E176 whole-proof examination: independent grade

Subject, independence and decision

Grade: PASS for each of the nine assigned conclusions and PASS for the unchanged report as a whole. The report's mathematical verdict is refutation-failed on each surface. This grade finds its review warrant valid within the commissioned scope. It makes no Lean, kernel, compiler, public standing, problem-status or publication claim.

Examination date: 2026-10-15 UTC. Role: independent grader, with disclosed model profile gpt-6-astra, in a fresh isolated context distinct from both the subject author and the original reviewer. The grader neither constructed the mathematical subject nor participated in the reviewed examination. No sibling grade, author inspection account, expected result or commissioner independence ruling was supplied. The commission charged this assignment to find grounds to void the report. No excluded exposure occurred during this grading; no incoming steering was consumed.

The subject is the exact returned report, its literal driver, independent finite program, positive and negative inputs, all retained results and witnesses, and the complete unchanged subject snapshot under reviewed/subject. The mathematical snapshot was frozen on 2026-10-15 at 13:17:10.123503 UTC; the returned report's freeze record is dated 15:29:39 UTC; the grading packet was prepared at 16:42:24 UTC. The commissioned later grading pin, rather than either older identity alone, was checked against the rendered charter and resolved. The operating instructions were separately identity-checked. Those custody facts and the exact source map are retained privately, separate from this mathematical account.

All 39 returned delivery files are preserved unchanged at reviewed/. All 31 delivered subject files agree byte-for-byte with their original-subject counterparts. Every substantive report, keeper and subject text was read completely, including both analytic source slices, all five small-case Lean files, all 17 runtime import files, and the complete finite data. The full original-record inventory was inspected to the depths stated below and in the private custody account. Original instructions and communications were read as historical evidence only. The report has not been repaired or rewritten, and no finding here supplies a missing part on behalf of its author.

The grading standard is the unabridged docs/verification.md, including “Report contract,” “Independent routes and repeated review,” the canonical failure modes and named patterns, and the Erdős-specific whole-claim contract and checklist; together with the complete campaign/guides/verification.md adapter. docs/evidence.md was read before the fresh mathematical runs. No repository battery or formal execution was used as evidence.

Separate conclusion grades

Use the report's exact convention: for each positive integer length $n$, every coloring $f:\{0,\ldots,n-1\}\to\{-1,1\}$ must admit integers $a\geq0,d\geq1$ with $a+(k-1)d<n$ and

$$\left|\sum_{j=0}^{k-1}f(a+jd)\right|\geq\ell.$$

$N(k,\ell)$ is the least positive successful length, when one exists. Set

$$T_2(k)=\frac{2(k^2-1)(k-1)(2k+1)}{k-3},\qquad T_s(k)=\frac43(k^2-k+1)(k-1)(k(k-1)+1).$$

The divisions are real divisions before the floor. Translation to $\{1,\ldots,n\}$ changes neither the property nor its minimum. The target correspondence is ell_two and ell_sqrt_k; ell_ck is not covered.

These are grades of the returned examination. They do not extend to any unseen correction, another version of a source, or an expanded claim.

Original independence warrant

The independence ruling rests on the primary records, not the report's self-description. The retained launch, original operating, native-session, continuation, residual-execution and historical Git records were assessed together. Derived navigation and equality checks locate and corroborate those records; they do not replace them.

The original launch created a separate local, blind, sealed, no-sync assignment with zero descendant authority and a refutation charge. Its exact mathematical extraction and separately frozen operating inputs were present before startup. The native inventory contains a generic readiness probe and the review's own three-step context. The probe received the root instructions and a content-free response request; it performed no tool calls. The main context's visible incoming user material consists of the root instructions and the three commissioned step prompts. Their assignment and step bodies match the frozen original charter and steps. The visible developer material is general operating guidance, not mathematical steering. The visible replacement contexts at both compactions preserve those same received messages; they introduce no other review or author advocacy. Opaque internal fields were not decoded.

All 91 recorded tool-call commands were inspected as data for their actual read, execution and write scopes. The substantive source-reading outputs corroborate the reported complete reading, including the full 1702-line Basic slice, 964-line Sqrt slice, finite files and 17 runtime files. The recorded commands do not open an inbox, read a feed or sibling context, extract canonical mathematics, merge a parent, browse, launch descendants, run Lean, or execute a repository battery. Large report-writing commands write the reviewer's own report and continuation; they are not incoming evidence. The original reviewer consulted only the permitted operating context, frozen subject, own outputs and authorized boundary commands. Repository utilities defined in the supplied import closure were read but were not invoked by the finite checker.

Preflight, post-start and settled-return subscription receipts are empty. The startup sent receipt is empty; the settled receipt contains the reviewer's outbound delivery notice. These facts concern subscriptions and outbound traffic respectively. They do not establish an empty inbox. The original private inbox dump is unavailable. The relevant independence question is whether excluded material reached the review context. The visible received-context record, exact assignment comparisons, complete command inventory, lack of inbox/read ingress, and the corroborating isolation receipts jointly supply sufficient evidence of separation here. No excluded received text, mathematical encouragement or directed attack was identified. Nothing in the report requires an excluded source to explain its content or refutation direction.

Sufficiency and materiality ruling: the unavailable inbox is an explicit evidentiary limit, not evidence that it was empty and not an established contamination. On the supplied primary record, independence is sufficiently supported. No exposure requiring a materiality waiver or fresh reviewer is shown. This ruling does not certify unrecorded private state, and a contrary primary record would require reassessment; it is not a favorable commissioner ruling imported into this grade.

The historical Git record supplies the original review's complete post-input change inventory. Its single iteration-end automatic commit adds 45 files: one own-memory file, 39 delivery files and five own-work files, all under the reviewer's own memory or plans. The headless record and settled return corroborate that order and the clean completed return. No corpus or input path occurs in that change inventory. Under the adapter's express automatic-continuation rule, this is an engine action bearing on neither independence nor scope. Operator setup and input preparation before review startup are not misattributed to reviewer tool actions. Neither that commit nor a finish receipt supplies a mathematical verdict.

The original reviewer is therefore eligible on the supplied record. Using the same disclosed model profile for separate fresh assignments does not make the grader the author or reviewer. This assignment assesses one review and its internal disjoint computational route; it claims no second independent review cycle.

Mathematical adjudication and adversarial checks

Common Fourier, Fejér and model interfaces

The report rederives the common identities rather than citing an unexamined Cassels or Turán result. I checked the actual normalization: with $\widehat f(t)=n^{-1}\sum_a f(a)\omega^{-ta}$, geometric orthogonality gives inversion and $\sum_t|\widehat f(t)|^2=1$. Consequently the cyclic second moment is

$$M_d=\sum_t|\widehat f(t)|^2 \left|\sum_{j=0}^{k-1}(\omega^t)^{jd}\right|^2.$$

There is no extra factor of $n$. The geometric proof uses $n>0$, not primality or invertibility of $d$. Repeated cyclic residues do not defeat the identity, and $n=1$ has just the diagonal orthogonality case.

For a unit $z$, expansion of the report's pair energy over $p,q=0,\ldots,k-1$ and $r,s=0,\ldots,D$ gives

$$Q=k^2+2\sum_{d=1}^{D}\left(1-\frac d{D+1}\right)|G_d(z)|^2 \geq k(D+1).$$

For each positive difference $d$, exactly $D+1-d$ pairs have $r-s=d$, and the same number have $r-s=-d$. Their magnitudes agree by conjugation. Each $p=q$ term contributes $D+1$ after normalization; coincident powers do not remove this contribution. Thus both the factor two and the diagonal lower bound survive. Multiplication by the nonnegative Fourier masses preserves the bound with total mass one. The unweighted inequality follows because the weights lie in $[0,1]$. The report also exposes a nontrivial numerical check at $k=2,D=4,z=-1$: weighted energy $16/5$, pair energy $52/5$. This is a normalization attack, not a finite substitute for the identity.

The decisive transfer is the wrap estimate. If $m=(k-1)d$, the number of starts with $a+m\geq n$ is exactly $\min(n,m)$. Under failure of the ordinary target, the non-wrapping squared sum is at most $b$, while all cyclic squared sums are at most $k^2$. Therefore

$$M_d\leq b+(k^2-b)\frac{\min(n,(k-1)d)}n \leq b+(k^2-b)\frac{(k-1)d}n.$$

The second inequality remains valid when every start wraps. Its coefficient is nonnegative on the used domains. The endpoint $a+(k-1)d=n$ wraps; it is never admitted as an ordinary AP. Failure of target 2 gives integer cap $b=1$; failure of the non-strict square-root target gives $S^2<k$, hence integer cap $b=k-1$. Squaring is reversible because both sides are nonnegative. In particular a four-term sum of 2 meets $\sqrt4$; it must not be excluded as an equality case.

For a non-wrapping start every term is its canonical representative in $[0,n)$. The Fin/ZMod bijection preserves the sums and normalized averages, and the squared complex norm of an embedded integer is its real square. Translation by one is a bijection of ordinary APs with the same positive difference. These bridges are actual implications, not an identification of cyclic and ordinary progressions.

The report's strongest attack is substantive: the alternating coloring at $k=5,n=6$ has ordinary five-term sums of magnitude one, while even cyclic differences give magnitude five. It refutes the shortcut from large cyclic energy to an ordinary conclusion. It does not refute the stated bound, whose wrap loss is paid in full. The period-four example $(-1,-1,1,1)$ similarly tests the first admissible difference-four endpoint. The original report already contains both witnesses and the reason the full proof survives them.

The two polynomial conclusions and minimum selection

For discrepancy 2, put $D=2k$. The lower average is $(k+1)/4$. The upper average under failure is

$$1+\frac{(k^2-1)(k-1)(2k+1)}{2n}.$$

Their difference is exactly

$$\frac{k-3}{4}\left(1-\frac{T_2(k)}n\right).$$

This is positive for every integer $k\geq5$ and $n>T_2(k)$. At $k=5$, $T_2=1056$: equality gives no contradiction, whereas the prescribed length 1057 gives gap $1/2114$. This verifies that the report's floor-plus-one step preserves the needed strict inequality.

For the square-root target let $K=k(k-1)$, $A=(k^2-k+1)(k-1)$, and $D=2K$. Direct finite summation gives

$$\sum_{d=1}^{D}w_d=D/2,\qquad \sum_{d=1}^{D}d w_d=D(D+2)/6.$$

Subtracting the weighted wrap upper bound from the weighted moment lower bound yields

$$\frac K2-\frac{2AK(K+1)}{3n} =\frac K2\left(1-\frac{T_s(k)}n\right)>0.$$

Here $K>0$ is guaranteed by $k\geq2$. At $k=2$, the threshold is 12 and the gap at 13 is $1/13$. The report does not consume the discrepancy-2 polynomial theorem at this step: only the common identities and model bridge are reused. Natural subtractions and coercions agree with the displayed real differences on their expressly stated domains.

For each positive threshold $T$, $B=\lfloor T\rfloor+1$ is a positive integer strictly greater than $T$, including integral $T$. The contradiction first establishes a successful length $B$. Only then does well-ordering give a least positive successful length at most $B$. The source properties require positivity, so the in-range Nat.find selection agrees with that minimum. N176 below its $k\geq5$ guard and N176Sqrt below its $k\geq2$ guard return fallback zero; neither branch proves existence or a positive minimum. The report distinguishes these meanings explicitly.

For surface 1 the report's three weakest-step groups are normalized Fourier moments; the pair-energy identity and multiplicities; and wrap, strict gap and minimum transport. For surface 2 they are the weighted identity; integer cap with exact weight sums and gap; and ordinary, non-strict, floor/minimum transport. Each group is rederived in the frozen report and composes correctly with its neighbors. These are not gaps filled by this grade.

Finite minima and structurally disjoint reproduction

The report proves the Boolean-code/sign-coloring bridge. Encoding the positive signs as binary digits gives a bijection with codes below $2^n$; every sign coloring is represented. For $k\geq2$, an ordinary AP contained in $[0,n)$ necessarily has $1\leq d<n$, so the source's bounded difference enumeration loses none. Its endpoint guard ensures every term is in range. The zero-length case contains one coloring and no AP and is recorded as a failure only; it is not a positive minimum.

The author route enumerates sign tuples, constructs APs as equally spaced subsets, and compares them to bounded start/difference enumeration. The independent route changes both the objects counted and the arithmetic primitive. For $k=2,3$, failure of target 2 is exactly absence of a monochromatic AP. Endpoint interpolation generates the constraints. The product of their complementary indicators expands into inclusion-exclusion over subsets of constraints. Each chosen subset imposes equality classes on vertices and admits $2^c$ assignments, where $c$ is its number of classes. Aggregating signed coefficients by canonical partitions is exact; merging labels by their minimum preserves that canonical representation. The program performs this recurrence over the complete finite edge list, not over sign tuples.

Conditional counts fix prefix signs, reject inconsistent class assignments, and contribute $2^{\text{free classes}}$. They therefore construct the first lexicographic witness without the author's sign-loop stopping rule. The full endpoint inventories, coefficient polynomials and branch counts are retained. At the upper lengths the polynomials are

$$q^3-3q^2+2q$$

and

$$q^9-16q^7+21q^6+42q^5-101q^4+58q^3+5q^2-10q.$$

Both evaluate to zero at $q=2$. This evaluated identity is warranted by the full finite recurrence and its mathematical reduction, not by quoting those coefficients alone.

For $k=4$, missing target 2 forces every four-term AP sum to be zero. Subtracting consecutive length-four blocks forces $f(i+4)=f(i)$. For lengths 4 through 6, the six balanced four-symbol seeds are all possibilities and only difference one occurs. At length 7 the progression $0,2,4,6$ forces opposite entries two positions apart, so the seed is $(u,v,-u,-v)$, with four choices. Through length 12 every admissible difference is at most 3: odd differences visit all four residues and difference two visits an opposite pair twice. Every such sum is zero. At length 13, $0,4,8,12$ has sum $4u\ne0$. Thus the program's empty seed list at 13 encodes a proved obstruction, not an assumed answer. No four-term AP exists below length 4.

The complete fresh census is:

The endpoint inventory is independently checked against $mn-(k-1)m(m+1)/2$, where $m=\max(0,\lfloor(n-1)/(k-1)\rfloor)$. This follows by summing the $n-(k-1)d$ legal starts over $1\leq d\leq m$. The first avoiding witnesses at lengths 2, 8 and 12 are respectively $(-1,1)$, two repetitions of $(-1,-1,1,1)$, and three repetitions of that block. Every earlier positive length has a positive avoiding count and a retained witness; no appeal merely to the immediately preceding length is needed.

The frozen report's weakest steps for surface 3 are code coverage, pigeonhole upper bound and opposite-sign lower bound. For surface 4 they are the monochromatic equivalence, partition inclusion-exclusion, and zero/positive counts with witnesses. For surface 5 they are the period-four recurrence, difference-two seed reduction, and the exact first difference-four obstruction. All are explained and independently derived in the report. Its attempted smaller minima fail with counts 2, 6 and 4; the possible surviving upper-length colorings are excluded by the separate partition or recurrence reasoning. This tests the original sign-enumeration failure class by a structurally different route.

One-term results and consequence dependencies

For surface 6, a one-term sum is always a single sign. Increasing the interval length or difference cannot make its magnitude 2. For surface 7, $a=0,d=1$ at length one has endpoint zero and magnitude one, which meets the non-strict target. Requiring a second distinct point for this one-term AP or substituting a guarded source zero changes the subject. These are the weakest deductions and strongest attempted convention attacks in the frozen report; their elementary direct proofs cover every stated sign and length.

For surface 8 the report uses, for $k\geq5$,

$$T_2(k)\leq18k^3,\qquad \lfloor T_2(k)\rfloor+1\leq19k^3\leq16^k.$$

The factor estimates use $k-3\geq k/3>0$, and the final comparison uses $k\leq2^k$ and $19\leq2^k$ only where $k\geq5$. The induction $k+1\leq2k\leq2^{k+1}$, with base one, justifies the first inequality uniformly in the parameter. The verified minima 3, 9 and 13 cover exactly $k=2,3,4$. The strongest domain attack would leave one of these bases unsupported or include $k=1$; neither happens.

For surface 9 the separate estimates are

$$T_s(k)\leq\tfrac43 k^5,\qquad \lfloor T_s(k)\rfloor+1\leq3k^5\leq64^k \quad(k\geq2).$$

Here the three factors are bounded by $k^2,k,k^2$, and $3\leq2^k$ is used only for $k\geq2$. The actual minimum at $k=1$ gives $1\leq64$. The report never uses either the false inequality $3\leq2^1$ or fallback zero to fill that base. For each exponential its three weakest groups are the appropriate polynomial existence result, rounding and uniform comparison, and the exact remaining bases. Their dependencies are tested before the consequences below.

Execution and reproduction claims

The original records corroborate actual author, independent and negative executions, including the exact package-relative command and its direct independent counterpart. The residual products include their concrete results and import traces; they agree with the returned data. I did not substitute these historical successes for a new execution.

The complete 39-file returned directory was reconstructed byte-exact in the grader's authorized scratch. From that directory I executed both shell blocks under the report's “Reproduction” heading literally, with REVIEW_SCRATCH set to a separate absolute scratch directory. All environment clearing, nice -n 10, one-thread variables, uv run --no-sync, -B, file arguments and script contents were preserved. The source map, real command, working directory, times, exits and stdout/stderr are retained privately. Execution on 2026-10-15 UTC produced:

Fresh mathematical products are retained at fresh/author_results.json, fresh/independent_results.json, fresh/negative_results.json and fresh/direct.json. Required code and exact input files remain unchanged at reviewed/; the report's commands run from that directory in an ordinary clone with the declared prepared Python environment. The commands select disposable output explicitly and do not write into the reviewed snapshot. No scratch-only mathematical code is needed to reconstruct the results.

The driver executes the generators before comparing their outputs; it does not accept stored success. Its explicit require checks survive optimization. The independent program uses exact integers, rejects unsupported input domains, and has explicit nonzero failure paths. Its retained negative input is genuinely false in each of the three cases, and the optimized run demonstrably rejects it. The false-input controls exercise the minimum claims, not every internal consistency guard.

The author's re-evaluation of its own chosen witness uses the same AP predicate as its discovery loop. That internal diagnostic has no additional independence weight. The report explicitly distinguishes the two author enumerations from independent reproduction, gives hypothetical invalid inventory/witness data as failure conditions rather than claiming those mutations were executed, and retains the separate partition and recurrence derivations and direct witness validation. I do not read its 12-check banner as 12 independent proofs. The finite warrant rests on the complete census, its reductions, the disjoint reproduction and the actual false-minimum controls. No tautological diagnostic is needed to establish any of the nine conclusions.

Premises, reading depth and order

The premise set was derived from the frozen statement, proof, source interfaces and consequence sentences, not accepted from an author's list. Dependency reading for this grade, 2026-10-15 UTC: no native claim or ledger row is consumed. A live ledger read is both unnecessary and outside the commission. No pending external grade is treated as an established premise. The source's conditional scalar helpers are discharged internally before the unconditional bounds; an unexecuted native_decide is not a premise of the finite warrant.

The grading order is common identities/model bridges, then surfaces 1 and 2 separately, then surfaces 3–7, followed by surface 8 using 1 and 3–5, and surface 9 using 2 and 7. Every consumed internal conclusion passes before its consumer. There is no arrow from surface 1 to surface 2, and the finite minima consume neither polynomial bound. Neither numerical comparison in the exponentials would alone validate this composition.

Complete report-contract adjudication

Canonical failure modes

These rulings apply to all nine surfaces; the applicability distinctions identify which operations occur in each proof.

Named patterns

The neighboring correlated-sample discipline is inapplicable: there is no sampling or statistical confidence argument. The evidence-coverage discipline is satisfied by the explicit separation of static proof reading, bounded computation and formal execution. This is the first examination; the second-cycle distinct-attack requirement is inapplicable. The separate structural-disjointness requirement for its computational batch is met.

Erdős-specific whole-claim checklist

Limits, credits and disposition

No essential mathematical premise or required report-contract component remains unprovided within this commission. The historical inbox remains unavailable; the explicit sufficiency ruling above is the basis for passing independence despite that limit. The retained private primary-record account states the scope of that ruling and permits its reassessment.

Kenta Kitamura / KitaKen1 owns the original source mechanism. The supplied Codex 5.5 and ChatGPT 5.5 Pro preparation-assistance credits are preserved. Plasma AI receives the supplied extraction, finite probe and exponential deduction credits, and the independent derivation, reproduction and grading work here. The report's external source versions and qualifications remain unchanged in its subject and provenance records.

This grade does not establish equality with the unlisted second archive's Basic file, fidelity of complete source archives, inspection of license texts, Lean elaboration, native_decide execution, compiler or kernel trust, or a reported build. It does not certify uncalled runtime utilities, other finite parameters, stronger exponential domains, sharpness of the general bounds, or bytes written after the frozen delivery.

The private frozen handoff is complete for grading. Public normalization, durable corpus filing and any authorized standing decision belong to the responsible integrator and acceptance process. Private records, paths, hashes and chronological diagnostics remain private. No repaired or expanded subject inherits this PASS.