Dimensional Mechanics

This series proposes that the universe's shape, quantum behavior, and fundamental constants emerge from a framework organized around a single quadratic equation: the bronze metallic mean, x²=3x+1. The core principle is basically the path integral idea applied to existence itself under the guise of treating dimensions as higher-level harmonic structure. The framework doesn't choose dimension 3 or the Standard Model gauge group or the values of the constants. It lets everything unstable destroy itself and derives what's left.

The model treats space as a three-dimensional shell embedded in six dimensions, where curvature creates a surface tension that matches the cosmological constant without vacuum energy. Quantum behavior emerges as interference between waves in the embedding dimensions rather than fundamental randomness. A filtering process governed by spin and parity filters dimensional states until dimension 3 emerges as the only nontrivial stable attractor under the recursion. The golden ratio is algebraically excluded from the bronze layer by Fibonacci growth in the metallic constant, and the real dimensions form the triangular number sequence 1, 3, 6, 10, 15, …, reproducing the critical numbers of string theory — T(4)=10, corpse sum 24, and 26=2×13 — from recursive dimensional selection rather than conformal anomaly cancellation.

Paper V ties together these results, among others. From the same bronze equation and the geometric/topological structure built around it, the framework derives the Standard Model gauge group, the electroweak mixing angle to about 0.2σ, the Higgs boson mass to about 0.2σ, the fine-structure constant to 0.03%, both independent charged lepton mass ratios to 0.004%, seven quark and quark-lepton mass ratios from 0.035% to 0.535%, including the exact bronze identity m_s/m_d=20, the QCD one-loop structure 33−12=21 with 33=3×((x*)²+(x**)²), the full cosmological energy budget within about 1.5σ of Planck observations, and a fixed dark-energy exponent that slightly improves the DESI DR2 BAO fit over ΛCDM at the same parameter count. The primary numerical outputs are algebraic functions of √13 with no fitted parameters.

V8 - Paper V updated, exploring a bounded seesaw/ghost-echo correction mechanism across the numerical residuals and deriving the correction capacities from the Jacobi spectrum and winding structure. The target-aware echo itself is an audit mechanism; the nontrivial result is that the independently derived capacities sort the residuals into physical depth tiers.

V9 - KK addition in Section 8 of DMV introduces a theorem, two propositions, a corollary, and a spectral table connecting n* and α to the compactified transverse S²x*, with the discussion and abstract updated accordingly.

V10 - Corrections across DM II, IV, and V: restored the missing ψ(x) factor in DM II's nonlocal field equation and corrected the Born-rule derivation so the |χ|² exponent comes from quadratic field energy rather than the bronze polynomial; replaced the global dψ construction in DM IV with the locally exact/globally non-exact angular form dθ and finished the Finkelstein-Rubinstein sign-representation argument; corrected the Planck comparison in DM V and removed the 2^-74/16.8σ ghost-echo significance claim, replacing it with depth-tier sortation from the independently derived C1 and C2 capacities.

V11 - Expanded Paper V’s mathematical foundations, deriving the exact Jacobi zero-mode projector, the full harmonic representation tower under diagonal SU(2), and the explicit gauge–shell mixed variation. Added a finite-cutoff Schur-complement noncompensation theorem, the vacuum Poincaré formulation of the Yang–Mills mass-gap problem, and a gauge-covariant curvature filter. Extended the bronze arithmetic through Q(√13) and added neutrino mass-splitting and candidate CKM/PMNS tests, separating the exact geometric identities from the remaining physical assumptions. Also introduced Dimensional Mechanics VI: The Hyperbolic Attractor Dynamics of the Metallic Mean Family, establishing that the realized and ghost roots of x² = nx + 1 occupy reciprocal and Lorentzian hyperbolic structures, with an exact U(1) phase representation. The ghost residual becomes the contracting eigenvalue of a reversible integer transfer matrix, yielding exact entropy, periodic-orbit counts, a dynamical zeta function, and arithmetic connections to Dirichlet L-values and modular geodesics. DM VI then connects the same bronze dynamics back to Paper V’s S³ Jacobi spectrum and gauge–shell coupling, extending the framework from recursive dimensional selection into explicit spectral, arithmetic, and operator dynamics.