← NS_O / 27 / C3-Y: Derivative-Chain / Intermittency Tradeoff, Direct-vs-Chain Gap Closure, and Joint-Concentration Routing

NS · 27 / C3-Y C3-Y · The C3 Series Closes Out, 25/25 2026-08

27 / C3-Y: Derivative-Chain / Intermittency Tradeoff, Direct-vs-Chain Gap Closure, and Joint-Concentration Routing

The final round of the C3 series opens with a Source Audit Correction (§0): re-anchoring the external theorem to its formal journal version — Grujić & Xu, Asymptotic Criticality of the Navier–Stokes Regularity Problem, Journal of Mathematical Fluid Mechanics 26, Article 53 (2024-07-27) — explicitly listing the theorem numbers cited (3.5 the direct criterion, 3.7 the energy a priori, 3.8/3.9 the ascending/descending chain, 3.14 chain-assisted). It re-derives three scales: the energy-a-priori scale $R_{\rm apr}^{(k)}\sim A_k^{-1/(k+3/2)}$, the direct finite-$k$ regularity scale $R_{\rm dir}^{(k)}\sim A_k^{-3/2/(k+3/2)}$, and the chain-assisted scale $R_{\rm chain}^{(k)}\sim A_k^{-1/(k+1)}$, with $R_{\rm dir}

Re-anchors the external theorem to its formal 2024 journal version. Obtains a core identity: the ratio of the active-volume exponents required by the direct route and the chain-assisted route is exactly $k+1$. Honestly corrects the previous round's (C3-X) $1/7$ label — the theorem-ready direct exponent at $k=2$ is $3/7$, and $1/7$ is only the weaker formal exponent requiring an additional chain-ascension condition. Proves that ancestry-local intermittency is not the same as globally uniform-local intermittency (No-Go 19.1); the gap shifts from "is it intermittent enough" to "can it be globalized, and can it recur across the specified time window." The C3 series closes out at 25/25, handing off to C4. — the bundle's own self-declared stage status, given as-is.

Connections

Relationship to the rest of the series, stated as closely as possible in the document's own words, not my interpretation.

“ancestry-local φ ⇏ uniform-local Φ — the missing link is globalization and time-gate recurrence, not just 'is it intermittent enough'.” — quoted from the paper's own Sections 19 and 47.

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