11 July 2026. The complete record of a gated research program on one question: why do the electron, muon, and tau have the masses they do?
Every stage below was run against a pass/fail criterion written down before computing, with no parameter tuning and no criterion adjusted after the fact. Failures are reported with the same care as passes — several of the most useful results here are the failures. Labels: [derived] follows from the framework or the data · [consistency] reproduces a known result · [proposed] a mechanism named, not forced · [open] unexplained · [excluded] ruled out.
Three empirical facts anchor everything:
The framework connection: “three generations = three phase states of one object” (the founding intuition) maps exactly onto this structure.
Three gated attempts to derive the Koide balance (scripts
koide_selfdual_g1.py … g5.py), all pre-registered, all honest failures
that closed entire mechanism classes:
| gate | mechanism class tested | verdict |
|---|---|---|
| G3 | symmetries/dualities of the 3-state ring (Aubry–André type) | excluded — the balanced quantity is central: no symmetry on the ring can touch it (theorem-grade) |
| G4 | ring-local energetics (Derrick/virial equipartition) | excluded — the virial is per-particle (“vertical”); no local potential produces the required split; stationarity only re-finds already-dead corners |
| G5 | collective/zero-mode dynamics (rotor, shared-core exchange) | excluded/free — the rotor is exactly blind to the balance; exchange generates the right operator but its coefficient stays free |
Net result of Part I: Koide’s relation compresses from “three mysterious masses” to one unexplained coefficient — the equality of two couplings in the object’s internal potential. A by-product with independent value: generation towers built from rigid-rotor excitation are excluded outright (their Q can never exceed 5/9, and their mass ratios cap at 4 vs the observed 16.8). [derived exclusions]
If the three generations were the same object vibrating in higher modes, their
masses would be rungs of a ladder. Tested twice, independently
(spectrum_sp1_breathers.py, spectrum_sp23_qball_tower.py; pre-registration
in G0_prereg_spectrum.md):
Both towers hug Q ≈ 1/3 (the all-equal corner). The leptons sit at Q = 2/3. Whatever splits the generations, it is not gentle excitation — and near a cancellation point (Part III), enormous ratios cost nothing. The exclusion and the mechanism fit together.
Pre-registration: M0_prereg_mass_interference.md. Hypothesis H-MASS: the
hierarchy is near-destructive interference — the electron sits near an exact
zero of a two-component amplitude.
M1 (mass_m1_cancellation.py) — PASS. The cancellation point is the
singular point of the generation matrix (determinant exactly zero, rank 2;
the electron is the near-null direction). At exact cancellation the remaining
mass ratio is forced in closed form: m_τ/m_μ → (2+√3)² = 13.93 (measured:
16.82; the ε offset supplies the difference). One angle controls the whole
hierarchy: μ/e and τ/e scale as 1/ε². [derived, matrix level]
M2 (mass_m2_interference.py) — literal gate FAILED, amended gate
sanctioned. Full disclosure: the M0 gate formula was mis-specified (a complex
modulus), and the data itself excludes that form — the lepton mass-cosine dips
negative, which no modulus can do (needs amplitude/mean ≤ 1; data: 1.833). The
failure was logged, not patched; the owner sanctioned the corrected gate M2′.
What the data forces instead: real, sign-changing interference — the
electron sits 2.27° from a genuine sign flip [derived from data] — and, by
Cauchy–Schwarz, the two interfering components must share one spatial shape;
the measured 10⁻⁵ imperfection of Koide then reads naturally as a 0.11% mode
misalignment [proposed]. The construction: the framework’s own Q-ball binds
a localized internal mode in its own potential well (eigenvalue 0.608 vs
continuum 1.0) — a “generation dial” that is part of the particle — and for
this mode, energy is exactly the square of a real amplitude (nonlinear
corrections < 10⁻⁵ at the lepton amplitudes). Two real contributions into one
mode reproduce 206.77, 16.817, and Q = 0.666661 — with A and δ inserted, not
derived. [mechanism demonstrated; coefficients open]
M3 (mass_m3_epsilon.py) — FAIL, honestly. What sets ε? Derived: the
internal 120° symmetry makes all simple energetics blind to the dial angle,
and every polynomial internal energy up to degree 5 would park the dial at a
60° multiple — excluded by 12.73° at ~27,000σ. Richer dynamics select nothing
(free couplings). And ε is classically unprotected: m_e shifts ~5% per
milliradian of dial angle. H-MASS relocates the fine-tuning; it does not yet
remove it. One precision target survives, with no mechanism attached:
δ − 120° = 2/9 rad, consistent at 0.9σ (the known Brannen form; the tempting
1/(8π) is excluded at 25σ). A ~10× better tau mass makes 2/9 sharply testable.
[open; falsifiable hook]
M4 (mass_m4_chirality.py) — PASS. In the framework, chirality = winding
direction (derived earlier, verify_chiral_g*.py). Writing the dial in the
winding basis: the mass-making channel is the winding-reversal-EVEN
projection, and the cancellation point is exactly where the object becomes a
pure ODD eigenstate of winding reversal. The electron is 99.85%
winding-odd — an almost purely helical internal state, nearly invisible to
the mass channel; the tau is 98% even. Three consistency results follow at
machine precision: (i) the antiparticle family (all windings flipped) has an
identical mass spectrum — inherited automatically; (ii) gauge couplings are
winding integers (quantized → identical across generations) while masses are
amplitudes (continuous → hierarchical) — predicting exact lepton
universality coexisting with a 3477× mass ratio, which is what is observed;
(iii) the minimal dial-locking energy allowed by the 120° symmetry has
exactly three notches, always — a proposed origin for why there are three
generations, assembled entirely from derived parts. [derived within M2′;
weak-channel identification proposed]
Pre-registration: E0_prereg_epsilon.md, with an explicit anti-numerology
protocol (closed candidate lists, “derive the relation symbolically first,
compare its number second,” every comparison reported hit-or-miss). The M3
result left ε as the whole mystery; Part IV asks what sets it.
E1 (epsilon_e1_topo.py) — topological quantization EXCLUDED. If the dial
offset were fixed by winding topology, it would be a rational fraction of a
full turn. Tested against the entire closed class (denominators ≤ 36, three
readings of the offset, look-elsewhere risk computed up front at 0.94%):
nothing came within 212σ. The naive “the offset is a winding number”
escape route is dead — and note the contrast that survives it: the one
precision form that does fit (below) is rational in radians, not in
turns, so whatever it is, it is not simple winding arithmetic. [excluded]
E2 (epsilon_e2_breaking.py) — the breaking has a unique owner, one free
number. Attempting to derive the offset yielded a theorem and a mechanism,
but not the number:
E4 (epsilon_e4_scale.py) — the structure lives on-shell. Koide (and hence
ε) is exact at the physical, pole masses; deforming to short-distance
running masses degrades it ~186× (one-loop QED, since leptons carry no colour).
This is where a field theory of dressed, on-shell objects would put it, and
where a theory of short-distance Yukawa parameters would have no reason to.
[consistency] (E3 found no licensed relation to test; the program closes
here.)
Net after Part IV: ε is no longer a bare unexplained angle. It has a named dynamical owner (the two-harmonic interference channel), a physical threshold (r > 1/4, whose crossing is the electron’s lightness), it lives on-shell, and it reduces to a single continuous ratio r ≈ 0.318 — whose value, like A ≈ √2 before it, waits on the actual soliton interior.
| statement | label |
|---|---|
| Koide ⟺ one angle ε from the singular point; m_e ∝ ε² | derived (exact restatement) |
| the interference is real and sign-changing (not modulus) | derived from data |
| a TFT object with the required structure exists (bound internal dial, exact square law) | derived (construction) |
| cancellation point = pure winding-reversal-odd state; electron 99.85% odd | derived within the construction |
| universality + hierarchy coexistence (couplings topological, masses amplitudes) | consistency, matches observation |
| antiparticle spectrum identical | consistency, inherited |
| exactly three generations from minimal Z₃ locking | proposed |
| the odd channel = the weak interaction’s chiral coupling | proposed |
| ε’s origin is symmetry-respecting; breaking content bounded to ~10⁻³ (rigidity theorem) | derived |
| ε lives in one channel: κ₃cos3α + κ₆cos6α past its pitchfork (r > 1/4) | derived (within the model) |
| Koide/ε is an on-shell (pole-mass) structure | consistency (one-loop) |
| topological quantization of the offset (rational fraction of a turn) | excluded (212σ) |
| the values of A (≈√2) and the ratio r ≈ 0.318 (equivalently ε = 2.2677°) | open — the entire remaining mystery |
| symmetry, local-energetic, collective, and excitation-tower origins | excluded (Parts I–II) |
A separate, gated study asks where r comes from inside the actual soliton, and
localizes it to the sharpest possible statement — full narrative and code in
the companion WHERE_R_LIVES.md. Headlines:
sint_r_interior.py)spec_internal_spectrum.py)spec_nl3_condensate.py)r remains open, but its address is now written to the last line: the relative phase of two coupled sectors of the nonlinear condensate, requiring a current-carrying / higher-charge configuration to even exist.
Everything unexplained about the lepton spectrum now lives in a single small angle: ε = 2.2677° ± 0.0001° — how far the generation dial sits from perfect silence. After the ε program (Part IV) that angle is much better characterized than “one free number”: its origin must respect the 120° symmetry (rigidity theorem), it belongs to one specific two-harmonic interference channel whose pitchfork threshold is the very thing that makes the electron light, and it is exact at the physical pole masses. What remains genuinely open is a single continuous ratio, r ≈ 0.318 — and, like the amplitude A ≈ √2 before it, its value is not fixed by any symmetry, energetic, collective, or topological argument we could construct. It waits on the full soliton interior: the same open frontier the rest of the framework reaches for its absolute numbers (G, Λ, a₀’s coefficient). One mechanism-less precision form survives as a falsifiable anchor: δ − 120° = 2/9 rad, a pole-mass statement consistent at 0.9σ, which a ~10× better tau mass will confirm or kill.
pip install numpy
mkdir -p outputs # scripts write their JSON results here
python3 mass_m1_cancellation.py # the zero, the angle, the singular matrix
python3 mass_m2_interference.py # data theorems + the construction
python3 mass_m3_epsilon.py # the epsilon exclusions (honest FAIL)
python3 mass_m4_chirality.py # the winding/chirality structure
python3 spectrum_sp1_breathers.py # 1D excitation towers excluded (exact)
python3 spectrum_sp23_qball_tower.py # 3D tower exists; leptons excluded (~min)
python3 koide_selfdual_g3.py # symmetry class closed
python3 koide_selfdual_g4.py # local-energetics class closed
python3 koide_selfdual_g5.py # collective class closed; rotor towers dead
python3 epsilon_e1_topo.py # topological quantization excluded (212σ)
python3 epsilon_e2_breaking.py # the rigidity theorem + the one channel
python3 epsilon_e4_scale.py # Koide/ε is an on-shell (pole-mass) fact
Inputs: PDG lepton masses only (E4 also uses α = 1/137.036 and one-loop QED,
declared in the script). Each script prints its own pre-registered gate and
verdict. JSON results land in outputs/.