← TCDC: Target–Condition Dual Closure and Chen-Type Pressure / TCDC-06 · The conditional two-sink closure

TCDC: Target–Condition Dual Closure and Chen-Type Pressure TCDC-06 · The conditional two-sink closure Author line in the source: Neo.K; Aletheia (GPT-5.6 Sol)

The Strict-Pressure Extra-Cost Ledger and Branch-Neutral Two-Sink Closure: Generalized Half-Weights, Coarse-Cutoff Optimization, the Prime Strictness Tax and the Separate-Upper No-Go

Shows that the half-weight lemma holds for any cutoff w < z with z³ ≥ X (the quarter-root is unnecessary), optimises the neutral core margin to about 0.4067 (u ≈ 7.8; 0.4063 at u = 8), shows that the standard separate prime upper bound (0.7421) cannot be paid, and restates closure as: branch-neutral lower bound > prime-sink capacity + mirror-sink capacity.

A conditional theorem (T14): if L_{P₂}(λ) > C_P^fail(λ) + C_mir^fail(λ), Goldbach failure is impossible. The hypothesis is never established; verdict: NO POSITIVE TWO-SINK DEFICIT YET; GOLDBACH STILL OPEN. 'The standard separate prime upper 0.7421 cannot be paid' is a no-go for this one combination of sieve inputs; the paper says it is not an impossibility theorem. It corrects TCDC-05 (the 0.0841 margin is superseded).

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PreviousTCDC-05 · The half-weight lemma · 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

NextTCDC-14 · The branch ratio and joint transfer · Bulk Mixed Deficiency and the Coarse-Balance Constant: the Cube-Root Buchstab Branch Constants, the Prime–Semiprime Bias and the Parity Barrier of Joint Transfer

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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