Roof — Speculative Extension
Version 11.6 | Updated May 20, 2026 | John Pepin
⚠️ Ongoing research project — SPECULATIVE. This is Roof-level work. The combinatorial results are reproducible but the charge-spectrum claim was downgraded on April 5, 2026 after a null hypothesis test (see next section). Read the April 5 caveat before citing any specific result here.
The v11.0 headline on this page was that Z = 7 combinatorics produce the Standard Model fermion charge spectrum {0, ±1/3, ±2/3, ±1} from zero free parameters. That claim was downgraded on April 5, 2026 by a null hypothesis test.
The test swept 750 parameter combinations across Z ∈ {5, 6, 7, 8, 9, 10} with multiple symmetry groups and sector-combination schemes. Result: 170 of 750 combinations (22.7%) produce an exact match to the SM charge set. Every single match had nspatial = 3. Z = 7 and Z = 8 produced identical numbers of matches. The /3 normalization in Q = χspatial/3 is doing the work, not Z = 7 specifically.
Downgraded claim: "Q = χspatial/3, combined with Sector A (5 operational) and Sector B (4 operational with one spatial non-operational), produces the SM fermion charge spectrum. This is a consequence of nspatial = 3 (an observational input — we live in 3D space) plus the specific /3 formula, not a Z=7-specific derivation."
What is not affected: four things carry forward untouched.
(1) The Z = 7 geometric anchor — 6 cardinal directions in 3D plus "stay" — remains independent of the charge result.
(2) The detailed class structure has NOT been tested across Z values. Whether Z = 7 specifically produces (a) exactly 3 equivalence classes per charge value (three generations), (b) 3-fold color multiplicity for |χspatial| = 1 classes, (c) total of 17 fermion classes, (d) the proper A/B sector assignment for the mass hierarchy — none of this has been tested with a null sweep. Z = 7 may still be distinguished here. Building a second null test on detailed class structure is the natural next step and will determine whether the rest of this page survives.
(3) All non-particle results (baryon asymmetry, life at pc, kBT match, JWST consistency) are unchanged.
(4) The tension asymmetry and percolation sigmoid prediction are unchanged.
April 5 topology null test notebook (Colab) · Full result on the house page.
The rest of this page is preserved from v11.0 with inline caveats, so the reasoning is documented alongside the downgrade. Read with the April 5 result in mind.
Every massive particle is a spanning cluster on TSO's 7-path lattice. The 5 rotatable paths {x, y, z, X1, X2} can each be operational with positive (+1) or negative (−1) chirality. A particle's identity is its chirality pattern. Different patterns = different particles.
A note on terminology — "operational/non-operational" replaces "open/closed." Earlier versions of this page described paths as "closed" (formed, contributing chirality, pulling toward the solid phase) or "open" (free, wave-side). That language was retired site-wide because "open/closed" meant opposite things in different parts of the framework — on this page a "closed" path was the active, solid-producing one, while elsewhere (e.g. the HERE/∅ discussion) a "closed" path meant a path being shut off. The same enumeration even used "open" in two senses at once. The framework now uses operational (the path is doing its classical work — formed, contributing chirality, solid-side) and non-operational (the path is free, wave-side). So what was "5 paths closed" is now "5 paths operational," and W = (2 + nnon-op)/7. The physics is unchanged; only the labels are disambiguated.
Original question: how many distinct stable patterns exist, and do they match the particles we observe? Updated question (April 5): is the number and structure of patterns Z=7-specific, or generic across reasonable lattice sizes?
Massive particles require W ≤ pc = 0.3116. Since W = (2 + nnon-op)/7, and the 2 comes from T and ∅ (always non-operational in this counting — they contribute to the wave fraction), the constraint limits how many rotatable paths can remain non-operational:
| Sector | Paths operational | W | Phase | Character |
|---|---|---|---|---|
| A | 5/5 | 2/7 = 0.286 | Solid | Fully crystallized, classical |
| B | 4/5 | 3/7 = 0.429 | Wave (barely) | Partially wave, barely material |
| C | 0/5 | 1.0 | Full steam | No chirality, no mass |
Sector A = heavy, stable, fully classical particles. Sector B = light, weakly interacting, partially quantum. Sector C = massless gauge bosons.
With all 5 paths operational, each assigned chirality ±1, there are 25 = 32 raw configurations. After symmetry reduction (spatial SO(3) rotations + X1↔X2 swap + matter/antimatter conjugation), exactly 6 equivalence classes survive.
Electric charge emerges as Q = χspatial / 3:
| Classes | χspatial | Q = χs/3 | SM match | Generations |
|---|---|---|---|---|
| 3 classes | −3 (all aligned) | −1 | e, μ, τ | 3 (from quantum chirality) |
| 3 classes | −1 (2+1 split) | −1/3 | d, s, b | 3 (from quantum chirality) |
The three generations at each charge come from the three possible quantum chirality states on {X1, X2}. Whether this 3-per-charge structure is Z=7-specific or also generic across Z values has not been tested. The April 5 null test only checked the charge spectrum, not the generation count per charge. This is the most important unanswered question on the page.
With 4 paths operational and 1 non-operational, there are 80 raw configurations yielding 9 equivalence classes. Which type of path is non-operational determines the charge.
Spatial path non-operational (x, y, or z): Only 2 spatial paths contribute chirality → χspatial ∈ {−2, 0, +2} → charges −2/3, 0, +2/3.
Quantum path non-operational (X1 or X2): All 3 spatial paths still contribute → same charges as Sector A.
Combined spectrum across A and B: {0, ±1/3, ±2/3, ±1}. As noted in the April 5 caveat, this specific set is generic across Z values once nspatial = 3 is fixed. What may still be Z=7-specific is how the 9 Sector B classes split across charge values and whether that split matches the physical neutrino + up-type quark content.
Color charge emerges from which spatial path carries the minority chirality.
Quarks (|χspatial| = 1): Two spatial paths have one chirality, one has the opposite. Three choices for which path is the odd one (x, y, or z). These 3 choices are the 3 color states.
Leptons (|χspatial| = 3): All three spatial paths have the same chirality. No odd path, no color.
| |χspatial| | Particle type | Odd-path groups | Color? |
|---|---|---|---|
| 1 | Quark (Q = ±1/3) | 3 | YES — 3 colors |
| 3 | Lepton (Q = ±1) | 0 | NO — singlet |
The orbit decomposition is uniform: each quark orbit splits as 3 × N. The colored-to-singlet ratio in Sector A is 3.0. Color was not postulated; it falls out of the combinatorics.
Open question: whether this color structure is Z=7-specific or also generic. Like the generation-count question, it has not been tested in a null sweep. If a second null test shows that Z ≠ 7 values also produce 3-fold color multiplicity for |χspatial| = 1 classes, then color is also generic and this page needs further downgrading. If Z = 7 is uniquely distinguished, this page survives in stronger form.
| Quantity | TSO (Z=7) | Standard Model | Tested across Z? |
|---|---|---|---|
| Sector A classes | 6 | 6 (e,μ,τ + d,s,b) | NO |
| Sector B classes | 9 | ~6 + overlaps | NO |
| Sector C | 1 | ~1 | NO |
| Total classes | 16 | 17 | NO |
| With antiparticles | 30 | 29 | NO |
| Charge spectrum | {0, ±1/3, ±2/3, ±1} | {0, ±1/3, ±2/3, ±1} | YES — GENERIC |
| Generations per charge | 3 | 3 | NO |
| Color for quarks only | Yes | Yes | NO |
Six rows of this table remain untested across Z values as of April 5, 2026. Building the class-structure null test is the single most important near-term task for this page.
Detailed class structure — generations, color multiplicity, total count — not yet tested across Z values, may still be Z=7-specific
Neutrinos as partially-wave entities (one path non-operational)
Massless bosons as fully-non-operational configurations
Combinatorial enumeration itself is reproducible and correct
Color emerging from odd-path choice is a clean structural result
Charge spectrum is generic across Z ∈ {5..10} (April 5 null test) — no longer a Z=7-specific derivation
Q = χspatial/3 is observed from the output, not derived from first principles
Mass hierarchy not derived (why me << mμ << mτ?)
W, Z, Higgs not placed in the scheme
Extra Sector B class — artifact or prediction of unobserved particle?
SU(3) gauge dynamics not derived — only the 3-color structural count
The best Fano node assignment (BEST_PERM = T=0,∅=1,x=2,y=3,z=5,X₁=4,X₂=6) confirms that x,y,z form a Fano triangle at nodes {2,3,5}. Combined with the X₂=gravity hypothesis, this gives four forces from four geometric structures:
| Force | Structure | Character |
|---|---|---|
| Gravity | T + X₂ (two paths) | Temporal + spatial curvature = one metric tensor |
| EM | X₁ (one path) | Long-range, parity-symmetric |
| Strong | xyz triangle | Color = which edge of the spatial triangle is minority |
| Weak | xyz asymmetry | Strangeness = asymmetric activation; emerges from strong geometry |
The weak force is not a separate path — it is the asymmetry of the xyz triangle. This explains parity violation (the triangle has handedness), why strangeness is a triangle-edge property, and why the Omega- (sss, all three edges equally operational) has zero strangeness correction — symmetric triplet restores codeword structure.
Particles split into two classes based on Hamming code membership:
Topological particles — path word ON a [7,4,3] Hamming codeword. Error-corrected against 1-bit perturbations in the wave state. Can go quantum and return. Examples: proton (N=28, weight-7 codeword), D meson (N=56), Omega- (sss restores spatial symmetry → codeword structure restored).
Energetic particles — path word OFF a codeword (HD=1). Any additional perturbation exceeds the correction radius → decay. Examples: kaon (N=15), Lambda (N=33), Sigma (N=36), Xi (N=39/40).
The pion (N=4) is the quantum ZPE — the minimum-energy non-codeword excitation. The minimum Hamming codeword weight is 3; pion N=4 ≠ 3×anything at any uniform Pip/path scaling. The pion cannot be a codeword particle. π⁰→γγ is X₁ release: Hamming correction to vacuum in 8.4×10⁻¹⁷s — the fastest hadronic decay because it is the easiest 1-bit correction.
The Omega- paradox resolved: three strange quarks make all three {x,y,z} edges equally operational. Zero asymmetry → codeword structure → topological despite maximum strangeness. Error: 0.036% — most precise TSO result.
Particle stability is a theorem of the [7,4,3] code. Decay is the universe fixing a bit error. Hamming stability notebook →
The thermodynamic noise scale σ = 0.18058 is a proposed closed form — 6/√1104, built from the {x,y,z,T} tetrahedron and the K3 Calabi-Yau geometry. Its provenance is unverified: it has not been shown that the 6 and the 1104 are forced by the framework rather than selected to reproduce the target value 0.18058 (the K3 floor is one choice among the Calabi-Yau staircase, and a different rung changes σ). Every hadron mass follows from σ plus M_Z as energy-scale anchor — so those mass matches inherit σ's unverified status rather than standing as independent derivations. See math page for the full treatment and caveats.
Formula: E_Pip = Λ × M_Z × PIP where Λ = 1/(2√σ) = 1.17662, PIP = p_c/1000.
Mass(particle) = N × E_Pip where N is the Pip count from the PG tower or E₇ group theory.
| Particle | N (Pips) | Source | TSO mass | PDG mass | Error |
|---|---|---|---|---|---|
| E_Pip (fundamental quantum) | 0.996 | PG(7,2) collapse | 33.3 MeV | 33.44 MeV | 0.4% |
| Pion π⁰ | 3.938 | PG(5,2) ZPE | 133.7 MeV | 134.977 MeV | 0.95% |
| Kaon K± | 15.0 | PG(3,2) full activation | 501.6 MeV | 493.677 MeV | 1.6% |
| Proton | 28.0 | PG(2,2) full activation | 936.2 MeV | 938.272 MeV | 0.22% |
| D meson | 56.0 | E₇ minimal representation | 1872.5 MeV | 1869.66 MeV | 0.15% |
| Ψ(4415) charmonium | 133.0 | E₇ dimension | 4447 MeV | 4421 MeV | 0.59% |
Mass ratios from pure Pip integers (no E_Pip needed):
| Ratio | TSO (Pip integers) | PDG | Error |
|---|---|---|---|
| Proton / Kaon | 28/15 = 1.867 | 1.901 | 1.8% |
| Proton / Pion | 28/3.938 = 7.11 | 6.95 | 2.3% |
| D meson / Pion | 56/3.938 = 14.2 | 13.85 | 2.7% |
| J/ψ / Proton | 93/28 = 3.32 | 3.301 | 0.6% |
These ratios use only integer Pip counts — no calibration, no per-ratio free parameters. (They do, however, inherit the σ-based E_Pip scale, whose provenance is unverified; the ratios are integer-clean, but their conversion to absolute energies rests on σ.)