Reproducible spectral computations on the Sierpiński gasket (SG) and related structures. This is the computational backbone of the RFT (Reality Fractal Theory) research programme.
Status: v2.0 — all core mathematics verified; selection rule for fermion masses remains an open problem.
- Spectral decimation ×5 — eigenvalues scale by factor ~5 between consecutive levels (exact for small λ).
- Weyl law — counting function N(λ) ~ λ^{d_s/2} with d_s/2 = ln(3)/ln(5) ≈ 0.6826. Finite-level bias ~3% at level 6, ~2.8% at level 8.
- Log-periodic oscillations — period ln(5), signature of decimation.
- Fukushima–Shima gaps — forbidden zones cluster at decimation fixed points λ ≈ 2.5 and λ ≈ 3.
- Full π-flux through all holes opens a mass gap: first eigenvalue jumps from ~0.006 to ~0.46 at level 5.
- Partial twists also gap the spectrum.
- Selection-rule test: gap-selected indices follow the lacunary
3^k − 1structure. The mass indices{1, 12, 17, 40}are not selected by gap rules.
- All
2^5 = 32generation-twist configurations at level 5 tested. - Max simultaneous hits on {1, 12, 17, 40}: 1 of 4.
- Index 40 never appears as gap-selected under any configuration.
- This closes the twist-by-generation family as a candidate selection rule.
Using integer indices n = {1, 12, 17, 40} with a single scale A = 104.63 MeV:
| Fermion | n | M_pred (MeV) | M_exp (MeV) | Error |
|---|---|---|---|---|
| μ | 1 | 104.63 | 105.658 | 0.97% |
| τ | 17 | 1778.71 | 1776.86 | 0.10% |
| c | 12 | 1255.56 | 1275.00 | 1.52% |
| b | 40 | 4185.20 | 4180.00 | 0.12% |
| Mean | 0.68% |
This beats the fitted non-integer values (0.87% mean error). The decimals in earlier versions were noise around integers.
- Fitted scale
A = 104.63MeV (from integer fit). - Formula
A = v·α/Φ⁶gives 100.13 MeV — 4.3% below fitted. Discrepancy confirmed. - Relative spread of per-fermion implied scales: 0.58%.
The RFT-String v8.7 claim that η̄(0) = −4 for the boundary potential V(i) = 0.5·(4y/√3 − 1) is not reproduced:
- For the claimed potential: asymmetry = +1
- For random potential: −1
- For zero potential: −2
- Value −4 never observed. The asymmetry is an O(1) finite-size artifact, not a topological invariant.
The top quark is intentionally absent from the 4-fermion integer table above.
Reason (verified in test_top_quark_consistency.py): the historically
advertised n_t = 1680 = 8·7# was computed under an OLDER scale
(A ≈ 103.63 MeV), not the A = 104.63 MeV used everywhere else in this
section. Recomputing at the current A gives three genuinely different
candidates:
| Candidate | n_t | How it's obtained |
|---|---|---|
| Advertised ("8×7#") | 1680 | v/(√2·A) at the old A ≈ 103.63 |
| Formula, current A | 1664 | v/(√2·A) at A = 104.63 |
| Best integer fit, current A | 1654 | round(m_top / A) at A = 104.63 |
Using the advertised n_t = 1680 with the current A gives a top-quark
error of 1.55% — more than double the 4-fermion mean (0.68%).
Including it would raise the reported mean to ~0.85%, back to the
level of the old fitted decimals the integer fit is presented as
improving on. No single n_t is asserted as correct here; the
inconsistency itself is the finding, guarded against regressing
silently by the three tests in test_top_quark_consistency.py.
Verified in test_sg_index_triviality.py on the actual SG graph (not
the abandoned carpet branch): under thousands of random on-site
potentials (unrelated to any specific flux/twist mechanism), index 1
appears in the gap-selected set ~95% of the time, regardless of
what the potential is. For comparison: index 12 ~37%, index 17 ~22%,
index 40 ~3.4%.
This means the μ↔n=1 correspondence (section 4) and the "1 of 4" hit reported for the exhaustive Möbius twist search (section 3) carry much less evidential weight than they appear to at face value — index 1 is close to a structural artifact of the ratio-based gap-selection rule itself, not a signature of the π-flux mechanism specifically. An index 40 hit would be far more meaningful (only ~3.4% base rate) than an index 1 hit (~95% base rate) — the four "hits" in this repo are not equally strong evidence, and should not be reported as if they were.
The formula m = A·n is numerically accurate, but n is not yet derived from spectral properties. We need a boundary condition or operator on SG × S¹ that selects n ∈ {1, 12, 17, 40} independently of experimental input.
Untested families:
- Scalar vertex potentials V(x) on SG
- Mixed boundary conditions (Dirichlet on some holes, Neumann on others)
- Other fractals: Sierpiński carpet, random dendrites
- SG × S¹ with different S¹ topologies and fluxes
- Higher-genus or branched coverings of SG
Naive Weyl fit: ~2.8% error at level 8 (9843 vertices). Phase-averaged sliding windows can dip below 1%, but the result depends on window width and step — not yet a unique correction.
Path forward: level 9+ (needs optimization), or analytical finite-size correction from decimation theory.
Electron, up, down, strange quarks not yet addressed. A fractal NJL mechanism is sketched but not implemented.
phi_genesis/
__init__.py — module exports
sg_laplacian.py — SG graph, Laplacian, decimation, Weyl
dirac_eta.py — Graph Dirac operator, η-invariant test
mass_check.py — Scale consistency audit
mobius.py — π-flux signed Laplacian (Z₂ cocycle)
mobius_selection_search.py — Exhaustive twist search (32 configs)
fukushima_shima.py — Gap structure verification
spectral_convergence.py — Dense solver, Weyl fits, window sensitivity
test_honest.py — 5 core tests (~4s)
test_mobius.py — 3 twist/integer tests (~0.1s)
test_fukushima_shima.py — 1 gap test (~1.6s)
test_convergence_and_selection.py — 4 deep tests (~13s, level 8 skipped in CI)
pip install numpy scipy
python -c "import phi_genesis; print('OK')"# Fast suite (~6s)
python test_honest.py
python test_mobius.py
python test_fukushima_shima.py
# Deep suite (~13s, excludes level-8 dense solver)
python test_convergence_and_selection.py
# Level-8 dense solver manually (~60s, 9843 vertices)
python -c "from phi_genesis import dense_laplacian_spectrum; dense_laplacian_spectrum(8)"| Quantity | Value | Source |
|---|---|---|
| A (integer fit) | 104.63 MeV | Best fit to μ/τ/c/b with integer n |
| A (formula) | 100.13 MeV | v·α/Φ⁶ — 4.3% below fitted |
| d_s/2 (theory) | 0.6826 | ln(3)/ln(5) |
| d_s/2 (naive, level 6) | 0.7051 | 3.3% above theory |
| d_s/2 (naive, level 8) | 0.7019 | 2.8% above theory |
| Mean error (integer n, 4 fermions) | 0.68% | Better than fitted decimals (0.87%) |
| Mean error (fitted n, 5 fermions) | 0.87% | Old non-integer values |
| η-asymmetry (claimed potential) | +1 | Not −4 |
- Selection rule search — scalar potentials, mixed BCs, other fractals, twisted S¹
- Deeper levels — level 9+ via sparse optimization or GPU
- Light fermions — implement fractal NJL integral
- Physical bridge — connect A = 104.63 MeV to electroweak scale via RG
If you use this code, please cite the RFT research programme and acknowledge that the mass formula m = A·n currently lacks an independent selection rule.
All computations are reproducible. All claims are tested against code. No black boxes.