TFP Section 15 (v9)

SECTION 15 — SCALE-DEPENDENT COHERENCE AND PHYSICAL CUTOFFS (v9.6)

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

15.0 Overview

Scale-dependent phenomena arise from the recursive organization of stable motifs within the emergent 3+1 geometry (Section 3). Ultraviolet/IR cutoffs, running couplings, and the mass hierarchy are determined by:

• Cluster coherence \( C(l) \) (Section 12.5)
• Geometric misalignment \( \beta(l) = 1 - C(l)^2 \) (Section 11.6)
• Coherence exponent \( \Omega \) (Section 4.4)

The 3D spatial manifold emerges from adjacency correlations (Section 3.3.1), while time arises from irreversible coarsening (Section 2.6.2). All dynamics occur within this emergent structure — no fundamental spacetime is assumed.

15.1 Cluster Coherence and Misalignment

At scale \( l \) (recursion depth), define amplitude and phase coherences:

• \( C_A(l)^2 \): Alignment of flow magnitudes (Section 10.1)
• \( C_\theta(l)^2 \): Alignment of correlation lags \( \tau_i \) (Section 3.1.2)

Total coherence:

\[ C(l)^2 = C_A(l)^2 \cdot C_\theta(l)^2 \tag{15.1} \]

Residual misalignment:

\[ \beta(l) = 1 - C(l)^2 \tag{15.2} \]

This modulates gauge couplings (Section 12.6) and gravitational strength (Section 7.4) in the emergent geometry.

15.2 Recursive Coherence Scaling

For nested clusters in the emergent 3D space, coherence decays with scale as:

\[ C(l)^2 = C_0^2 \left( \frac{l_0}{l} \right)^d \tag{15.3} \]

where \( d \) is the effective embedding dimension (Section 3.3). This power law arises because deeper recursion requires more update steps to maintain phase alignment in the 3D spatial manifold, increasing susceptibility to decoherence (Section 13.1).

15.3 Discrete-to-Continuum Mapping

Coarse-grained flow difference (Section 12.2):

\[ \langle \Delta F \rangle(\mathbf{r}) = \frac{1}{N_R} \sum_{i \in R} \Delta F_i \tag{15.4} \]

Gravitational potential from curvature (Section 7.4):

\[ \Phi_{\text{grav,phys}}(\mathbf{r}) = \frac{\xi_{\text{spatial}}^2}{\xi_{\text{temporal}}^2} \cdot \text{Tr}[R_{\mu\nu}(\mathbf{r})] \tag{15.5} \]

Electromagnetic potential from phase gradient (Section 12.3):

\[ \mathbf{A}(\mathbf{r}) = \kappa_A \cdot \langle \nabla \theta \rangle(\mathbf{r}) = \kappa_A \cdot \omega \langle \nabla \tau \rangle(\mathbf{r}) \tag{15.6} \]

All fields are defined on the emergent 3D spatial manifold with time as a separate causal parameter.

15.4 Minimal Scales and Continuum Validity

The discrete minimal scale is the lattice spacing \( a_{\text{phys}} \) (Section 5.5). The continuum is valid when:

\[ \xi_{\text{spatial}}(l) \gg \frac{1}{\beta_{\text{misalign}}(l)} \tag{15.7} \]

where \( \beta_{\text{misalign}}(l) \) is the misalignment cost (Section 15.9). Wave-particle duality (Section 10.7) depends on the ratio:

• Wave-like: \( \xi_{\text{spatial}} / \xi_{\text{temporal}} \approx c_{\text{pred}} \)
• Particle-like: \( \xi_{\text{spatial}} / \xi_{\text{temporal}} \gg c_{\text{pred}} \) and high reflection fraction

These criteria are evaluated in the emergent geometry.

15.5 Running Couplings

From Section 12.6, the electromagnetic coupling is:

\[ \alpha_{\text{EM}}(l) = C_{\text{EM}}^* \cdot C_\theta(l)^2 \tag{15.8} \]

For SU(3), the strong coupling is:

\[ \alpha_s(l) = C_S^* \cdot C_A(l)^2 \tag{15.9} \]

Effective couplings include misalignment corrections:

\[ \alpha_{\text{EM,eff}}(l) = \frac{\alpha_{\text{EM}}(l)}{1 + \beta_\theta(l)} \tag{15.10} \]

\[ \alpha_{s,\text{eff}}(l) = \frac{\alpha_s(l)}{1 - \beta_A(l)} \tag{15.11} \]

Confinement occurs when \( \beta_A(l) \to 1 \), causing \( \alpha_{s,\text{eff}} \to \infty \) (Section 14.5).

15.6 Lattice Action and Calibration

The emergent action unit is \( \hbar_c = E_c T_c \) (Section 5.3.1). Per-node action is:

\[ S_{\text{node}} = F_0 \cdot a_{\text{phys}} \cdot \tau_{\text{phys}} \tag{15.12} \]

Calibration requires \( S_{\text{node}} = \hbar_c \), so:

\[ F_0 = \frac{\hbar_c}{a_{\text{phys}} \tau_{\text{phys}}} \tag{15.13} \]

For coherent clusters, the effective action is enhanced by \( C(l)^2 \):

\[ S_{\text{effective}} = F_0 a_{\text{phys}} \tau_{\text{phys}} C(l)^2 = \hbar_c C(l)^2 \tag{15.14} \]

Thus, the physical force scales as:

\[ F_0^{\text{eff}} = \frac{\hbar_c}{a_{\text{phys}} \tau_{\text{phys}} C(l)^2} \tag{15.15} \]

15.7 Universal Mass Law

From Section 4.4, mass is determined by the coherence exponent \( \Omega \):

\[ M_{\text{dim}} = \exp\left[ -\pi (\Omega - 1) \right] \tag{15.16} \]

For hierarchical fermionic clusters in the emergent 3+1 geometry, \( \Omega \) depends on recursion depth \( d \):

\[ \Omega(d) = \Omega_0 + \gamma d + \delta d^2 \tag{15.17} \]

This quadratic form arises because topological complexity in the 3D spatial manifold grows quadratically with depth. The mass law becomes:

\[ M(d) = M_e \exp\left[ \chi_0 + \alpha d + \beta d^2 \right] \tag{15.18} \]

where \( \chi_0 = -\pi(\Omega_0 - 1) \).

15.8 Three-Generation Theorem

Theorem: Stable fermionic clusters exist only for recursion depths \( d \in \{1, 2, 3\} \).

Proof: Stabilization of a depth-\( d \) cluster requires \( N_{\text{updates}}(d) \sim k^d \) coherent update steps (Section 4.3.3). Substrate decoherence imposes a maximum lifetime \( T_{\text{decoherence}} \) (Section 11.4). Stability requires:

\[ N_{\text{updates}}(d) < \frac{T_{\text{decoherence}}}{\tau_0} \tag{15.19} \]

For typical substrate parameters (Section 5.5), \( k \approx 2 \) and \( T_{\text{decoherence}} / \tau_0 \approx 10 \), so \( d_{\text{max}} = \lfloor \log_2(10) \rfloor = 3 \). Clusters with \( d \geq 4 \) cannot stabilize in the emergent geometry. ∎

15.9 Misalignment Cost

The misalignment cost quantifies tension from reflection bottlenecks (Section 2.1.1):

\[ \beta_{\text{misalign}}(l) = \sum_{i \in \text{cluster}(l)} \beta |\Delta F_i| \cdot \mathbb{I}(\sigma_i = -1) \tag{15.20} \]

This sets the decoherence timescale \( T_{\text{decoherence}} \sim 1 / \beta_{\text{misalign}} \) (Section 11.4).

15.10 Continuum Validity Conditions

The continuum approximation holds when:

• \( \xi_{\text{spatial}}(l) \gg a_{\text{phys}} \)
• \( T_{\text{cluster}}(l) \gg \tau_{\text{phys}} / C(l)^2 \)
• \( \xi_{\text{spatial}}(l) \gg 1 / \beta_{\text{misalign}}(l) \)

These ensure sufficient averaging over discrete structure in the emergent 3D spatial manifold (Section 2.3).

15.11 Axiomatic Closure

Phenomenon	Substrate Origin	Axiom
Running Couplings	Scale-dependent coherence \( C(l) \)	Section 12.6
Mass Hierarchy	Recursion depth \( d \) → \( \Omega(d) \)	Section 4.4
Three Generations	Computational depth limit \( d_{\text{max}} = 3 \)	Section 15.8
UV Cutoff	Lattice spacing \( a_{\text{phys}} \)	Section 5.5

15.12 Bridge to Section 16 — Validation

Section 15 provides:

• Running couplings from recursive coherence in emergent 3+1 geometry
• Mass hierarchy from recursion depth in 3D spatial manifold
• Three-generation limit from computational stability
• UV/IR cutoffs from substrate granularity

Section 16 validates these predictions against experimental data, confirming the parameter-free structure of TFP.