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Grade and roles

Grader: the distinct contract grader, who commissioned the review. Date: 15 September 2026. PASS for the exact whole-claim report contract and reviewer independence. Native integration and durable retention were pending at grading; the grade itself promoted no corpus claim.

The argument's author and the fresh independent reviewer were distinct contexts. The grader commissioned the review and read the author and review material but did not construct the argument. The grader is distinct from both author and reviewer and grades the report, without claiming a second blind mathematical review. The current filing was prepared by the exposed author and does not independently review its own correspondence.

Exact subjects and authentication

The subject is the byte-exact reviewed proof. The graded original report's full substantive native rendering is retained separately; the original report, the original review manifest and the original grader authentication record remain in working storage.

That authentication record reports regular nonsymlink inputs, stable streams, identity and duplicate-key checks, and exact subject/report joins. It expressly describes identity checking, not a second mathematical review. The grader read the complete subject, review and review manifest; complete reads resolved an earlier truncated aggregate output. The original grade itself remains in working storage; the native rendering does not claim byte identity with it.

Basis for PASS

The report restates one infinite positive-integer set, the unordered representation convention including diagonals, both fixed constants, every sufficiently large real cutoff, and the same-set consequence for each smaller power. It correctly limits the improvement to the loss term, without claiming a larger main exponent, a public record or resolution of Problem158.

All ten checklist items receive explicit verdicts. Independent rederivations cover digital transfer and injectivity; fixed-modulus degeneracies and signed factorization; the collision lattice's line term; conditional expectation and gcd sums; the non-strict critical endpoint; a single probability outcome; simultaneous hypergraph selection; and the every-real-cutoff conversion. The strongest attack tries singular and resonant divisor-count cases, including negative offsets, and separately tests a short collision-plane direction and shell ratios arbitrarily near the critical endpoint. These are substantive checks of the frozen whole argument, not author-run agreement.

No native L-claim is consumed as an established premise. The admitted L4 index is only a comparison and standing reference. Bertrand interval nonemptiness, prime-field cyclicity, the prime number theorem and the probability interfaces are exact external premises with versions, theorem locators, specializations and actual reading depth. The review checked their identified statements and does not recursively certify their proofs. It reconstructs the necessary local Markov/tail-union deductions without assuming shell-event independence. Cilleruelo attribution, L2 standing, historical author reading and novelty remain outside the authenticated mathematical coverage, not silent premises.

The fresh reviewer records no construction history, prior use, author advocacy, sibling verdict or substantive reconciliation. Its narrow comparison read was expressly admitted and was not used as proof. Required instructions are operating context, not mathematical evidence. The assignment, administrative messages and exact actual reading list support PASS on independence. The separate native-trust design-lane exposure did not reach this reviewer.

The frozen subject wiki/research/leads/balanced_auxiliary_prime_density/evidence/assets/loss_refinement_reviewed.txt as it stood on 2026-09-15T05:40:27Z carries the author's own standing sentences at lines 21-22, 661-662 and 665-667 (author-recorded, without independent review or acceptance; L4 and L2 standing at tier 0), and the admitted L4 comparison card as it stood on 2026-09-15T02:18:25Z carries L4's tier: 0 frontmatter and "Current standing" section; a separately spawned materiality grader (model: Claude Fable 5.1) ruled this exposure immaterial by the content test on 18 September 2026, because the text neither states nor implies the refutation-failed verdict and the review's reasoning did not lean on it.

Verdict and retained limitations

The verdict is refutation-failed for the unconditional existence theorem. The original grade recorded compliance with the bounded review admission and its unchanged clock. Nine full visual source pages and conservative PDF and repeated-instruction accounting were reported. Raw web transfer counts were not measured: conservative artifact-size charges were used. No computational reproduction was necessary or claimed. These are the original grade's reported facts, not independently repeated filing checks.

Before any tier-bearing use, preserve the exact reviewed subject, substantive review, scoped grade, source versions/locators and necessary retained inputs under the native owner's evidence/verify home, with ordinary-clone-resolvable links. Reconcile the new result with L4's existing statement and historical evidence without rewriting the old assessment to cover new mathematics. Decide current statement/tier only as part of that exact integration. No mathematical execution, formal verification, claim promotion or landing is performed by this grade.

The separate research owner retains the balanced-schedule argument without changing L4. The present author-prepared retention is not a correspondence verdict, and the historical PASS does not extend itself to an unchecked transformation or a later repair.