Section 15 — Scale-Dependent Coherence and Physical Cutoffs (TFP, v11.1)

SECTION 15 — SCALE-DEPENDENT COHERENCE AND PHYSICAL CUTOFFS (v11.3)updated

Renormalization from Recursive Cluster Stability in Emergent 3+1 Geometry
By John Gavel

15.0 Overview

Scale-dependent phenomena emerge from the recursive organization of stable flow motifs. There is no prior, independent notion of “energy scale”; rather, running couplings, effective masses, and physical cutoffs are bookkeeping consequences of maintaining consensus across recursive motifs under the finite 132-bit concurrency budget established in Section 3.1.

Fundamental Hardware Invariants:

• Concurrency Depth (\(H\)): 132 bits (Ref: Section 3.1)
• Vector Backbone (\(K\)): 12 neighbors (Ref: Section 14.3)
• LCD Substrate Frequency (\(f_s\)): 0.217 GeV (Ref: Section 14.4)

15.1 Cluster Coherence and Misalignment

The stability of a motif at recursion scale \(l\) is defined by the total coherence:

\( C(l)^2 = C_A(l)^2 \cdot C_\theta(l)^2 \)

where \(C_A\) is amplitude coherence and \(C_\theta\) is phase coherence. The residual misalignment \(\beta(l)\) represents the probability that a node fails consensus during a single update cycle:

\( \beta(l) = 1 - C(l)^2 \)

15.2 The Shannon Entropy Limit of Coherence

The coherence decay function is not an empirical fit; it is derived directly from the Shannon entropy of the 132-bit substrate. Each spatial dimension consumes a fixed 10-bit update cache to resolve motion, as derived from the decimal-log mapping of the fundamental phase cycle (Section 14.14).

This fixes the effective embedding dimension at \(d = 3\), yielding the power-law decay:

\( C(l)^2 = C_0^2 \cdot (l_0 / l)^3 \)

This “bit-squeeze” logic dictates the decay of consensus as motifs become more complex, with the exponent fixed by hardware constraints rather than phenomenology.

15.3 Recursive Scaling and the Three-Generation Limit

Nested clusters decay in coherence as recursion depth \(l\) increases. For the hardware-fixed embedding dimension \(d = 3\):

\( C(l)^2 = C_0^2 \cdot (l_0 / l)^3 \)

Hardware concurrency imposes a strict upper bound. In three dimensions, each dimension consumes a 10-bit update cache.

Dimensional saturation follows directly: \(H = 132 \Rightarrow d_{\max} = 3\) (Ref: Section 14.7). Any attempt at a fourth generation (\(d = 4\)) requires a bit demand of \(122 + 40 = 162\) bits. Since \(162 > 132\), the address cannot be resolved by the substrate, forcing prompt decay.

15.4 Discrete-to-Continuum Mapping

The continuum description is an effective projection of discrete flows.

The gravitational potential \(\Phi_{\mathrm{grav}}(r)\) tracks amplitude residue (bookkeeping) within the substrate.

The electromagnetic potential \(A(r)\) tracks phase gradients and couples directly to \(C_\theta\).

Because gravity tracks ledger residue rather than active bandwidth, it does not renormalize like gauge forces. This hardware partitioning explains the hierarchy problem without invoking new degrees of freedom.

15.5 Running Couplings and the Resolution Gap

As probe energy \(Q\) increases, the substrate is sampled at higher recursion depths, increasing \(\beta(l)\).

Strong coupling:

\( \alpha_s(l) = C_s^\ast \cdot \dfrac{C_A(l)^2}{1 - \beta_A(l)} \)

Electromagnetic coupling:

\( \alpha_{\mathrm{EM}}(l) = C_{\mathrm{EM}}^\ast \cdot \dfrac{C_\theta(l)^2}{1 + \beta_\theta(l)} \)

Confinement occurs as \(\beta_A(l) \rightarrow 1\), corresponding to the Nyquist limit of the 132-bit budget, driving \(\alpha_s \rightarrow \infty\). Threshold jumps arise at integer bit residues of the update cycle (Ref: Section 14.14).

15.6 Metabolic Cost of Action (\(h\))

Planck’s constant emerges as the minimal action quantum set by the substrate frequency:

\( h = E_{\mathrm{update}} \cdot T_{\mathrm{update}} = f_s \cdot (1 / f_s) = 1 \)

The per-node action is therefore a hardware invariant, independent of the number of nodes within the horizon (\(N_{\mathrm{bits}}\)).

15.7 Universal Mass Law

Fundamental fermions obey:

\( m_{\mathrm{fund}} \propto \exp[\chi(d)] \cdot \Omega_{\mathrm{eff}}(l) \cdot C(l) \)

Composite hadrons are governed by motif count \(N\) and tension \(F_{\mathrm{tension}}\), remaining stable against coherence decay due to internal saturation (Ref: Section 14.3).

Thus, inter-generational mass hierarchy arises from recursion depth, while composite masses remain effectively static.

15.8 Scale-Dependent Anomalies

The muon \(g-2\) anomaly directly measures substrate-scale jitter:

\( a_\mu(l) \propto \frac{\alpha}{2\pi} \cdot \frac{m_\mu}{f_s} \cdot \frac{1}{H \cdot 11} \cdot \frac{1}{C(l)} \)

This confirms the anomaly as a consequence of consensus damping in high-frequency motifs, rather than virtual loop corrections.

15.9 Bridge to Section 16 — Validation

Section 15 completes the scale-dependent mechanics of the TFP manifold. All running constants are now derived from the Shannon entropy of the 132-bit budget and its fixed update-cache allocation.

Section 16 will present the empirical audit, comparing these derived values against 2024 CODATA measurements for \(\alpha\), \(\sin^2\theta_W\), \(g-2\), and the mass hierarchy.