65 / C6-P: Ancient Defect-State Classification, Record-Peak Derivative Rigidity, and Peak-Local Pressure Closure

C6-O left behind a three-scale picture: mass scale $\gg$ peak scale $\gg$ derivative scale, with a Derivative-Tower Restart triggered whenever some derivative order's amplitude diverges under peak normalization. C6-P gives a clean correction: record-peak boundedness by itself already eliminates every fixed-order derivative tower. Citing the Koch–Nadirashvili–Seregin–Šverák bounded-mild-N–S parabolic-smoothing theorem, C6-P.1 Fixed-Order Record-Peak Derivative Rigidity proves that for every fixed $k$ there is a constant $C_k$, depending only on $k,\nu$, such that $\|D^kv_n(0)\|_\infty\le C_k$, which in the original variables reads $A_{k,n}\le C_kA_n^{k+1}$. C6-P.2 Fixed-Order Derivative-Tower No-Go therefore directly overturns C6-O's hypothesis: $\widehat A_{k,n}^{peak}\to\infty$ at fixed $k$ simply cannot happen at the record peak, and C6-P.3 No Fixed-Order Subpeak Scale further gives the uniform lower bound $b_{k,n}/a_n\ge C_k^{-1/(k+1)}>0$ — no fixed order can ever produce a physical scale smaller than the peak. The only escape route still standing is the order-index itself diverging ($k_n\to\infty$), but citing the spatial-analyticity theorems of Xu and of Wang–Gao–Xue, C6-P.4 Raw High-Order Growth Is Not a Physical Tower proves that record-peak boundedness by itself already gives a uniform positive radius of analyticity, so the factorial growth of raw high-order derivatives is just an ordinary analytic baseline and cannot be read as a new physical concentration scale — a genuine high-order escape must instead use a factorially normalized root (reusing C5-H's all-order no-go technique). C6-O's three-layer tower "mass $\gg$ peak $\gg$ derivative" is thus corrected back to a two-layer "mass $\gg$ peak," plus one independent order/geometry-escape option. It then addresses whether the peak itself degenerates: if the first-order ratio $\widehat A_{1,n}^{peak}\to0$, the Ancient Flattening Rigidity Theorem (C6-P.5) cites Lei–Yang–Yuan's newest (2026) backward-uniqueness theorem for bounded mild 3D N–S, proving that the ancient profile must equal a single nonzero constant vector over the entire backward time interval — not merely flat at one instant, but a thoroughly Galilean-trivial solution; C6-P.6 therefore explicitly states that this branch carries no TS/GP/HF defect at all. The non-flat branch is gathered into the Fixed-Order Carrier Dichotomy (C6-P.7): either local capture (P-KVIS) or Derivative-Carrier Translation Escape (P-KESC, where the fixed-order derivative is itself bounded but escapes spatially) — this formally replaces C6-O's derivative-tower branch. On the pressure side, the round continues and closes out C6-O's breakthrough: the Peak-Local Pressure Hessian Closure Theorem (C6-P.9/10) fully proves that, in a bounded peak frame, the pressure Hessian is entirely determined by a finite radius, and the far-away mass tail cannot sustain an independent $O(1)$ pressure contribution; C6-P.11 Conditional Ancient GP Inheritance shows that, under five conditions, the local GP metadata legitimately carries over to the ancient limit. But this same pressure argument cannot be transplanted directly to TS's operator channel: the $P_{st}$ projection kernel is a general order-zero Calderón–Zygmund type ($|x-y|^{-3}$, not absolutely integrable in three dimensions), unlike the pressure-Hessian kernel ($|x-y|^{-5}$), which is naturally integrable — this opens a new Projected-Operator Tail Gate, and TS's ancient inheritance formally becomes conditional. The round gathers everything into C6-P.12 Ancient Peak-State Reduction: FLAT, $GP_{\rm loc}^{anc}$, $HF_{k,\rm geom}^{anc}$, $TS_{\rm loc/proj}^{anc}$, Spatial Carrier Escape, or Order-Geometry Escape — explicitly emphasizing that no fixed-order physical derivative-tower branch survives any longer. It also once again warns that ancient inheritance ≠ ancient recurrence. Formally hands off to C6-Q — Ancient Defect Rigidity, Projected-Operator Tail Control, and Spatial Carrier Escape, with ten constituent obligations directly attacking: whether the $P_{st}$ tail can be controlled, and whether the ancient GP/HF states can genuinely recur or be excluded — this will be the final round of the C6 series.