12 July 2026. A companion to GENERATIONS_PROGRAM.md.
The generations program compressed the entire mystery of the three lepton masses into a single continuous number: r ≈ 0.318, the ratio that fixes how far the “generation dial” sits from the point where the electron would be massless. Everything else — the 207× and 17× mass gaps, Koide’s exact balance, the electron’s anomalous lightness — follows from it. This note is the honest record of three gated attempts to compute r from the actual soliton, and what they establish. r is not solved here. What is delivered is the sharpest possible statement of where it lives, a validated computation of the object that carries it, and a rigorous argument for why it has resisted every method.
Labels as before: [derived] rigorous · [computed] numerical, validated ·
[consistency] · [excluded] · [open]. Every stage has a pass/fail
gate fixed before running (SINT0_, SPEC0_, SPEC0b_ pre-registrations).
SINT0_prereg_r.md, sint_r_interior.py.
The dial angle δ is a gauge-invariant loop quantity — the phase of the coupling that ties the three generations together, a flux, not a property of any one of them. Two facts pin this down [derived]:
I then computed that overlap energy directly from the real soliton profile. The result [computed]: the offset-producing harmonics are exponentially tiny when two lumps merely touch, and reach the size needed for r ≈ 0.3 only in deep overlap — where the lumps have effectively merged into one object. So the “three separated lumps” (molecule) picture of the generations is excluded: it gives r ≈ 0 wherever it is self-consistent, and r ≈ 0.3 only where it has collapsed. r’s seat is a single merged soliton with internal structure — the same hard object that holds the amplitude A ≈ √2 and every absolute scale in the framework.
SPEC0_prereg_spectrum.md, spec_internal_spectrum.py.
I computed the actual internal fluctuation spectrum of the Q-ball — Bogoliubov–de Gennes, resolved by angular momentum ℓ — with the background relaxed on the computation grid so the exact zero-mode identities hold. The solver is validated: the U(1) Goldstone appears at ℓ=0 (frequency 5×10⁻¹⁴, machine-exact) and the translation zero mode at ℓ=1, both as they must.
At the relevant ball the internal spectrum is clean [computed]:
| ℓ | bound internal mode |
|---|---|
| 0 | only the Goldstone (no bound breathing mode) |
| 1 | only the translation zero mode |
| 2 | one bound quadrupole shape mode |
| 3 | one bound triangular shape mode ← the generation dial |
| 4 | none |
So the three-fold structure the generations program inferred is a genuine, computed object: the soliton’s triangular surface oscillation. Honest qualification: it is bound only for large (thin-wall) balls — it unbinds for compact ones — so the generation structure is a feature of the large-charge regime, not of every soliton. The dial now has a face; it is not an abstraction.
SPEC0b_prereg_nl3.md, spec_nl3_condensate.py.
The natural next step — condense that triangular mode and read off the dial potential — was attempted, and it fails for a clean, rigorous reason [derived + computed]. Rotating a single triangular deformation on a round ball is just rotating the whole thing; it costs zero energy. I built the finite-amplitude condensate from the real computed mode and swept the dial: the energy is flat to 3 parts in 100,000. The orientation of a single mode is a rotational Goldstone, not the dial. r is in no single-mode condensate.
This is the decisive localization. The dial potential has both a three-fold and a six-fold harmonic; for the energy to depend on δ at all, two different angular sectors (the three-fold and the six-fold) must both be present, with an independent relative phase between them. r is that relative phase — a flux between sectors. This is the tightest place a number can hide, and it retroactively explains the whole program: r survived symmetry arguments, local energetics, collective dynamics, topological quantization, and the single-mode condensate, because it was never in any of them. Two independent routes — the flux picture (§1) and the Goldstone argument (§3) — converge on the same answer: the dial is a relative phase between sectors, not any one mode’s orientation.
A physical by-product [computed]: the triangular mode’s self-interaction is defocusing (positive quartic) — the ball does not spontaneously become triangular. The three-generation structure is a sustained, coupled structure, not a spontaneously broken one.
| statement | label |
|---|---|
| δ is a gauge-invariant flux; mass sums are exactly δ-independent | derived |
| the loose three-lump “molecule” picture of the generations | excluded |
| r’s seat is a single merged soliton (not separated lumps) | derived |
| the generation dial = the soliton’s triangular (ℓ=3) surface mode | computed, validated |
| that mode is bound only in the large-charge (thin-wall) regime | computed |
| a single-mode condensate has a flat dial (rotational Goldstone) | derived + computed |
| r is a two-sector (three-fold × six-fold) relative-phase / flux quantity | derived |
| the soliton does not spontaneously become triangular (defocusing) | computed |
| the value of r ≈ 0.318 | open |
r is not computed — and after this, the reason is precise, not vague. Its value is the relative phase of two coupled angular sectors of the nonlinear condensate: a quantity absent from any single mode or static shape of one soliton, requiring a current-carrying / higher-charge configuration to even exist. That is the concrete next object. Like the amplitude A ≈ √2 beside it, and like Newton’s G, |Λ|, and the a₀ coefficient elsewhere in the framework, r is an absolute internal number that bottoms out on the full nonlinear soliton interior — the framework’s one recurring frontier, reached here by the narrowest and best-characterized path in the whole program.
pip install numpy
mkdir -p outputs
python3 sint_r_interior.py # r is a flux; molecule excluded; seat located
python3 spec_internal_spectrum.py # the internal BdG spectrum; l=3 dial mode (validated)
python3 spec_nl3_condensate.py # single-mode dial is flat -> r is two-sector
Inputs: PDG lepton masses and the repo Q-ball potential only. Each script
prints its pre-registered gate and verdict; JSON lands in outputs/.