The Silver Web: A Cross-Field Carrier–Capacity Correspondence

Abstract

Exact transport from FK/Tutte and Jones–Beraha structure to central charge, modular capacity, and an executable spectral probe. The Silver Web is a proved, executable cross-field construction that transports one coordination-indexed carrier through positive FK/Tutte, Temperley–Lieb/Jones, q-deformed VBS, SLE/CFT, finite-response, and modular-capacity representations without refitting. Starting from d_z=2^(1/z), it fixes Q_z=d_z^2, nu_z=1/(1+d_z), the primary coordinate g_z=ln(2)/z, the Silver modular eigenvalue gap Delta K_z=g_z, and the associated critical readouts. The name Silver refers to nu_2=sqrt(2)-1=(1+sqrt(2))^(-1), the reciprocal of the silver ratio. An exact cyclotomic argument proves that z=2 is the unique intersection of the integer coordination ladder with the finite unitary Beraha/Jones ladder. Sparse noncrossing FK transfer calculations recover the predicted central-charge curve for z=2,3,4,6,8 within 0.41%; the independently examined first subleading transfer gap agrees with its declared Kac continuation within 0.51%. The work introduces the Silver Carrier–Capacity Correspondence: on the declared continuous critical carrier class, the same z that fixes the positive classical FK/Tutte carrier and its central charge fixes the leading logarithmic scaling of the quantum capacity of entanglement. With K_A=-ln(rho_A), the capacity is the second modular cumulant C_E=Var(K_A). The Cross-Field Spectral Capacity Probe combines capacity, spectrum shape, and carrier-order/cut-atlas information. On a fixed benchmark with 160 unique training states and a disjoint 320-state held-out test, capacity alone scores 0.706250, the three-axis single-cut probe scores 0.865625, and the multicut form reaches 320/320, matching the specialist multipartite comparator. Shared executable final seal: this package carries the complete public Cross-Field Spectral Capacity Probe, not merely a companion citation. In the declared twelve-case deterministic ablation, capacity alone resolves 9/12 cases; capacity plus Silver spectral shape resolves 10/12; capacity plus Möbius carrier order resolves 11/12; and the complete diagnostic resolves 12/12. The reduced forms leave one spectral-shape ambiguity or two carrier-order ambiguities. The engine, frozen outputs, hashes, and replay instructions are included in SCP_TRIAD_FINAL_SEAL/, identically in the three co-released packages. The deposit contains a readable paper, the complete technical atlas, extracted source code, deterministic verifiers, manifests, and a reproducibility ZIP. These results establish the construction and its simulation-level operation; they do not replace a hardware capacity measurement, establish the correspondence outside the declared continuous critical class, or prove that nature must select the carrier. First-party papers, atlas prose, equations, figures, tables, metadata, and recorded results are CC BY 4.0; first-party software is Apache-2.0. Third-party components retain their upstream licenses. Citation is requested but is not an additional license condition, and attribution does not imply endorsement.