Examination of the conditional Problem 824 implication

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conditional_grade_reasoning: Grading reproduction of the conditional examination's signed, extraction and counting keepers, preserving their exact adverse and surviving scopes.

counting_keeper: Independent proof of the full real-variable squarefree-pair bound from Hypothesis A+, including overlap deletion and gcd fiber multiplicity.

extraction_keeper: Independent reconstruction of the successor extraction, its balanced sieve, and the failure of an unrestricted reading of its dyadic bound.

grade: Independent whole-contract grade of the conditional Problem 824 examination, retaining its dyadic refutation, accepted balanced scope and unaccepted antecedents.

interface_disposition: Exact conditional implications permitted by the separate elementary and conditional-chain grades, with both adverse scope findings retained.

premises: Exact conditional assumptions, classical theorem interfaces, and source-reading limits of the squarefree-pair examination.

report: Whole-statement adversarial examination of the conditional smooth-successor implication, with a counterexample to its unrestricted auxiliary dyadic formulation.

signed_keeper: Independent written reconstruction of the signed Type II implication under its explicit shift-cancellation and elementary hypotheses.

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The report, premises, signed derivation, extraction and counting derivation examine the supplied text frozen and examined on 2026-10-15 UTC. The report maps all seven retained inputs under evidence/assets/conditional_chain_subject/ to their native subjects.

The examination records refutation-failed for the implication from P and the operative E estimates to, for every $\varepsilon>0$ and all sufficiently large real $x$, $h(x)\ge h_{\rm sf}(x)>x^{2-\varepsilon}$, with simultaneous strict $1\le a<b<x$, coprimality, equal divisor sums and squarefreeness. The independent grade, examined on 2026-10-16 UTC, is PASS for the complete conditional examination and its adverse unrestricted-dyad finding. Its reproduction account assesses all three keepers. The graded interface comparison uses exactly the separately accepted E fragments, leaving the full P endpoint hypothesis and quoted classical premises explicit. It supplies neither P acceptance nor an unconditional Problem 824 conclusion.

The isolated dyadic estimate with only a product-size hypothesis is refuted-as-stated. Its $m=1$ witness loses a roughness factor. The proof of the balanced specialization actually consumed survived examination. The current extraction states that balanced range and its box-error use explicitly; these statement clarifications are not the frozen wording.

The source method is OpenAI's unrefereed Weighted dilation graphs, smooth shifted primes and totient fibers, 24 September 2026 release. Its supplied excerpt is retained under Apache-2.0, identified by the source card. The successor application and reduction are Li and Zhuang's, following Pollack and Pomerance after Erdős. Project exposition and examination are credited to Plasma AI. No original classical proof, original Li–Zhuang PDF, or PDF-fidelity comparison is certified.

Mechanical maintenance consists of moving subject inputs to the assets home, adding .md.txt suffixes to text snapshots and maintaining their links; removing the private revision from the read map and adjusting its filenames; replacing one operational role label; and restoring six bare carriage-return \rm labels (one in the signed derivation, five in the extraction). The report's preservation wording reflects these changes. No mathematical assertion, proof step, hypothesis, scope or verdict is changed. No executable mathematical evidence is part of this record.

Display titles, page separators and wiki metadata are formatting maintenance.

The original reviewer's bootstrap settings were displayed before their identity comparison; the grade records why their operational content supplied no material mathematical exposure. The grading parser's disclosed wrong-directory write was corrected before successful execution; the initial intended-path invocation failed without executing. Filing confirmed the permitted configuration-only trust append and preserved both the scripts and actual-read records. No excluded mathematical input was received through these incidents, and no exposure waiver is relied on.

The grade and its reproduction account maintain links to the already filed canonical subjects and keepers; the interface disposition's intended filing paths are updated to these homes. Its preservation wording reflects the declared metadata maintenance. Page separators and wiki metadata are formatting only. No scope, hypothesis, date, mathematical finding or verdict changed.