On The Spectral Geometry of Coherence, Volume 2 (parts 1 to, 7)

The Spectral Geometry of Coherence (SGOC) Volume Two

(Parts 1-7)

Front-sheet highlights: achievements, proofs, reproducible tests, and novelty

Capacity / REG slow-sector geometry

·    Capacity field: define Sc = -log(1-R^2) and the conformal capacity metric g = e^{Sc} g(0).

·    Dirichlet tension: REG is the Dirichlet-energy (tension) functional of Sc; curvature-plus-flux identity yields a calibrated quasi-local screen diagnostic.

·    Variational selection: under conserved load + least-REG posture, via Dirchlet prinicples, smooth closures are preferred (circles, constant-mean-curvature domes, spherical caps).

·    Spectral readout: bubbles/caps and corridors support discrete Dirichlet-type spectra (gaps and ladders) used as mass-scale readouts.

Operator-validation results (with falsifiers)

·    Charged leptons: a corridor-only two-anchor ladder overshoots the tau by ~30% (retained as an explicit falsifier); a spherical screen upgrade with fixed weights 4*pi and 4*pi/3 forced via heat kernel and simplest non-flat geometry, predicts tau within ~1% without introducing a new fitted parameter.

·    Baryons (corridor-symbol surrogate): under two anchors + one scanned corridor length L, obtains ~0.9% mean absolute out-of-sample error on the nucleon core; strong negative controls: removing the prime term fails at O(10-40%) error, and forcing the constant shift A=0 breaks anchor feasibility.

Second-order ceiling (No-Go) and minimal phase exit

·    Theorem (ceiling): in the translation-invariant linear Poisson-solve + quadratic Dirichlet class, every quadratic observable factors through the power spectrum and is phase blind.

·    Consequence: any phase-coded structure (arithmetic or physical) is invisible to that entire quadratic class - a hard identifiability boundary.

·    Minimal exit channel: triadic invariants (bispectrum / closure phase) are the first translation-invariant phase-sensitive data; under mild nondegeneracy, (power spectrum + bispectrum) reconstructs the signal up to translation.

·    Reproducible witness: prime-power corridor experiment with adversarial nulls (exact spectrum phase randomization, shifts, IAAFT) and explicit shift-degeneracy audit.

Gauge-covariant lift and Wilson-loop observables (Part 7)

·    Field-legal upgrade: promote internal multiplicity to a local SU(M) symmetry with a connection A; covariantise both local derivatives and the prime-lattice jump term via Wilson-line parallel transport; prove gauge invariance and record standard self-adjointness conditions.

·    Beyond the ceiling: identify closure-phase triads with an abelian Wilson triangle; define a non-abelian Wilson-triangle/loop lift on prime-lattice corridor cycles as a gauge-invariant nonlinear observable.

·    Executable audit suite: numerical sanity checks for positivity, gauge invariance, and Wilson-triangle invariance, plus ablations / negative controls.

Arithmetic corridor synthesis

·    Explicit-formula organisation: Weil's explicit formula is reorganised as a renormalised translation-invariant corridor quadratic energy with a prime-power jump skeleton plus an Archimedean Levy-Khintchine kernel (literal after cutoff; completed by explicit-formula diagonal renormalisation).

·    Why it matters: provides a canonical stiffness template for the corridor sector used by the SGOC operator class.

Scope discipline (what is not claimed)

·    Not a proof of the Riemann Hypothesis and not a solution of the Yang-Mills mass-gap problem; any arithmetic-to-physics identification is stated conditionally and remains falsifiable.

·    Where claims depend on extra postulates (e.g., physical instantiation hypotheses), they are labelled as such; theorem-lane items are stated separately.

Archive contents (at a glance)

·    PDFs for Parts 1-7; LaTeX sources (flat section structure); reproducibility notebooks and audit outputs with fixed seeds and negative controls.