Temporal Flow Physics — Master Notation & First-Principles Derivations
By John Gavel
Preface — how I present this
I write from first principles: start only with two primitive temporal flows, show what they produce, and name every derived symbol so nothing is hand-waved. Below I give the notation, its direct physical meaning, and the short derivation or construction step that connects it to the primitives F+, F-.
1. Primitive temporal flows
Notation → Meaning
\(F_+(x,t),\;F_-(x,t)\) — the two counter-propagating discrete temporal flows. These are the only dynamical primitives I assume.
First principles / update rule (canonical):
I define local updates that include a source/asymmetry term, local dissipation, and optional noise:
\[ F_{\pm}(x,t+1) = F_{\pm}(x,t) + \sigma_{\pm}(x,t)\,k - \lambda\,\Delta F_{\pm}(x,t) + \xi_{\pm}(x,t) \]
Derivation note: this is not an additional assumption — it is the simplest local discrete update that (a) allows asymmetry \(\sigma_\pm\), (b) dissipates via a discrete Laplacian \(\Delta\), and (c) permits open-system effects \(\xi\).
2. The DeltaF cluster — coherence unit
Notation → Meaning
\(\Delta F(x,t)\) — the local asymmetry, the seed of coherence.
Definition (algebraic):
\[ \Delta F(x,t) \;\equiv\; F_+(x,t) - F_-(x,t) \]
Derivation / role: whenever \(|\Delta F|\) persists across updates the site becomes a coherent unit (a DeltaF cluster). Operationally I detect clusters by thresholding the magnitude of \(\Delta F\) and requiring temporal persistence (see Section 5 for the formal depth definition).
3. Coherence potential \(\Phi\)
Notation → Meaning
\(\Phi\) — scalar coherence potential of a DeltaF cluster; it measures resistance to dissipation and encodes alignment, compressional stiffness, and recursive survival.
Canonical construction:
\[ \Phi \;=\; A\,\langle F_+F_- \rangle_{\rm loc} \;+\; B\,\Delta_{\rm eff}^{-1} \;+\; C\sum_{n=1}^{d} \eta^n\,F^{(n)}_{\rm persist} \]
How this follows from primitives:
- The alignment term \(A\langle F_+F_-\rangle\) is the local inner product of forward/backward flows — if they reinforce, the site stores more coherence (this is the origin of charge-like behavior).
- The compressional term \(B\Delta_{\rm eff}^{-1}\) comes from the second spatial difference of \(\Delta F\) (low curvature → strong compressional support).
- The recursive term weights how many consecutive update steps the cluster survives: \(F^{(n)}_{\rm persist}\) is the amplitude after n iterations. This term creates hierarchical depth.
4. Effective dissipation / curvature
Notation → Meaning
\(\Delta_{\rm eff}(x,t)\) — a discrete local curvature / dissipation strength.
Definition (finite difference):
\[ \Delta_{\rm eff}(x,t) \;=\; \left|\,\Delta F(x+1,t) - 2\,\Delta F(x,t) + \Delta F(x-1,t)\,\right| \]
Derivation / role: \(\Delta_{\rm eff}\) is directly computable from the primitive \(\Delta F\) values. Small \(\Delta_{\rm eff}\) means the cluster is smooth and resists dissipation; \(B\,\Delta_{\rm eff}^{-1}\) then increases \(\Phi\).
5. Recursive depth \(d\)
Notation → Meaning
\(d\) — integer depth: the number of sequential updates a cluster survives above the coherence threshold.
Operational definition:
\[ d \;=\; \max\{ n \;|\; |\Delta F^{(t-n+1)}| > \epsilon_{\rm coh} \} \]
Derivation / role: This is measured directly from the time history of \(\Delta F\). It is the generational index: \(d=1\) → electron class in the model, \(d=2\) → muon class, \(d=3\) → tau class (subject to stability constraints).
6. Charge proxy \(\rho_{\rm TFP}\)
Notation → Meaning
\(\rho_{\rm TFP}\) — local proto-charge, derived from net flow alignment.
Definition:
\[ \rho_{\rm TFP}(x,t) \;=\; \langle F_+(x,t) - F_-(x,t) \rangle_{\rm loc} \;=\; \langle \Delta F(x,t)\rangle_{\rm loc} \]
Derivation / role: Because \(\Delta F\) quantifies net directional bias, local averaging of \(\Delta F\) yields a divergence source for emergent electric fields (see Section 7–8). This is the minimal bridge from flows to gauge charge.
7. Emergent potentials \(\Phi_{\rm EM}\) and \(\mathbf{A}\)
Notation → Meaning
\(\Phi_{\rm EM}\) — emergent scalar potential; \(\mathbf{A}\) — emergent vector potential.
Construction from \(\Phi\) and \(\Delta F\):
\[ \Phi_{\rm EM} \;=\; \kappa_{\Phi}\,\nabla \Phi \qquad\text{and}\qquad \mathbf{A} \;=\; \kappa_A\,\langle \Delta F \rangle_{\text{directional}} \]
Derivation / role: Gradients of the coherence potential and directed flow asymmetry coarse-grained over patches produce potentials whose derivatives obey Maxwell-like relations in the continuum limit. No fields are assumed — they emerge from the flows.
8. Emergent fields \(\mathbf{E},\mathbf{B}\)
Notation → Meaning
Electric and magnetic like fields built from the emergent potentials.
Definitions:
\[ \mathbf{E} \;=\; -\nabla \Phi_{\rm EM} - \partial_t \mathbf{A}, \qquad \mathbf{B} \;=\; \nabla \times \mathbf{A} \]
Derivation / role: Standard relations appear because \(\Phi_{\rm EM}\) and \(\mathbf{A}\) are constructed as a scalar gradient and a directed average respectively. The emergent fields are coarse-grained consequences of the primitive flows.
9. Mass law parameters: \(\alpha\) and \(\beta\)
Notation → Meaning
\(\alpha\) couples \(\Phi\) to mass; \(\beta\) couples recursive depth \(d\) to mass.
Phenomenological mass law (log form):
\[ \ln m \;=\; \ln M_0 \;+\; \alpha\,\Phi \;+\; \beta\,d, \] or equivalently \[ m \;\propto\; \exp(\alpha\Phi + \beta d). \]
Derivation / role: \(\alpha\) and \(\beta\) are response coefficients: \(\alpha\) measures how compressional coherence shifts energy scales; \(\beta\) measures how recursive nesting produces exponential hierarchies. In practice we fit or derive them from the primitive update coefficients via renormalization (see implementation notes in code).
10. Generational mass structure
Notation → Meaning
Particle masses as a function of depth
\[ m(d) \sim \exp\big(\alpha + \beta d\big) \]
Role: This is the working hypothesis: depth orders generations and the exponential form converts small integer differences into large mass ratios (when \(\beta\) is tuned appropriately).
11. Local time-shift operator
Notation → Meaning
\(T_{\Delta t}\) — the discrete update operator.
Definition:
\(T_{\Delta t}[F] = F(t+\Delta t)\)
Role: iterate \(T_{\Delta t}\) to measure recursion and compute \(d\).
12. Spatial difference operators
Notation → Meaning
Discrete gradient and Laplacian on the chain.
Definitions:
\(\nabla F(x) = F(x+1) - F(x)\)
\(\Delta F(x) = F(x+1) - 2F(x) + F(x-1)\)
Role: these appear in dissipation terms and in constructing \(\Delta_{\rm eff}\).
13. Coherence threshold
Notation → Meaning
\(\epsilon_{\rm coh}\) — the cutoff for declaring a local site coherent.
Rule:
\(|\Delta F| > \epsilon_{\rm coh}\) implies the site is part of a coherent domain (subject to temporal persistence checks).
14. Scale-dependent coherence exponent \(\beta(\ell)\)
Notation → Meaning
Running exponent describing how node weighting or coherence coupling changes under coarse graining.
Example form:
\[ \beta(\ell) = \beta_0\left(\frac{\ell}{\ell_0}\right)^{-\gamma} \]
Role: use when you renormalize your lattice to larger blocks; it tells you how \(\beta\) should flow with scale.
15. ΔF emission from horizon-like objects
Notation → Meaning
\(\Delta F_{\rm emit}\) — coherent flow emission from high curvature regions (black-hole analogues).
Empirical relation:
\(\Delta F_{\rm emit} \sim \dfrac{1}{\Delta_{\rm eff}^{\rm horizon}}\)
Role: heuristic: higher curvature (small \(\Delta_{\rm eff}\)) → larger emission of coherent flow packets; used in cosmological modeling in TFP.
16. Domain coherence measure and corrected merging
Notation → Meaning
\(S_D\) — relational boundary divergence of domain \(D\); \(\Omega_D\) — domain coherence exponent.
Operational S_D (chain):
\[ S_D \;=\; \sum_{\langle i,j\rangle \in \partial D} \big|\Delta F_i - \Delta F_j\big| \] (sum over boundary neighbor differences or, equivalently, sum of active edge tensions inside the domain.)
Corrected merging identity (adopted as the consistent algebraic rule):
\[ \boxed{\,1 + S_{D_{12}} \;=\; (1 + S_{D_1})(1 + S_{D_2})\,} \]
Link to Ω and Σ:
\[ \Omega_D \;=\; \log\big(1 + S_D\big), \qquad \log\Sigma \;=\; \sum_D \Omega_D \]
Interpretation: taking \(\Omega_D=\log(1+S_D)\) makes \(\sum\Omega_D\) additive and consistent with multiplicative composition of \(1+S_D\). This removes the earlier inconsistency and leaves \(\Sigma\) interpretable as a global coherence product when appropriate.
17. Σ as an emergent equilibrium tendency (soft)
Notation → Meaning
\(\Sigma\) — global coherence aggregate; not a strict conserved quantity under internal updates. It instead expresses a statistical tendency that can be biased by a soft penalty.
Soft penalty form:
\[ \sigma_{\rm dev} \;=\; \frac{\Sigma}{\Sigma_0} - 1 \quad\Rightarrow\quad E_\lambda \;=\; \tfrac{\lambda}{2}\,\sigma_{\rm dev}^2 \]
Role: use this quadratic term in your objective to nudge the system toward target coherence without enforcing exact conservation. This matches open-system physics and empirical numerics.
18. Mass–Σ connection (consistent final form)
With \(\Omega_D=\log(1+S_D)\) we have
\[ \sum_D \Omega_D = \log\Sigma. \]
If total mass suppression follows the additive coherence exponent, the total mass factor is
\[ M_{\rm total} \;\propto\; \exp\!\Big(-\pi\sum_D \Omega_D\Big) \;=\; \Sigma^{-\pi}. \]
Role: this gives a consistent route from domain edge tensions \(S_D\) to emergent particle masses via \(\Omega_D\) and \(\Phi_D\).
Final remarks — why this ordering matters
Everything here is constructed directly from the primitives \(F_+,F_-\): \(\Delta F\) is algebraic, \(\Delta_{\rm eff}\) is a finite difference, \(\Phi\) is a small linear combination of local measurements and a recursive tally, \(d\) is read from the time history, and the mass law is a local linear model in the natural variables \(\Phi\) and \(d\). The corrected merging rule and the soft Σ bias close the logical gaps that previously produced contradictions. This is the minimal, consistent axiomatization you can use to implement numerics and then test the hypotheses against data.
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