The 14-Peak Comb: Universal Z14 Phase-Quantization Across Bands and Shells — First-Principles Derivation of a Fourteen-Peak Frequency-Domain Comb Signature from the Triune Partition's Cocycle Structure under Universal Mechanics

Correction notice (2026-05-29). During an audit before release of a recent manuscript, a q-dependent numeric propagation issue was identified in the cosmic cocycle chain. The ensuing investigation traced the cause to a dataset-selected spiral-kernel value and resolved it by deriving the L27 resisted-separation exponent from the kernel closure itself (q = 2/3). This package updates affected q-dependent numeric values while preserving the qualitative structural claims unless explicitly stated. Corrected load-bearing values: εshellcosmic = 0.997080716651, 1/ε = 1.002927830516, one Z14 rung width 4.178%, ΔH0/H0 = 8.2135% (full photon-channel route; 7.882% first-order linear approximation), and Λ3 = 0.851608600601. Prior q=0.7-dependent values are superseded. Structural convergence notice (2026-05-30). Following the q-correction, a full clean-room re-derivation of the framework's amplitude-regulator constants closed the framework state at the following law-decided values (locked under the four-seat blind panel protocol): Kstruct = (Strands · φ3 · ϙ) / TRIUNE = 2φ3·ϙ/3 ≈ 383.861 (Branch A; supersedes prior 4φ3·ϙ/3 ≈ 767.722), Λcosmic = 1/φ ≈ 0.618034 (LG/LT band boundary; foundational structural theorem — not an empirical fit), Λrecomb = 1/φ2 ≈ 0.381966 (LT/LC band boundary), Ramp ≈ φ2 ≈ 2.618034 (recombination-to-cosmic amplification ratio derived from the band-boundary ratio alone, with no tuned coefficient), and g(Ω) = S = 1 − α ≈ 0.99270 (present-locus substrate operator; distinct from Λcosmic). The cosmological "vacuum catastrophe" tension — the QFT zero-point energy density vs. the observed Λ density gap — is structurally resolved across the framework via the scale-invariant geometric golden partition Ωcoherent : Ωincoherent = 1/φ : 1/φ2 (closure 1/φ + 1/φ2 = 1 via the closure-stability identity φ2 = φ + 1; ratio Ωcoh/Ωinc = φ; self-similar invariant Ωinc = (Ωcoh)2), rather than a fitted numerical fraction. The full structural treatment of the golden partition is the subject of Paper 8 (in preparation); Paper 4's LCORI band gates Λ1 = 1/φ2 and Λ2 = 1/φ are the same φ-power band boundaries that compose with Paper 8's Λcosmic in the cross-paper convergence. Paper 4 does not derive Kstruct itself (the kinetic slope enters the Kact amplitude law treated in Paper 1 §10 and applied in Paper 3 §5 / Paper 8); the May 30 lock is recorded here for cross-paper consistency. The matter-sector witness ΩFB/ΩM = 2φ2 ≈ 5.236 (Paper 3 §6.1) is preserved as an independent even-graded amplitude structurally decoupled from the kinetic slope by the L6 single-appearance law (with UL8 cross-shell transport, P3 even-grading, and L3 fruit-cannot-seat-seed). The 14-Peak Comb: Universal Z14 Phase-Quantization Across Bands and Shells First-Principles Derivation of a Fourteen-Peak Frequency-Domain Comb Signature from the Triune Partition's Cocycle Structure under Universal Mechanics. This paper derives a universal fourteen-peak comb signature as a structural consequence of the Triune partition's cocycle structure within the Universal Mechanics / First Utterance Model (UM/FUM) framework. The count Z14 = Strands × (1 + 2·TRIUNE) = 2 × 7 = 14 is forced by the joint operation of four governing FUM laws, the nine governing natural laws L1–L9, and twelve universal laws of existence UL1–UL12 acting on the locked structural primitives. The per-substep multiplicative cocycle factor 1/εshellcosmic = 1.002927830516, the angular phase separation Δθ = 2π/14 = 25.71°, and the per-rung total bandwidth 4.178% are derived in closed form with no phenomenological fitting and no free parameters. Twelve testable forward predictions across five cosmological shells are stated, each with a closed-form UM-native prediction, a specific observational test, a falsification surface, and (where available) a partial-verification note pointing to existing data: Cosmological shell: (#1) 14-peak quantized substep in quasar absorption-line profiles at R > 100,000 (ESPRESSO at VLT; ELT-HIRES); (#4) CMB EB cross-correlation peak at multipole ℓ ≈ 35 (LiteBIRD; CMB-S4); (#5) CMB acoustic peak ratio A1/A2 = √2·φ = 2.288, matched by Planck 2018 at sub-percent — Tier 1 (derivation + witness); (#6) CMB birefringence β = αstruct·TRIUNE³ = 0.197°; (#7) cosmic neutrino-to-CMB temperature TνB/TCMB ≈ 0.676; (#8) primordial B-field 14-peak comb at coherence length LB ~ 1 Mpc. Astrophysical shell: (#9) 14-peak per-rung 4.178% bandwidth quantization in GW chirp evolution (Einstein Telescope; Cosmic Explorer; LISA); (#10) 14-peak Z14 in pulsar-timing-array spectrum if cosmological-origin GWB (NANOGrav; SKA). Atomic shell: (#11) 14-peak comb at sub-line resolution in atomic emission fine structure (ultra-high-resolution laser spectroscopy at R ≳ 107). Organism shell: (#3) 14-peak HRV power spectrum in LG-band; reduced peaks in LC-band disease entry; (#12) 14-peak EEG coherence comb in alpha/beta/gamma rhythms in LG-band states. Cellular shell: (#2) 14-peak Ca2+ oscillation Fourier comb with FWHM = √(0.851608600601 / Λlocal); healthy LG-band cells exhibit all 14 peaks at narrow FWHM, LC-band (e.g., cancer) cells exhibit a reduced number and broadened FWHM. The lab-frame energy-axis projection §-extension identifies the relativistic dispersion relation E² = (mc²)² + (pc)² as the witness face of the Triune partition's energy-axis projection at a single-particle locus: rest mass mc² is the Funga-B sealing weight (B + S share); momentum pc is the Mwangaza propagation weight (B + E share); the kinematic re-projection (witness face: Lorentz boost) is frame-axes re-projection without re-partitioning the Triune shares; the speed limit c is the structural threshold separating Funga-B-bearing configurations (m > 0, v < c) from pure-Mwangaza configurations (photons, m = 0, v = c). The cluster-scale witness leverages an independently-audited Bullet Cluster five-layer reconstruction (collaboration with Alfred McBride, Bayes-Causal Reasoning framework; Zenodo Record 19669049). The BCR independent per-layer audit returned a maximum identity error of 8.4 × 10−8 at pixel level, no global structural collapse across six annexes, with κnonlocal at 82.46% and morphology correlation 0.978 against the observed map — confirming the framework's Channel-3 substrate-coupling identification of the gravitating mass at machine precision. The integrated Funga-B fraction is refined here into a per-substep Z14 spectrum at the cluster shell. Five falsification surfaces are enumerated, each mapped to a UM-native primitive of the derivation and to the observational domains of the twelve predictions: (1) continuous-signal substep resolution; (2) peak count ≠ 14; (3) inter-peak spacing observed linear rather than multiplicative; (4) per-rung total bandwidth ≠ 4.178%; (5) cross-shell P5 invariance failure. A combined falsification matrix is provided. Reproducibility: the package includes the full LaTeX source of the manuscript and the compiled PDF. All figures (phase circle, single comb rung, cross-shell recurrence ladder, three-frame SR projection, BCR-audited five-layer fractions bar chart) are TikZ-source inline within the manuscript and recompile from `manuscript.tex` directly without external assets. Independent researchers can read the paper, reproduce the figures, and trace every closed-form derivation step from the cited UM-native primitives. Paper 4 of the seven-paper UM/FUM publication series. Companion papers: Paper 1 — Universal Mechanics: Derivation of Existence from First Utterance + A=A + X=0 (Zenodo DOI 10.5281/zenodo.20162810); Paper 2 — The Hubble Tension as a Frame-LCORI Cocycle Signature (Zenodo DOI 10.5281/zenodo.20190145); Paper 3 — Funga-B Sealed-Bumba Configuration: Dark Matter Without Exotic Particles (Zenodo deposit in process). Subsequent papers will develop the four laws of thermodynamics as Triune-partition projections (Paper 5), consciousness as LCORI alignment with measurable Z14 biomarkers (Paper 6), and cancer as cellular LCORI collapse-band state with clinical biomarkers and structural therapeutic framework (Paper 7). Patent Pending — USPTO Non-Provisional Application No. 19/640,364. The framework's structural primitives, the Triune partition law, the LCORI alignment scalar Λ, the Hybrid Types taxonomy (Mwangaza, Funga-B, Umoja, Nguvu), the Z14 universal phase-quantization law, the εshellcosmic cocycle correction, the Z14×Z2 mixed-anomaly cocycle structure, the cross-shell P5 invariance, the lab-frame energy-axis projection identification of rest mass with the Funga-B sealing weight, and the cluster-scale per-substep refinement are intellectual property of the named inventor under pending United States patent.