2}{2} -\frac{(\delta-t_0)^2}{\log(\delta/t_0)} \asymp\delta^2. On a positive cell I=[a,a+\ell], it is \asymp\ell^3/a; summing the regular relative mesh gives \asymp\eta^2. Consequently, uniformly over permitted meshes, \begin{equation} V_n\asymp\delta^2+\eta^2 \asymp(\delta+\eta)^2. \label{eq:smooth-arithmetic-cell-variance-size} \end{equation}

We now transfer this exact distance to the covariance form; this step is needed because an arithmetic center is not a continuum center. If \mathfrak d_n(b)>0, let \mu_* be its unique minimizer and put \begin{equation} c_p:=b_{j(p)}-\mu_*t_p. \label{eq:smooth-minimizing-prime-coefficients} \tag{8.53} \end{equation} Differentiation of the finite sum gives the exact normal equation \begin{equation} \sum_{W<p\leq y}\frac{t_pc_p}{p}=0. \label{eq:smooth-arithmetic-physical-normal-equation} \tag{8.54} \end{equation} Normalize for the moment so that \mathfrak d_n(b)=1. Equation (8.52), together with A_{\alpha,n}\asymp1 and V_n\asymp(\delta+\eta)^2, gives \|q\|_D\leq1,\qquad |\lambda|+|\mu_*|\ll_{\delta,\eta}1. In particular |q_0|\leq H_0^{-1/2}, \alpha_0\ll_{\delta}H_0^{-1}, and t_0\to0.

Define the piecewise affine continuum coefficient f_n(t):=b_j-\mu_*t\qquad(t\in I_j), and extend it on (0,t_0) by f_n(t_0)t/t_0; denote the resulting function by \widetilde f_n. The cellwise Mertens argument in (8.38)– (8.39) gives \begin{equation} \|\widetilde f_n\|_{\mathcal H}^2=1+o_n(1),\qquad \langle\widetilde f_n,t\rangle_{\mathcal H}=o_n(1). \label{eq:smooth-minimizer-continuum-norm-gauge} \tag{8.55} \end{equation} Here the second relation is (8.54) after quadrature. For clarity about the moving low cell, the only term that can retain the fixed-cutoff Mertens constant is |b_0|^2(H_0-H_0^{(c)})=O_W(|b_0|^2)=o_n(1); every term carrying a factor t converges by weighted Mertens summation. Also |f_n(t_0)|=o_n(1), so the linear extension has o_n(1) squared norm and o_n(1) t-pairing. Thus no convergence D\to D_c has been assumed.

For prime coefficients c=(c_p), write the leading discrete form as \begin{equation} \mathscr Q_n^{(0)}(c):= \sum_{W<p\leq y}\frac{h(t_p)c_p^2}{p} +\sum_{\substack{W<p,r\leq y\\p\ne r}} \frac{K(t_p,t_r)c_pc_r}{pr}. \label{eq:smooth-leading-minimizer-form} \end{equation} The continuum double integral contains its diagonal as a measure-zero set, whereas the finite prime double sum in the quadrature temporarily contains p=r. The difference is \begin{equation} \Delta_{\rm diag}(c):= \sum_{W<p\leq y}\frac{K(t_p,t_p)c_p^2}{p^2}=o_n(1) \qquad\bigl(\mathfrak d_n(b)=1\bigr). \label{eq:smooth-minimizer-kernel-diagonal} \end{equation} To verify this uniformly, on the positive cells p\geq y^\delta, so \sum_{t_p>\delta}\frac{|K(t_p,t_p)|c_p^2}{p^2} \ll y^{-\delta}\sum_{t_p>\delta}\frac{c_p^2}{p}=o(1). On I_0, c_p=b_0-\mu_*t_p, where |b_0|=O(H_0^{-1/2}) and \mu_*=O_{\delta,\eta}(1). Since \sum_{p>W}\frac{t_p^2}{p^2} =O_W((\log y)^{-2}),\qquad \sum_{p>W}\frac{t_p^4}{p^2} =O_W((\log y)^{-4}), and K(t,t)=O(t^2), the low-cell contribution is o(1) as well. The two-index quadrature proved in (8.42)– (8.46) applies also to f_n=J_{\mathcal P}b-\mu_*t: the step part is covered by the displayed operator comparison, while the t-part follows from the same Stieltjes summation, and its coefficient is bounded under the present normalization. Together with (8.55), it gives \mathscr Q_n^{(0)}(c) =\langle\widetilde f_n,A\widetilde f_n\rangle_{\mathcal H}+o_n(1). All o_n(1) terms in these comparisons are uniform on the normalized class \mathfrak d_n(b)=1: q is bounded in D-norm, \lambda,\mu_*=O_{\delta,\eta}(1), the positive-cell dimension is fixed, and the low coordinate tends to zero. In particular, the resulting threshold for n does not depend on b. The Poincaré inequality (8.12) now applies: the two relations in (8.55) say that the squared distance of \widetilde f_n from \operatorname{span}\{t\} is 1+o_n(1). Hence, by homogeneity, there is a mesh-uniform \kappa_1>0 such that, for every b and all sufficiently large n, \begin{equation} \mathscr Q_n^{(0)}(c) \geq\kappa_1\mathfrak d_n(b)^2 \label{eq:smooth-arithmetic-physical-Poincare-transfer} \tag{8.56} \end{equation} when c is defined by (8.53). If \mathfrak d_n(b)=0, then (8.52) gives b=0, so the assertion is trivial.

It remains to verify that every arithmetic error is relative to this same minimizing distance. If E_{pr}=E_{rp} is any prime covariance error with \sup_p\sum_rp|E_{pr}|\leq\varepsilon, symmetry and 2|c_pc_r|\leq c_p^2+c_r^2 give the exact estimate \begin{equation} \left|\sum_{p,r}E_{pr}c_pc_r\right| \leq\varepsilon\sum_p\frac{c_p^2}{p} =\varepsilon\mathfrak d_n(b)^2. \label{eq:smooth-minimizer-prime-row-error} \tag{8.57} \end{equation} The proof of Corollary 7.6, before band averaging, gives the corresponding prime-level guard estimate \begin{equation} \sup_{W<p\leq y}\sum_{W<r\leq y} p|E^{\rm guard}_{pr}| \ll_{B,W}\frac{y^2L^{A+3}}n=o_n(1) \label{eq:smooth-guard-prime-row-error} \end{equation} for the fixed exponent A in that corollary. Explicitly, a deleted integer satisfies the two score envelopes (7.68). Taking the signs of the covariance differences as the coefficient vector b in (7.69), the exact conditional covariance identity (7.67) gives O_{B,W}(y^2L^{A+2}/n), which is stronger than the displayed bound. Hence (8.57) applies to guarded deletions as well as to the analytic covariance remainders. Therefore the squarefree and prime-power remainders in (8.36) and (8.47), including the guarded-deletion remainder from Corollary 7.6, contribute at most \left\{\frac{C_{\rm sf}+C_{\rm pow}}W +o_{n;B,W,\mathcal P}(1)\right\}\mathfrak d_n(b)^2. The main constants here are independent of the subsequently fixed tilt box. Equivalently, if \mathscr E_n(c) denotes the sum of these actual covariance errors, then \begin{equation} \operatorname{Var}_\xi\!\left(\sum_pc_pv_p\right) =\mathscr Q_n^{(0)}(c)+\mathscr E_n(c),\qquad |\mathscr E_n(c)| \leq\left\{\frac{C_{\rm sf}+C_{\rm pow}}W +o_{n;B,W,\mathcal P}(1)\right\}\mathfrak d_n(b)^2. \label{eq:smooth-actual-minimizer-covariance-ledger} \end{equation}

Finally, the one-mark head estimates and (7.61) give \left\|\operatorname{Cov}_\xi \left(Z,\sum_pc_pv_p\right)\right\| \ll_{B,W}\frac1L\sum_p\frac{|c_p|}{p} \ll_{B,W}\frac{\sqrt{\log L}}L\,\mathfrak d_n(b). Together with (8.48), the nuisance Schur subtraction is consequently O_{B,W}\!\left(\frac{\log L}{L^2}\right) \mathfrak d_n(b)^2. Indeed, the definition (8.23) and the exact score identity (8.51) give \frac1{q_n}b^T\mathcal C_\xi^Zb =\left\|P_{Z^\perp}\{(b\cdot\Omega)^{\circ,\xi}\}\right\|_2^2 =\left\|P_{Z^\perp} \left\{\left(\sum_pc_pv_p\right)^{\circ,\xi}\right\}\right\|_2^2. First choose W so that the box-independent 1/W terms are smaller than half the Poincaré gap, and then take n sufficiently large for the fixed box and mesh. Equations (8.56) and (8.52) then yield \begin{equation} \frac1{q_n}b^T\mathcal C_\xi^Zb \geq\kappa\,\mathfrak d_n(b)^2 \geq\kappa\|q\|_D^2. \label{eq:smooth-full-quotient-from-minimizer} \end{equation} Every band vector has a unique decomposition b=q+\lambda\alpha, q\in\mathcal G_n, and \|q\|_D^2=\inf_\lambda\|b-\lambda\alpha\|_D^2. Thus the last display is exactly (8.25); no gap on \mathcal G_n has been used as a substitute for the full quotient inequality.

The inverse assertion is a separate projected statement. The gauge isomorphism (8.44), the pulled comparison (8.45), and the two successive Neumann steps following that display give the projected \ell^\infty-inverse for the actual un-Schur band covariance C_\xi. Its norm is bounded independently of the later box and mesh; only the threshold for the vanishing arithmetic remainder depends on them. Apply Lemma 8.3 with P=P_{\alpha,n} and E=U_\xi\Gamma_\xi^{-1}U_\xi^T. Equation (8.49) and the symmetric band-row calculation preceding (8.47) give the required operator and quadratic perturbation bounds O_{B,W}(\log L/L^2)=o_{n;B,W}(1). The lemma therefore preserves the projected inverse, proving (8.26). This proves (8.25)– (8.26) with the constant row removed and the head/physical block jointly, rather than separately, inverted. ◻

Proof. Step 1: a sharp continuum inverse. We first prove the continuum assertion, using only the continuum gauge \mathcal G_c, centers \alpha^{(c)}, and harmonic matrix D_c. For a band vector define the linear reconstruction \begin{equation} (J_{\mathcal P}^\sharp q)(t):=\frac{t}{\alpha_j^{(c)}}q_j \quad(t\in I_j). \label{eq:smooth-linear-cell-reconstruction} \end{equation} Thus \begin{equation} \|J_{\mathcal P}^\sharp q\|_{X_t}=\|q\|_{\sharp,c}, \label{eq:smooth-sharp-reconstruction-isometry} \end{equation} including on the moving low cell. If a uniform continuum inverse did not exist, there would be a sequence of moving cutoffs and permitted meshes, numbers \varepsilon_\nu\downarrow0, and q_\nu\in\mathcal G_{c,\nu} such that \|q_\nu\|_{\sharp,c}=1,\qquad \max_j\frac{|[P_{\alpha,c}B_{t_0,\mathcal P}q_\nu]_j|} {\alpha_j^{(c)}}\leq\varepsilon_\nu. As in (8.28), there is a bounded scalar \beta_\nu such that \begin{equation} (B_{t_0,\mathcal P}q_\nu)_j =\beta_\nu\alpha_j^{(c)}+r_{\nu,j},\qquad |r_{\nu,j}|\leq\varepsilon_\nu\alpha_j^{(c)}. \label{eq:smooth-weighted-unprojected-equation} \tag{8.60} \end{equation} Here one may take r_\nu=P_{\alpha,c}B_{t_0,\mathcal P}q_\nu,\qquad \beta_\nu= \frac{(\alpha^{(c)})^TD_cB_{t_0,\mathcal P}q_\nu} {(\alpha^{(c)})^TD_c\alpha^{(c)}}. The denominator is bounded below uniformly by the cells meeting [1/2,1]. Also \|q_\nu\|_\infty\leq\|q_\nu\|_{\sharp,c}=1, \sum_jH_j^{(c)}\alpha_j^{(c)}=1-t_0, and the compressed continuum operators are uniformly bounded. Hence |\beta_\nu|\leq C, with C independent of the moving cutoff and mesh. For the local estimates below write q=q_\nu, \beta=\beta_\nu, and suppress \nu from the cell data. Put x_j=q_j/\alpha_j^{(c)} and, on I_j, set d(t):=(J_{\mathcal P}^\sharp-J_{\mathcal P})q(t) =x_j(t-\alpha_j^{(c)}). Then \|x\|_\infty=\|q\|_{\sharp,c} and, exactly on every cell, \begin{equation} \int_{I_j}d(t)\,\frac{dt}{t}=0. \label{eq:smooth-sharp-cell-centering} \end{equation} For the multiplier part the constant value h(\alpha_j^{(c)}) therefore cancels. On a positive relative cell, Lipschitz continuity of h, the mesh bounds, and \alpha_j^{(c)}\asymp\inf I_j give \frac1{H_j^{(c)}}\left| \int_{I_j}h(t)d(t)\,\frac{dt}{t}\right| \leq C\eta\alpha_j^{(c)}\|q\|_{\sharp,c}. On I_0, use h(t)=1+O(t) and the two explicit estimates \frac1{H_0^{(c)}}\int_{I_0}(t-\alpha_0^{(c)})\,\frac{dt}{t}=0, \qquad \frac1{H_0^{(c)}}\int_{I_0}|t-\alpha_0^{(c)}|\,\frac{dt}{t} \leq2\alpha_0^{(c)}. They give an O(\delta\alpha_0^{(c)})\|q\|_{\sharp,c} low multiplier error. For the integral part, |K(s,t)|\leq Cst gives \left|\frac1{H_i^{(c)}}\int_{I_i}\int_{t_0}^1 K(s,t)d(t)\,\frac{dt}{t}\frac{ds}{s}\right| \leq C\alpha_i^{(c)}\|q\|_{\sharp,c} \sum_k\int_{I_k}|t-\alpha_k^{(c)}|\,dt. The low summand is O(\delta^2). On the regular positive cells the sum is O(\eta): each summand is O(\eta^2(\inf I_k)^2), and comparison with the corresponding geometric Riemann sum gives O(\eta). Hence, uniformly over all permitted meshes, \begin{equation} \max_j\frac{\left| [E_{\mathcal P}A^{(t_0)}(J_{\mathcal P}^\sharp-J_{\mathcal P})q]_j \right|}{\alpha_j^{(c)}} \le Cw\|q\|_{\sharp,c}. \label{eq:smooth-step-linear-reconstruction-error} \end{equation} If G(s)=A^{(t_0)}J_{\mathcal P}^\sharp q(s)/s, then (8.19), h\in C^1, and the fact that J_{\mathcal P}^\sharp q(t)/t=x_j is constant on each cell give \operatorname{osc}_{I_0}G\leq C\delta\|q\|_{\sharp,c},\qquad \operatorname{osc}_{I_j}G\leq C\eta\|q\|_{\sharp,c}\quad(j\geq1). Moreover, since H_j^{(c)}\alpha_j^{(c)}=|I_j|, for every cell \begin{equation} (E_{\mathcal P}A^{(t_0)}J_{\mathcal P}^\sharp q)_j =\alpha_j^{(c)}\frac1{|I_j|}\int_{I_j}G(s)\,ds. \label{eq:smooth-sharp-average-to-pointwise} \end{equation} In particular, on the low cell, (E_{\mathcal P}A^{(t_0)}J_{\mathcal P}^\sharp q)_0 =\alpha_0^{(c)}\frac1{\delta-t_0} \int_{t_0}^{\delta}G(s)\,ds. Consequently (8.60) implies \begin{equation} \left\|\frac{A^{(t_0)}J_{\mathcal P}^\sharp q}{t}-\beta\right\|_\infty \ll w+o(1). \label{eq:smooth-weighted-reconstructed-equation} \tag{8.61} \end{equation} Extend the low linear formula to (0,t_0), and denote the resulting function by \widetilde f. Since |K(s,t)/s|\ll t, the omitted input changes the weighted output by O(t_0^2). For the extended function, its quotient by t is constant on (0,\delta]. The formulas for h and K(s,t)/s, with F\in C^2[0,2], therefore give \operatorname{osc}_{0\leq s\leq\delta} \frac{A\widetilde f(s)}s=O(\delta). At s=0, the quotient kernel is interpreted by (8.19). Hence (8.61) holds on all of (0,1] for the full operator, with the same O(w)+o(1) right side. Put e(t)=A\widetilde f(t)/t-\beta. Self-adjointness and At=0 give the exact identity 0=\langle t,A\widetilde f\rangle_{\mathcal H} =\frac{\beta}{2}+\int_0^1t\,e(t)\,dt. Hence |\beta|\leq\|e\|_\infty=O(w)+o(1). The X_t inverse (8.14) then puts the reconstruction within O(w)+o(1) of \lambda t. Since (\alpha^{(c)})^TD_cq=0, this means exactly 0=\sum_jH_j^{(c)}(\alpha_j^{(c)})^2 \frac{q_j}{\alpha_j^{(c)}}. The total weight in this sum is bounded below by a structural positive constant: the cells contained in [1/2,1] have centers at least 1/2 and total harmonic mass bounded below. Averaging q_j/\alpha_j^{(c)}=\lambda+O(w)+o(1) against these positive weights therefore gives \lambda=O(w)+o(1). Once the structural threshold w_0 is sufficiently small, this contradicts \|q\|_{\sharp,c}=1. Equivalently, with \begin{equation} \Pi_c:=S_c^{-1}P_{\alpha,c}S_c,\qquad \mathcal R_c:=\operatorname{Ran}\Pi_c,\qquad \mathcal T_c:=\Pi_cS_c^{-1}B_{t_0,\mathcal P}S_c\Pi_c, \label{eq:smooth-continuum-sharp-operator} \end{equation} there is a mesh- and t_0-uniform constant C_\sharp^{(0)} such that \begin{equation} \left\|(\mathcal T_c|_{\mathcal R_c})^{-1}\right\|_{\infty\to\infty} \le C_\sharp^{(0)}. \label{eq:smooth-continuum-sharp-inverse} \tag{8.62} \end{equation}

Step 2: arithmetic cell data and sharp h,K quadrature. The arithmetic transfer must be relative. If \|b\|_\sharp\leq1, then |b_{j(q)}|\leq\alpha_{j(q)}\ll t_q on positive cells, while on the low cell |b_0|\leq\alpha_0. The pointwise form of (7.14) and (7.10) is, for p\ne q, \begin{equation} p|E_{pq}^{\rm sf}|\ll_{B,W}\frac1{qL}. \label{eq:smooth-sharp-pointwise-squarefree-error} \tag{8.63} \end{equation} The diagonal has the same L^{-1} error, in addition to the Bernoulli correction treated below. Therefore the exact band average (8.37) gives \begin{equation} \begin{split} |(\mathsf E_nE^{\rm sf}J_nb)_0| &\ll_{B,W} L^{-1} \ll_{B,W}\alpha_0\frac{\log L}{\delta L},\\ |(\mathsf E_nE^{\rm sf}J_nb)_i| &\ll_{B,W}\alpha_i\frac1{\delta L}\quad(i\ge1),\\ \max_i\frac{|(\mathsf E_nE^{\rm sf}J_nb)_i|}{\alpha_i} &\ll_{B,W}\frac{\log L}{\delta L}. \end{split} \label{eq:smooth-sharp-squarefree-transfer} \end{equation} For the low row, the first inequality is the explicit calculation \frac1{H_0}\sum_{p\in\mathcal P_0}\frac1p \sum_q O_{B,W}\!\left(\frac{|b_{j(q)}|}{qL}\right) \ll_{B,W}\frac1L, because \sum_q|b_{j(q)}|/q\ll H_0\alpha_0+\sum_{j\ge1}H_j\alpha_j\ll1. Thus the numerator is genuinely o(\alpha_0), since \alpha_0\asymp\delta/\log L. The active cells have the numerical guards deleted. By (7.63), that deletion contributes an operator G_n satisfying, for \|b\|_\sharp\leq1, \begin{equation} \max_i\frac{|(G_nb)_i|}{\alpha_i}=o_{n;B,W,\mathcal P}(1), \label{eq:smooth-sharp-guard-deletion} \end{equation} because its band output is o_n(\alpha_0), \|b\|_\infty\leq1, and \alpha_i\gg\alpha_0 on every positive cell. Thus guard deletion is part of the vanishing sharp remainder.

We next state the sharp quadrature which replaces an unspecified additive discretization error. Let \widehat B_n^{h,K} be the band operator with no marked error, defined by \begin{equation} (\widehat B_n^{h,K}b)_i:=\frac1{H_i} \sum_{p\in\mathcal P_i}\frac1p\left\{ h(t_p)b_i+\sum_{\substack{W<q\le y\\q\ne p}} \frac{K(t_p,t_q)}q b_{j(q)}\right\}. \label{eq:smooth-discrete-hK-operator} \end{equation} Then, for every fixed permitted mesh, \begin{equation} \left\|S_n^{-1}\widehat B_n^{h,K}S_n -S_c^{-1}B_{t_0,\mathcal P}S_c \right\|_{\infty\to\infty} =o_{n;W,\mathcal P}(1). \label{eq:smooth-sharp-main-quadrature} \tag{8.64} \end{equation} Here is the low-cell calculation needed for this relative statement. Since F\in C^2[0,2] and F(0)=1, the function \varkappa(s,t):=\frac{K(s,t)}{st} has a continuous extension to [0,1]^2. The (i,k)-entry of the conjugated discrete kernel, with the diagonal temporarily inserted, is \begin{equation} \frac{\alpha_k}{H_i\alpha_i} \sum_{p\in\mathcal P_i}\frac{t_p}{p} \sum_{q\in\mathcal P_k}\frac{t_q}{q} \varkappa(t_p,t_q). \label{eq:smooth-sharp-kernel-entry} \tag{8.65} \end{equation} To see every normalization, introduce the probability measures \begin{align*} \rho_{j,n} &:=\frac1{H_j\alpha_j} \sum_{p\in\mathcal P_j}\frac{t_p}{p}\,\delta_{t_p},& \rho_{j,c} &:=\frac1{H_j^{(c)}\alpha_j^{(c)}} \mathbf 1_{I_j}(t)\,dt, \end{align*} and retain \omega_{n,j}=H_j\alpha_j^2, \omega_{c,j}=H_j^{(c)}(\alpha_j^{(c)})^2. Because H_j\alpha_j=\sum_{p\in\mathcal P_j}t_p/p and H_j^{(c)}\alpha_j^{(c)}=|I_j|, these are indeed probability measures. Formula (8.65) is exactly \begin{equation} \omega_{n,k}\iint\varkappa(s,t)\, d\rho_{i,n}(s)d\rho_{k,n}(t), \label{eq:smooth-sharp-kernel-probability-entry} \tag{8.66} \end{equation} and its continuum counterpart is the same expression with n replaced by c.

The moving-low convergence follows from the uniform weighted prime count \begin{equation} \sup_{t_0\leq x\leq1}\left| \sum_{W<p\leq y^x}\frac{t_p}{p}-(x-t_0)\right| =O_W((\log y)^{-1}). \label{eq:smooth-weighted-prime-distribution} \tag{8.67} \end{equation} This is \sum_{p\leq X}(\log p)/p=\log X+O(1), divided by \log y. Stieltjes summation and uniform continuity of \varkappa therefore give \rho_{j,n}\Rightarrow\rho_{j,c}, uniformly against all sections of \varkappa, on every positive cell and also on I_0=[t_0,\delta]. On the low cell, \omega_{n,0}=(H_0\alpha_0)\alpha_0=o_n(1),\qquad \omega_{c,0}=(\delta-t_0)\alpha_0^{(c)}=o_n(1), whereas \omega_{n,k}-\omega_{c,k}=o_n(1) on every positive cell. Thus (8.66) converges in all four low/positive orientations: a low input carries the vanishing factor \omega_{0,\bullet}, and a low output is averaged against a probability measure. In particular neither H_0 nor \alpha_0^{-1} is lost. Removing the inserted diagonal changes the conjugated operator by at most \max_i\frac1{H_i}\sum_{p\in\mathcal P_i} \frac{|K(t_p,t_p)|}{p^2}=o_{n;W,\mathcal P}(1). Indeed |K(t,t)|\ll t^2. On a positive cell its primes tend to infinity, while on the low cell t_p^2=(\log p)^2/(\log y)^2 and \sum_{p>W}(\log p)^2/p^2<\infty. For the diagonal multiplier on I_0, write h(t)=1+t\widetilde h(t), where \widetilde h is continuous. The constant part has average exactly one in both the arithmetic and continuum rows; (8.67) handles the remaining weighted average, together with H_0/H_0^{(c)}\to1. Ordinary Mertens quadrature applies on every positive cell. The dimension is fixed after the mesh is chosen, so the entrywise convergence is exactly (8.64). Together with (8.41), this also verifies the sharp convergence of the arithmetic and continuum gauges; no convergence of D to D_c is used.

Step 3: the remaining sharp arithmetic errors and the nuisance Schur block. The Bernoulli diagonal correction, kept out of E^{\rm sf}_{pq} in the preceding off-diagonal calculation, satisfies \begin{equation} |E_i^{\rm sf,diag}| \ll\frac1{H_i}\sum_{p\in\mathcal P_i} \frac{|b_{j(p)}|}{p^2} \le\frac{\alpha_i}{W}. \label{eq:smooth-sharp-squarefree-diagonal} \end{equation} For the prime-power part we use the product-weighted aggregate bounds, rather than an unavailable unweighted comparison of H_0^{-1}\sum_{p\in\mathcal P_0}p^{-2} with \alpha_0. For \|b\|_\sharp\le1, harmonic centering and the regular mesh give \sum_q\frac{t_q|b_{j(q)}|}{q}\ll1,\qquad \sum_q\frac{t_q|b_{j(q)}|}{q^2}\ll\frac1W. Indeed, on a positive cell \alpha_j\asymp t_q, while on the low cell |b_0|\sum_{q\in\mathcal P_0}\frac{t_q}{q} \le H_0\alpha_0^2\ll1. The unweighted vanishing remainders in the three aggregate estimates are also harmless at the moving low cell. More precisely, for the fixed regular mesh, \begin{equation} \sum_q\frac{|b_{j(q)}|}{q^2}\ll_W\alpha_0+o(\alpha_0), \qquad \frac1{H_i\alpha_i}\sum_{p\in\mathcal P_i}\frac1{p^2} =O_{W,\delta,\eta}(1), \label{eq:smooth-sharp-unweighted-power-bands} \tag{8.68} \end{equation} uniformly in i. For the first estimate, the low-cell contribution is |b_0|\sum_{p>W}p^{-2}\ll_W\alpha_0, while every positive cell starts at a power of y and is o(\alpha_0); the second estimate is the same reciprocal calculation after division by the cell’s harmonic mass. Also \sum_q|b_{j(q)}|/q\ll1, as in the squarefree calculation above. Thus the unweighted error coefficient in (7.52) remains o(\alpha_i) in each orientation. Denote by E_i^{JI},E_i^{IJ},E_i^{JJ}, and E_i^{\rm diag} the four prime-power contributions to the normalized band output. We write every orientation rather than invoke symmetry. The box-independent main terms satisfy \begin{align} |E_i^{JI}|_{\rm main} &\ll \frac1{H_i}\sum_{p\in\mathcal P_i}\frac{t_p}{p^2} \sum_q\frac{t_q|b_{j(q)}|}{q} \leq\frac{\alpha_i}{W} \sum_q\frac{t_q|b_{j(q)}|}{q} \ll\frac{\alpha_i}{W},\notag\\ |E_i^{IJ}|_{\rm main} &\ll \frac1{H_i}\sum_{p\in\mathcal P_i}\frac{t_p}{p} \sum_q\frac{t_q|b_{j(q)}|}{q^2} \ll\frac{\alpha_i}{W},\notag\\ |E_i^{JJ}|_{\rm main} &\ll \frac1{H_i}\sum_{p\in\mathcal P_i}\frac{t_p}{p^2} \sum_q\frac{t_q|b_{j(q)}|}{q^2} \ll\frac{\alpha_i}{W^2},\notag\\ |E_i^{\rm diag}|_{\rm main} &\ll \frac{|b_i|}{H_i}\sum_{p\in\mathcal P_i}\frac1{p^2} \leq\frac{\alpha_i}{W}. \label{eq:smooth-sharp-four-power-main-directions} \end{align} Let \epsilon^{(0)}_{B,W}(n) be the single pair remainder in (7.52). The four unweighted remainders, divided by \alpha_i, are bounded respectively by \begin{align} &\frac{\epsilon^{(0)}_{B,W}(n)}{H_i\alpha_i} \sum_{p\in\mathcal P_i}\frac1{p^2} \sum_q\frac{|b_{j(q)}|}{q},\notag\\ &\frac{\epsilon^{(0)}_{B,W}(n)}{\alpha_i} \sum_q\frac{|b_{j(q)}|}{q^2},\notag\\ &\frac{\epsilon^{(0)}_{B,W}(n)}{H_i\alpha_i} \sum_{p\in\mathcal P_i}\frac1{p^2} \sum_q\frac{|b_{j(q)}|}{q^2},\notag\\ &\frac{\epsilon^{(0)}_{B,W}(n)|b_i|}{H_i\alpha_i} \sum_{p\in\mathcal P_i}\frac1{p^2}. \label{eq:smooth-sharp-four-power-error-directions} \end{align} Each is o_{n;B,W,\mathcal P}(1) uniformly in i, by (8.68), \sum_q|b_{j(q)}|/q\ll1, and the fact that every positive-cell \alpha_i is bounded below by a fixed multiple of \delta, whereas the low-cell value is \alpha_0. We also keep the literal endpoints separate. For \|b\|_\sharp\leq1, the endpoint ledger (7.51) gives, orientation by orientation, \begin{align} \frac{|E_i^{JI,\mathrm{end}}|}{\alpha_i} &\ll_{B,W}\frac Ln\frac{\#\mathcal P_i}{H_i\alpha_i} \sum_q|b_{j(q)}| \ll_{B,W,\delta,\eta}\frac{y^2L}{n},\notag\\ \frac{|E_i^{IJ,\mathrm{end}}|}{\alpha_i} &\ll_{B,W}\frac Ln\frac{\#\mathcal P_i}{H_i\alpha_i} \sum_q|b_{j(q)}| \ll_{B,W,\delta,\eta}\frac{y^2L}{n},\notag\\ \frac{|E_i^{JJ,\mathrm{end}}|}{\alpha_i} &\ll_{B,W}\frac {L^2}n\frac{\#\mathcal P_i}{H_i\alpha_i} \sum_q|b_{j(q)}| \ll_{B,W,\delta,\eta}\frac{y^2L^2}{n},\notag\\ \frac{|E_i^{\mathrm{diag,end}}|}{\alpha_i} &\ll_{B,W}\frac {L^2}n\frac{\#\mathcal P_i}{H_i} \frac{|b_i|}{\alpha_i} \ll_{B,W,\delta,\eta}\frac{yL^2}{n}. \label{eq:smooth-sharp-four-endpoint-directions} \end{align} Indeed \#\mathcal P_i\leq y, \sum_q|b_{j(q)}|\leq y, and |b_i|\leq\alpha_i. Moreover, H_i\alpha_i=\sum_{p\in\mathcal P_i}t_p/p tends to \delta on the low cell and to |I_i| on each positive cell, while H_i is bounded below on every permitted cell. Hence the four displays imply the common relative bound \begin{equation} \max_i\frac{|E_i^{\rm endpoint}|}{\alpha_i} \ll_{B,W,\mathcal P}\frac{y^2L^2}{n} =o_{n;B,W,\mathcal P}(1). \label{eq:smooth-sharp-four-power-endpoints} \end{equation} Thus all box dependence in the four directions is contained in one remainder \epsilon^{\sharp}_{B,W,\mathcal P}(n)\to0 for each fixed mesh, and each direction separately is relative at the moving low row. Consequently \begin{equation} |E_i^{\rm pow}| \le\alpha_i\left\{\frac{C_{\rm pow}^{\sharp}}W +\epsilon^{\sharp}_{B,W,\mathcal P}(n)\right\}, \label{eq:smooth-sharp-power-transfer} \tag{8.69} \end{equation} where C_{\rm pow}^{\sharp} is independent of the compact tilt box and of the subsequently chosen fine mesh. Finally (8.48), the marked head/physical rows, and the exact removal in (8.50) give, for \|b\|_\sharp\leq1, |(U_\xi)_{i\nu}|\ll_{B,W}\frac{H_i}{L},\qquad \sum_jH_j|b_j|\leq\sum_jH_j\alpha_j\ll1, \qquad \|\Gamma_\xi^{-1}\|\ll_W1. Consequently \begin{equation} |E_i^{\rm head/phys}| \ll_{B,W}\frac1{L^2} \leq\alpha_i\,O_{B,W}\!\left(\frac{\log L}{\delta L^2}\right) =o_{n;B,W}(\alpha_i). \label{eq:smooth-sharp-head-physical-transfer} \end{equation} For reference, the relative estimates just proved can be read in the following ledger; the displayed quantity is always the supremum over \|b\|_\sharp\leq1: \begin{array}{c|c} \text{source}& \displaystyle\max_i |(E_nb)_i|/\alpha_i\\ \hline \text{leading $h,K$ quadrature} &o_{n;W,\mathcal P}(1)\\ \text{squarefree marked off-diagonal} &O_{B,W}(\log L/(\delta L))\\ \text{guard deletion} &o_{n;B,W,\mathcal P}(1)\\ \text{squarefree Bernoulli diagonal} &O(1/W)\\ \text{prime powers} &C_{\rm pow}^{\sharp}/W+ \epsilon^\sharp_{B,W,\mathcal P}(n)\\ \text{head/physical Schur block} &O_{B,W}(\log L/(\delta L^2))\\ \text{arithmetic/continuum projection} &o_{n;W,\mathcal P}(1). \end{array} Thus every vanishing row error is o(\alpha_i), including the low row i=0; no additive o(1) is divided by \alpha_0 after the fact. It follows that the sharp discrete operator is a perturbation, in the sharp norm, whose nonvanishing part is (C_{\rm sf}+C_{\rm pow}^{\sharp})/W, with every dependence on the later compact box and fixed mesh in a remainder tending to zero with n. Equivalently, before either gauge projection is applied, \begin{equation} \left\|S_n^{-1}\mathsf B^Z_\xi S_n -S_c^{-1}B_{t_0,\mathcal P}S_c\right\|_{\infty\to\infty} \leq\frac{C_{\rm sf}+C_{\rm pow}^{\sharp}}W +o_{n;B,W,\mathcal P}(1). \label{eq:smooth-unprojected-sharp-operator-comparison} \tag{8.70} \end{equation}

Step 4: identification of the varying sharp gauges and transfer of the inverse. We finish by identifying the two quotient spaces explicitly. Put \begin{align} \Pi_n&:=S_n^{-1}P_{\alpha,n}S_n, &\mathcal R_n&:=\operatorname{Ran}\Pi_n,\notag\\ \mathcal T_{n,\xi} &:=\Pi_nS_n^{-1}\mathsf B^Z_\xi S_n\Pi_n, &\mathcal T_c&:=\Pi_cS_c^{-1}B_{t_0,\mathcal P}S_c\Pi_c. \label{eq:smooth-two-sharp-projected-operators} \end{align} Equations (8.41) and (8.70), together with the uniform boundedness of the continuum operator, give the explicit projected comparison \begin{equation} \|\mathcal T_{n,\xi}-\mathcal T_c\|_{\infty\to\infty} \le \frac{C_{\rm sf}+C_{\rm pow}^{\sharp}}W +o_{n;B,W,\mathcal P}(1). \label{eq:smooth-two-sharp-operator-comparison} \tag{8.71} \end{equation} All operators in this display are extended to the full coordinate space by the displayed projections. Put \varepsilon_n:=\|\Pi_n-\Pi_c\|_{\infty\to\infty}=o_n(1). The exact formulas preceding (8.41) show that both projections have kernel \operatorname{span}\{\mathbf1\} and hence rank s. For x\in\mathcal R_c, the restriction U_n:=\Pi_n|_{\mathcal R_c}:\mathcal R_c\longrightarrow\mathcal R_n satisfies \|U_nx-x\|_\infty =\|(\Pi_n-\Pi_c)x\|_\infty \leq\varepsilon_n\|x\|_\infty. Thus U_n is injective when \varepsilon_n<1, and equality of the two ranks makes it onto. Moreover, \begin{equation} \|U_n\|\leq1+\varepsilon_n,\qquad \|U_n^{-1}\|\leq(1-\varepsilon_n)^{-1}. \label{eq:smooth-sharp-range-isomorphism} \tag{8.72} \end{equation} If y=U_nx\in\mathcal R_n, then, under the ambient coordinate identification, \|U_n^{-1}y-y\|_\infty=\|x-\Pi_nx\|_\infty \leq\frac{\varepsilon_n}{1-\varepsilon_n}\|y\|_\infty. For x\in\mathcal R_c, the exact expansion inside U_n^{-1} is \mathcal T_{n,\xi}U_nx-U_n\mathcal T_cx =(\mathcal T_{n,\xi}-\mathcal T_c)U_nx +\mathcal T_c(U_n-I)x +(I-\Pi_n)\mathcal T_cx. The last two terms are each at most \varepsilon_n\|\mathcal T_c\|\,\|x\|_\infty, up to the harmless factor 1+\varepsilon_n in the first one. Equations (8.72) and (8.71) therefore give the pulled-back comparison \begin{equation} \left\|U_n^{-1}(\mathcal T_{n,\xi}|_{\mathcal R_n})U_n -\mathcal T_c|_{\mathcal R_c}\right\|_{\infty\to\infty} \leq\frac{C_{\rm sf}+C_{\rm pow}^{\sharp}}W +o_{n;B,W,\mathcal P}(1). \label{eq:smooth-pulled-sharp-operator-comparison} \end{equation} We enlarge the structural threshold W_0 once and for all, before the actual W is selected, so that the box-independent 1/W term is smaller than (4C_\sharp^{(0)})^{-1} for every W\geq W_0. Only after this choice are the compact box and mesh fixed and n taken large enough to absorb the remainder. The Neumann series and (8.62) now give \left\|(\mathcal T_{n,\xi}|_{\mathcal R_n})^{-1}\right\|_{\infty\to\infty} \le 2C_\sharp^{(0)}+o(1). Finally put x=S_n^{-1}q and v=S_n^{-1}u. Because q,u\in\mathcal G_n, the exact projection formula gives x,v\in\mathcal R_n, and (8.58) becomes \mathcal T_{n,\xi}x=v. The hypothesis gives \|v\|_\infty\leq Cw, while \|x\|_\infty=\|q\|_\sharp. Applying the preceding inverse bound therefore yields \|q\|_\sharp\leq C_\sharp w, which is (8.59). This also makes the order of the choices W, box, mesh, and n explicit. ◻

Proof. Mertens’ theorem and (8.20) give the first assertion with uniform constants. In the continuum low cell, \begin{equation} \int_{t_0}^{\delta}(t-\alpha_0^{(c)})^2\frac{dt}{t} =\frac{\delta^2-t_0^2}{2} -\frac{(\delta-t_0)^2}{\log(\delta/t_0)} =\frac{\delta^2}{2}+o(1). \label{eq:smooth-low-V-constant} \tag{8.80} \end{equation} For a positive cell I=[a,a+\ell], direct integration and the two-sided regular-mesh bounds give \begin{equation} \int_I(t-\alpha_I)^2\frac{dt}{t} \asymp\frac{\ell^3}{a},\qquad \int_I|t-\alpha_I|\frac{dt}{t}\asymp\frac{\ell^2}{a}. \label{eq:smooth-positive-cell-moments} \tag{8.81} \end{equation} The merged terminal cell satisfies the same bounds. Summing the geometric mesh and applying Mertens gives \sum_p|g_p|/p\ll\delta+\eta,\qquad V\asymp\delta^2+\eta^2\asymp w^2.

We next justify the sharp right-side estimate for the non-step coefficient g_p=\alpha_{j(p)}-t_p. This does not follow merely by applying the preceding step-input operator estimate, since on the moving low cell one can have |g_p|\gg\alpha_0. All prime sums in this paragraph range over W<p\leq y. Put \begin{equation} B_1(g):=\sum_p\frac{|g_p|}{p},\qquad G_1(g):=\sum_p\frac{t_p|g_p|}{p},\qquad B_2(g):=\sum_p\frac{|g_p|}{p^2},\qquad G_2(g):=\sum_p\frac{t_p|g_p|}{p^2}. \label{eq:smooth-raw-g-moment-ledger-definition} \end{equation} The cell moment calculations in (8.80)– (8.81), followed by weighted Mertens summation, give \begin{align} \|g\|_\infty&\ll w,& B_1(g)&\ll w,&G_1(g)&\ll w,\notag\\ G_2(g)&\leq W^{-1}G_1(g)\ll w/W, \label{eq:smooth-raw-g-first-moment-ledger}\tag{8.82}\\ \frac1{H_i}\sum_{p\in\mathcal P_i} \frac{t_p|g_p|}{p}&\ll w\alpha_i \qquad(0\leq i\leq s). \label{eq:smooth-raw-g-local-product-moment}\tag{8.83}\end{align} For the two unweighted reciprocal-square quantities needed below one has, for the fixed regular mesh, \begin{align} B_2(g)&=O_W(\alpha_0)+o_n(\alpha_0), \label{eq:smooth-raw-g-unweighted-square-sum}\tag{8.84}\\ d_i(g):=\frac1{H_i}\sum_{p\in\mathcal P_i}\frac{|g_p|}{p^2} &=o_{n;W,\delta,\eta,\mathcal P}(w\alpha_i) \quad\hbox{uniformly in }i. \label{eq:smooth-raw-g-local-diagonal}\tag{8.85}\end{align} Here is the low-row calculation, where these assertions are not a consequence of a step bound. Since |g_p|\leq\alpha_0+t_p on \mathcal P_0, \begin{equation} d_0(g)\leq \frac{\alpha_0}{H_0}\sum_{p>W}\frac1{p^2} +\frac1{H_0\log y}\sum_{p>W}\frac{\log p}{p^2} =o_n(w\alpha_0). \label{eq:smooth-raw-g-low-diagonal-calculation} \end{equation} Indeed, after division by w\alpha_0, the two terms are respectively O_W((wH_0)^{-1}) and O_W((w\delta\log y)^{-1}). The same estimate before division by H_0 proves the low-cell contribution to (8.84). Every positive cell starts at y^\delta, so its contribution to either display is exponentially small relative to \alpha_0. This proves (8.84)– (8.85) without losing a factor \alpha_0^{-1}.

We now contract every covariance term against g. The leading squarefree band output is R_i^{h,K}(g):=\frac1{H_i}\left\{ \sum_{p\in\mathcal P_i}\frac{h(t_p)g_p}{p} +\sum_{p\in\mathcal P_i}\sum_{q\ne p} \frac{K(t_p,t_q)g_q}{pq}\right\}. Exact harmonic centering gives \sum_{p\in\mathcal P_i}g_p/p=0. Since h(t)-1=O(t), |K(s,t)|\ll st, and H_i^{-1}\sum_{p\in\mathcal P_i}t_p/p=\alpha_i, (8.82)– (8.83) give \begin{equation} \left|R_i^{h,K}(g)\right| \ll \frac1{H_i}\sum_{p\in\mathcal P_i} \frac{t_p|g_p|}{p} +\alpha_iG_1(g) \ll w\alpha_i. \label{eq:smooth-raw-g-leading-band-row} \tag{8.86} \end{equation} In particular this includes the low-row estimate that was formerly implicit. The squarefree marked error (8.63) contracts to \begin{equation} |R_i^{\rm sf,err}(g)| \ll_{B,W}\frac{B_1(g)}L\ll_{B,W}\frac wL, \qquad \max_i\frac{|R_i^{\rm sf,err}(g)|}{w\alpha_i} \ll_{B,W}\frac{\log L}{\delta L}=o(1). \label{eq:smooth-raw-g-squarefree-error} \end{equation} The Bernoulli diagonal is bounded by d_i(g), hence is o(w\alpha_i) by (8.85). Guard deletion is also relative: apply (7.63) to g/\|g\|_\infty and use \|g\|_\infty\ll w. Its output is therefore o(w\alpha_0)=o(w\alpha_i) on every row.

For completeness, decompose V_p=I_p+J_p and use (7.34)– (7.36). The box-independent main terms in the four orientations satisfy the following bounds, where R_i^{JI},R_i^{IJ},R_i^{JJ},R_i^{\rm diag} denote their respective contributions to (D^{-1}\operatorname{Cov}_\xi(\Omega,S_g))_i: \begin{align} |R_i^{JI}(g)|_{\rm main} &\ll \frac1{H_i}\sum_{p\in\mathcal P_i}\frac{t_p}{p^2}\,G_1(g) \ll\frac{w\alpha_i}{W},\notag\\ |R_i^{IJ}(g)|_{\rm main} &\ll \frac1{H_i}\sum_{p\in\mathcal P_i}\frac{t_p}{p}\,G_2(g) \ll\frac{w\alpha_i}{W},\notag\\ |R_i^{JJ}(g)|_{\rm main} &\ll \frac1{H_i}\sum_{p\in\mathcal P_i}\frac{t_p}{p^2}\,G_2(g) \ll\frac{w\alpha_i}{W^2},\notag\\ |R_i^{\rm diag}(g)|_{\rm main} &\ll d_i(g)=o_n(w\alpha_i). \label{eq:smooth-raw-g-four-power-main-directions} \end{align} Let \epsilon_0=\epsilon^{(0)}_{B,W}(n) be the pair remainder in (7.52). After division by w\alpha_i, its four contractions are bounded, respectively, by \begin{equation} \epsilon_0\frac1{H_i\alpha_i} \sum_{p\in\mathcal P_i}\frac1{p^2}\frac{B_1(g)}w, \quad \epsilon_0\frac{B_2(g)}{w\alpha_i}, \quad \epsilon_0\frac1{H_i\alpha_i} \sum_{p\in\mathcal P_i}\frac1{p^2}\frac{B_2(g)}w, \quad \epsilon_0\frac{d_i(g)}{w\alpha_i}. \label{eq:smooth-raw-g-four-power-error-directions} \end{equation} The first expression tends to zero by (8.68) and (8.82); the third and fourth do so by (8.84)– (8.85). For the second expression, the low row is O_W(\epsilon_0/w), while on every positive row \alpha_i\gg\delta; it too tends to zero after the already chosen positive w, W, box, and mesh have been fixed. This is threshold dependence only and creates no box-dependent main constant. Finally, the literal endpoint terms obey \begin{equation} \max_i\frac{|R_i^{\rm endpoint}(g)|}{w\alpha_i} \ll_{B,W,\delta,\eta,\mathcal P}\frac{y^2L^2}{n}=o(1). \label{eq:smooth-raw-g-four-power-endpoints} \end{equation} Indeed \#\mathcal P_i\leq y, \sum_q|g_q|\ll wy, and H_i\alpha_i tends to \delta on the low cell and is bounded below on every fixed positive cell. Thus all JI,IJ,JJ, diagonal, tail, endpoint, squarefree, and guard terms have now been contracted against the actual non-step score.

The finite head and physical Schur subtraction is relative as well. We record its raw nuisance vector here so it is not inferred from the later post-regression calculation. Before guard deletion, (7.64) and centered component-weight cancellation give |\operatorname{Cov}_\xi(H_\nu^\circ,S_g)| \ll_{B,W}\frac1L\sum_p\frac{|g_p|}{p} \ll_{B,W}\frac wL. The physical coordinate has the same bound by (7.61). Since \|S_g\|_\infty\ll_WwL, (7.65) and (7.67) show that guard deletion changes either covariance by O_{B,W}(wyL^{A+1}/n)=o_{n;B,W}(w/L). Hence \begin{equation} \|\operatorname{Cov}_\xi(Z,S_g)\| \ll_{B,W}\frac wL+o_{n;B,W}(w/L). \label{eq:smooth-raw-g-nuisance-vector} \end{equation} Together with \|(U_\xi)_{i,*}\|\ll_{B,W}H_i/L, (8.48), and \|\Gamma_\xi^{-1}\|\ll_W1, this gives \begin{equation} \frac1{H_i}\left| \bigl(U_\xi\Gamma_\xi^{-1} \operatorname{Cov}_\xi(Z,S_g)\bigr)_i\right| \ll_{B,W}\frac w{L^2}+o_n(w/L^2) =o_{n;B,W}(w\alpha_i). \label{eq:smooth-raw-g-nuisance-Schur-row} \tag{8.87} \end{equation}

Combining (8.86)– (8.87), we have proved the explicit sharp ledger \begin{equation} \max_i\frac{|(D^{-1}b^Z)_i|}{w\alpha_i} \leq C_{\rm rhs}+\frac{C_{\rm pow}^{\rm raw}}W +r_{n;B,W,\mathcal P}, \qquad r_{n;B,W,\mathcal P}\longrightarrow0. \label{eq:smooth-raw-g-complete-sharp-ledger} \end{equation} Here C_{\rm rhs},C_{\rm pow}^{\rm raw} depend only on the structural Dickman and regular-mesh bounds; in particular they are independent of the later box radius and of the permitted mesh. All dependence on the fixed box and mesh is confined to the vanishing remainder and its n-threshold. Moreover, (8.41) shows that S_n^{-1}P_{\alpha,n}S_n is uniformly bounded, so projecting the right side in (8.74) preserves this sharp estimate. Increase only the corresponding n-threshold until r_{n;B,W,\mathcal P}\leq1. The right-side constant passed to the weighted inverse can therefore be chosen from C_{\rm rhs}, the already fixed structural 1/W_0 bound, and one spare unit, independently of the actual W, box radius, or permitted mesh. Applying Lemma 8.5 gives \begin{equation} |q_0^{\rm reg}|\le C_{\sharp}w\alpha_0,\qquad |q_j^{\rm reg}|\le C_{\sharp}w\alpha_j\quad(j\geq1). \label{eq:smooth-regression-weighted-bound} \tag{8.88} \end{equation} We next prove, rather than assume, the sharper nuisance estimate needed for the lower Schur bound. Put P_{\rm med}:=S_g-q^{\rm reg}\!\cdot\Omega=\sum_pc_pV_p, \qquad c_p=g_p-q_{j(p)}^{\rm reg}. Equations (8.73) and (8.88), together with \sum_jH_j\alpha_j=O(1), give \begin{equation} \|c\|_\infty\ll w,\qquad \sum_p\frac{|c_p|}{p}\leq \sum_p\frac{|g_p|}{p} +\sum_jH_j|q_j^{\rm reg}|\ll w. \label{eq:smooth-postband-coefficient-first-moment} \end{equation} Moreover, the quadratic coefficient bound used below is the direct calculation \begin{equation} \sum_p\frac{c_p^2}{p} \leq2V+2\sum_jH_j(q_j^{\rm reg})^2 \leq2V+2C_\sharp^2w^2\sum_jH_j\alpha_j^2 \ll w^2. \label{eq:smooth-postband-coefficient-second-moment} \end{equation}

First omit the numerical guards and decompose the full tilted mixture into its finitely many exact component cells C_a, with weights \lambda_a. By (7.64), for a fixed reference component one may write \mathbb E_{C_a,\xi}V_p=m_p+e_a(p), \qquad |e_a(p)|\ll_{B,W}\frac1{pL}, uniformly in a,p. Each head indicator H_\nu^\circ is constant on C_a; write that value as h_{\nu,a} and its mixture mean as \bar h_\nu. The law of total covariance and \sum_a\lambda_a(h_{\nu,a}-\bar h_\nu)=0 give the exact cancellation \begin{align} \operatorname{Cov}_\xi(H_\nu^\circ,P_{\rm med}) &=\sum_a\lambda_a(h_{\nu,a}-\bar h_\nu) \sum_pc_p\{m_p+e_a(p)\}\notag\\ &=\sum_a\lambda_a(h_{\nu,a}-\bar h_\nu) \sum_pc_pe_a(p) \ll_{B,W}\frac1L\sum_p\frac{|c_p|}{p} \ll_{B,W}\frac wL. \label{eq:smooth-explicit-head-postband-cancellation}\tag{8.89}\end{align} The physical coordinate satisfies the same bound directly by (7.61): \begin{equation} |\operatorname{Cov}_\xi(R,P_{\rm med})| \leq\sum_p|c_p|\,|\operatorname{Cov}_\xi(R,V_p)| \ll_{B,W}\frac wL. \label{eq:smooth-explicit-physical-postband-cancellation} \tag{8.90} \end{equation}

It remains to check that numerical guards do not destroy these cancellations. Every coordinate of Z is bounded by a constant depending only on W, while \|P_{\rm med}\|_\infty\leq\|c\|_\infty \sum_{W<p\leq y}V_p\ll_WwL. Equations (7.65) and (7.67) therefore show that guard deletion changes each covariance in (8.89)– (8.90) by O_{B,W}\!\left(\frac{wyL^{A+1}}n\right) =o_{n;B,W}(w/L). Since \dim Z is fixed after W, we have proved the vector estimate \begin{equation} \left\|\operatorname{Cov}_\xi(Z,P_{\rm med})\right\| \ll_{B,W}\frac wL+o_{n;B,W}(w/L). \label{eq:smooth-explicit-postband-nuisance-vector} \tag{8.91} \end{equation} Together with \|\Gamma_\xi^{-1}\|\ll_W1, this gives \|a_Z\|=\|\Gamma_\xi^{-1}\operatorname{Cov}_\xi(Z,P_{\rm med})\| \ll_{B,W}\frac wL+o_n(w/L)\leq C_Ww after increasing only the n-threshold for the already fixed box. Thus the nonvanishing comparison constant may still be chosen independently of the box radius. Equations (8.73) and (8.88) imply all three bounds in (8.76). The constants used for the medium-prime vector are the Mertens main-term constants and the continuum weighted inverse underlying C_{\sharp}. The preceding relative display shows that finite nuisance regression changes the medium vector only by a vanishing relative term; its nonvanishing finite-dimensional part is recorded separately in a_Z. Hence C_{\rm cmp} is fixed from the continuum regression before W, the tilt-box radius, or the final mesh is chosen.

For the lower Schur bound we again keep the continuum and arithmetic objects separate. Define \begin{equation} V_c:=\int_{t_0}^1 |\alpha_{\mathcal P}^{(c)}(t)-t|^2\,\frac{dt}{t}, \qquad \alpha_{\mathcal P}^{(c)}(t):=\alpha_j^{(c)}\quad(t\in I_j). \label{eq:smooth-continuum-cell-variance} \end{equation} The moment calculations above and weighted Mertens summation give \begin{equation} V=V_c+o_{n;W,\mathcal P}(w^2),\qquad V_c\asymp w^2. \label{eq:smooth-two-cell-variances} \tag{8.92} \end{equation} To transfer the actual fitted vector q^{\rm reg}\in\mathcal G_n, put \begin{equation} x_n:=S_n^{-1}q^{\rm reg},\qquad x_c:=\Pi_cx_n,\qquad q^{(c)}:=S_cx_c. \label{eq:smooth-regression-continuum-identification} \end{equation} Then q^{(c)}\in\mathcal G_c, and (8.41) together with (8.88) gives \begin{equation} \max_j\left| \frac{q_j^{(c)}}{\alpha_j^{(c)}} -\frac{q_j^{\rm reg}}{\alpha_j}\right| =o_{n;W,\mathcal P}(w). \label{eq:smooth-regression-two-gauges-relative} \tag{8.93} \end{equation}

Let q_{\mathcal P}^{(c)}(t)=q_j^{(c)} on I_j. Harmonic centering of t-\alpha_j^{(c)} on each cell gives the exact continuum identity \begin{align} &\inf_{\lambda\in\mathbb R} \int_{t_0}^1\left|\alpha_{\mathcal P}^{(c)}(t) -q_{\mathcal P}^{(c)}(t)-\lambda t\right|^2\frac{dt}{t} \notag\\ &\qquad=\inf_{\lambda\in\mathbb R} \left\{\|(1-\lambda)\alpha^{(c)}-q^{(c)}\|_{D_c}^2 +\lambda^2V_c\right\} \geq \frac{A_cV_c}{A_c+V_c}\gg V_c, \label{eq:smooth-Schur-geometric-lower}\tag{8.94}\end{align} where A_c:=(\alpha^{(c)})^TD_c\alpha^{(c)}\asymp1, \qquad (\alpha^{(c)})^TD_cq^{(c)}=0. The second relation is the exact continuum gauge. Thus no arithmetic orthogonality is inserted into a continuum integral.

The exact finite-n identity \begin{equation} \sum_{W<p\leq y}t_pv_p(m) =\frac{L+R(m)-\sum_{p\leq W}v_p(m)\log p}{\log y} \label{eq:smooth-exact-log-identity} \tag{8.95} \end{equation} shows that, after row centering, the physical and head columns subtract the function t exactly at finite n. Hence the distance from the kernel direction in (8.94) is the one relevant after the joint nuisance Schur complement.

We spell out the discrete-to-continuum quadratic transfer for the actual non-step coefficient. On I_j\cap[t_0,1], put c_{\mathcal P}^{(c)}(t):=\alpha_j^{(c)}-t-q_j^{(c)}, and extend its value at t_0 linearly to (0,t_0), exactly as in (8.30); denote the resulting function on (0,1] by \widetilde c_{\mathcal P}^{(c)}. Cellwise Mertens quadrature, (8.92), and (8.93) first give \begin{equation} \begin{aligned} \sum_{W<p\le y}\frac{ |(\alpha_{j(p)}-t_p-q_{j(p)}^{\rm reg}) -c_{\mathcal P}^{(c)}(t_p)|}{p} &=o_{n;W,\mathcal P}(w),\\ \sum_{W<p\le y}\frac{ |(\alpha_{j(p)}-t_p-q_{j(p)}^{\rm reg}) -c_{\mathcal P}^{(c)}(t_p)|^2}{p} &=o_{n;W,\mathcal P}(w^2). \end{aligned} \label{eq:smooth-actual-continuum-coefficient-comparison} \end{equation} Here W,\delta,\eta, and hence w>0, have already been fixed before n\to\infty. Thus the relative center convergence in (8.40) is legitimately o_n(w) at this stage; no estimate uniform as w\downarrow0 is being asserted. The first line controls the marked off-diagonal error, and the second, together with boundedness of the continuum covariance operator and Cauchy–Schwarz, changes its quadratic form by o(w^2). The two-index marked formula then gives, uniformly on each fixed box, \begin{equation} \operatorname{Var}_\xi\!\left(\sum_{W<p\le y}c_pI_p\right) =\langle \widetilde c_{\mathcal P}^{(c)}, A\widetilde c_{\mathcal P}^{(c)}\rangle_{\mathcal H} +O_{B,W}(w^2/L)+O(w^2/W) +o_{n;B,W,\mathcal P}(w^2). \label{eq:smooth-nonstep-quadratic-transfer} \end{equation} Here the O(w^2/W) term is precisely the squarefree diagonal correction; it is retained rather than hidden in the little-oh term. At the low boundary, |c_{\mathcal P}^{(c)}(t_0)| \leq \alpha_0^{(c)}+t_0+|q_0^{(c)}| =O(\alpha_0^{(c)}+t_0)=o_n(w). Consequently the low extension changes both the squared norm and the quadratic form by o_n(w^2), by the calculation in (8.32). Therefore (8.12), (8.94), and (8.92) give a fixed positive multiple of w^2 before the following relative perturbations are subtracted. The finite nuisance Schur loss is, by (8.91) and (8.48), \operatorname{Cov}_\xi\!\left(Z, S_g-q^{\rm reg}\!\cdot\Omega\right)^T\Gamma_\xi^{-1} \operatorname{Cov}_\xi\!\left(Z, S_g-q^{\rm reg}\!\cdot\Omega\right) =O_{B,W}(w^2/L^2)+o_{n;B,W,\mathcal P}(w^2). Thus the continuum Poincaré gap is transferred at the relative w^2 scale before the prime-power replacement is made. The upper bound in (8.77) follows directly from the unprojected score and (8.73).

At the w^2 scale every transfer is relative. From the marked squarefree errors and the coefficient bounds already proved, \begin{align} E_{\rm sqfree} &\ll_{B,W}\frac1L\left\{ \left(\sum_p\frac{|c_p|}{p}\right)^2 +\sum_p\frac{c_p^2}{p}\right\} =O_{B,W}(w^2/L),\notag\\ E_{\rm sf,diag} &\ll\sum_p\frac{c_p^2}{p^2} \leq\frac1W\sum_p\frac{c_p^2}{p} =O(w^2/W),\notag\\ E_{\rm disc} &\leq \varepsilon_{W,\delta,\eta}(n) \left\{\left(\sum_p\frac{|c_p|}{p}\right)^2 +\sum_p\frac{c_p^2}{p}\right\} =o_n(w^2), \label{eq:smooth-relative-squarefree-errors}\tag{8.96}\end{align} where quantitative Mertens summation gives \varepsilon_{W,\delta,\eta}(n)\ll 1/\log L+\log L/L+e^{-c\sqrt L}. The endpoint terms in the four-mark formula are included in this \varepsilon; the cofactor margin n/y^4=n^{1/9} makes their total o_n(w^2) for the fixed mesh. For the nuisance block, (8.91), the gap \Gamma_\xi^{-1}=O_W(1), and the exact physical removal in (8.50) give \begin{equation} E_{\rm head/phys}=o_n(w^2). \label{eq:smooth-relative-head-physical-error} \tag{8.97} \end{equation} These displays include respectively the off-diagonal squarefree, squarefree-diagonal, discretization, and head/physical errors; none is being replaced by a generic additive o_n(1).

It remains to justify the relative prime-power statement. Write I_p=1_{p\mid m} and J_p=v_p-I_p. The product-weighted aggregate estimate (7.34), with its transposed and two-extra-power analogues, gives, for p\ne q, \begin{equation} |\operatorname{Cov}(J_p,I_q)| \le C_{\rm pow}\frac{t_pt_q}{p^2q} +\frac{\epsilon_{B,W}(n)}{p^2q} +O_{B,W}(L/n), \label{eq:smooth-JI-bound} \end{equation} with product-weighted analogues for the diagonal, transpose, and \operatorname{Cov}(J_p,J_q). Hence \begin{align} &\left|\operatorname{Var}\left(\sum_pc_pv_p\right) -\operatorname{Var}\left(\sum_pc_pI_p\right)\right| \notag\\ &\quad\le \frac{C_{\rm rel}(C_{\rm cmp})}{W}w^2 +\epsilon^{\rm rel}_{B,W,C_{\rm cmp}}(n)w^2. \label{eq:smooth-power-variance-calculation} \end{align} This is (7.60), applied with (8.76). Together with the O(w^2/W) squarefree diagonal in (8.96) and (8.97), this proves (8.79). For clarity, the complete relative-error ledger is therefore \begin{equation} \begin{array}{c|c} \text{contribution}&\text{bound}\ \\ \hline E_{\rm sf,off}&O_W(w^2/L)\\ E_{\rm sf,diag}&O(w^2/W)\\ E_{\rm pow}&C_{\rm rel}(C_{\rm cmp})w^2/W +\epsilon^{\rm rel}_{B,W,C_{\rm cmp}}(n)w^2\\ E_{\rm disc}&o_{n;W,\mathcal P}(w^2)\\ E_{\rm head}&o_{n;B,W,\mathcal P}(w^2)\\ E_{\rm phys}&o_{n;B,W,\mathcal P}(w^2). \end{array} \label{eq:smooth-complete-relative-error-ledger} \end{equation} The last two entries are listed separately: the head term follows from the finite-dimensional gap (8.48) and the marked head rows, while the physical leading term is removed exactly by (8.50) and its remaining Stieltjes error is o_n(w^2). The constant C_{\rm cmp}, and hence C_{\rm rel}(C_{\rm cmp}), was fixed from the continuum regression before W, the mesh, or the tilt box; in particular it is independent of all three. Choose W using the box-independent constant C_{\rm rel}(C_{\rm cmp}), and only after the compact box is known take n large enough to absorb \epsilon^{\rm rel}_{B,W,C_{\rm cmp}}(n); the lower Schur bound is then preserved.

Finally we verify the marked row without suppressing an orientation, diagonal, or endpoint term. Fix W<p\leq y; all covariances in the following calculation are under \nu_{n,\xi}. Since C_g=\sum_qc_qV_q-a_Z^TZ, first consider the prime part. The squarefree diagonal is O(|c_p|/p)=O(w/p). For q\ne p, (8.35) and (8.36) give \begin{align} \sum_{q\ne p}|c_q|\, |\operatorname{Cov}(I_p,I_q)| &\ll \frac{t_p}{p}\sum_q\frac{t_q|c_q|}{q} +\frac{\|c\|_\infty}{p} \left\{\frac{C_{\rm sf}}W+\epsilon^{\rm sf}_{B,W}(n)\right\} \ll_{B,W}\frac wp. \label{eq:smooth-compensated-squarefree-marked-row}\tag{8.98}\end{align} Here and below the threshold for the vanishing remainder may depend on the already fixed box. The coefficient estimates (8.76) also imply \begin{equation} \sum_q|c_q|\ll wy,\qquad \sum_q\frac{t_q|c_q|}{q}\ll w,\qquad \sum_q\frac{t_q|c_q|}{q^2}\ll\frac wW. \label{eq:smooth-compensated-marked-row-coefficient-ledger} \end{equation} Using V=I+J, the four estimates (7.34)–(7.36) now give the following separate contractions: \begin{align} \sum_{q\ne p}|c_q|\,|\operatorname{Cov}(J_p,I_q)| &\ll \frac w{p^2}+\epsilon_{B,W}(n)\frac w{p^2} +\frac{wyL}{n},\notag\\ \sum_{q\ne p}|c_q|\,|\operatorname{Cov}(I_p,J_q)| &\ll \frac w{pW}+\epsilon_{B,W}(n)\frac w{pW} +\frac{wyL}{n},\notag\\ \sum_{q\ne p}|c_q|\,|\operatorname{Cov}(J_p,J_q)| &\ll \frac w{p^2W}+\epsilon_{B,W}(n)\frac w{p^2W} +\frac{wyL^2}{n},\notag\\ |c_p|\,|\operatorname{Var}(V_p)-\operatorname{Var}(I_p)| &\ll \frac w{p^2}+\epsilon_{B,W}(n)\frac w{p^2} +\frac{wL^2}{n}. \label{eq:smooth-compensated-power-marked-row-ledger}\tag{8.99}\end{align} For the last line, J_p>0 implies I_p=1, whence |\operatorname{Var}(I_p+J_p)-\operatorname{Var}(I_p)| \leq3\mathbb E J_p^2; this is exactly the diagonal estimate in (7.36). All endpoint terms in (8.99) are uniformly o(w/p), since \begin{equation} \sup_{p\leq y}\frac{pyL^2}{n} \leq\frac{y^2L^2}{n}=n^{-5/9}L^2=o(1). \label{eq:smooth-compensated-marked-row-endpoints} \end{equation} Thus the full prime part of C_g contributes O_W(w/p), uniformly on the fixed box once its n-threshold is chosen. We also spell out the nuisance marked family used here and later. Before guard deletion, (7.64) and the same centered component-weight cancellation as in (8.89) give |\operatorname{Cov}(V_p,H_\nu^\circ)|\ll_{B,W}\frac1{pL} for every head coordinate, while (7.61) gives this estimate for the physical coordinate. Guard deletion changes either covariance by at most O_{B,W}(\delta_\xi L), by (7.67). Uniformly for p\leq y, \delta_\xi L\,(pL)\ll_{B,W}\frac{y^2L^{A+2}}n=o(1), so, in the fixed-dimensional Euclidean norm, \begin{equation} \|\operatorname{Cov}_\xi(V_p,Z)\|\ll_{B,W}\frac1{pL}. \label{eq:smooth-explicit-nuisance-marked-family} \tag{8.100} \end{equation} Since \|a_Z\|\ll_Ww, it follows that |\operatorname{Cov}(V_p,a_Z^TZ)|\ll_{B,W}\frac w{pL} \ll_W\frac wp after increasing the threshold for the fixed box. Adding this nuisance row to (8.98) and (8.99) proves (8.78). ◻

Proof. We define one finite-dimensional map and solve all moments simultaneously. For u\in\mathcal G, put \mathscr B(u)=\sum_ju_j\Omega_j. Use the fixed statistic S_g=\sum_pg_pv_p and the nuisance vector Z from (8.23). A gauge-fixed parameter is \xi=(u,a,\lambda)\in\mathcal G\times\mathbb R^{e+1}\times\mathbb R, with exponential score \mathscr B(u)+a^TZ+\lambda S_g. The gauge coordinates are concrete. If e_0,\ldots,e_s are the standard band vectors, use, for 1\leq j\leq s, \begin{equation} q^{(j)}:=e_j-\frac{H_j\alpha_j}{H_0\alpha_0}e_0. \label{eq:smooth-concrete-gauge-basis} \tag{8.106} \end{equation} Since H_0\alpha_0=\delta+o(1), these vectors are well defined and form a basis of \mathcal G=\{u:\alpha^TDu=0\}. Primal coordinates mean coefficients in this basis; dual coordinates mean evaluation on the same basis. On the nuisance block we use the ordinary Euclidean norm. The intrinsic quotient-dual norm is specified below and is the one used for all mesh-uniform estimates. This is merely a nonsingular change of coordinates in (8.21), by (8.95). For a statistic F, put m_F(\xi)=\sum_m z_m(\xi)F(m). To avoid concealing a coordinate factor, regard the quotient moment as an element of the finite dual \mathcal G^*: its value at v\in\mathcal G is the moment of \mathscr B(v). The normalized moment map is \begin{equation} \mathcal M(\xi):=\frac L{q_n}\left( \big[v\mapsto m_{\mathscr B(v)}(\xi)-m_{\mathscr B(v)}(0)\big], m_Z(\xi)-m_Z(0), m_{S_g}(\xi)-m_{S_g}(0)\right). \label{eq:smooth-normalized-moment-map} \end{equation} Its prescribed endpoint is \begin{equation} \tau:=\frac L{q_n}\left( \big[v\mapsto v^T\Delta\big],\ 0,\ \alpha^T\Delta\right). \label{eq:smooth-normalized-moment-target} \end{equation} To specify the quotient norm without a coordinate convention, put d_D:=\frac L{q_n}D^{-1}\Delta. Then \tau_{\mathcal G}(v)=\langle v,d_D\rangle_D,\qquad \|d_D\|_\infty\ll_W1, by (8.102); the actual gauge representative is P_{\alpha,n}d_D. We define \|\tau_{\mathcal G}\|_\infty:=\|P_{\alpha,n}d_D\|_\infty. The projection is uniformly bounded by (8.22). This target is exact: if the Z-moments do not change, then (8.101) says that changing the band moments by \Delta changes the S_g-moment by \alpha^T\Delta. The envelopes (8.102) and (8.103) give \begin{equation} \|\tau_{\mathcal G}\|_\infty+\|\tau_Z\|\ll_W1,\qquad |\tau_g|\ll_Ww. \label{eq:smooth-target-coordinate-sizes} \end{equation}

We display the full inverse used at an arbitrary point of the path. Let \mathscr Q denote the joint vector of the quotient band scores (\mathscr B(q^{(j)}))_{1\leq j\leq s} and Z, in the concrete coordinates (8.106), and write \begin{equation} A_\xi:=\operatorname{Cov}_\xi(\mathscr Q,\mathscr Q),\quad b_\xi:=\operatorname{Cov}_\xi(\mathscr Q,S_g),\quad d_\xi:=\operatorname{Var}_\xi(S_g),\quad \sigma_\xi:=d_\xi-b_\xi^TA_\xi^{-1}b_\xi. \label{eq:smooth-full-covariance-blocks} \end{equation} In these primal/dual gauge coordinates, the Jacobian of \mathcal M is exactly the covariance block \begin{equation} D\mathcal M(\xi)= \begin{pmatrix}A_\xi&b_\xi\\ b_\xi^T&d_\xi\end{pmatrix}. \label{eq:smooth-full-moment-Jacobian} \tag{8.107} \end{equation} For a right side (v_{\mathscr Q},v_g), its inverse is not a sequential nonlinear fit but the single block formula \begin{align} \dot\lambda&=\sigma_\xi^{-1} \{v_g-b_\xi^TA_\xi^{-1}v_{\mathscr Q}\},\notag\\ \dot{\mathscr Q}&=A_\xi^{-1} \{v_{\mathscr Q}-b_\xi\dot\lambda\}. \label{eq:smooth-full-block-inverse}\tag{8.108}\end{align} Thus the quotient and nuisance coordinates are continuously refitted whenever the slow coordinate moves.

We now identify both terms involving A_\xi^{-1} with the operators estimated in the preceding lemmas. This is an exact finite-dimensional calculation. Let \mathbf Q:\mathbb R^s\to\mathcal G be the matrix whose columns are the vectors q^{(j)} in (8.106); thus a concrete quotient coordinate x represents the actual band vector q=\mathbf Qx. Set \begin{align*} C_\xi&:=\operatorname{Cov}_\xi(\Omega,\Omega),& U_\xi&:=\operatorname{Cov}_\xi(\Omega,Z),& \Gamma_\xi&:=\operatorname{Cov}_\xi(Z,Z),\\ h_\xi&:=\operatorname{Cov}_\xi(\Omega,S_g),& z_\xi&:=\operatorname{Cov}_\xi(Z,S_g),& \overline C_\xi&:=C_\xi-U_\xi\Gamma_\xi^{-1}U_\xi^T. \end{align*} Thus \overline C_\xi=q_n^{-1}\mathcal C_\xi^Z. Literally in the concrete primal/dual coordinates of (8.106), \begin{equation} A_\xi= \begin{pmatrix} \mathbf Q^TC_\xi\mathbf Q&\mathbf Q^TU_\xi\\ U_\xi^T\mathbf Q&\Gamma_\xi \end{pmatrix}, \qquad b_\xi=\binom{\mathbf Q^Th_\xi}{z_\xi}. \label{eq:smooth-concrete-full-blocks} \end{equation}

Let a joint right side be (r_{\mathcal G},r_Z), and choose any r^\flat\in\mathbb R^{s+1} representing its quotient functional: r_{\mathcal G}(v)=v^Tr^\flat for v\in\mathcal G. Directly eliminating the nuisance variable in A_\xi(x,a)=(r_{\mathcal G},r_Z), with q=\mathbf Qx, gives the equivalent pair of equations \begin{align} a&=\Gamma_\xi^{-1}(r_Z-U_\xi^Tq),\notag\\ v^T\overline C_\xi q &=v^T(r^\flat-U_\xi\Gamma_\xi^{-1}r_Z) \qquad(v\in\mathcal G). \label{eq:smooth-exact-joint-block-elimination}\tag{8.109}\end{align} Since the Euclidean annihilator of \mathcal G is \operatorname{span}\{D\alpha\}, the second line is, in turn, equivalent to the intrinsic projected equation \begin{equation} P_{\alpha,n}D^{-1}\overline C_\xi P_{\alpha,n}q =P_{\alpha,n}D^{-1} (r^\flat-U_\xi\Gamma_\xi^{-1}r_Z), \qquad q\in\mathcal G. \label{eq:smooth-exact-projected-block-elimination} \tag{8.110} \end{equation} This equivalence uses the actual band values of q; no norm of the coordinate matrix \mathbf Q or its inverse is introduced.

Take first (r^\flat,r_Z)=(h_\xi,z_\xi). Equations (8.109)– (8.110) are then exactly (8.74) and (8.75). If q^{\rm reg}=\mathbf Qx^{\rm reg}, they prove, without an implicit coordinate identification, \begin{equation} A_\xi^{-1}b_\xi=(x^{\rm reg},a_Z),\qquad \sigma_\xi=\operatorname{Var}_\xi (S_g-q^{\rm reg}\!\cdot\Omega-a_Z^TZ) =\operatorname{Var}_\xi(C_g). \label{eq:smooth-exact-full-regression-identification} \tag{8.111} \end{equation} The variance identity follows also by expanding the square and using the two normal equations in (8.109); thus it is an ordinary finite Schur complement, not an additional probabilistic assumption.

For the actual fast target, v_{\mathscr Q}=(\tau_{\mathcal G},0), choose r^\flat=\frac L{q_n}\Delta=Dd_D. Write A_\xi^{-1}v_{\mathscr Q}=(x^{(0)},a^{(0)}) and put u^{(0)}=\mathbf Qx^{(0)}\in\mathcal G. The same exact elimination now reads \begin{equation} P_{\alpha,n}D^{-1}\overline C_\xi P_{\alpha,n}u^{(0)} =P_{\alpha,n}d_D,\qquad a^{(0)}=-\Gamma_\xi^{-1}U_\xi^Tu^{(0)}. \label{eq:smooth-fast-block-elimination} \tag{8.112} \end{equation} By (8.26) and \|P_{\alpha,n}d_D\|_\infty\ll_W1, \begin{equation} \|u^{(0)}\|_\infty\ll_W1. \label{eq:smooth-fast-quotient-bound} \tag{8.113} \end{equation} Moreover |(U_\xi)_{i,*}|\ll_{B,W}H_i/L, \sum_iH_i\ll\log L, and \|\Gamma_\xi^{-1}\|\ll_W1. Therefore \|a^{(0)}\|\ll_{B,W}\frac{\log L}{L}=o_{n;B,W}(1). In particular the fast nuisance coefficient is bounded by an absolute multiple depending only on W, after increasing the threshold for the already fixed box; its bound does not enter the choice of that box.

We may now apply Lemma 8.6 through the exact identification (8.111). It gives \begin{equation} \sigma_\xi\asymp_Ww^2,\qquad |q_j^{\rm reg}|\ll w\alpha_j,\qquad \|a_Z\|\ll_Ww,\qquad \|c\|_\infty=\max_p|g_p-q_{j(p)}^{\rm reg}|\ll w. \label{eq:smooth-full-regression-effective-bounds} \tag{8.114} \end{equation} The only nontrivial target pairing in the numerator of (8.108) is now an exact dual pairing: \begin{align} |b_\xi^TA_\xi^{-1}v_{\mathscr Q}| &=|(A_\xi^{-1}b_\xi)^Tv_{\mathscr Q}| =\left|\frac L{q_n}\sum_jq_j^{\rm reg}\Delta_j\right|\notag\\ &\ll_Ww\sum_jH_j\alpha_j\ll_Ww. \label{eq:smooth-exact-slow-target-pairing} \end{align} Here the nuisance target is zero, (8.102) bounds \Delta_j, and \sum_jH_j\alpha_j=\sum_{W<p\leq y}t_p/p=O(1). Together with |v_g|\ll_Ww, the first line of (8.108) therefore gives |\dot\lambda|\ll_Ww^{-1}. Its second line, interpreted through (8.111) and (8.112), is the exact pair \dot u=u^{(0)}-\dot\lambda q^{\rm reg},\qquad \dot a=a^{(0)}-\dot\lambda a_Z. Consequently, prime by prime, \begin{equation} \dot u_{j(p)}+\dot\lambda g_p =u^{(0)}_{j(p)}+\dot\lambda (g_p-q_{j(p)}^{\rm reg}) =u^{(0)}_{j(p)}+\dot\lambda c_p. \label{eq:smooth-exact-effective-prime-coefficient} \end{equation} Equations (8.113), (8.114), and the bounds for \dot\lambda,a^{(0)} now prove \begin{equation} |\dot\lambda|\ll_Ww^{-1},\qquad \max_{W<p\le y}|\dot u_{j(p)}+\dot\lambda g_p| +\|\dot a\|+w|\dot\lambda|\leq C_*. \label{eq:smooth-effective-block-inverse-bound} \tag{8.115} \end{equation} The leading continuum blocks give a constant C_{{\rm inv},W} independent of the tilt box and the regular mesh. Before this constant or the ODE box is introduced, W has already made the box-independent terms C_{\rm pow}/W, C_{\rm pow}^{\sharp}/W, and C_{\rm rel}(C_{\rm cmp})/W smaller than the fixed continuum gaps. All dependence on a later fixed tilt box is in \epsilon_{B,W}(n), its explicitly rescaled relative version \epsilon^{\rm rel}_{B,W,C_{\rm cmp}}(n), or the analogous fixed-mesh remainders, all of which tend to zero with n. Fix C_*=2C_{{\rm inv},W}. After the box below is prechosen, only the threshold for n is increased so that all these B_*,W-dependent vanishing errors use the spare factor two. Thus (8.115) holds with this noncircular C_*.

We now remove the possible bootstrap circularity. Define the effective norm \|\xi\|_{\rm eff}:= \max_{W<p\le y}|u_{j(p)}+\lambda g_p| +\|a\|+w|\lambda|. Choose in advance B_*=4C_*, after W but before the ODE. On the prechosen box \|\xi\|_{\rm eff}\leq B_*, the logarithm of the Radon–Nikodym derivative is bounded by \frac{B_*}{L}\sum_{p>W}v_p(m)+O_W(B_*/L) \leq\frac{B_*}{\log W}+o_W(1). This gives a fixed dominating density. More sharply, the tilted marked lemmas show that the leading h,K blocks are unchanged and that the box-dependent local errors are O_{B_*,W}(1/L). After increasing the fixed-mesh threshold for n, all covariance gaps and row estimates used in (8.115) remain within the reserved factor two. In particular its right side is at most 2C_{{\rm inv},W}=C_*<B_*/2 throughout this box. The box was chosen before the path, so this conclusion is not being used to define its own domain.

The constants C_* and B_* are uniform over all permitted regular meshes. They may therefore be fixed after W and before a particular mesh is instantiated; after the mesh and the box are fixed, only the threshold for n is increased. In particular, the choice of W never depends on B_*. Fix such a mesh and such an n. In the gauge coordinates the parameter space is finite-dimensional and \mathcal M is C^\infty. On the closed effective box \mathcal K_*:=\{\xi:\|\xi\|_{\rm eff}\leq B_*\} the preceding covariance gaps make D\mathcal M(\xi) invertible and \|D\mathcal M(\xi)^{-1}\tau\|_{\rm eff}\leq C_* \qquad(\xi\in\mathcal K_*). Solve the straight-target ODE \begin{equation} \xi'(t)=D\mathcal M(\xi(t))^{-1}\tau,\qquad \xi(0)=0. \label{eq:smooth-moment-map-ODE} \tag{8.116} \end{equation} Let [0,T_{\max}) be its maximal solution interval while the path remains in the interior of \mathcal K_*. Local existence follows from (8.107). More explicitly, D\mathcal M is smooth and invertible on a neighborhood of the closed box, so \xi\mapsto D\mathcal M(\xi)^{-1}\tau is locally Lipschitz there; the finite-dimensional Picard–Lindelöf theorem gives both local existence and uniqueness. For t<\min(T_{\max},1), integration gives \|\xi(t)\|_{\rm eff}\leq C_*t\leq C_*=B_*/4. If T_{\max}\leq1, the same derivative bound makes \xi(t) Cauchy as t\uparrow T_{\max}. Its limit lies in the strictly smaller box \|\xi\|_{\rm eff}\leq B_*/4, where the Jacobian remains invertible. The local existence theorem therefore restarts the solution past T_{\max}, a contradiction. Hence T_{\max}>1. The chain rule gives \frac d{dt}\mathcal M(\xi(t))=\tau, \qquad \mathcal M(\xi(1))=\tau. To check the converse coordinate implication, let \Delta^{\rm out} be the resulting vector of band-moment increments. Equality against every v\in\mathcal G implies \Delta^{\rm out}-\Delta=cD\alpha for some scalar c, because the ordinary annihilator of \mathcal G is \operatorname{span}(D\alpha). Preservation of the Z-moments preserves the physical and head moments, so the exact logarithmic identity turns the slow equation into \alpha^T\Delta^{\rm out}=\alpha^T\Delta. Hence c\,\alpha^TD\alpha=0, and c=0. Thus \mathcal M(\xi(1))=\tau is precisely all band, head, physical, and mass equations. Existence, rather than strict convexity, has now been proved; strict convexity only supplies uniqueness in the fixed gauge.

For later use we record the primewise row bound for every block of this inverse. Along (8.116), \begin{align} \left|\operatorname{Cov}_\xi(v_p,\dot{\mathscr Q}_{\rm quot})\right| &\ll_Wp^{-1},\notag\\ \left|\operatorname{Cov}_\xi(v_p,\dot a^TZ)\right| &\ll_W(pL)^{-1},\notag\\ \left|\dot\lambda\operatorname{Cov}_\xi(v_p,C_g)\right| &\ll_Ww^{-1}\frac wp\ll_Wp^{-1}. \label{eq:smooth-each-block-prime-row}\tag{8.117}\end{align} For the first line one must include the leading row, not only its arithmetic error. If u_j are the bounded effective quotient coefficients, then \left|\operatorname{Cov}_\xi\left( v_p,\sum_ju_j\Omega_j\right)\right| \le\frac{C_W}{p} +C_W\frac{t_p}{p}\sum_{W<q\le y}\frac{t_q}{q} +o(1/p) \ll_W\frac1p. This uses the diagonal multiplier, the product kernel (8.11), and the box-uniform full-valuation transfer (8.47). The second line follows from the full head/physical marked family (8.100) (whose physical coordinate is (7.61)), and the third from (8.78). Multiplication by q_n/L\asymp N/L and integration over 0\leq t\leq1 show that the bridge changes every marked moment by O_W(N/(pL)). Combined with (6.55), this proves (8.105).

Only z^{\rm act} was reweighted. The effective box bounds its density distortion between two fixed positive constants. Hence, for a constant C_{B_*,W} independent of m,n, \begin{equation} 0\leq z_m(\xi(1))\leq C_{B_*,W}z_m^0 \leq\frac{C'_{B_*,W}}L. \label{eq:smooth-final-active-coordinate-bound} \end{equation} On a shared clean coordinate the only frozen summand is the protected layer, also O_W(1/L); the top support is disjoint. Therefore the complete coordinate weight is between zero and (C'_{B_*,W}+C''_W)/L<1 for all sufficiently large n. The positive zero-head pool persists. The disjoint top support and the frozen additive protected summand are untouched, proving the remaining margin assertions. ◻

Proof. Choose \eta_0 below the continuum inverse thresholds in Lemma 8.4 and Lemma 8.5. The quotient and nuisance rows in (8.117) are O_W(1/p), with constants independent of the permitted mesh. The slow row is uniform for the nontrivial reason \begin{equation} |\dot\lambda|\, |\operatorname{Cov}_\xi(v_p,C_g)| \ll_W\frac1w\frac wp\ll_W\frac1p, \label{eq:smooth-uniform-slow-row-cancellation} \end{equation} so its constant is also independent of w. Integrating the three rows for 0\leq t\leq1 and multiplying by q_n/L\asymp N/L gives a mesh- and w-independent addition to the rough-stage marked-rate constant. Enlarge the previously fixed C_{\rm tan} by this amount. All box- and mesh-dependent errors tend to zero after the mesh is fixed and are absorbed by increasing only n_0. This proves the stated quantifiers. ◻

Proof. Put \delta_{\rm hd}=\varphi(P_{\rm hd})/P_{\rm hd}. The interval length is |I_{uv}|=\frac nu \left(2-\frac uv-\frac{Kh}{n}\right)+O(1). Counting the fixed reduced residue classes modulo P_{\rm hd} gives \begin{equation} \#\{a\in I_{uv}:(a,P_{\rm hd})=1\} =\left\{\delta_{\rm hd}\left(2-\frac uv\right)+o(1)\right\} \frac nu. \label{eq:tangent-raw-list} \tag{9.4} \end{equation} The error is uniform because n/u\geq n/y=n^{7/9}.

We next remove the exceptional rows. Write a=Rb, where R=R_y(a) and P^+(b)\leq y. If X_R=2n/R<X_0=n^{\delta_*}, then \begin{equation} R>\frac{2n}{X_0},\qquad b<\frac{2X_0}{u}. \label{eq:tangent-exceptional-factorization} \tag{9.5} \end{equation} For an upper bound, discard the additional lower cutoff R>2n/X_0. For fixed b, the full physical interval forced by a\in I_{uv} has location and length both \asymp_{r_0}n/(ub); whenever it can contain an exceptional multiplier, (9.5) makes this length \gg_{r_0}n/X_0. The interval Selberg sieve (2.11) applies uniformly: its lcm remainder level is y^4=n^{8/9}, while n/X_0=n^{1-\delta_*}, \qquad \delta_*<1/18. For a fixed sieve constant C_{\rm sv}>0, it follows that the number E_{uv} of exceptional multipliers is \begin{align} E_{uv} &\ll\frac{n}{u\log y} \sum_{b\leq2X_0/u}\frac1b +\frac{X_0}{u}\frac{y^4}{(\log y)^2}\notag\\ &\leq \left\{C_{\rm sv}\frac{\delta_*}{\theta}+o(1)\right\} \frac nu. \label{eq:tangent-exceptional-loss}\tag{9.6}\end{align} Indeed the second term is o(n/u), since \frac{X_0y^4}{n(\log y)^2} =n^{\delta_*+8/9-1+o(1)}=o(1). If u>2X_0, this sum is empty. Notice that the loss is a small fixed fraction, not o(n/u).

It remains to check the actual guards. We give the complete census of those guards that can meet a lower endpoint ua or va. Every modified prefix anchor is Pq, where P>y is its unique rough marker and every prime factor of q is at most 2R+1\leq W; hence it contains no medium label. A row-zero anchor is a prime greater than n, so it contains no medium label either. A promoted anchor is 2^{k_p}p^{e_p}. If p>y, its only smooth prime is 2; if p\leq y, it is one of at most \pi(y) promoted smooth anchors. Consequently only O(\pi(y)) anchors can be divisible by a prime in (W,y].

There are O(yL) component factors in the fully reserved bank by (5.14); taking both states changes only the implied constant. These are the only remaining lower guards. Indeed, G_{\rm fix}\subset(2n,M] by (6.37), and every member of E_{\rm donor} lies above 2n, so neither class can equal a lower endpoint. Thus the number of endpoint-relevant guards is O(\pi(y))+O(yL)=O(yL). For a fixed label u, the equation ua=g determines at most the one integer multiplier a=g/u; the same statement holds for v. Therefore the guard deletion is, uniformly in the permitted pair, \begin{equation} O(yL)=o(n/u), \label{eq:tangent-guard-loss} \tag{9.7} \end{equation} because u\leq y and y^2L/n=o(1). The dedicated-row condition costs zero because \mathscr D_{\rm row}=\varnothing. More generally, the quantitative data in (6.42) would bound its loss by O\{yL(y/u+1)\}=o(n/u). Thus the definition cannot call on a row whose flexible quota was set to zero. Terminal exceptional rows were already removed in (9.6). Forced exceptional upper factors lie above 2n and cannot occur in a lower list.

Combining (9.4), (9.6), and (9.7), the one-time inequality (6.1) gives C_{\rm sv}\frac{\delta_*}{\theta} <\frac14\delta_{\rm hd}(2-r_0), \qquad \delta_{\rm hd}(2-r_0) -C_{\rm sv}\frac{\delta_*}{\theta} >\frac34\delta_{\rm hd}(2-r_0)>0. Thus, without changing \delta_*, one may take \kappa=\frac12\left\{\delta_{\rm hd}(2-r_0) -C_{\rm sv}\frac{\delta_*}{\theta}\right\}>0. ◻

Proof. Fix one band and write its consecutive prime cells as \mathcal Q_1,\ldots,\mathcal Q_m. Put s_i:=\sum_{p\in\mathcal Q_i}r_p,\qquad F_i:=\sum_{h\leq i}s_h\quad(1\leq i<m),\qquad F_0=F_m=0. The last equality is exactly the band-balance assertion in (9.1). If the boundary between \mathcal Q_i and \mathcal Q_{i+1} is Q=n^t, band balance also gives |F_i| =\left|\sum_{Q<p\leq n^b}r_p\right|. Using the pointwise bound in (9.1) and subtracting the two instances of (2.9), at their actual endpoints, we obtain uniformly \begin{equation} |F_i|\ll C_{\rm tan}\frac NL \left\{\log\frac bt+\frac1{tL}\right\}. \label{eq:tangent-cut-load} \tag{9.17} \end{equation} The term 1/(tL) is the endpoint error O(1/\log Q); this remains valid in the moving first band, where t\geq\log W/L. Thus no fixed-positive lower exponent is being assumed.

We next give the promised flow algorithm. Across boundary i, send the signed load F_i from \mathcal Q_i to \mathcal Q_{i+1}: if F_i<0, reverse the direction. On either side distribute its absolute value uniformly among all primes of that cell. The two resulting lists of nonnegative port masses have the same total |F_i|; the usual greedy matching of their first still-positive entries realizes this transport with at most |\mathcal Q_i|+|\mathcal Q_{i+1}|-1 positive edges.

Let d_p be the boundary-flow divergence at p, with outgoing minus incoming convention. Since the signed boundary load on the right of cell i is F_i, while that on its left is F_{i-1}, \sum_{p\in\mathcal Q_i}d_p=F_i-F_{i-1}=s_i. Consequently q_p:=r_p-d_p has sum zero in each cell. Greedily match the positive and negative q_p’s inside that cell, directing flow from the former to the latter. This uses at most |\mathcal Q_i|-1 positive edges and gives divergence exactly q_p. Adding the boundary and internal flows proves (9.11). Endpoints of an internal edge lie in one cell, and endpoints of a boundary edge lie in adjacent cells. The possible merged terminal cell was allowed for in the exponent \rho^3, so (9.12) follows.

This construction also proves the quantitative claims without an unproved distribution step. Put m_i:=|\mathcal Q_i|. The total boundary traffic incident to p\in\mathcal Q_i is at most b_p:=\frac{|F_{i-1}|+|F_i|}{m_i}. The internal traffic incident to p is |q_p|, and |d_p|\leq b_p; hence \begin{equation} f_p\leq |r_p|+2b_p. \label{eq:tangent-port-load-identity} \tag{9.18} \end{equation} For p\asymp_\rho Q=n^t, (9.10) says m_i\asymp_\rho p/(tL). Substitution of (9.17) into (9.18), together with |r_p|\leq C_{\rm tan}N/(pL), proves (9.15).

There are only a bounded number of greedy matchings incident to a cell: one internal matching and at most two boundary matchings. Summing their edge counts proves (9.13). Moreover, the cross-boundary traffic is \sum_i|F_i|, while the internal traffic is \frac12\sum_p|q_p| \leq\frac12\sum_p|r_p|+\sum_i|F_i|. Therefore \begin{equation} \sum_{u,v}f_{uv} \leq\frac12\sum_{W<p\leq y}|r_p|+2\sum_{\rm boundaries}|F_i|. \label{eq:tangent-traffic-ledger} \tag{9.19} \end{equation} The first term is \begin{equation} \frac12\sum_{W<p\leq y}|r_p| \ll C_{\rm tan}\frac{N\log L}{L}=o(N) \label{eq:tangent-direct-traffic} \end{equation} by (2.9). Consecutive cell boundaries have exponent spacing \asymp_\rho1/L, so (9.17) and an upper Riemann sum give \begin{align} \sum_{\rm boundaries}|F_i| &\ll_\rho C_{\rm tan}N\left\{ \int_0^{\delta_b}\log\frac{\delta_b}{t}\,dt + \sum_{j=1}^s\int_{a_{j-1}}^{a_j} \log\frac{a_j}{t}\,dt\right\}+o(N)\notag\\ &\ll_\rho C_{\rm tan}N(\delta_b+\theta\eta)+o(N). \label{eq:tangent-path-traffic}\tag{9.20}\end{align} Here the accumulated 1/(tL) endpoint terms are O(N\log L/L)=o(N). Also \int_0^{\delta_b}\log(\delta_b/t)\,dt=\delta_b, while \int_a^b\log(b/t)\,dt \leq\frac{(b-a)^2}{a} \leq\eta(b-a) for every regular positive band; summing gives O(\theta\eta). Equations (9.19)–(9.20) prove (9.14). Finally, t\log(b/t)\leq b-t; it is at most \delta_b in the first band and at most \theta\eta in a positive regular band. This proves (9.16) and completes the construction. ◻

Proof. The row identities are unchanged by the tangent and were integer identities before it. Primes p\leq W were exact after the head fit, primes W<p\leq y are exact by (9.30), and primes p>y were exact by the complete rough-signature equations. This proves the valuation identity simultaneously at all primes. Coordinate feasibility was proved above. ◻

Proof. Freeze every coordinate as soon as it becomes integral. At an intermediate stage let F be the set of strictly fractional coordinates and put m=|F|. A row meeting F contains at least two members of F: after the frozen integral coordinates in that row are removed, the sum of the remaining coordinates is an integer, which one strictly fractional coordinate cannot supply. Hence at most m/2 row equations are active.

Retain every active row equation and every column having more than 4d members of F. There are at most dm coordinate–column incidences, so fewer than m/4 column equations are retained. The retained system is a finite real linear system and has rank less than 3m/4<m. It therefore has a nonzero null vector z=(z_a)_{a\in F}. Its complete feasible movement interval is I_z:=\{t\in\mathbb R: 0\leq x_a+tz_a\leq1\ \text{for every }a\in F\}. Because every x_a, a\in F, is strictly between zero and one, 0 is an interior point of I_z. Since z\ne0, this is a bounded closed interval with two finite endpoints. Move to either endpoint. Every retained equation is preserved and at least one additional coordinate reaches 0 or 1; it is then frozen. Consequently the procedure stops after at most |\mathcal A| iterations with an integral vector.

Row equations are never dropped, which proves (10.5). At the first stage when a column is not retained, it has at most 4d fractional coordinates. Its value was preserved at every earlier stage, and only those then-fractional coordinates can subsequently change. Each changes by at most one, so its total change from the original vector is at most 4d. A column retained through the last stage is preserved exactly. This proves (10.6). ◻

Proof. Use Lemma 10.1. Equations (10.8)–(10.9) put the integer error e in the universal box. Apply Lemma 5.3 to -e. It supplies a subcollection of whole paths to toggle. Set \varepsilon_g=1 exactly for those paths and \varepsilon_g=0 otherwise. The operation is the state replacement \begin{equation} G_{\rm bank}^0=\bigsqcup_{g\in\mathcal B}G_g^0 \quad\rightsquigarrow\quad G_{\rm bank}^{\rm fin} =\bigsqcup_{g\in\mathcal B}G_g^{\varepsilon_g}; \label{eq:bank-state-replacement} \end{equation} no bank factor is multiplied onto the already charged base. An untoggled path remains literally in its base state, whereas a toggled path is replaced once, simultaneously in all of its components.

By universality and the definition of a path change, \begin{equation} v(G_{\rm bank}^{\rm fin})-v(G_{\rm bank}^0) =\sum_{\varepsilon_g=1} \{v(G_g^1)-v(G_g^0)\}=-e. \label{eq:bank-final-valuation-change} \tag{10.10} \end{equation} Likewise, for every complete signature S, m_S(G_{\rm bank}^{\rm fin})-m_S(G_{\rm bank}^0) =\sum_{\varepsilon_g=1} \{m_S(G_g^1)-m_S(G_g^0)\}=0 by (5.2). Thus every fixed row token charged in the base remains one fixed row token after exactification, and the flexible row equations preserved by floating rounding remain valid.

The rounded flexible change is +e, while (10.10) is -e. Starting with (10.3), their sum leaves every prime valuation equal to v(Y). Finally, both possible states of every path were jointly guarded before the choice of the toggle subplan. Therefore an arbitrary subplan, including the one just selected, is collision-free internally and disjoint from G_{\rm fix}, the anchors, and all flexible candidates. This proves both the product identity and the asserted distinctness. ◻

Proof. We first discharge, in order, the five hypotheses of Proposition 10.2.

1. Proposition 9.3 gives 0\leq x_a\leq1 and every integer complete-signature row sum.

2. The joint guarding required there is supplied by Lemmas 5.1 and 6.5, together with the anchor avoidance in Lemma 5.2. In particular, donor occurrences are not separately reinserted.

3. The charged base is exactly (10.1); the simultaneous combined charge is (6.50); and the valuation identity after that charge is (10.3).

4. The donor-backed row-token construction in Section 5.5 gives nonnegative quotas (6.40), and every complete path obeys the all-factor identity (5.2).

5. The required two-sided box universality is precisely Lemma 5.3.

Thus there is no additional assumption at the final interface.

Take X, the state indices (\varepsilon_g), and G_{\rm bank}^{\rm fin} from Proposition 10.2, and define \begin{equation} \mathcal G_{\rm res}:= G_{\rm fix}\ \cup\ G_{\rm bank}^{\rm fin}\ \cup\{a\in\mathcal A:X_a=1\}. \label{eq:final-residual-set} \end{equation} The union is disjoint by that proposition. Since v(B_0)=v(G_{\rm fix})+v(G_{\rm bank}^0) and the rounded-minus-fractional convention (10.7) gives \sum_{a\in\mathcal A}X_av(a) =\sum_{a\in\mathcal A}x_av(a)+e, the state-replacement identity (10.10) and the exact post-tangent certificate give, at every prime, \begin{align} v\left(\prod_{a\in\mathcal G_{\rm res}}a\right) &=v(B_0)+\sum_{a\in\mathcal A}x_av(a)+e-e \notag\\ &=v(Y). \label{eq:final-residual-valuations} \end{align} Equality at every prime proves the integer identity \begin{equation} \prod_{a\in\mathcal G_{\rm res}}a =Y=\frac{T_n}{D'}. \label{eq:final-residual-product} \end{equation}

The full central-anchor construction, with the guarded modifications of Section 5.4, gives a set \mathcal H'\subset(n,2n], disjoint from \mathcal G_{\rm res}, and \prod_{a\in\mathcal H'}a=C_nD' by (5.17). Every factor in \mathcal G_{\rm res} is legal and lies in (n,M]: this holds for G_{\rm fix} and the two bank states by their construction and for the flexible set by the definition of \mathcal A. Hence \begin{equation} \mathcal S:=\mathcal H'\cup\mathcal G_{\rm res} \label{eq:final-complement-subset} \end{equation} consists of distinct integers in (n,M] and satisfies \begin{equation} \prod_{a\in\mathcal S}a =(C_nD')\frac{T_n}{D'} =C_nT_n =Q(n,M). \label{eq:final-complement-product} \end{equation} ◻