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Phase 13 · The P₂ constant ledger
Legendre's Conjecture: The Base-Space Prime-Fitting Program
Phase 13 · The P₂ constant ledger
Author line in the source: Neo.K × AI collaboration (no AI named)
The Square-Anchored P₂ Constant Ledger: the Rough-Lower Gate, the Bilinear Semiprime Upper Bound and the Parity-Signed Reformulation
A constant ledger for S₃ − E₂: heuristic Buchstab constants (prime 1, semiprime log 2 with bands log(4/3) and log(3/2), rough 1 + log 2), so the main-term margin is +1 ('PASS'); but the crossing of the current black-box provider is 'FAIL / NOT PROVED' (the standard linear rough lower bound is 0 at s = 3/2); the exact parity-signed identity P = (S₃ − L₃)/2; gap ranking: rough lower bound PRIMARY, semiprime upper bound SECONDARY, signed Liouville control PARITY-DIRECT.
The heuristic main-term constant ledger passes and the crossing of the current black-box providers fails: the paper writes that failure 'comes from provider strength, not from the architecture's main-term constant'. Legendre's conjecture remains unproved.
'MAIN TERM CONSTANT LEDGER = PASS' is an ideal-asymptotic (heuristic) statement that the same paper labels 'algebraic / heuristic main term'; it is not a theorem. The page also carries the 'P₃ frontier' framing shared by Phases 12–14 that the Errata corrects.
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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.