← Legendre's Conjecture: The Base-Space Prime-Fitting Program
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Phase 5 · The fit-ordered fibre (an honest negative)
Legendre's Conjecture: The Base-Space Prime-Fitting Program
Phase 5 · The fit-ordered fibre (an honest negative)
Author line in the source: Neo.K × AI collaboration (no AI named)
Canonical Fit-Ordered Fiber, Fixed-Point Depth and the Low/High Obstruction Decomposition
The 'V3' position formula (a truncated Cipolla-type expansion of p_N) puts a fit centre within O(log n) of the left square edge, but the nearest 6k±1 candidate is prime only 17.1477% of the time (10 ≤ n ≤ 20000), and V3 ordering gives the same fixed-point depth as scanning from the left edge (median 4, max 55): 'no distinct advantage'; preconditioning at z = (ln(n+1))² lifts the rank-1 hit rate to 45.94%; 'sublinear fit-depth ⇒ Legendre' is stated, not proved.
An honest negative result: the rank-1 prime section is falsified (17.15%) and the position formula has no advantage over scanning from the left edge.
The shipped Legendre_Phase5_experiment.py never terminates at its default n = 10 (order() waits for 1000 candidates while the n = 10 cell has 7); with a one-line guard the pre-publication read-through reproduced every figure of the paper exactly. The fit centre lies within O(ln n) of n² by construction, so the operator carries no cell-scale prime-location information — the paper itself says 'Prime-Fitting Base Space ≠ Position-Fitting Base Space'. 'Sublinear depth ⇒ Legendre' is much stronger than Legendre's conjecture itself (Phase 12 admits this).
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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.