SECTION 14 — EMERGENT GAUGE SYMMETRIES AND RECURSIVE MASS GENERATION (v11.3)updated
Finalized for Geometric Unity and Empirical Closure
By John Gavel
14.0 Overview
All Standard Model structure emerges from recursive alignment and misalignment of multi-component causal flow motifs in emergent 3+1 geometry. The concurrency budget \(H = 132\) arises from the icosahedral coordination number \(K = 12\) via \(H = K \cdot (K - 1)\), as derived from finite relational capacity and the non-redundancy constraint in Section 3.1.
This section maps the fixed hardware budget directly to gauge symmetries, fermionic matter, and the universal mass hierarchy—without free parameters or external fields.
14.1 Multi-Component Flow Multiplets
A node supporting \(n\) independent causal recursion paths carries an \(n\)-component multiplet:
\( \Psi_i = (F_i^{(1)}, F_i^{(2)}, \ldots, F_i^{(n)}) \)
Each component \(F_i^{(k)} \in \{+1, -1\}\) represents a signed causal flow along path \(k\).
Phase accumulation occurs as:
\( \theta_i^{(k)} = \omega^{(k)} \cdot \tau_i^{(k)} \;\; \mathrm{mod}\; 2\pi \)
Higher-order harmonics provide internal degrees of freedom required for non-Abelian closure:
\( \theta_i^{(k)} \rightarrow \{\theta_i^{(k)}, 2\theta_i^{(k)}, 3\theta_i^{(k)}\} \)
14.2 Emergent Lie Algebra
Local rephasings act as:
\( \delta \Psi_i = i \, \varepsilon_a \, T_a \, \Psi_i \)
The generators \(T_a\) permute internal path indices. Non-commutativity of sequential phase adjustments produces:
\( [T_a, T_b] = i \, f_{abc} \, T_c \)
The structure constants \(f_{abc}\) are fixed by path-intersection topology:
- \(n = 1 \Rightarrow U(1)\) (Abelian loops)
- \(n = 2 \Rightarrow SU(2)\) (Doublet interference)
- \(n = 3 \Rightarrow SU(3)\) (Triplet harmonic closure)
14.3 Recursive Mass Generation — The Memory-Lag Law
Mass is the physical manifestation of recursive latency. Each stable motif partitions the 132-bit budget as follows:
- Lag \(L_r = 122\) bits: persistent address memory (inertia), defined as \(H - U_{1D} = 132 - 10\).
- Update \(U_r = 10\) bits per spatial dimension: transition cache (kinetic potential), derived from decimal-log phase resolution (Section 14.14).
Top-quark threshold: In three dimensions, motion requires \(3 \times 10 = 30\) bits. Total demand becomes \(122 + 30 = 152\) bits. Since \(152 > 132\), the motif exceeds the hardware budget at recursion depth \(D = 32\), explaining the top quark’s prompt decay and the absence of a fourth generation.
14.4 Substrate Resolution and LCD Frequency
The Least Common Denominator (LCD) substrate frequency sets the fundamental resolution:
\( f_{\mathrm{substrate}} = 0.217 \;\mathrm{GeV} \)
Action emerges as:
\( \hbar = E_c \cdot T_c = f_{\mathrm{substrate}} \cdot (1 / f_{\mathrm{substrate}}) = 1 \)
The muon anomalous magnetic moment arises from substrate update jitter:
\( a_\mu^{\mathrm{TFP}} = \frac{\alpha}{2\pi} \cdot \frac{m_\mu}{f_{\mathrm{substrate}}} \cdot \frac{1}{132 \cdot 11} \cdot \frac{1}{64} \approx 2.95 \times 10^{-9} \)
Here, \(132\) is total concurrency, \(11 = K - 1\) non-self neighbors, and \(64 = 8^2\) arises from transverse bifurcation of the \(SU(3)\) motif structure.
14.5 Neutrino Mass and Oscillations
Neutrinos are parity residues of a \(720^\circ\) spinor update spanning two full 132-bit cycles.
The absolute mass scale follows:
\( M_\nu = M_e \cdot (1 / H)^2 \cdot (1 / (2H)) \approx 0.11 \;\mathrm{eV} \)
Hierarchical splittings arise from angular deviations \(\delta_i\) in projections of icosahedral face centers, yielding \(m_1 < m_2 < m_3\), consistent with oscillation and cosmological constraints.
14.6 The Geodesic Shadow (\(\delta_{\mathrm{spec}}\))
Signals traverse geodesic edges of the 3D icosahedral shell rather than Euclidean chords, creating a spectral deficit.
For a unit-edge icosahedron:
\( R = \sqrt{(5 + \sqrt{5}) / 8} \approx 0.9511 \)
\( \rho = (1 + \sqrt{5}) / 4 \approx 0.8090 \)
The ratio:
\( \rho / R \approx 0.8507 \)
Thus:
\( \delta_{\mathrm{spec}} = \ln(\rho / R) \approx -0.1618 \;\text{bits} \)
14.7 Electroweak Unification and Parity Splitting
The geodesic shadow splits under \(SU(2) \times U(1)\) parity:
\( \delta_{\mathrm{EM}} = \tfrac{1}{2} \ln(\rho / R) \approx -0.0809 \)
\( \delta_{\mathrm{weak}} = +0.0809 \)
The weak mixing angle follows:
\( \cos\theta_W = \exp(\delta_{\mathrm{weak}}) = \sqrt{R / \rho} \approx 0.9223 \)
14.8 Gauge Boson Resonance — The Damped Z-Lock
The \(W\) and \(Z\) bosons correspond to inner (\(\rho\)) and outer (\(R\)) shells of the icosahedral lattice.
Substrate viscosity introduces damping:
\( \delta = |\delta_{\mathrm{spec}}| / 2 \approx 0.0405 \)
Including higher-order corrections:
\( \cos\theta_{W,\mathrm{obs}} = \cos\theta_{W,\mathrm{ideal}} \cdot e^{-\delta} \Rightarrow \sin^2\theta_W \approx 0.231 \)
14.9 Exact Derivation of \(\alpha^{-1} = 137.036\)
The ideal clearance budget is:
\( H + K = 132 + 12 = 144 \)
Indexing cost:
\( \log_2(132) \approx 7.044 \;\text{bits} \)
Including the EM geodesic correction:
\( 7.044 - 0.0809 = 6.9631 \;\text{bits} \)
Therefore:
\( \alpha^{-1} = 144 - 6.9631 = 137.0369 \)
14.10 The Higgs as a Junction-Reset Resonance
The Higgs boson (125 GeV) is the ground-state resonance of 132-bit pressure summation. The \(1/12\) backbone deficit permits the 10-bit update buffer to clear without corrupting the 122-bit address memory, acting as a cooling channel that prevents ledger overwrite during high-load operations.
14.11 Unified Hardware Audit Table
| Phenomenon | Substrate Origin | Hardware Logic / Hard Numbers |
|---|---|---|
| Mass (\(m\)) | Network latency | \( m = \hbar \cdot (122 / 132) \) |
| Higgs (125 GeV) | Buffer reset | 132-bit pressure resonance |
| Uncertainty (\(\hbar\)) | Rounding error | \( 0.106 = 1 - e^{-1/12} \) |
| Top quark limit | Buffer overflow | \( 122 + 30 = 152 > 132 \) |
| Fine structure (\(\alpha\)) | Geodesic shadow | \( \alpha^{-1} = 144 - (\log_2 132 - 0.0809) \) |
| Weak mixing angle | Parity-split shadow | \( \sin^2\theta_W \approx 0.231 \) |
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