Section 8 — Unified Classical Fields (v11.0)

Section 8 — Unified Classical Fields (v11.0)

Geometry, Electromagnetism, and Scalar Dynamics from Coarse-Grained Alignment
By John Gavel

8.0 Overview

At macroscopic scales, the discrete update dynamics of the substrate admit a coarse-grained description. The primary observable is the alignment field \(A(x,t)\), a statistical summary of binary flow coherence. All classical fields arise as bookkeeping constructs describing how handshake capacity is allocated across space and time.

Three effective fields emerge:

• Scalar field \(\phi(x,t)\): local density of handshake utilization (pressure).
• Vector field \(A_\mu(x,t)\): directional routing bias of alignment (current).
• Metric \(g_{\mu\nu}(x,t)\): correlation distance between updates (map).

Classical dynamics correspond to configurations minimizing substrate tension subject to the fixed update budget: \[ H_\text{total} \le 132 \] No independent classical constants are introduced. All couplings are inherited from handshake accounting and saturation limits.

8.1 Alignment Tensor as the Classical Observable

The alignment tensor encodes how binary flows cohere over multiple ticks. It is a ledger for handshake allocation:

• Phase structure: relative ordering of updates, producing a U(1)-like symmetry supporting electromagnetic behavior.
• Magnitude structure: total handshake density, measuring update cost and producing metric strain.

Classical fields are therefore summaries of how alignment consumes the update budget.

8.2 Scalar Field as an Order Parameter

Define the scalar field \(\phi(x,t)\) as the coarse-grained magnitude of aligned flows within a coherence cell. Variations in \(\phi\) represent accumulated update cost. The substrate minimizes gradients in this cost subject to saturation constraints: \[ \nabla^2 \phi = D_\text{eff}^2 \, \phi \] where \(D_\text{eff}\) is the effective correlation depth of the alignment network. For static modes, this reproduces the Klein-Gordon structure. The associated mass \(m_\phi\) is operational: it measures the cost of maintaining persistent alignment.

8.3 Electromagnetic Coupling as Routing Bias

Electromagnetic behavior arises from directional imbalance in alignment:

• Charge density \(\rho\): net alignment bias, proportional to \(\phi\).
• Current \(J\): transport of alignment, given by \(\phi \, v_C\), where \(v_C\) is the coherence propagation velocity.

The physical vector potential \(A_\mu\) is the coarse-grained convolution of these biases over the coherence length \(L_c\). Gauge freedom reflects redundancy in bookkeeping, not physical indeterminacy.

8.4 Metric Emergence from Correlation Distance

The metric \(g_{\mu\nu}\) measures correlation delay between updates at separated sites. Regions of high handshake density require more update cycles to resolve differences. This increases correlation distance and produces metric strain. Curvature is secondary: it records how much update time is consumed locally. Geometry is inferred from synchronization cost, not imposed.

8.5 Coarse-Graining Validity

Classical description holds only when handshake demand remains below saturation. If local alignment requires more than the per-tick capacity, coarse-grained fields fail to track substrate dynamics faithfully. This sets a hard domain of validity for classical field theory within TFP.

8.6 The 132-Bit Impedance (Hardware Constraint)

Alignment fields are constrained by saturation: \[ D_n \le 11 \] With \(K=12\) adjacency slots but only 11 available per tick, one relational channel is always mechanically unavailable. This produces an intrinsic impedance to alignment flow. Vacuum impedance is not resistive; it is the permanent cost of routing updates under saturation. The fine-structure constant \(\alpha\) measures this relational friction. Its observed value \(\alpha^{-1} \approx 137.036\) follows from saturation geometry and efficiency factors.

8.7 Budget Conservation (Master Constraint)

The defining conservation law of TFP is the fixed update budget: \[ H_\text{total} = H_\text{mass} + H_\text{field} + \Delta H_\text{gravity} = 132 \]

• \(H_\text{mass}\): handshakes locked into closed loops (persistent motifs, particles)
• \(H_\text{field}\): handshakes used to route gradients (forces)
• \(\Delta H_\text{gravity}\): unresolved remainder propagated forward in time

If total demand exceeds 132, coarse-graining fails. This is not a singularity but a substrate freeze.

8.8 Unified Action (TFP Form, Updated)

The classical action encodes handshake accounting directly: \[ S = \int d^4x \, \sqrt{-g} \, \Bigg[ \frac{1}{2} (\nabla \phi)^2 + V(\phi) + \frac{1}{4\eta} F_{\mu\nu} F^{\mu\nu} + \frac{\Delta H}{16\pi} \frac{R}{R_U} \Bigg] \] Terms arise from substrate constraints:

• \(\frac{1}{2} (\nabla \phi)^2\): cost of maintaining persistent alignment
• \(V(\phi)\): saturation penalties and self-interactions within coherence cells
• \(\frac{1}{4\eta} F_{\mu\nu} F^{\mu\nu}\): routing cost of directional alignment, \(\eta = H / K^2 = 11/12 \approx 0.9167\)
• \(\frac{\Delta H}{16\pi} R / R_U\): residual handshake remainder, \(\Delta H = 1/12\), \(R\) = Ricci scalar, \(R_U\) = cosmic dilution radius

The \(1/16\pi\) factor reflects spherical dilution of finite handshake flux, derived from inverse-square shell counting: \(N_\text{shell}(r) \approx 10 r^2 + 2\).

8.9 Interpretation

• Electromagnetism is strong because it uses routed handshakes.
• Mass is heavy because it locks handshakes in loops.
• Gravity is weak because it relies exclusively on the unresolved remainder, diluted across the largest available scale \(R_U\).

Weakness is not smallness—it is scarcity.