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TCDC-05 · The half-weight lemma
TCDC: Target–Condition Dual Closure and Chen-Type Pressure
TCDC-05 · The half-weight lemma
Author line in the source: Neo.K; Aletheia (GPT-5.6 Sol)
Quarter-Root Weighted Pressure and an Adaptation of Chen's Lemma: Half-Weights for Strict Semiprimes, Medium-Divisor Multiplicity, the Extra Prime-Branch Cost and a Constant Gate
Proves by finite case analysis a pointwise Chen-type weight inequality for strict-semiprime pressure with half-weights (the coefficient 1/2 is minimal under a natural linear ansatz) and computes that at level 3/5 the coarse term minus the half medium-divisor term is only about 0.0841, before the prime-branch, exactly-one-medium and triple-factor costs are paid.
The one genuine sieve-style lemma in TCDC: a pointwise half-weight inequality proved by finite case analysis; 0.0841 is a thin margin before the prime-branch and other costs are paid. Verdict: HSP STILL NOT PROVED.
The inequality was recomputed on 581,148 triples (y, w, X) with zero violations (pre-publication read-through): that checks the lemma itself, not whether it yields HSP. TCDC-06 later shows the quarter-root is unnecessary and the margin rises to 0.4067. The verdict box of this paper has no GOLDBACH line, but its status line says Goldbach is still unproved.
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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.