FQFT Scale Transition Model

Public status boundary. This route is a finite multiscale surrogate for review. It does not assert accepted physics or proof status.

PROPOSED COMPUTATIONAL MODEL

This page specifies a source-bounded numerical testbed for an authorial research framework. It is not an accepted physical law, a proof of the underlying theory, or experimental validation. Every new equation below is a computational proposal for review. Standard mathematical constructs are identified as such, and failure conditions are part of the model.

Purpose

Fractal Quantum Field Theory (FQFT) is represented in the current corpus as a proposed scale-dependent extension of standard quantum-field reasoning. The present route does not assume that such an extension is already formalized. It defines a finite multiscale scalar-field surrogate whose sole purpose is to test whether a proposed scale-transition rule can preserve declared observables, converge under refinement, and recover an ordinary lattice-field baseline in a stated limit.

The model therefore operationalizes four corpus obligations:

1. Scale dependence must be explicit.
2. Admissible transformations and invariants must be declared.
3. A standard comparison and recovery limit must exist.
4. Failure must be measurable rather than narrated away.

Source basis

Canonical source route	Evidence ID	Route	publicText	proposed computational mapping
/02-foundations/fractal-quantum-field-theory	FQFT-S1A	/04-mathematics/simulations/fqft-scale-transition-model	Field interactions are proposed to exhibit fractal, scale-dependent architecture that preserve fundamental symmetries across different observational magnitudes.	Source-bounded multiscale field surrogate
/02-foundations/fractal-quantum-field-theory	FQFT-S1B	/04-mathematics/simulations/fqft-scale-transition-model	Formalization targets require defining objects and morphisms, admissible transformations, invariants and preservation obligations, trace functions and observer context, domain restrictions and counterexample classes, routing mathematical claims to Lean/HoTT/Coq or proof-paper development, routing physics claims to simulation, observable, and falsifiability registry, and failure modes include undefined terms, unsupported strength, missing review route, media substitution, and source drift.	Source-bounded multiscale field surrogate
/02-foundations/science-of-fabric-reality	FQFT-S2-GRAMMAR	/04-mathematics/simulations/fqft-scale-transition-model	Every physics-facing extension must declare domain, observable, transformation law, invariant, trace, observer, boundary, and failure mode.	Source-bounded multiscale field surrogate
/02-foundations/science-of-fabric-reality	FQFT-S2-FAILURE	/04-mathematics/simulations/fqft-scale-transition-model	Public failure modes include metaphor inflation, physics overreach, proof inflation, project inflation, and totalization.	Source-bounded multiscale field surrogate
/09-library/physica-nova	FQFT-S3	/04-mathematics/simulations/fqft-scale-transition-model	PHYSICA should support simulation protocols; a simulation route should include model equations, parameter definitions, discretization choices, stability criteria, error bounds, sensitivity analysis, code version, input data, output data, and reproduction instructions; if numerical claims change after correction, the public page should make the correction visible; proposed extensions should include a standard contact point, recovery limit, new parameter, observable, prediction, error analysis, and failure interpretation.	Source-bounded multiscale field surrogate
/02-foundations/invariant-engineering	FQFT-S4	/04-mathematics/simulations/fqft-scale-transition-model	A minimal formal invariant-engineering object can be expressed as state space, transform set, invariant set, trace function, observer role, and boundary class; admissible transform T is valid when invariant_set(S) is preserved and trace(T) is inspectable; failure modes include undefined invariant, broken provenance, observer ambiguity, boundary collapse, and runtime drift.	Source-bounded multiscale field surrogate

State space

Let the simulation contain levels

$$\ell =0,1,\dots ,L,$$

where each level carries a finite graph or lattice

$$G_{\ell }=(V_{\ell },E_{\ell })$$

with (N_\ell=|V_\ell|) nodes, positive quadrature weights (w_i^{(\ell)}), and a weighted positive-semidefinite Laplacian (L_\ell).

The real benchmark field at level (\ell) is

$${\varphi }_{\ell }\in {\mathbb{R}}^{N_{\ell }}.$$

This real scalar field is a numerical surrogate. It is not presented as a quantum state or as the final field content of FQFT.

Between adjacent levels, define:

• (R_\ell:\mathbb{R}^{N_\ell}\rightarrow\mathbb{R}^{N_{\ell+1}}), a restriction or coarse-graining operator;
• (P_\ell:\mathbb{R}^{N_{\ell+1}}\rightarrow\mathbb{R}^{N_\ell}), a prolongation or reconstruction operator.

The operators must be stored with the run record. A scale transition is not reproducible when its (R_\ell) and (P_\ell) are unspecified.

Benchmark dynamics

A reviewable first testbed is the discrete gradient flow

$${\varphi }_{\ell }^{n+1}={\varphi }_{\ell }^n-\mathrm{\Delta }t[{\kappa }_{\ell }{L}_{\ell }{\varphi }_{\ell }^n+m_{\ell }^2{\varphi }_{\ell }^n+{\lambda }_{\ell }{\left({\varphi }_{\ell }^n\right)}^{\odot 3}].$$

Here:

Symbol	Meaning
(\Delta t)	numerical time step
(\kappa_\ell)	level-dependent spatial coupling
(m_\ell^2)	benchmark quadratic coefficient
(\lambda_\ell)	benchmark quartic coefficient
(\odot 3)	componentwise cube

This is a standard scalar-field numerical scaffold, introduced only to test scale-transition logic. It is not an extracted FQFT field equation.

The corresponding diagnostic functional is

$$E_{\ell }[\varphi ]=\frac{{\kappa }_{\ell }}{2}{\varphi }^{\mathsf{T}}{L}_{\ell }\varphi +\sum _{i=1}^{N_{\ell }}{w}_i^{(\ell )}[\frac{m_{\ell }^2}{2}{\varphi }_i^2+\frac{{\lambda }_{\ell }}{4}{\varphi }_i^4].$$

For positive coefficients and a sufficiently small step, this benchmark should not exhibit unexplained numerical energy growth.

Scale transition

The coarse state is defined by

$${\varphi }_{\ell +1}=R_{\ell }{\varphi }_{\ell }.$$

The round-trip reconstruction is

$${\hat{\varphi }}_{\ell }=P_{\ell }{R}_{\ell }{\varphi }_{\ell }.$$

Define the normalized reconstruction defect

$$D_{\ell }^{\mathrm{rec}}=\frac{{\Vert {\varphi }_{\ell }-P_{\ell }{R}_{\ell }{\varphi }_{\ell }\Vert }_{W_{\ell }}}{{\Vert {\varphi }_{\ell }\Vert }_{W_{\ell }}+\epsilon },$$

where

$$\parallel x{\parallel }_{W_{\ell }}^2=\sum _i{w}_i^{(\ell )}{x}_i^2$$

and (\varepsilon>0) prevents division by zero.

For an observable (O), define a scale-consistency defect

$$D_{\ell ,O}^{\mathrm{scale}}=\frac{|O_{\ell +1}(R_{\ell }{\varphi }_{\ell })-O_{\ell }({\varphi }_{\ell })|}{|O_{\ell }({\varphi }_{\ell })|+\epsilon }.$$

No universal preservation threshold is assumed. Every run must declare its tolerances before execution.

Observables

The minimum observable set is:

1. Weighted norm$$$$

$$2.\ast \ast Benchmarkenergy\ast \ast$$

E_\ell[\phi].

$$3.\ast \ast Reconstructiondefect\ast \ast$$

D_\ell^{\mathrm{rec}}.

$$4.\ast \ast Observable-specificscaledefect\ast \ast$$

D_{\ell,O}^{\mathrm{scale}}.

$$\text{5. **Spectral occupancy** Expand (phi\_ell) in eigenmodes of (L\_ell) and record the fraction of norm in declared spectral bands. 6. **Grid-refinement difference** Compare a run against a finer lattice after projecting both to a common level. \#\# Proposed scale law test A candidate scale flow may be parameterized by}$$

g_{\ell+1}=g_\ell+\Delta s,\beta_g(g_\ell,\ell),

$$\text{for each coupling (gin{kappa,m^2,lambda}). This equation is not an extracted FQFT beta function. It is a generic discrete flow interface. A proposed FQFT law becomes testable only after (beta\_g) is explicitly defined and its dimensions, fixed points, recovery behavior, and prediction differences are stated. \#\# Numerical scheme \#\#\# Spatial representation Use either: - a regular one- or two-dimensional periodic lattice; or - a finite weighted graph with symmetric adjacency. The baseline implementation should begin with a periodic one-dimensional lattice because the spectrum and refinement map can be checked directly. \#\#\# Time integration Start with explicit Euler for transparency, then reproduce selected runs with Heun or fourth-order Runge–Kutta. For a bounded field amplitude (|phi\_i|le A), a conservative local step bound is}$$

\Delta t < \frac{2}{ \kappa_\ell\lambda_{\max}(L_\ell) + m_\ell^2 + 3\lambda_\ell A^2 }.

$$ This is a numerical stability guide for the benchmark update—not a theorem about FQFT. ### Restriction and prolongation The default pair is: - restriction by local weighted averaging; - prolongation by linear interpolation. At least one alternative pair must be tested. A claimed scale effect that disappears when the operator pair changes is likely a discretization artifact. ## Test regimes | Regime | Configuration | Intended test | |---|---|---| | Smooth baseline | low-frequency initial field | round-trip reconstruction | | Ultraviolet-rich | high-frequency mode mixture | information loss under restriction | | Coupling flow | nonconstant \(\kappa_\ell,m_\ell,\lambda_\ell\) | scale-law sensitivity | | Refinement | repeated \(N,2N,4N\) runs | numerical convergence | | Operator stress | alternate \(R_\ell,P_\ell\) | dependence on arbitrary coarse graining | | Recovery | scale flow switched off | return to ordinary lattice baseline | ## Recovery requirement The model does not establish a standard-QFT recovery theorem. It provides a minimal numerical acceptance condition: $$

\max_{\ell,O}D_{\ell,O}^{\mathrm{scale}}\rightarrow 0

$$ under the declared recovery limit and grid refinement. Failure to approach the baseline is not evidence for new physics. It is evidence that the proposed transition rule is incomplete, inconsistent, or numerically under-resolved. ## Failure conditions The run fails when any of the following occurs: - `non_finite_state`: NaN or infinite field values; - `energy_blowout`: unexplained growth outside the declared integrator tolerance; - `no_grid_convergence`: refinement differences do not decrease; - `operator_dependence`: the qualitative result changes under reasonable \(R/P\) alternatives; - `no_recovery_limit`: the baseline model is not recovered when scale modification is disabled; - `unbounded_transition_defect`: scale or reconstruction defects exceed preregistered tolerances; - `dimensionally_undefined_flow`: a proposed coupling flow has no consistent units; - `untraceable_run`: code version, seed, operators, or parameters are missing. ## Falsifiability boundary This simulation can weaken a proposed FQFT scale rule by showing that it: - lacks a stable continuum or refinement trend; - cannot recover its declared baseline; - depends entirely on arbitrary coarse-graining choices; - violates its own stated invariants; - produces no observable distinction from the benchmark after numerical error is controlled. It cannot by itself validate FQFT as a physical theory. ## Reproducibility record Every run must store: ```json { "simulationId": "fqft-scale-transition-model", "modelVersion": "", "codeCommit": "", "seed": 0, "levels": [], "latticeOrGraph": {}, "restrictionOperator": "", "prolongationOperator": "", "couplings": {}, "timeStep": 0, "steps": 0, "tolerances": {}, "observables": [], "failures": [] } ``` ## Review gate Promotion beyond `NEW_PROPOSAL_FOR_REVIEW` requires: 1. explicit coupling-flow definitions; 2. dimensional analysis; 3. grid and step convergence; 4. comparison with a standard lattice-field baseline; 5. uncertainty and sensitivity analysis; 6. independently reproducible code; 7. a stated observable or formal theorem target. See the [Researcher Workbench](/03-research/researcher-workbench) for the active review state. $$