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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.19593222 |
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- <p class="MsoNormal">This note translates a logarithmically divergent prime-labeled partition profile into concrete four-dimensional observables on a scalar/conformal gravity lane. Using Hartnoll and Yang’s conformal primon gas as the source model, it assumes the induced scalar profile S_c ~ nu log(1/delta) as delta = s - 1/2 approaches zero from the positive side and then derives explicit asymptotics for the effective optical index, the Newtonian-potential analogue, and the Unruh-type temperature. It further proves a sharp optical-delay threshold: the induced travel-time integral remains finite for nu < 2, becomes logarithmically divergent at nu = 2, and diverges by a power law for nu > 2. Finally, the same large-S_c regime feeds the anomaly-generated Yang-Mills gauge weight from the companion coherence/Yang-Mills note, yielding a logarithmic weakening of the effective non-abelian coupling. The paper is a translation note, not a black-hole theorem, holographic dictionary, or replacement for critical string theory.</p> <p class="MsoNormal">CORE CONTRIBUTION<br>The paper takes a mathematically sharp prime-partition critical profile and pushes it through the scalar/conformal gravity dictionary to obtain equally sharp optical and gauge-facing consequences. Its main gain is not a new source model by itself, but a disciplined receiving translation: a logarithmic prime criticality becomes explicit asymptotics for gravity-facing observables, a clean threshold for horizon-like optical delay, and a concrete large-S_c source for the anomaly-generated weakening of the effective non-abelian coupling. That gives the prime-critical regime a precise four-dimensional observable reading without inflating it into a full black-hole or holographic claim.</p> <p class="MsoNormal">MAIN TECHNICAL OUTPUTS<br>1. Logarithmic critical source profile S_c ~ nu log(1/delta) extracted as the scalar-lane input.<br>2. Explicit asymptotics for n_g, Psi_N, and T_U in the associated scalar/conformal gravity model.<br>3. Sharp optical-delay threshold separating finite, logarithmic, and power-law divergence regimes.<br>4. Translation of the same critical scalar regime into the anomaly-generated Yang-Mills gauge weight.<br>5. Logarithmic weakening law for the effective non-abelian coupling in the critical regime.<br>6. Clean separation between what the note actually proves and stronger black-hole or holographic claims.<br>7. Concrete four-dimensional comparison framework linking prime-partition criticality to optical and gauge-facing observables.</p> <p class="MsoNormal">CLAIM BOUNDARY<br>1. The note does not prove a full black-hole theorem.<br>2. It does not supply a full continuum phase/cochain bridge.<br>3. It does not claim a holographic dictionary.<br>4. It does not replace the standard extra-dimensional consistency conditions of critical string theory.<br>5. Its gauge-weight consequence is downstream of the companion four-dimensional coherence/Yang-Mills effective coupling note.</p> <p class="MsoNormal">KEYWORDS<br>primon gas; prime partition criticality; scalar/conformal gravity; optical delay; horizon threshold; effective gauge weight; Hartnoll-Yang; SGOC.</p> <h2>Related SGOC papers</h2> <p>Primal Gravity: From Prime-Zeta Sc to Tensorial Plebanski-Einstein Geometry 10.5281/zenodo.19647370</p> <p>Primal Quantum Gravity: Beyond-first-order predictive laws, the Jacobson--Schrodinger bridge via capacity geometry, and common-source zeta structure. DOI: 10.5281/zenodo.19592981</p> <p>Branch Structure, Restricted Arithmetic Profiles, and Phase-Sensitive Exits in the Jacobson--Schrodinger Bridge 10.5281/zenodo.19601147</p> <p class="PlainListing">Capacity Geometry: REG, Dirichlet Tension, Curvature Identity, and Quasi-Local Mass. The Spectral Geometry of Coherence, Volume Two, Part 1, v2.0. DOI: 10.5281/zenodo.19156067</p> <p class="PlainListing">The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge. The Spectral Geometry of Coherence, Volume Two, Part 1A, v2.0. DOI: 10.5281/zenodo.19156316</p> <p class="PlainListing">Mass as Spectral Data in Capacity Geometry: Bulk Corridors and Bubble Caps. The Spectral Geometry of Coherence, Volume Two, Part 2, v2.0. DOI: 10.5281/zenodo.19161229</p> <p class="PlainListing">An Operational Verification Suite for a Capacity-Field Gravity Readout: Dual-Engine Poisson-Bridge Gates, Relativity Coherence Checks, and REG Micro-Audits. The Spectral Geometry of Coherence, Volume Two, Part 3, v2.0. DOI: 10.5281/zenodo.19161141</p> <p class="PlainListing">Arithmetic Source to Recovered Capacity Scalar. Carrier-mediated prime-built routes, restricted scalar closure, and exact stopping criterion. The Spectral Geometry of Coherence, Volume Three, Part 5. DOI: 10.5281/zenodo.19580874</p> <p class="PlainListing">Arithmetic Source Scalars, Recovered Capacity Geometry, and the Restricted Gravity Sector. The Spectral Geometry of Coherence, Volume Three, Part 6. DOI: 10.5281/zenodo.19593161</p> <p class="PlainListing">Gauge-Covariant Coherence Coupling of the Capacity Scalar to Yang--Mills in Four Dimensions: No-go results, conformal triviality, and the first viable effective coupling. DOI: 10.5281/zenodo.19593005</p>