Phi-Genesis

Verified Results and Evidence Boundaries

RFT-SIRM Phi-Genesis Evidence standard

This page summarises the results recorded in Phi-Genesis v2.2.1. “Verified” here means supported by the stated derivation, numerical computation or regression test within its stated scope. It does not mean that every claim is machine-checked or that finite-level evidence proves a universal theorem.

1. Sierpiński-gasket spectral structure

These statements concern the graph family, operator convention and finite-level checks documented in the source. Start with source code and the README.

2. Weyl exponent and finite-level estimates

For the Sierpiński gasket, the theoretical exponent is

[ \frac{d_s}{2}=\frac{\ln 3}{\ln 5}\approx0.6826. ]

The repository reports that a window-free chord estimator is within 0.1% of its stated comparison expression at level 9 for (N\ge1000). This is a result for that estimator and range, not proof that every finite-level estimator converges uniformly.

The inertia-based count path uses Sylvester’s law of inertia with sparse LU rather than computing the full spectrum. The documented level-10 graph has 88,575 vertices. The reported resource figures and window studies are implementation- and parameter-specific; see Open Problem #2 and the reproduction guide.

3. Möbius / π-flux boundary conditions

For the tested level-5 graph, the documented full π-flux configuration raises the first eigenvalue from approximately 0.006 to approximately 0.46, opening a spectral gap in that instance. Partial twists also produce gaps in the tested cases.

This does not establish a selection mechanism for ({1,12,17,40}). In the exhaustive search of all (2^5=32) generation-twist configurations, the maximum simultaneous target hit was 1 of 4 and index 40 was never selected.

4. Integer mass fit — a fit, not a prediction

The repository records a single-scale fit (m=A n), with (A=104.63) MeV and integer indices (n={1,12,17,40}), matching four selected masses with a mean error of about 0.68%.

The index set was chosen with the masses in view. Because no independent spectral rule has derived those indices, this numerical agreement is not a prediction and must not be described as a confirmed mass law. The alternative scale expression documented in the README gives approximately 100.13 MeV, about 4.3% below the fitted value.

The top-quark index is unresolved: the repository documents candidates 1654, 1664 and 1680 arising from different procedures/scales. No single candidate is endorsed.

5. η-invariant claim — not reproduced

The RFT-String v8.7 claim (\bar\eta(0)=-4) for the stated boundary potential was not reproduced. The documented test reports asymmetry +1 for the claimed potential, −1 for a random potential, and −2 for zero potential; −4 was not observed in the tested cases. The repository interprets the observed asymmetry as a finite-size effect, not as a demonstrated topological invariant.

6. How to read the evidence { #how-to-read-the-evidence }

Label Interpretation
Theoretical identity Follows from specified mathematical definitions and assumptions.
Reproducible numerical result A computation can be rerun for the documented finite instance and implementation.
Fit / estimator output Depends on model choice, parameter range and estimator design.
Negative search result The tested family did not meet its predeclared criterion; it does not prove no mechanism exists anywhere.
Open problem Current evidence is incomplete, inconclusive, or sensitive to a remaining methodological choice.

Phi-Genesis’ numerical results are not automatically formal proofs. In particular, the Lean-verified status of Evgeny’s Theorem does not transfer to this repository: the operator families, observables and verification methods differ.

Primary sources

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