← TCDC: Target–Condition Dual Closure and Chen-Type Pressure / TCDC-04 · The Chen-type architecture and the cube-root barrier

TCDC: Target–Condition Dual Closure and Chen-Type Pressure TCDC-04 · The Chen-type architecture and the cube-root barrier Author line in the source: Neo.K; Aletheia (GPT-5.6 Sol)

High-Suffix P₂ Pressure and Chen-Type Weighting: the Pressure Scale Corrected to m/log³ m, the Exact Prime/Semiprime Split and the Cube-Root Linear-Sieve Barrier

Corrects the main scale of HSP/MRC to m/log³ m, proves the exact split Π = S − P for cube-root-rough complements and the identity Π = S(w) − B(w, z) − P, shows that a direct cube-root lower linear sieve needs distribution level θ > 2/3 (so 3/5 and 5/8 do not suffice), and sets a contract of 'quarter-root coarse lower bound − Buchstab/Chen loss − prime branch'.

Corrects the scale of TCDC-03 (m/log² m → m/log³ m) and shows that a direct cube-root lower linear sieve needs level > 2/3. HSP and MRC remain NOT YET PROVED; GOLDBACH STILL OPEN. The paper states that this is not a direct instance of Chen's theorem ('a prime+P₂ existence theorem does not by itself prove HSP'): 'Chen-type' means similar in polarity, not that the theorem can be invoked directly.

Connections · Connections

PreviousTCDC-01 · The failure transport polytope · The Prime-Square Target and the Failure Fixed-Surface Construction: the Root-Bounded Prime Terminal Domain, the Goldbach Failure Transport Polytope, Capacity Deficits and Cut Certificates

NextTCDC-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

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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