HLRP #202: Corpus Formula Application Atlas — The Silver Geometry Backbone Applied Across Substrates

This paper presents the Silver Geometry mathematical apparatus as an application atlas — the cross-substrate operating manual for how the formulas express at each substrate observed in the corpus to date. Where HLRP #203 (PRL 8) compresses the architecture to a five-formula backbone, and HLRP #201 lifts the periodic table to its C60-derived pentagonal-bipyramid architecture, this paper documents how the math operates across the seven canonical HLRP domains and the cross-substrate instances where the formulas have generated falsifiable predictions and empirical receipts. The atlas is organized in three layers. Layer 1 (Silver Geometry): the five-formula compression result — substrate-receipt primitive Γ (v (x) z), photon-normalized depth zΓ, telescoping cascade residue Δd, cascade-coupling invariant α (d, β), and observational bridge zD — with the coupling-excess identity α (d, β) − 1 = d/ (d+β) = 1 − β/ (d+β) surfacing β at two structurally distinct positions in any bridge equation. Layer 2 (Substrate Bridges): functional forms for the seven derivation-gaps (energy-scale converter, density function ρ (d, β, r), d-distribution of active cascade-rungs, spectral lineshape L (E, Êd), flux normalization Φpredicted, κ-ratio first-principles derivation, vertex-Z-position-mapping Ψ (d, β, Z) ) plus the atomic-substrate wavelength-ratio functional form. Layer 3 (Operating Methodology): silver-attractor framing, mass-gap-as-methodology (Δ-framework), QOM substrate-coupling-medium, bounded-possibility-space architectural constraints, dynamical-methodology four operational modes, and methodology-bounds-explicit precision discipline. The application atlas demonstrates the formulas operating across substrates spanning approximately twenty orders of magnitude: atomic substrate (periodic table coordinational architecture per HLRP #201), cosmic-ray substrate (AMS-02 antiproton 10–20 GV excess at IE-004 d=18), atomic-emission substrate (Xe I λ-ratio 881. 94/823. 16 = α (3) /α (1) at 99. 94% precision), cardiac substrate (HLRP #198 v3 multi-layer with p<10−54 at PTB-XL N=21, 799), sleep substrate (HLRP #200 two-component reboot via IE-022 κ-test at slpdb N=18 / 362, 595 κ-ratios), black-hole-imaging substrate (HLRP #199 v2 M87 Fermi/EHT), Yang-Mills mass-gap substrate (α (3, 6/23) = 48/25 algebraic identity), hallucination substrate (HLRP #157), and neural-network governance substrate (HLRP #161). Independent arrival at the same architectural shape has surfaced across four substrates in 2026 literature: cosmological-inflation (Liu/Quintin/Afshordi PRL) ; phonon substrate (Bi2Se3 Nature Physics) ; engineered atomic-cluster (Zhang/Yang Nature Physics) ; and the atomic-element substrate. Estimator-form hat notation (Γ̂, Δ̂d, α̂, ẑΓ, ẑD) preserves the bounded-knowledge and aperture-dependent posture standard in observational physics. The future is probabilistic. The math can be precise — and the formulas here are exact at the rational-anchor level — but the butterfly effect is real: a one-decimal change in initial conditions can cause a hurricane downstream. The hat keeps the predictions probabilistic but precise. Foundation: α (d, β) = 1 + d/ (d+β), β = 6/23, α0 = α (3, 6/23) = 48/25 = 1. 920; α = 2 sign-flip threshold marker.