← Legendre's Conjecture: The Base-Space Prime-Fitting Program / Phase 3 · Operator audit and self-correction

Legendre's Conjecture: The Base-Space Prime-Fitting Program Phase 3 · Operator audit and self-correction Author line in the source: Neo.K × AI collaboration (no AI named)

Operator Audit, Closed-Loop Canonicalization and the Historical Obstruction Theorem

Shows that the open-loop rollout of Phase 2 is not the original closed-loop predictor (the position ratio stalls at about 0.7904); audits the original closed-loop fit for 1000 ≤ N ≤ 200000 (p_200000 = 2750159): same square cell as the true p_N 91.07%, δ < 1 82.24%, 6k±1-snapped candidate prime only 21.79%, equal to p_N only 0.58%; and proves the 'historical obstruction theorem': a composite q ≡ ±1 (mod 6) in (n², (n+1)²) has a least divisor ≤ n, so q is prime iff gcd(q, H_n) = 1.

An explicit self-correction: the open-loop operator of Phase 2 is falsified as the canonical fitter. The 'historical obstruction theorem' is in essence a restatement of trial division. The restored original operator uses the true prime history (A_N = the mean of the actual p_k), so it is not an autonomous generator. The closed-loop audit for N ≤ 200000 ships no code or data and was not reproduced in the pre-publication read-through; Phase 2 itself carries no 'superseded' note (see the download table below).

Connections · Connections

PreviousPhase 1 · The canonical consecutive-square space · Canonical Algebraic Consecutive-Square Base Space
NextPhase 4 · The obstruction residue field and the exact sieve identity · Square-Anchored Obstruction Residue Field, Atomic Reduction and the Exact Sieve Identity

This is a full-text translation of the Chinese source, which remains authoritative for its research claims. Translation is not independent mathematical verification and does not upgrade the original evidence level.

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