Endpoint cancellation: whole-statement examination

---

Subject and independence

Examination date: 2026-10-15 UTC. Subject freeze: 2026-10-15, 14:29 UTC; the supplied neutral extraction was prepared at 15:09:22 UTC. Reviewer: independent mathematical examiner, disclosed model OpenAI gpt-6-astra with max reasoning. The examiner was neither an author of this subject nor a collaborator on its construction. No earlier review was supplied.

The exact retained subjects are the following paths relative to this report; they must remain beside it when the package is filed under wiki/research/erdos_824_uniformity/evidence/verify/:

The mathematical source is OpenAI, Weighted dilation graphs, smooth shifted primes and totient fibers, release of 24 September 2026, an unrefereed manuscript. Source-owned mathematics retains that attribution. The project reconstruction and the independent examination are prepared for Plasma AI. The shared model provider is disclosed; independence here is separation from authorship and mathematical communications, not a claim of different providers.

The inputs were supplied as a neutral extraction with explicit [cut] markers. They were read from that extraction, remained unchanged, and are retained with the mechanical maintenance listed on the record index. Their identity and the operating-file identities were checked against the delivered manifest before use. The permitted mathematical read set was exactly these three files. No original source repository, PDF, outside paper, web search, author context, previous report, other private memory, inbox, or feed was read. The subscription tables inspected before and after startup and at the examination step were empty. There was no excluded mathematical or communication exposure. Private identity and isolation receipts are separate from this mathematical record.

Operating reading included the complete docs/verification.md and campaign/guides/verification.md, README.md, campaign/AGENTS.md, docs/evidence.md, docs/approach_vetting.md, docs/math_authoring.md, and docs/weaving.md; the required claims, evidence, tiers and project-claim passages of docs/anatomy.md; and the permitted reading, lifecycle and configuration instructions. Those pages supplied procedure, not additional mathematical premises. Output truncations in some large reads were addressed by smaller reads of the needed passages; no verdict rests on unseen truncated text. No mathematical computation, statistical sample, formal proof, or repository gate is offered as evidence.

An administrative format check initially misidentified two displayed absolute-value formulas as table rows. The checker was corrected; its original and successful rerun receipts, together with a rejected private checkpoint edit, are retained in the private operating accompaniment for the grader. No mathematical conclusion rests on those operations.

The independent reasoning is retained in memory transfer, ideal comparison, and Mellin cancellation, together with the lift and application calculations below. reconstruction gives the noncomputational reconstruction route. This is a whole-statement examination of P, not merely containment of the application pair in a narrower theorem. It does not accept the separate E estimates or the final catalog conclusion.

Restatement with all quantifiers

Let $L=\log x$, $T=\log L$, and $W=\exp(\sqrt L)$, with $x\to\infty$. Fix

$$0<a<b<c<d<0.47,\quad c_0,C_0,v_-,v_+>0,\quad 0<q<1,\quad \ell\in\mathbb Z_{\ge1}.$$

For each sufficiently large $x$ take pairwise disjoint finite prime groups $\mathcal P_g$, indexed by disjoint sets $\mathcal S,\mathcal B$, with $c_0T\le|\mathcal S|,|\mathcal B|\le C_0T$. Small groups are arbitrary subsets of primes in $[\exp(L^a),\exp(L^b)]$. Each big group is the set of all primes in an interval contained in $[\exp(L^c),\exp(L^d)]$. Every group has harmonic mass $V_g=\sum_{p\in\mathcal P_g}p^{-1}\in[v_-,v_+]$. The choices may vary with $x$ subject to these bounds; no conclusion assumes that an inadmissible collection exists.

For a positive integer $n$, let $n_*$ be its largest divisor coprime to every group prime, and let $\omega_g(n)$ count distinct group primes dividing $n$. With $(y)_\ell=y(y-1)\cdots(y-\ell+1)$, set

$$W_\ell(n)=\prod_g q^{\omega_g(n)-\ell}\frac{(\omega_g(n))_\ell}{V_g^\ell}.$$

A factor is zero when $\omega_g(n)<\ell$. Multiplicities in $n$ do not increase $\omega_g(n)$. For fixed $\ell,q,v_-$ each group factor is bounded, hence $0\le W_\ell(n)\le L^{O(1)}$ uniformly in $n$.

Fix $C>0$ and $0<\tau<\eta<1/4$. An endpoint has the form

$$F_0(n)=\sum_{mpr=n_*} \alpha_m\chi_0(m)m^{i\sigma_0} \mathbf1_{p\in I_p,\ P^-(p)>W}p^{i\sigma_1}c_r, \tag{R1}$$

where all factors are positive integers, $p$ is an integer rather than a prime variable, $P^-(1)=\infty$, the coefficients may be complex, $|\alpha_m|,|c_r|\le L^C$, $\alpha_m=0$ unless $P^-(m)>W$, and $I_p$ is any interval contained in $[x^\tau,x^\eta]$. The character $\chi_0$ has modulus at most $L^C$, and $\sigma_0,\sigma_1$ are real with absolute values at most $L^C$. There is no roughness, multiplicativity, positivity, or regularity condition on $c_r$. The two endpoints may have unrelated sequences, intervals, characters and twists within these common bounds. All these data may depend on $x$.

The property $(\mathrm D_B)$ for the original sequence $\alpha$ is

$$\left|\sum_{m\in I}\frac{\alpha_m\chi(m)m^{iu}}m\right| \le L^{-B} \tag{R2}$$

for every interval $I\subset[1,x^2]$, every Dirichlet character of modulus at most $L^B$, including principal and imprimitive characters, and every real $|u|\le L^B$. Any original interval restriction is already part of $\alpha$; no replacement convolution inherits (R2) without proof.

For fixed $D_0>0$, a fixed compact interval $[c_\Psi,C_\Psi]\subset(0,\infty)$, and fixed implied constants in the derivative bounds below, P asserts that a fixed $B$ exists such that, for every sufficiently large $x$, uniformly over all the preceding data and

$$0<|k|\le L^C\ (k\in\mathbb Z),\qquad xL^{-C}\le Z\le xL^C,\qquad |a_1|,|a_2|\le L^C,$$

with $a_1,a_2\in\mathbb R$, every smooth $\Psi$ supported in $[c_\Psi,C_\Psi]$ with $\|\Psi^{(j)}\|_\infty\ll_jL^{C(j+1)}$, and endpoints $F_0,G_0$ whose first sequence satisfies (R2),

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

The constants and threshold depend only on the fixed bounds and choices, not the individual admissible groups, sequences, intervals, characters, frequencies or shift. No discrepancy is required of the second endpoint. The assertion includes both signs of $k$ and conjugation of the discrepancy-bearing endpoint or its coefficients. All constants are fixed before $x\to\infty$; no effective numerical threshold is asserted.

Three weakest steps and independent derivations

1. Exact memory transfer and restoration of distinct births

Memory transfer, (M1)–(M14), derives the single-prime lifespan expansion directly from its residue expectation. A prime with $r$ active runs has factor $V_g^{-r}/(p b_p)$, paying for its root residue once. Pending residues use counting measure. The factorial identities $\binom{k+t}{t}/(k+t)!=1/(k!t!)$ and $(j)_a/j!=1/(j-a)!$ give the ghost and edge adjoints even on numerical diagonals. This independently addresses normalization, ghost endpoints, return visits and reversed edges.

Weighted Schur estimates give a visit exponent $C_v$ independent of the comparison construction. The memory tail at $B_{\rm mem}=\lceil L^2\rceil$ is $\exp(-\Omega(L^2))$. A dirty edge has norm at most $2L^{C_a}\sqrt{B_{\rm mem}/\binom{J+\ell}{\ell}^{|\mathcal S|}}$: the omitted small-label products are distinct integers below every pending big prime, and each pending residue allows at most one of them. Reversal handles stores. On clean edges the small-state contraction has exponent $C_s$ independent of $J$ and comparison complexity; Fourier fibers are the ideal operator or its adjoint, with negligible density and free-label exclusion errors.

The collision correction is not discarded. At equality rank $r$ its weighted count is at most $L^{C'r}$. Chronological fresh births give $L^{C_h(N+1)}(C/P_*)^r$, where $P_*=\exp(L^c)$. Phases inserted at the birth addresses give instead $O(L^{(-G+C_v)N+C_v+2r(G+C_a)})$; at most $2r$ edges lose their signed estimate, even if a pivot has already died. Splitting at $r_0\asymp N\log L/L^c=o(N)$ proves the required total correction. Relative residue replacement factors out the possibly tiny deterministic active-prime damping before Bonferroni and CRT estimates. Finally, the positive semidefinite Gram trace and spectral Hölder work with the unnormalized constant mark vectors $u_n$. This proves the transfer actually needed by mark-independent endpoints for either sign, without assuming P.

2. Ideal comparison for arbitrary tuple tests

Ideal comparison, (I1)–(I11), checks the full operator test space. For a product fiber at $n$, its probability is at most $L^\kappa/n$, including repetitions. Fiberwise Cauchy–Schwarz therefore collapses an arbitrary tuple test to square norm $L^\kappa\|f\|_2^2/U$ on a product dyad; it never assumes the test is itself a function of the product.

Conditional character Gram matrices on retained log cells have norm at most two by the absolute, short-interval Siegel–Walfisz specialization. This has no inverse-cell-mass operator loss. For discarded cells the correct estimate is a bilinear $L^1$ bound on independent tuple spaces, $\mathbb P(E)^{1/2}\|f\|_2\|g\|_2$, not a false small-restriction $L^2$ operator claim. Large rational denominators use the integer bilinear estimate. Small denominators use all inducing primitive characters, pointwise reproduction, and the change of both matched log products, of total size at most $2h$.

The full-probe fraction for a symmetric test is exactly $(j)_m(M-j)_\ell/(M)_{m+\ell}$. Maximization in $j/M$ bounds it by $2\ell^\ell m^{-\ell}$, and the bound tensorizes across groups even for entangled tests. The asymmetric kernels leave separated conditional tests. Alternating nonempty freed subsets cancel the raw pairing exactly; the exceptional full-probe pieces are summed before taking absolute values. Kernel precision and the coefficient exponent are consequently fixed before $J$. This provides the uniform ideal bound used in step 1.

3. Full-endpoint local Mellin energy

Mellin cancellation, (F1)–(F18), uses one small marked prime, not a property peculiar to the application pair. Prime-square errors are $Y L^{O(1)}e^{-L^a}$. The surviving residual is the arbitrary bounded coefficient $c_{r_*}W_{\mathbf t-\mathbf e}^{\mathbf b}(r)$. At a rational arc, split $r$ by its gcd with the denominator; only the residual acquires that condition. The original $\alpha$ is unchanged.

The local Fourier-to-Mellin conversion has normalization $Y^{-1}$; the logarithmic-kernel replacement costs $(H/Y)^2\sum|a_w|^2$. Smooth product separation, including the exact conversion to reciprocal coefficients, reduces to $\int|P_sP_lR|^2$. Ordinary times use the small-prime bound and mean square because $H/(LP)\to\infty$. The exceptional small-prime set occupies at most $\exp(O(L^b+L^{1-a}\log L))=Y^{o(1)}$ unit intervals. Their number is not multiplied by a logarithmic bound: the residual's Gram kernel and its scale $U\ge Y^{1-\eta-o(1)}>Y^{0.6}$ give $\sum_I\sup_I|R|^2\ll L^{O(1)}$ for the entire $\eta<1/4$ range.

The long rough-integer factor saves every fixed logarithmic power on every allowed subinterval, including very short ones. The block sieve is reconstructed with its coefficient, level and growing-truncation bounds. First and second derivative tests, iterated finite differencing, and zero extension on short intervals handle positive and negative $t+\sigma_1$ up to $x^2$. Low times alone use (R2), on an interval inside $[1,x^2]$ with a character and frequency of previously fixed logarithmic sizes. The result is

$$\int|P_sP_lR|^2 \ll L^{-2A_s+K_1}+L^{-2A_l+K_2} +L^{-2B+B_0+K_3}+o(L^{-Q}), \tag{R3}$$

where the $K_i$ are fixed without $B$ or proxy precision. Choose $A_s,A_l$, then $B_0$, then $B$. No relation between the two endpoints, smoothness of $\alpha$, or roughness of $c_r$ enters this reasoning.

Strongest attempted refutation and other attacks

The strongest attack was a collision between two births separated by the death of the first particle. An absolute one-collision estimate really is insufficient: $\exp(-L^c)$ cannot absorb $L^{CN}=\exp(CN\log L)$ for $N=2\lceil L^{0.52}\rceil$. Moreover, a state no longer remembers a dead pivot, so a purported state-local equality constraint would lose information. This identifies an exact dangerous inference, not a numerical near miss.

It does not refute the frozen reconstruction. Equality Fourier phases can be attached to the two births themselves; ghost phases can be symmetrized within their factorial batches. Unaffected edges retain their full signed pattern sums. The low-rank estimate and the chronological high-rank estimate above cover complementary ranges with $r_0/N\to0$. Equations (M11)–(M13) retain the complete response to this attack. No single-collision shortcut is used in the verdict.

Additional attacks and their outcomes were:

These are analytic attacks, not independent random samples or finite evidence for a universal assertion. No counterexample or unsupported essential inference survived them within the declared external boundaries.

Physical lift, overlap errors, and major arcs

This section checks the bridge between the three retained derivations. For a fixed physical source list $S'$ with $M=J+\ell$ distinct labels per group, let $P(S')$ be its product. The state mass is $\prod_gV_g^{-M}$. On that state, $n_*\mid n/P(S')$ and $|F_0(n)|\le L^{2C}\tau(n_*)^2$. A fixed fourth divisor moment gives

$$\sum_{\substack{n\asymp X\\P(S')\mid n}}|F_0(n)|^2 \ll \frac X{P(S')}L^{O(1)},\qquad \sum_{S'}\frac1{P(S')}\prod_gV_g^{-M}\le1. \tag{R4}$$

Here $P(S')=x^{o(1)}$ and $X=x^{1+o(1)}$, so endpoints of the count do not alter the bound. The damping $q^{(\omega(n)-Ms)/2}\le1$ on physical states. Thus both endpoint norms are at most $X^{1/2}L^{C_6}$ with $C_6$ independent of $J$. The two rough divisor choices in (R1) number at most $\exp(O(\sqrt L))$, because each uses primes above $W$; hence the required pointwise bound also holds even if $c_r$ is arbitrary. The vectors are mark independent. Positions outside the positive supported dyads can be assigned zero, avoiding logarithms of nonpositive integers.

Use the conjugate-linear-first pairing with $f(n)=n^{-ia_1}F_0(n)q^{(\omega(n)-Ms)/2}$ and the corresponding $g(n)=n^{ia_2}G_0(n)q^{(\omega(n)-Ms)/2}$. Insert $\Psi(n/(ZD))/n$ and $D^{-i(a_1+a_2)}$. Separating the smooth function of $\log(n/Z)-\log D$ puts its first-variable factor into $f$ with its conjugate, and supplies ideal frequency $\zeta=-a_1-a_2-v$. The bounded factor $X/n$ is absorbed in $f$, and the pairing is divided by $X$. A fixed exponent bounds the retained $v$ before the eventual tail accuracy. Dyadic and Fourier total mass costs $L^{C_F}$, with $C_F$ fixed before comparison kernels.

For a pattern with $c_g$ free slots and $t_g=\ell+c_g$ unshared endpoint slots, split $D=D_RD_C$ into the shared and free products and write $n=D_Rw$, $n'=D_Rw'$. The joint list normalization is

$$V_g^{-M-t_g}=V_g^{-(J-c_g)-2t_g}.$$

The factor $1/n=1/(D_Rw)$ turns each shared label into a harmonic draw $1/(pV_g)$. Away from overlaps, $\omega_g(n)-M=\omega_g(w)-t_g$ and the graph/vector half-dampings give the full marked weights at $w,w'$. The phase identity is

$$(D_Rw)^{ia_1}(D_Rw')^{ia_2}(D_RD_C)^{-i(a_1+a_2)} =(w/D_C)^{ia_1}(w'/D_C)^{ia_2},\quad w'=w+kD_C. \tag{R5}$$

Endpoints are invariant under multiplication by group primes, so no endpoint data change. The raw pattern has $D_C=1$, $t_g=\ell$ and coefficient one. It gives exactly (P) after the negligible exclusions below, not a different shifted convolution.

For fixed $w,w'$, only $O_J(L)$ prime values can violate the shared draw exclusions. Each harmonic atom is at most $v_-^{-1}\exp(-L^a)$; there are $O_J(T)$ draws. Overlap damping can cost $q^{-2Ms}=L^{O_J(1)}$, which is harmless against that exponential. The endpoint absolute sums use (R4). If a free big prime divides an endpoint, it divides both $w,w'$ because it divides $D_C$; writing $w=pv$ gives an additional $1/p$ in the harmonic sum. Endpoint invariance and fixed divisor moments bound this contribution by $L^{O(1)}/p$. These estimates cover all forbidden coincidences after summing patterns and hold for either sign. In particular there is no assumption that $c_r$ is small on multiples of a group prime.

Expanding the comparison coefficients into their log-cell and character factors leaves bounded factors $b_1(D_A),b_2(D_B)$ and endpoint mark functions independent of the free products. Each free product is nonempty and has size at least $\exp(L^c)$; their product has cell scale $H'$ and $Y=ZH'=x^{1+o(1)}$. The Fourier multiplier is

$$M(\theta)=\mathbb E b_1(D_A)b_2(D_B)e(-k\theta D_AD_B). \tag{R6}$$

Product collapse and the integer bilinear estimate control (R6) off major arcs. More precisely, approximate $k\theta$ with maximum denominator $\lfloor H'/L^{C'}\rfloor$. A denominator at most $L^{C'}$, after division by $k$, produces an arc for $\theta$ of denominator at most $|k|L^{C'}$ and width at most a fixed logarithmic power divided by $H'$. Every other approximant has both denominator and complementary quotient large enough for (I2). Choose $C'$ for the required saving, then $C_{\rm arc}$ to contain all the first arcs. The number of arcs is polynomial in $L$; changing the sign of $k$ changes only a numerator.

Parseval and the fixed endpoint square norms handle minor arcs. On an arc, put $H=H'/L^{C_{\rm arc}}$. The first endpoint's local energy is $O(YL^{-A})$ by (F1)–(F18), whereas the second endpoint needs only its $O(YL^{C_8})$ full square norm. Cauchy–Schwarz, division by $Y$ and the polynomial arc, pattern and separation costs give arbitrary saving after choosing $A$ large enough. Thus the transfer pairing, raw identity and comparison bounds compose into (P) for the full class (R1).

Exact signed application interface

This comparison examines what the supplied application consumes as P. It does not use the application to prove P, and does not award acceptance to E or to the application's final catalog use.

The band's distinguished integer always belongs to $h$ in the split $u=eh$. For a fixed assignment pattern and Fourier component the resulting coefficient is

$$a_h(h)=h^{i\varrho}W_\ell(h)H(h_*),\qquad H(t)=\sum_{pr=t}\mathbf1_{p\in I_{**},\ P^-(p)>W}p^{i\sigma}c_r. \tag{R7}$$

The interval $I_{**}=[x^{\tau_*},x^{\eta_*}]$ is the original common band, with $0<\tau_*<\eta_*<b_*/4<1/4$. For the square of a fixed component, the two copies use the same $H$, hence the same interval and residual $c_r$. No restriction on the prime factors of $r$ is needed.

The early exponent assertions used by this interface can be checked without P. At each slot the remaining logarithmic budget $R_i$ is nonnegative. Taking a prime reduces both budget and available logarithms by its log; skipping requires $R_i<(c_2+1)s_j$. The next band has at least $r_0-1$ remaining pure slots with total log at least $(c_2+1)s_j$. The last-band slack is at most $(c_2+1)s_{j_x}$. Initially the available log is at least $(1-b_*/2)L-O(1)$, exceeding $\log E\le(1-b_*)L-K_0T$. This proves $EL^{-C_s}\le e\le E$, with $C_s=(c_2+1)c_+$ independent of $K_0$.

There are $N_s=O(T)$ transition factors with fixed smooth compact arguments. Their Fourier mass over all patterns is at most $(2C_F)^{N_s}=L^{O(1)}$. Cutting each frequency at $L$ leaves a tail bounded by $2^{N_s}N_sC_MC_F^{N_s-1}L^{-M}$, and the slot twist is

$$\sigma_t=-\xi_t/s_{j(t)} -\delta_t\sum_{i>t}\xi_i/s_{j(i)}=O(L/T).$$

Pure-prime assignments at a fixed product number at most $(r_0!)^{j_x+1}=L^{O(1)}$. Thus the coefficient and Fourier exponents in (R7) are fixed independently of $K_0$ and of tail accuracy. The absolute bound against which these tails are taken uses fixed divisor moments, not discrepancy. For the Cauchy diagonal, $e\mid(mn+1)/2$ and $\sum_{e\mid(mn+1)/2}\tau((mn+1)/(2e))^2\le\tau(mn+1)^3$. Consequently $\sum_{v\ll x}\tau(v)\tau(v+1)^3\ll xL^{C_5}$, with $C_5$ independent of $K_0$. This verifies the specific independence needed to fix $K_0$ before the off-diagonal parameters.

For an off-diagonal pair, direct subtraction gives

$$mn=2eh-1,\quad m'n=2eh'-1, \qquad m'-m=-2ek,\quad h'-h=-kn,\quad mh'-m'h=k. \tag{R8}$$

The first equation makes $n$ a unit modulo $2e$, so $2e\mid m'-m$; $k$ is a nonzero integer of either sign. Set $z=m'h$, so $z+k=mh'>0$. The supported range gives $|k|\ll L^{K_0+C_s}$ and $Z=xL^{O(1)}$.

The converse does not lose a divisibility condition. Suppose $0<|k|<W$, $m,m'$ are rough above $W$, $m'\equiv m\pmod{2|k|}$, $e=(m-m')/(2k)>0$, and $m'h=z$, $mh'=z+k$. Then $e$ is a positive integer and

$$m(h'-h)=-k(2eh-1).$$

Since $(m,k)=1$, $n=(2eh-1)/m$ is a positive integer and $m'n=2eh'-1$. These operations invert (R8). Both $m,m'$ are units modulo $2|k|$, so the congruence is exactly $\varphi(2|k|)^{-1}\sum_{\chi\bmod2|k|}\chi(m)\overline{\chi(m')}$, with all characters included.

The pulled-back argument is $n/V=((m-m')z/(kmm')-1/m)/V$, with the minus sign retained. On a fixed box in $\log(m/U),\log(m'/U),\log(z/Z)$, derivatives of $e/E$ and $n/V$ have sizes $O(U/(|k|E))$ and $O(Z/(|k|UV)+1/(UV))$. After $K_0$ is fixed these are fixed logarithmic powers. Smooth separation therefore supplies fixed logarithmic frequency and total-mass exponents. Sharp restrictions of the original $\alpha$ remain in that sequence; they are not differentiated. Absolute convolution square norms $ZL^{O(1)}$ pay for separation tails before any use of P.

For each separated character and frequency triple, the actual pair is

$$\begin{aligned} F_0(v)&=\sum_{mpr=v_*}\alpha_m\chi(m)m^{i(\varrho-\xi_2)} \mathbf1_{p\in I_{**},\ P^-(p)>W}p^{-i\sigma}\overline{c_r},\\ G_0(v)&=\sum_{mpr=v_*}\alpha_m\chi(m)m^{i(\xi_1+\varrho)} \mathbf1_{p\in I_{**},\ P^-(p)>W}p^{-i\sigma}\overline{c_r}. \end{aligned} \tag{R9}$$

Indeed at $z=m'h$, $h^{i\varrho}=z^{i\varrho}(m')^{-i\varrho}$; at $z+k=mh'$, $\overline{h'^{i\varrho}}=(z+k)^{-i\varrho}m^{i\varrho}$. Conjugating the first line of (R9) gives the required $\overline{\alpha_{m'}}\overline{\chi(m')}$ and the unbarred $p^{i\sigma}c_r$ from $H(h_*)$. The second line gives $\alpha_m\chi(m)$ and the barred copy from $\overline{H(h'_*)}$. Thus the separated correlation is precisely $Z$ times (P), with

$$a_1=\varrho+\xi_3,\qquad a_2=-\varrho, \qquad \Psi(t)=t\psi_Z(t). \tag{R10}$$

Here $(z/Z)/z=1/Z$, which checks the harmonic-to-unweighted factor. The identity $z_*=m'h_*$ follows because $m'$ has no group prime; the analogous identity holds for $z+k$. Every part of (R9) is within the full class (R1), with a single enlarged fixed coefficient/frequency bound chosen before $D_0$ and $B$.

The closure under conjugation is also literal:

$$\sum_{m\in I}\frac{\overline{\alpha_m}\chi(m)m^{iu}}m =\overline{\sum_{m\in I}\frac{\alpha_m\overline{\chi(m)}m^{-iu}}m}. \tag{R11}$$

The character class and frequency range are unchanged. Conjugating an endpoint conjugates its coefficients and character and negates its twists. Further interval restriction preserves (R2) by intersection of intervals. No same-$B$ invariance under arbitrary additional twists is claimed: the required fixed twists are inside (R1), and the final $B$ exceeds their eventual modulus and low-time frequency demands.

Order of choices and uniformity

For P, fix group bounds, $C,\tau,\eta$, the cutoff family and $D_0$. The exponent $C_6$ in (R4) and the initial separation exponent $C_F$ fix $E>D_0+2C_6+C_F+3$ and a fixed ideal frequency bound. Choose $G>E+C_v+3$ and $E_{\rm id}>G+C_s+2$. The exponents $C_v,C_s$ are independent of comparison complexity. Next fix the probes, comparison accuracy, cell and character precisions, hence $m,C_1,C_a$. Only then choose $J$ so that

$$\tfrac12c_0\log\binom{J+\ell}{\ell}>C_a+G+3.$$

All comparison costs are now fixed. Choose arc and local-energy targets, then $A_s$, the long-factor saving $A_l$, its $B_0$, and finally $B$ above every modulus, frequency and low-time saving exponent in (R3). Arbitrarily small later Fourier tails use more derivatives, not a larger previously fixed frequency exponent. The inequalities $0.52+d<1$, $b<c$, $d<1/2$, and $\eta<1/4$ respectively provide the residue modulus margin, small-prime/time-scale separation, rough group exclusion and endpoint pointwise bounds, and the residual sparse mean-square margin. Each is used within its stated range.

In the signed application, first fix group and band data, the designated band, $b_*,C,D_*$. The transition and early norm exponents do not depend on $K_0$, as checked above; choose $K_0$ from the diagonal next. Then fix the separation bounds and $D_0$ to absorb the $k$ sum, Fourier mass, dyads and Cauchy loss. The normalized character average introduces no additional unbounded character-count loss. Invoke P to fix $B$. Final proxy precision comes after this, under the expressly required precision-independent coefficient bound. Establishing that a prospective proxy actually satisfies (R2) is an E obligation, not an inference from P.

Premises, reading depth, and deduction census

The census was derived from the frozen proof on 2026-10-15, including its consequence sentences. There is no established native L-claim premise. Accordingly no live claim-ledger row was authorized, requested or read; there is no unreported tier-0 premise or prospective promotion in this proof. Definitions 3.1–3.3 define finite label spaces and bounded operators by the absolute row/column estimates; their use does not presume their later cancellation assertions. P and source Theorem 5.1 are the target, never an assumed premise of their own reconstruction.

The following are the exact external classical boundaries. They were claims checked in the supplied transcription and specialized here; their original cited texts and proofs were unread. Reliance is on the commission's expressly designated classical statements. The report does not claim an independent proof of them, a visual transcription check, or an outside bibliographic verification.

1. Siegel–Walfisz, source Theorem 2.2. For fixed $A,C>0$, uniformly in $(a,r)=1$ and $1\le r\le(\log y)^C$, $\pi(y;r,a)=\operatorname{Li}(y)/\varphi(r) +O_{A,C}(y/(\log y)^A)$, with constants independent of $y,a,r$ and not necessarily effective. The source identifies Tao, 254A notes 2: Complex-analytic multiplicative number theory, 9 December 2014, Exercise 64, bibliography [22]. For the big-prime cells, $y\ge\exp(L^c)$ makes every required modulus $L^{O(1)}$ a fixed power of $\log y$. Summing over reduced classes cancels the main term for a nonprincipal character. Partial summation gives absolute harmonic bounds on arbitrary intersected intervals; it does not assert relative equidistribution on tiny intervals. The quotient of two distinct primitive characters has nonprincipal character on a common modulus at most $Q^2$. Choosing the Siegel–Walfisz precision after $Q$ and the cell threshold pays for residue-class and conditional-mass losses. The case $r=1$ also supplies any plain prime-number estimate needed; no independent stronger prime theorem is assumed.
2. Mertens, source (2.2). $\sum_{p\le y}p^{-1}=\log\log y+O(1)$ and $\prod_{p\le y}(1-p^{-1})\sim e^{-\gamma_E}/\log y$. The source identifies Tao, 254A notes 1: Elementary multiplicative number theory, 23 November 2014, Theorems 15 and 26, bibliography [21]. The first estimate gives the fixed divisor-moment Euler product and bounded mass in each sieve block. The second gives $\sum_d|\lambda_d|/d\le\prod_{p\le W}(1+p^{-1})\ll\sqrt L$. Only those bounds are used; no unquoted error rate is required.
3. Derivative tests, source Lemma 2.7. For real $f$ on an interval containing $H$ consecutive integers: if $f'$ is monotone in $[j+\lambda,j+1-\lambda]$, $j\in\mathbb Z$, $0<\lambda\le1/2$, then $\sum e(f(n))\ll\lambda^{-1}$; if $f\in C^2$ and $\lambda\le|f''|\le C\lambda$, then $\sum e(f(n))\ll_C H\sqrt\lambda+\lambda^{-1/2}$. Both hold on every subinterval satisfying their hypotheses, with $e(t)=\exp(2\pi it)$. The source identifies Montgomery–Vaughan, Multiplicative number theory II: Primes and sieves, undated author-hosted draft accessed 13 September 2026, Corollary 16.6 and Theorem 16.7, bibliography [15]. The first test is used on $z\log n/(2\pi)$ with derivative close to zero, using the interval $(-1,0)$ for negative derivatives. The second gives the sparse Gram kernel and the final differenced logarithmic phase. Higher derivative cancellation is derived by finite differencing in (F15)–(F16), not assumed as another external theorem.

The same-paper order is acyclic: elementary Section 2 deductions, Section 3 conditional transfer, Section 4 ideal construction, Section 5 physical and Fourier reductions, the three time ranges, P, then its application interface. The next table records actual proof inspection and findings. “Proof verified within this examination” means the supplied proof was read and its essential inferences checked against the independent reasoning cited; it is not a tier award or a claim to have checked an original PDF.

Theorem 2.3, the multiplicative large sieve, is not consumed in this P chain: its statement was read, its external proof was not verified, and the verdict does not rely on it. Historical references to Efron–Stein, Matomäki–Radziwiłł, Soundararajan, Ford–Halberstam, Helfgott–Radziwiłł and Pilatte are not extra premises: the particular projections, moments, sieve and memory constructions needed here were checked from their supplied proofs. Basic finite character orthogonality, CRT, Fourier inversion, Cauchy–Schwarz, Dirichlet approximation and Hilbert-space inequalities are used in their explicit elementary roles above; no omitted analytic theorem is hidden under those historical comparisons.

Checklist verdicts

These are findings for this frozen subject, not inferred standing from an earlier review. The canonical list, all named patterns, and the Erdos list are recorded separately because each is required by the commissioned contract.

Canonical failure modes

Named patterns

Erdos-specific checklist

The neighboring evidence-coverage discipline is met by identifying which keeper and calculation supports each essential deduction. There are no statistical samples for which correlation adjustment is needed. Administrative identity comparisons do not add mathematical strength to these analytic findings.

Verdict, limitations, and next ownership

Mathematical verdict: refutation-failed. The entire stated endpoint cancellation theorem P, its essential source-proof reconstruction through the order of choices, and every clause consumed as P in the supplied signed application survived this examination, relative to the three explicitly designated classical boundaries. The examination is not restricted to the special application endpoints: (F1)–(F18) allow every coefficient and interval in (R1), and the second endpoint may be unrelated to the first.

No essential P inference or quantified endpoint range is left as an unfinished obligation in this report. This is an adversarial mathematical finding, not an independent grade, a native tier award, a formal proof, or evidence of refereeing of the unrefereed source.

The limits are exact:

• The source was read as the supplied transcription. Its agreement with an original PDF was not inspected. The classical results were accepted only in the quoted forms; their original proofs and bibliographic fidelity were not independently checked.
• Source Section 6 was not supplied. Therefore the application's assertion that it is the source's Theorem 6.1 with precisely the listed sign changes remains an unverified source-fidelity assertion. The signed algebra and the actual P interface are verified directly above and do not depend on that comparison. If the historical equivalence is to be retained as an independently checked claim, the commissioning coordinator must supply its exact Section 6 input to a separately authorized examination.
• The report does not establish that any proposed proxy satisfies $(\mathrm D_B)$, accept the separate E estimates, prove the final catalog conclusion, or assess text written after the freeze. Those remain with their responsible project owners and the coordinator.

The coordinator owns intake, preservation of the entire package at a durable repository-relative location, and assignment to a distinct fresh grader. That grader decides contract validity, independence and accepted weight; the examiner does not grade or promote this report. The coordinator also owns the limited source-fidelity obligation just stated. Private delivery alone does not settle any mathematical standing.