The lifted shift cancellation on the successor endpoints: a reconstruction

E824 uniformity · The sign audit · The weighted-dilation manuscript

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This page tests premise P of the conditional route to Problem 824: the uniform shifted-correlation estimate that OpenAI's weighted-dilation manuscript states as its Theorem 5.1. The reconstruction follows its Sections 3–5 and closes the quantitative estimates for the endpoint class consumed by the Type II argument for $mn=2u-1$. This is an author-recorded source proof reconstruction by Plasma AI, with the classical inputs specified below. It supplies no independent acceptance of the unrefereed manuscript. The route's other premise, the elementary estimates (E), is not established here. Section numbers, equation numbers and result labels refer to OpenAI's September 24, 2026 release manuscript.

The reconstruction uses the retained library transcription of Sections 2–5, with the factorial adjoint formulas, endpoint statement and endpoint norm calculation also checked against the PDF, pages 18–19 and 35–36. The log-cell error bound was also read on PDF page 31. The remaining proof inspection is textual; this is not a visual comparison of every page. The same-paper lemmas below are reconstructed as part of the argument, rather than assumed from the statement of Theorem 5.1.

The consumed statement

Write $L=\log x$, $T=\log L$ and $W=\exp(\sqrt L)$. The prime groups, the marked weight $W_\ell$ with $\ell\ge1$, and the group-free part $n_*$ are those of the manuscript's Definition 3.1 and Equation (5.1): pairwise disjoint small groups of primes in $[\exp(L^a),\exp(L^b)]$ and big groups that are intervals of primes in $[\exp(L^c),\exp(L^d)]$, with $0<a<b<c<d<0.47$, harmonic masses in $[v_-,v_+]$, and between fixed multiples of $T$ groups of each kind.

More explicitly, $n_*$ is the largest divisor of $n$ prime to all group primes, $\omega_g(n)$ counts distinct divisors in group $g$, and, for a fixed $0<q<1$,

$$W_\ell(n)=\prod_g q^{\omega_g(n)-\ell}\frac{(\omega_g(n))_\ell}{V_g^\ell}, \qquad V_g=\sum_{p\in\mathcal P_g}\frac1p.$$

An endpoint is a function of $n_*$ of the form

$$F_0(n)=\sum_{mpr=n_*}\alpha_m\chi_0(m)m^{i\sigma_0}\, \mathbf 1_{p\in I_p,\ P^-(p)>W}\,p^{i\sigma_1}c_r,$$

where $p$ runs over integers, $|\alpha_m|,|c_r|\le L^C$, $\alpha_m=0$ unless $P^-(m)>W$, $I_p\subset[x^\tau,x^\eta]$ is an interval with fixed $0<\tau<\eta<1/4$, $\chi_0$ has modulus at most $L^C$, and $|\sigma_0|,|\sigma_1|\le L^C$. The sequence $\alpha$ satisfies $(\mathrm D_B)$ if

$$\Bigl|\sum_{m\in I}\frac{\alpha_m\chi(m)m^{iu}}m\Bigr|\le L^{-B}$$

for every interval $I\subset[1,x^2]$, every character $\chi$ of modulus at most $L^B$ (principal and imprimitive ones included) and every $|u|\le L^B$.

Premise P, on the consumed class. Fix the group data, $C$, $D_0$, $\tau$, $\eta$ and a compact interval $[c_\Psi,C_\Psi]\subset(0,\infty)$. There is a fixed $B$ such that, for all large $x$, all integers $0<|k|\le L^C$ of either sign, all $xL^{-C}\le Z\le xL^C$, all $|a_1|,|a_2|\le L^C$, every $\Psi$ supported in $[c_\Psi,C_\Psi]$ with $\|\Psi^{(j)}\|_\infty\ll_jL^{C(j+1)}$ with fixed implied constants, and all endpoints $F_0,G_0$ whose first sequence $\alpha$ satisfies $(\mathrm D_B)$,

$$\sum_{\substack{z>0\\ z+k>0}}\frac{\Psi(z/Z)}z\,z^{ia_1}(z+k)^{ia_2}\, \overline{F_0(z)}\,G_0(z+k)\,W_\ell(z)\,W_\ell(z+k)\ll L^{-D_0}.$$

For the successor Type II step the exact pair is

$$\begin{aligned} F_0(v)&=\sum_{mpr=v_*}\alpha_m\chi(m)m^{i(\varrho-\xi_2)} \mathbf1_{p\in I_{**},\ P^-(p)>W}p^{-i\sigma}\overline{c_r},\\ G_0(v)&=\sum_{mpr=v_*}\alpha_m\chi(m)m^{i(\xi_1+\varrho)} \mathbf1_{p\in I_{**},\ P^-(p)>W}p^{-i\sigma}\overline{c_r}. \end{aligned} \tag{S}$$

Here $\chi$ is modulo $2|k|$, $I_{**}=[x^{\tau_*},x^{\eta_*}]$ is the common band interval with $0<\tau_*<\eta_*<b_*/4<1/4$, and $c_r$ is the common residual coefficient from that band partition, with no roughness condition. The first factor is the original $\alpha$, including its interval restrictions. All frequencies, coefficient sizes and cutoff derivatives have fixed logarithmic bounds. Substitution in P uses $a_1=\varrho+\xi_3$, $a_2=-\varrho$ and $\Psi(t)=t\psi_Z(t)$; the separated unweighted correlation is $Z$ times the displayed harmonic one.

Thus it suffices to establish the stated wider endpoint class. Proving that class does not enlarge the application: (S) keeps the original sequence, common rough interval and common residual coefficient. The frequency and coefficient bounds are fixed before $D_0$ and $B$; the final proxy cell precision is chosen after $B$.

The proof chain

The proof has four components. Their estimates are composed below, with the quantifier order recorded at the end.

1. The physical lift (Section 5). The correlation is the raw-pattern term of a pairing $\langle f,\mathcal Ag\rangle$ for the physical dilation operator $\mathcal A$ of Definition 3.3. Positions move by $kD$, where $D$ is the product of the shared and free labels. The vectors $f(n)=n^{-ia_1}F_0(n)q^{(\omega(n)-Ms)/2}$ and $g$ are mark-independent. The added comparison patterns of Section 4 produce shifted correlations with shift $kD_AD_B$, where $D_A,D_B$ are free products of size $\exp(L^c)\le D_AD_B\le x^{o(1)}$.
2. Transference (Section 3). If the ideal operator $\mathcal T(\Theta)$ on independent big-group labels has norm at most $L^{-E_{\rm id}}$ uniformly in $\Theta$, then the root-averaged moment $\langle u_n,(\mathcal A\mathcal A^*)^Ru_n\rangle$ with $R=\lceil L^{0.52}\rceil$ is at most $L^{-2RE}$ (Theorem 3.5), and Corollary 3.11 turns this into $|\langle f,\mathcal Ag\rangle|\ll XL^{-E+2C_3}$ for a mark-independent $f$.
3. The ideal kernels (Section 4). Comparison patterns with log-cell and character kernels make $\sup_{\Theta,|\zeta|\le L^{C_4}}\|\mathcal T(\Theta)\|\le L^{-E_{\rm id}}$ (Theorem 4.1).
4. The comparison correlations (Section 5). Their multiplier $\mathbb E\,b_1(D_A)b_2(D_B)e(-k\theta D_AD_B)$ is small off major arcs by an integer bilinear estimate. On the arcs, the first endpoint's local Fourier energy reduces to a Mellin integral of $P_sP_l\mathcal R$: a small marked prime, the long rough integer, and $\alpha$ times the residual factor. The small-prime polynomial, an oscillation bound for the long rough factor, and $(\mathrm D_B)$ handle the three time ranges.

What was checked, and how

Each item below names the step and what was verified. "Re-derived" means that the argument was rebuilt independently of the source's wording; "checked" means that each inequality of the source argument was verified in order.

Corollary 3.11, re-derived. The Gram matrix $(\langle u_n,(\mathcal A\mathcal A^*)^Ru_m\rangle)$ over a block of positions is positive semidefinite, so its top eigenvalue is at most its trace. Spectral Hölder gives $|\langle f_j,\mathcal Ag_j\rangle|\le|g_j|,|f_j|^{1-1/R} \langle f_j,(\mathcal A\mathcal A^*)^Rf_j\rangle^{1/(2R)}$. Blocks of length $H$ with $\log H=O_J(TL^d)$ make the factor $(H\sup|f_n|^2)^{1/(2R)}$ bounded, because $d<0.47<0.52$ and $\sup|f_n|\le\exp(O(\sqrt L))$. Hölder with exponents $2$, $2R/(R-1)$ and $2R$ over the blocks gives the stated bound.

Lemma 3.6, re-derived. With $Q_{\rm act}$ the product of the active primes along a path, $\log Q_{\rm act}=O_J(NTL^d)=o(L)$ needs $0.52+d<1$. Bonferroni truncation at degree of order $NT$ has $\exp(o(L))$ terms whose moduli multiply to $\exp(o(L))$; the counting error $\exp(o(L)-\gamma L)$ is relative to the main term $Q_{\rm act}^{-1}\exp(-O(NT))$. The deterministic damping of active primes can be tiny, of order $q^{O(N^2Ms)}$, but it is the same factor in both models and cancels exactly, so the relative estimate stands.

Lemma 3.7, re-derived on the path weights. For an active big prime, the factor $p^{-1}\prod_{i:\,p\notin\Lambda_i}(1-\eta_iH_i)$ expands by the two telescoping identities into one lifespan, with an optional ghost birth and ghost termination at hits outside the active visits. For a never-active prime the factor is $b_p$ plus orphan intervals. Relative to the baseline $\prod_pb_p$, a lifespan with $r$ active runs carries exactly $V_g^{-r}/(pb_p)$. On the operator side a fresh birth costs $\widetilde\mu_g(p)$, each promotion $V_g^{-1}$, and a ghost birth $V_g\widetilde\mu_g(p)$ times $-\eta_j/\rho$. Following one particle through the visit and edge operations, its $\rho$ factors combine to $-\eta_j$ at a ghost birth, $q_t$ at each pending hit, and $-\eta_i$ at a ghost termination, and a nonzero residue gives no factor; this is the residue-side weight. The local choices correspond one to one with lifespan terms under the distinct-birth rule: dropping an unshared label ends a lifespan, storing it and later terminating it is the suffix term, a ghost birth followed by promotion is the prefix term, and the injection of promoted slots is forced because each pending prime has one birth. The path expression uses row actions only, so the factorial memory measure enters the norm bounds but not the identity. The adjoint formula (3.19) and the accounting (3.25) were checked as written.

Lemma 3.8 and the truncation. To obtain the ghost adjoint, partition an input memory into $k$ continuing and $t$ terminated particles. Its factorial measure times the subset count is $\binom{k+t}{t}/(k+t)!=1/(k!t!)$. A batch of $b$ fresh particles adds $1/b!$. Thus interchanging the terminated and fresh variables in the joint integral interchanges the birth and termination coefficients, with no reciprocal-prime factor. For an edge, retrieving $a$ ordered particles from memory size $j$ leaves measure $1/(j-a)!$. Reversal retrieves the $b$ stores from memory size $j-a+b$ and leaves exactly the same measure. Residues have counting measure; a prescribed residue has mass one. This derives (3.19) and (3.25), including numerical coincidences, rather than treating them as a probability normalization.

The ghost row ratio is $\exp(|C_i|v\,\widetilde\mu_g(\mathcal P_g))(A_i+|D_i|/v)^{z}$, and (3.21) makes the second factor at most one for both damping values; the column ratio swaps $C_i$ and $D_i$ and is covered by the factor $\max(1,v_+)$ in (3.21). Promotions are controlled by $\sup_z(z)_a\rho^{z-a}$, and on reversal stores become ordered retrievals controlled in the same way. The memory cap $\lceil L^2\rceil$ costs $\exp(-\Omega(L^2))$.

Lemma 3.9, re-derived. In each small group, source symmetrization omits a uniform $\ell$-subset of the $M$ distinct labels, independently of every big-group choice, because the coefficients use big labels only. The omitted products $U$ are distinct integers below $\exp(\ell|\mathcal S|L^b)<\exp(L^c)\le p$ for every pending big prime $p$, so they are distinct modulo $p$. A promotion needs $r+\sigma kD_{\mathcal B}F/U\equiv0\pmod p$, with $F$, $k$ and $D_{\mathcal B}$ invertible, which fixes $U$. So at most $\lceil L^2\rceil$ of the $\binom M\ell^{|\mathcal S|}$ choices allow any promotion, one for each pending particle under the memory cap. For stores, the same count uses the target-side symmetrization of the reversed edge. Schur's test gives (3.30).

Lemma 3.10, re-derived. On a clean edge, the small-state maps $P_{\mathbf d}$ factor the physical normalization $V_g^{-M-\ell}$ symmetrically, with $\sum_{\mathbf d}|P_{\mathbf d}w|^2\le L^{C_s}|w|^2$ and $C_s$ independent of $J$. Fourier transformation on the small torus and the pending residues turns translation by $\sigma kd'D_{\mathcal B}$ and the closure phase into $e(\sigma\Theta D_{\mathcal B})$ with $\Theta=kd'(\theta+\vartheta)$ real, where $\vartheta$ is the fiber's frequency. Each fiber is the ideal operator, up to the density change $1+O_J(sN/P_*)$ and the free-label ban, both of norm cost $o(L^{-A})$.

The equality-rank expansion, re-derived. The identity (3.41) is the sign sum over value-preserving permutations, and the rank-$r$ coefficients sum to at most the coefficient of $t^r$ in $\prod_{j<A_*}(1+jt)$, hence at most $A_*^{2r}$. Large rank uses one point mass $O(1/P_*)$ per dependent birth; with $r_0\asymp NT/L^c$ the tail is $o(L^{-EN})$. Small rank inserts each equality through phases at its two birth addresses, so at most $2r$ edges lose the signed bound, and $r_0=o(N)$ keeps the total at $L^{(-G+C_v+o(1))N}$.

Section 4, re-derived. The collapse (4.4) of tuple tests to integer sequences; the integer bilinear Lemma 4.2; Lemma 4.3 for denominators above $Q$ (bilinear bound for both terms, the kernels being operators of norm at most $2$) and at most $Q$ (character expansion of $e(uv/r)$ with $\sum|c_\psi|\le\sqrt Q$, the reproduction formula (4.18), discarded cells and the Lipschitz error of the log-phase); and the cancellation in Theorem 4.1. There, the symmetric Efron–Stein components that contain every probe of a block and avoid the last $\ell$ slots form a fraction at most $2\ell^\ell m^{-\ell}$ per group, and the alternating sum over nonempty freed subsets equals one exactly. Each of the two matched log products changes by less than $h$, so their sum changes by less than $2h$; this constant is absorbed by the error term in (4.17).

Section 5, re-derived or checked. The raw pattern's lifted pairing equals the target sum: removing the shared product $D_R$ turns $1/n$ into harmonic draws of mass one, leaves $W_\ell(w)W_\ell(w+k)$, and cancels the phases. The endpoint norm $X^{1/2}L^{C_6}$ has $C_6$ independent of $J$, through a fixed fourth divisor moment. The minor-arc claim follows from Dirichlet approximation of $k\theta$ and Lemma 4.2. On the arcs the factors $m,p_s,p$ are units modulo the arc denominator, and splitting $r$ by $(r,r_0)$ leaves a unit modulo $r_0/(r,r_0)$, so principal and imprimitive characters are kept. The Gallagher conversion (5.14)–(5.16), Lemma 5.2 (at most $\exp(O(L^b+L^{1-a}\log L))$ exceptional intervals), Lemma 5.3 (the Gram kernel bound and $U>Y^{0.6}$, which needs $\eta<0.4$), Lemma 5.4 (van der Corput with $K=\lceil4/\tau\rceil+3$ derivatives) and the completion were all re-derived. The low-time range uses $(\mathrm D_B)$ with modulus at most $L^{C+C_{\rm arc}}$, frequency at most a fixed power of $L$, and an interval inside $[1,x^2]$.

Section 2 inputs used by Sections 3–5. The block sieve (Lemma 2.9) and the rough-integer polynomial (Corollary 2.10), which Lemma 5.4 needs, were re-derived: the truncated inclusion–exclusion $\sum_{a\le h}(-1)^a\binom ta=\binom{t-1}h$ for even $h$ and $t\ge1$ (the sum is one when $t=0$), the disjoint supports giving coefficients of modulus one, and the level $z^{4h+2}$. The Siegel–Walfisz theorem, the divisor moments, the mean value and derivative tests, and Fourier separation are classical and are used in their stated forms.

Closing the quantitative estimate

The bounds above compose as follows. This calculation records the uniformity that is needed in addition to the sign check.

Let $C_6$ be the endpoint norm exponent and $C_F$ the logarithmic cost of the initial dyadic and Fourier separations. Both are fixed by the endpoint data before the comparison family. Choose a transfer exponent $E>D_0+2C_6+C_F+3$. The physical list normalization is essential here: if $S'$ is a list of $M$ primes in each group and $P(S')$ its product, then

$$\sum_{S'}\frac1{P(S')}\prod_g V_g^{-M}\le1.$$

Applying the fixed fourth divisor moment after fixing $S'$ bounds $\|f\|^2$ and $\|g\|^2$ by $XL^{2C_6}$ without a power of $L$ depending on $J$. The pointwise bound required in Corollary 3.11 follows because only $O(\sqrt L)$ prime factors above $W$ can divide an integer of size $x^{O(1)}$; the two rough divisor choices therefore number $\exp(O(\sqrt L))$.

Set $P_*=\exp(L^c)$ and $B_{\rm mem}=\lceil L^2\rceil$, keeping this cutoff distinct from the discrepancy exponent $B$. The memory argument gives, before restoring birth distinctness,

$$\|\mathcal E_i\| \le 2L^{-E_{\rm id}+C_s} +2L^{C_a}\sqrt{B_{\rm mem}/\binom{J+\ell}{\ell}^{|\mathcal S|}}.$$

Choose $G>E+C_v+3$ and $E_{\rm id}>G+C_s+2$. The ideal construction chooses its probes and all kernel precisions with this target. Only then choose $J$ so that $\tfrac12c_0\log\binom{J+\ell}{\ell}>C_a+G+3$. The last display is at most $L^{-G}$ eventually. Neither $C_v$ nor $C_s$ depends on the comparison complexity, so this order has no cycle.

For completeness, the rank correction does not replace distinct births by an absolute single-collision error. For rank $r$, the number of weighted equality patterns is at most $L^{C'r}$, and their two usable bounds are

$$L^{C_h(N+1)}(C/P_*)^r, \qquad O\!\left(L^{(-G+C_v)N+C_v+2r(G+C_a)}\right), \quad N=2\lceil L^{0.52}\rceil.$$

In the first bound, chronological dependent births each pay a fresh atom $O(P_*^{-1})$; remembering a pivot after its death only conditions future bounds on a fixed value. In the second, Fourier phases impose equalities at their birth operations. At most $2r$ edges lose their signed estimate. Splitting at $r_0=\lceil C''N\log L/L^c\rceil$, the first bound summed over $r\ge r_0$ is $o(L^{-EN})$ for large fixed $C''$. Since $r_0/N=o(1)$, the second summed below $r_0$ is $L^{(-G+C_v+o(1))N}=o(L^{-EN})$. The memory tail is $\exp(-\Omega(L^2))$ and residue replacement has arbitrary relative precision. Hence the root moment has the required saving, and Corollary 3.11 bounds the residual physical pairing.

Removing shared harmonic draws makes its raw term exactly the desired correlation. For a comparison term, put $Y=ZH'$ and $H=H'/L^{C_{\rm arc}}$, where $H'$ is the product scale of the two nonempty free products. The integer bilinear estimate controls the multiplier off the arcs. On an arc the character expansion retains every character of the quotient modulus, including the principal one; the residual gcd restriction goes into $d_r$, not into a replacement for $\alpha$.

The local Fourier-energy estimate reduces, with fixed logarithmic costs, to the following bound for each four-factor box:

$$I=\int_{|t|\le2LY/H}|P_s(t)P_l(t)\mathcal R(t)|^2\,dt.$$

The small prime has size $P\le\exp(L^b)$, the rough integer has $x^{\tau_*}/2\le P'\le x^{\eta_*}$, and the residual has size $U\asymp Y/(PP')>Y^{0.6}$. The boxed coefficients have squared norm $L^{O(1)}/U$ by a fixed divisor moment, uniformly in $c_r$. The preceding lemmas give constants $K_1,K_2,K_3$, fixed without $B$ or the proxy precision, such that

$$I\ll L^{-2A_s+K_1}+L^{-2A_l+K_2} +L^{-2B+B_0+K_3}+o(L^{-Q}) \tag{3}$$

for any prescribed fixed $Q$, with choices in the order described next.

For the first term use $|P_s|\le L^{-A_s}$ and ordinary mean square for $P_l\mathcal R$. Its size exceeds the time range because $H/(2LP)\to\infty$, using $b<c$. On the complementary set, the $j$-th moment with $j=\lfloor\log Y/\log(2P)\rfloor$ gives at most $\exp(O(L^b+L^{1-a}\log L))=Y^{o(1)}$ occupied unit intervals. The Gram kernel for the residual is bounded by $U/(1+|t-t'|)+Y^{1/2}$; its row sum on those intervals is $O(U\log Y)$ since $U>Y^{0.6}$. This bounds the sum of residual suprema by $L^{K_2}$ without multiplying by the number of intervals. Choose $A_l$ large enough, and then the long-rough-factor lemma gives $|P_l(t)|\ll L^{-A_l}$ for $L^{B_0}\le|t|$ throughout the retained time range. This proves the second term.

For $|t|\le L^{B_0}$, the first factor of $\mathcal R$ is

$$\sum_{m\in I}\frac{\alpha_m\chi_m(m)m^{i(t+\sigma_0)}}m, \qquad I\subset[1,x^2].$$

It is the same restricted sequence as in (S). The character modulus and the frequency are bounded by fixed powers of $L$ chosen earlier. Taking $B$ above those exponents gives a bound $L^{-B}$, and integration over an interval of length $2L^{B_0}$ gives the last term of (3). The other factors have harmonic bounds independent of $B$.

Given $Q$, first choose $A_s>(Q+K_1)/2+1$ and $A_l>(Q+K_2)/2+1$, then choose $B_0$ from the long-factor lemma, and finally choose $B>(Q+B_0+K_3)/2+1$ as well as the modulus and frequency requirements. Then $I\ll L^{-Q}$. Taking $Q$ large enough for all the already fixed separation costs proves the local energy bound $\int_{|\beta|\le1/H}|\widehat{\mathcal U}(u/r+\beta)|^2d\beta \ll YL^{-A}$. Cauchy--Schwarz with the second endpoint's norm controls each major arc; the polynomial number of arcs and comparison patterns is absorbed by choosing $A$ first. Combining the comparison bounds with the small residual pairing proves P at the asserted uniformity.

The argument uses no feature of the final proxy cells. It applies to every sequence in (S) satisfying the fixed coefficient bounds and $(\mathrm D_B)$, including arbitrary further interval restrictions. It does not prove that the proxies have that discrepancy: that is part of E. The use of $|k|$, translations, and units modulo primes larger than $|k|$ covers both signs without modifying the chosen exponents.

The order of choices

For the consumed class the order is consistent. The group and band data, $C$ and $D_0$ come first. The endpoint norm exponent, which is independent of $J$, fixes the transfer target $E$; then $G>E+C_v+3$ and $E_{\rm id}>G+C_s+2$. The probe count $m$, the comparison precision $A_0$ and the kernel precisions of Theorem 4.1 follow, hence $C_1$, and then $J$, large enough for the rare-transfer saving (3.32). Next come the arc exponent, the local-energy target, $A_s$, the long-polynomial saving and $B_0$. The discrepancy exponent $B$ is chosen last within the theorem. None of these depends on the proxy precision of the prime extraction, so that precision can be chosen after $B$, as the Type II application needs. Both signs of $k$ are covered: $k$ enters only through $|k|\le L^{O(1)}$, translation by $kD$, invertibility modulo primes above $|k|$, and the Dirichlet approximation of $k\theta$, whose sign changes only a numerator.

Standing and what remains

The reconstruction closes P on the consumed class at author-recorded standing. Its source is still an unrefereed manuscript, and no independent acceptance or tier is conferred. The retained derivation supports this conclusion at author-recorded standing. The classical boundaries are Siegel--Walfisz in Theorem 2.2, Mertens' estimates in (2.2), and the first and second derivative tests in Lemma 2.7. Divisor moments, mean squares, smooth separation and the block-sieve input are derived in Section 2 as used above; the multiplicative large sieve is not needed for P itself.

The whole P examination of 2026-10-15 records refutation-failed for the full supplied endpoint class and signed application. Its distinct grade is VOID for present acceptance, with zero accepted independent weight. A separate ruling found the grader's unrelated-status exposure immaterial, but expressly supplies no new whole-contract PASS. The source reconstruction therefore retains its author standing; the original classical proofs and PDF fidelity remain outside the examination.

Premise P does not settle Problem 824. The route also needs (E): the Section 2 estimates, the candidate mass of Proposition 7.1, the proxy discrepancy of Lemma 7.3 and the proxy densities of Lemma 7.4. Only the Section 2 inputs used inside the proof of P were checked here. With P and E, the conditional route gives Hypothesis $\mathrm A^+$ and then Li and Zhuang's bound $h(x)>x^{2-\varepsilon}$. The whole conditional implication has a separate examination, whose distinct grade is pending; it does not supply either antecedent.

The P examination's scrutiny includes:

• the factorial adjoint measures (3.19) and (3.25) of the memory space, on which the column bounds of Lemmas 3.8 and 3.9 rest;
• the completeness of the clean and dirty decomposition of each edge;
• the claim that every exponent fixed before $J$ is independent of $J$: $C_{\rm abs}$, $C_v$, $C_s$, $C_6$ and the cost of the comparison family.