SECTION 18 — CALIBRATION AND PHYSICAL ANCHORING IN TEMPORAL FLOW PHYSICS (TFP) (v11.2)

SECTION 18 — CALIBRATION AND PHYSICAL ANCHORING IN TEMPORAL FLOW PHYSICS (TFP) (v11.2)

By John Gavel

18.0 Purpose and Logical Role

All prior sections derive physics in dimensionless substrate variables. No physical constants appear inside the ΔF network update rules.

This section provides the unique and minimal anchoring procedure that maps dimensionless TFP observables → physical quantities measured in experiment. Calibration is required once, after which all predictions follow automatically. No tuning freedom exists beyond this mapping.

18.1 Origin of Dimensionless Variables

Substrate consists only of:

• Binary node states: \(F_i \in \{+1, -1\}\)
• Adjacency relations
• Recursive update ordering

All native quantities are dimensionless counts, ratios, or probabilities. Emergent dimensionless quantities (Sections 12–17):

• Characteristic length: \(L_c\)
• Characteristic recursion time: \(T_c\)
• Characteristic update energy: \(E_c\)
• Characteristic inertial scale: \(M_c\)
• Action: \(\hbar_c = E_c \cdot T_c\)
• Characteristic speed: \(c_{\text{char}} = L_c / T_c\)

None of these yet correspond to meters, seconds, joules, or kilograms.

18.2 General Physical Mapping

Introduce linear calibration factors:

• Length: \(\lambda_L\)
• Time: \(\lambda_T\)
• Energy: \(\lambda_E\)
• Mass: \(\lambda_M\)
• Action: \(\lambda_h\)

Physical quantities:

• \(L_{\text{phys}} = L_c \cdot \lambda_L\)
• \(T_{\text{phys}} = T_c \cdot \lambda_T\)
• \(E_{\text{phys}} = E_c \cdot \lambda_E\)
• \(M_{\text{phys}} = M_c \cdot \lambda_M\)
• \(\hbar_{\text{phys}} = \hbar_c \cdot \lambda_h\)

18.3 Action Consistency (First Principle Constraint)

From Section 14.5.4:

\(\hbar_c = E_c \cdot T_c\)

Physical action must satisfy the same:

\(\hbar_{\text{phys}} = E_{\text{phys}} \cdot T_{\text{phys}}\)

Substitute mappings:

\(\hbar_c \cdot \lambda_h = (E_c \cdot \lambda_E) \cdot (T_c \cdot \lambda_T) \implies \lambda_h = \lambda_E \cdot \lambda_T\)

18.4 Mass Consistency (Inertia Derivation)

Dimensionless mass (Section 17.2):

\(M_c = \frac{E_c \cdot T_c^2}{L_c^2}\)

Physical mass:

\(M_{\text{phys}} = \frac{E_{\text{phys}} \cdot T_{\text{phys}}^2}{L_{\text{phys}}^2}\)

Substitute mappings and cancel dimensionless terms:

\(\lambda_M = \frac{\lambda_E \cdot \lambda_T^2}{\lambda_L^2}\)

18.5 Gravitational Calibration

From Section 17.3, dimensionless gravitational coupling:

\(G_{\text{dim}}(l) = \frac{L_c^3}{M_c T_c^2} \cdot \frac{\delta_n(l)}{D_n(l)} \cdot \kappa_G\)

Choose anchoring scale \(l_\star\) with high cluster coherence \(C(l_\star)\) and small residual misalignment \(\beta(l_\star)\).

Physical gravitational constant:

\(G_{\text{phys}} = G_{\text{dim}}(l_\star) \cdot \lambda_G \implies \lambda_G = \frac{G_{\text{phys}}}{G_{\text{dim}}(l_\star)}\)

18.6 Node-Level Force Scale

Per-node action: \(S_{\text{node}} = F_0 \cdot L_c \cdot T_c\)

Dimensionless force: \(F_{0,\text{dim}} = \hbar_c / (L_c \cdot T_c)\)

Physical force:

\(F_{0,\text{phys}} = F_{0,\text{dim}} \cdot \lambda_F = \frac{E_c \cdot \lambda_E}{L_c \cdot \lambda_L} \implies \lambda_F = \frac{\lambda_E}{\lambda_L}\)

18.7 Electromagnetic Calibration

Dimensionless EM coupling (Section 17.4):

\(\alpha_{\text{em,dim}}(l) = \langle F_{\text{topo}} \cdot (\delta_n(l)/D_n(l)) \cdot C_\theta(l)^2 \rangle \cdot \kappa_{\text{EM}}\)

Physical EM coupling:

\(\alpha_{\text{EM,phys}} = \alpha_{\text{em,dim}}(l_\star) \cdot \lambda_{\text{EM}} \implies \lambda_{\text{EM}} = \frac{\alpha_{\text{EM,phys}}}{\alpha_{\text{em,dim}}(l_\star)}\)

18.8 Mass Calibration

Dimensionless mass: \(M_{\text{dim}}(d) = M_{e,\text{dim}} \cdot \exp(\chi_\Omega(d))\)

Physical mass: \(M_{\text{phys}}(d) = M_{\text{dim}}(d) \cdot \lambda_M\) with \(\lambda_M\) fixed as above. All fermion masses are therefore predictions, not adjustable.

18.9 Complete Calibration Workflow

1. Identify coherence anchoring scale \(l_\star\)
2. Compute \(G_{\text{dim}}(l_\star)\) and set \(\lambda_G = G_{\text{phys}} / G_{\text{dim}}(l_\star)\)
3. Compute \(\alpha_{\text{em,dim}}(l_\star)\) and set \(\lambda_{\text{EM}} = \alpha_{\text{EM,phys}} / \alpha_{\text{em,dim}}(l_\star)\)
4. Choose \(\lambda_L\) or \(\lambda_T\) as primary scale
5. Deduce \(\lambda_E, \lambda_M, \lambda_h\) from consistency relations:
• \(\lambda_h = \lambda_E \cdot \lambda_T\)
• \(\lambda_M = \frac{\lambda_E \cdot \lambda_T^2}{\lambda_L^2}\)
6. After this, no further calibration is allowed.

18.10 Dimensional Latency and Handshake Allocation (Lorentz Derivation)

Total handshake budget per tick: \(H = 132\)

Per-axis allocation: \(\lambda_x, \lambda_y, \lambda_z\)

Constraint:

\(\lambda_x^2 + \lambda_y^2 + \lambda_z^2 = (H \cdot \tau_0)^2\)

At rest (symmetric): \(\lambda_x = \lambda_y = \lambda_z = H \cdot \tau_0 / \sqrt{3}\), so \(t_{\text{rest}} = \sqrt{\lambda_x^2 + \lambda_y^2 + \lambda_z^2} = H \cdot \tau_0\)

Moving at velocity \(v\) along x-axis:

\(\lambda_x = \gamma H \tau_0 / \sqrt{3}, \quad \lambda_y = \lambda_z = H \tau_0 / (\sqrt{3} \gamma)\), where \(\gamma = 1/\sqrt{1 - v^2/c^2}\)

Check: \(\lambda_x^2 + \lambda_y^2 + \lambda_z^2 = (H \tau_0)^2\)

This recovers relativistic mass, length, and time dilation: \(m(v) = \gamma m_0\), \(L(v) = L_0 / \gamma\), \(t(v) = \gamma t_0\)

18.11 Interpretation and Non-Adjustability

Calibration constants are not tunable. Fixed once by matching \(G\) and \(\alpha\). All remaining predictions follow automatically. No physical constants are inserted into substrate dynamics. TFP is therefore axiomatic at the substrate level, predictive at the physical level, and falsifiable once calibrated.

18.12 Minimality Proof: Two Anchors Only

Anchor A — Scale: fixes \(f_{\text{substrate}}\) → sets \(L_c, T_c, E_c, M_c\) via \(\hbar\) and \(c_{\text{char}}\)

Anchor B — Interaction Strength: fixes \(\alpha_{\text{EM}}\) → sets phase coupling normalization and relative force strengths

Proof:

• One anchor insufficient: fixing only \(f_{\text{substrate}}\) sets units, but interaction strengths remain undefined → mass ratios and \(G\) cannot be predicted
• Two anchors sufficient: \(f_{\text{substrate}} + \alpha_{\text{EM}}\) → all masses, couplings, and \(G\) uniquely determined
• Three anchors overdetermined: any additional anchor is redundant or falsifies predictions

Conclusion: Calibration is minimal and complete. No further free parameters exist.

18.13 Bridge to Section 19

After Section 18, all dimensionless quantities have physical meaning. Masses, couplings, and scales are numerically fixed. Section 19 uses these calibrated values to generate direct experimental predictions and falsification tests.