Auditing the Wind Tunnel: the Twin-Enrichment Closed Form at Reachable ω, and the Marginal-versus-Per-Centre Distinction

Part XXIII extrapolated the twin-centre enrichment multiplier prod_{q|N} q/(q-2) to the primorial depth omega=15, obtaining x7.03, and flagged the step honestly as a closed-form self-portrait assuming local-factor multiplicativity persists past the omega ~ 5-6 range where it had been checked. We audit that assumption directly. Sieving all centres to 6N <= 2x10^8 (813,370 twin centres) we measure the actual enrichment, factor by factor. Three findings. (i) The microscopic marginal law is exact: the twin rate among centres divisible by q exceeds the global rate by q/(q-2), verified to <~0.3% for every q = 5,...,59; the divisor/non-divisor ratio is the complementary (q-1)/(q-3), reconciling the two forms used in Part I and Part XXIII as two views of one mechanism. (ii) Marginal multiplicativity holds within Poisson error for every prime combination with an adequate sample (<~1% through triples), with a faint systematic lean to the suppression side -- the sign of the Part IV second-order residual, but within our statistics. (iii) The wind-tunnel object prod q/(q-2), however, is a marginal factor and is not the per-centre likelihood. The correct per-centre enrichment of a centre with divisor set D is C * prod_{q in D} (q-1)/(q-3) with a global constant C = prod_{q>3} q(q-3)/((q-1)(q-2)) = 0.7216; this matches the data to <1% across the whole reachable range, whereas the wind-tunnel object over-predicts the per-centre rate by up to ~38% for typical factorisations. For the squarefree primorial centre the two nearly coincide (6.98 vs 7.03, a 0.66% gap), so the headline x7.03 is sound for the primorial centre and for the marginal divisibility statement -- but it is an upper envelope, not the typical omega-stratified enrichment, which sits far below it (<E> ~ 2.6 at omega=6 versus the 4.27 envelope). The wind-tunnel mechanism is thus vindicated microscopically; the two corrections are interpretive (marginal vs per-centre) and scope (envelope vs typical). As throughout, this is a measurement, not a theorem, and no claim is made about the infinitude of any constellation. Part XXVIII of the series "Arithmetic Geodynamics on the 6N Skeleton." Code, data, and figures: https://github.com/Ruqing1963/6N-windtunnel-audit

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Zenodo (CERN European Organization for Nuclear Research)
Published
2026-06-08
DOI
https://doi.org/10.5281/zenodo.20596630
Primary Topic
Analytic Number Theory Research
Type
preprint
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preprint

Auditing the Wind Tunnel: the Twin-Enrichment Closed Form at Reachable ω, and the Marginal-versus-Per-Centre Distinction

Ruqing Chen
Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
preprint

Auditing the Wind Tunnel: the Twin-Enrichment Closed Form at Reachable ω, and the Marginal-versus-Per-Centre Distinction

Ruqing Chen
preprint en

Abstract

Part XXIII extrapolated the twin-centre enrichment multiplier prod_{q|N} q/(q-2) to the primorial depth omega=15, obtaining x7.03, and flagged the step honestly as a closed-form self-portrait assuming local-factor multiplicativity persists past the omega ~ 5-6 range where it had been checked. We audit that assumption directly. Sieving all centres to 6N <= 2x10^8 (813,370 twin centres) we measure the actual enrichment, factor by factor. Three findings. (i) The microscopic marginal law is exact: the twin rate among centres divisible by q exceeds the global rate by q/(q-2), verified to <~0.3% for every q = 5,...,59; the divisor/non-divisor ratio is the complementary (q-1)/(q-3), reconciling the two forms used in Part I and Part XXIII as two views of one mechanism. (ii) Marginal multiplicativity holds within Poisson error for every prime combination with an adequate sample (<~1% through triples), with a faint systematic lean to the suppression side -- the sign of the Part IV second-order residual, but within our statistics. (iii) The wind-tunnel object prod q/(q-2), however, is a marginal factor and is not the per-centre likelihood. The correct per-centre enrichment of a centre with divisor set D is C * prod_{q in D} (q-1)/(q-3) with a global constant C = prod_{q>3} q(q-3)/((q-1)(q-2)) = 0.7216; this matches the data to <1% across the whole reachable range, whereas the wind-tunnel object over-predicts the per-centre rate by up to ~38% for typical factorisations. For the squarefree primorial centre the two nearly coincide (6.98 vs 7.03, a 0.66% gap), so the headline x7.03 is sound for the primorial centre and for the marginal divisibility statement -- but it is an upper envelope, not the typical omega-stratified enrichment, which sits far below it (<E> ~ 2.6 at omega=6 versus the 4.27 envelope). The wind-tunnel mechanism is thus vindicated microscopically; the two corrections are interpretive (marginal vs per-centre) and scope (envelope vs typical). As throughout, this is a measurement, not a theorem, and no claim is made about the infinitude of any constellation. Part XXVIII of the series "Arithmetic Geodynamics on the 6N Skeleton." Code, data, and figures: https://github.com/Ruqing1963/6N-windtunnel-audit

Zenodo (CERN European Organization for Nuclear Research)
Energoservis (Czechia) (CZ)
Affordable and clean energy
Analytic Number Theory Research
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