v2.0 — Core Mathematics Verified
Phi-Genesis
Spectral Core for Fractal Vacuum Substrate.
Reproducible spectral computations on the Sierpiński gasket. All claims tested against code. No black boxes.
104.63
MeV — Scale A
0.68
% Mean Error
4
Fermions
14
Tests Passing
Sierpiński tetrahedron — chaos game, 22,000 points, rotating live
Methodology
Research Pipeline
01
POSTULATE
→
02
GEOMETRY
→
03
SPECTRUM
→
04
COMPUTE
→
05
VERIFY
Verified Results
Reproducible via Code
All results below are generated directly from the repository test suite. One continuous scale A is fit to the data — the four spectral indices {1, 12, 17, 40} are chosen to match the masses too, not derived from an independent rule (Open Problem #1, below).
Verified
Sierpiński Gasket Spectral Theory
Spectral decimation ×5 — eigenvalues scale by factor ~5 between consecutive levels (exact for small λ). Weyl law N(λ) ~ λ^{d_s/2} with d_s/2 = ln(3)/ln(5) ≈ 0.6826. Log-periodic oscillations with period ln(5).
d_s/2 = 0.6826
Verified
Möbius (π-Flux) Boundary Conditions
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. Gap-selected indices follow lacunary 3^k − 1 structure.
Δλ₁ ≈ 0.454
Verified
Exhaustive Twist Search
All 2⁵ = 32 generation-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.
32 configs · 1/4 max
Verified
Integer Mass Formula
Using integer indices n = {1, 12, 17, 40} with single scale A = 104.63 MeV. Mean relative error 0.68% — beats fitted non-integer values (0.87%). The decimals in earlier versions were noise around integers.
0.68% mean error
Verified
Scale Consistency Audit
Fitted scale A = 104.63 MeV. Formula A = v·α/Φ⁶ gives 100.13 MeV — 4.3% below fitted. Discrepancy confirmed. Relative spread of per-fermion implied scales: 0.58%.
4.3% discrepancy
Rejected
η-Invariant Claim — REJECTED
RFT-String v8.7 claim that η̄(0) = −4 for boundary potential V(i) = 0.5·(4y/√3 − 1) is not reproduced. Value −4 never observed. Asymmetry is an O(1) finite-size artifact, not a topological invariant.
−4 never observed
Mass Spectrum
Integer Mass Formula
Four heavy fermion masses matched using a single scale A = 104.63 MeV with integer spectral indices — no non-integer fudge factors per fermion. The indices themselves are picked to fit, not predicted in advance (see Open Problems).
Mass Generation
m_f = A · n_f where A = 104.63 MeV and n_f ∈ {1, 12, 17, 40}
| Fermion | Integer n | M_pred (MeV) | M_exp (MeV) | Error |
|---|---|---|---|---|
| μ (muon) | 1 | 104.63 | 105.658 | 0.97% |
| c (charm) | 12 | 1255.56 | 1275.00 | 1.52% |
| τ (tau) | 17 | 1778.71 | 1776.86 | 0.10% |
| b (bottom) | 40 | 4185.20 | 4180.00 | 0.12% |
| Mean Relative Error | 0.68% | |||
0.68%
Integer n, 4 fermions
0.87%
Fitted n, 5 fermions (old)
Key Numbers
Quantities at a Glance
| Quantity | Value | Source / Note |
|---|---|---|
| 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. Value −4 never observed. |
Open Problems
What Remains Unsolved
The mass formula m = A·n is numerically accurate, but n is not yet derived from spectral properties. These are the priority research directions.
#1
Priority
Selection Rule for n_f
We need a boundary condition or operator on SG × S¹ that selects n ∈ {1, 12, 17, 40} independently of experimental input.
- Scalar vertex potentials V(x) on SG
- Mixed boundary conditions (Dirichlet/Neumann)
- Other fractals: Sierpiński carpet, random dendrites
- SG × S¹ with different S¹ topologies and fluxes
- Higher-genus or branched coverings of SG
#2
Open
Finite-Level Convergence of d_s/2
Naive Weyl fit shows ~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.
- Level 9+ via sparse optimization or GPU
- Analytical finite-size correction from decimation theory
#3
Open
Light Fermions
Electron, up, down, strange quarks not yet addressed. A fractal NJL mechanism is sketched but not implemented.
- Implement fractal NJL integral
- Connect light sector to heavy sector via RG flow
Roadmap
Planned Research
01
Selection Rule Search
Scalar potentials, mixed boundary conditions, other fractals, twisted S¹
02
Deeper Levels
Level 9+ via sparse optimization or GPU acceleration
03
Light Fermions
Implement fractal NJL integral for electron and light quarks
04
Physical Bridge
Connect A = 104.63 MeV to electroweak scale via renormalisation group
Repository
Repository Structure
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 — 5 deep tests (~110s, includes level-8 dense solver)
__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 — 5 deep tests (~110s, includes level-8 dense solver)
Infrastructure
The Repo Audited Itself
Same discipline the physics gets: when the test infrastructure broke, it went in the changelog instead of getting quietly deleted.
pytest -v — before the fix (commit aee047d)
collected 462 items
phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/
phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/
phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/
... 33 levels deep, same 14 tests each time ...
phi_genesis/phi_genesis/test_honest.py::test_decimation_factor_5 PASSED
======================== 462 passed in 2025.82s (0:33:45) ========================
phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/
phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/
phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/phi_genesis/
... 33 levels deep, same 14 tests each time ...
phi_genesis/phi_genesis/test_honest.py::test_decimation_factor_5 PASSED
======================== 462 passed in 2025.82s (0:33:45) ========================
Broken
Self-Referential Symlink
Commit
aee047d shipped a git-tracked symlink phi_genesis → . as an import workaround. pytest's directory walker recursed into it infinitely: the same 14 tests rediscovered 33 times over, inflating a ~2 minute run to 33m45s before the OS-level symlink-depth limit finally cut it off.462 collected
Fixed
Real Package Directory
Commit
a1a66c9 replaced the symlink with an actual phi_genesis/ package, relative imports, and a proper pyproject.toml — installable regardless of what the checkout folder is named, no recursion possible.14 collected
462 ÷ 14 = 33 — not a coincidence. If a future run ever collects anything other than 14 items, check for a reintroduced symlink first.
Quick Start
Installation & Tests
install.sh
# Clone and install (editable, so `phi_genesis` imports from anywhere)
git clone https://github.com/RFT-SIRM/Phi-Genesis.git
cd Phi-Genesis
pip install -e .
pip install pytest
# Verify import
python -c "import phi_genesis; print('OK')"
# Full suite — 14 tests, ~110s (level-8 dense solver included)
pytest -v
git clone https://github.com/RFT-SIRM/Phi-Genesis.git
cd Phi-Genesis
pip install -e .
pip install pytest
# Verify import
python -c "import phi_genesis; print('OK')"
# Full suite — 14 tests, ~110s (level-8 dense solver included)
pytest -v
test output
# pytest -v (actual output, phi_genesis v2.1)
test_honest.py::test_decimation_factor_5 ... PASSED
test_honest.py::test_weyl_exponent ... PASSED
test_honest.py::test_eta_claim_fails ... PASSED # −4 never observed; asym = +1
test_honest.py::test_a_formula_discrepancy_is_real ... PASSED # 4.3% gap, formula v·α/Φ⁶
test_honest.py::test_no_single_scale ... PASSED # 0.58% spread across fermions
test_mobius.py::test_mobius_gaps_the_spectrum ... PASSED
test_mobius.py::test_gap_selection_not_particles ... PASSED
test_mobius.py::test_integer_rounding_quality ... PASSED # 0.68% mean error
test_fukushima_shima.py::test_gap_structure ... PASSED
test_convergence_and_selection.py::test_dense_solver_matches_sparse_at_level_6 ... PASSED
test_convergence_and_selection.py::test_naive_fit_stays_above_one_percent_at_level_8 ... PASSED
test_convergence_and_selection.py::test_window_averaging_is_not_a_stable_correction ... PASSED
test_convergence_and_selection.py::test_no_generation_twist_selects_all_four_targets ... PASSED # 32 configs, index 40 absent
test_convergence_and_selection.py::test_forty_never_selected_by_generation_twists ... PASSED
======================== 14 passed in ~110s =========================
# All 14 tests reproducible from a clean checkout: pip install -e . && pytest -v
test_honest.py::test_decimation_factor_5 ... PASSED
test_honest.py::test_weyl_exponent ... PASSED
test_honest.py::test_eta_claim_fails ... PASSED # −4 never observed; asym = +1
test_honest.py::test_a_formula_discrepancy_is_real ... PASSED # 4.3% gap, formula v·α/Φ⁶
test_honest.py::test_no_single_scale ... PASSED # 0.58% spread across fermions
test_mobius.py::test_mobius_gaps_the_spectrum ... PASSED
test_mobius.py::test_gap_selection_not_particles ... PASSED
test_mobius.py::test_integer_rounding_quality ... PASSED # 0.68% mean error
test_fukushima_shima.py::test_gap_structure ... PASSED
test_convergence_and_selection.py::test_dense_solver_matches_sparse_at_level_6 ... PASSED
test_convergence_and_selection.py::test_naive_fit_stays_above_one_percent_at_level_8 ... PASSED
test_convergence_and_selection.py::test_window_averaging_is_not_a_stable_correction ... PASSED
test_convergence_and_selection.py::test_no_generation_twist_selects_all_four_targets ... PASSED # 32 configs, index 40 absent
test_convergence_and_selection.py::test_forty_never_selected_by_generation_twists ... PASSED
======================== 14 passed in ~110s =========================
# All 14 tests reproducible from a clean checkout: pip install -e . && pytest -v
Reference
Citation
citation.bib
@software{phi-genesis-2026,
title = {Phi-Genesis: Spectral Core for Fractal Vacuum Substrate},
author = {RFT-SIRM},
year = {2026},
url = {https://github.com/RFT-SIRM/Phi-Genesis},
note = {v2.0 — Reproducible spectral computations on the Sierpiński
gasket. The mass formula m = A·n currently lacks an
independent selection rule.}
}
title = {Phi-Genesis: Spectral Core for Fractal Vacuum Substrate},
author = {RFT-SIRM},
year = {2026},
url = {https://github.com/RFT-SIRM/Phi-Genesis},
note = {v2.0 — Reproducible spectral computations on the Sierpiński
gasket. The mass formula m = A·n currently lacks an
independent selection rule.}
}
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.