Masses Rung Table Structure Rung Eq Predicted Of Ne Nu3

The rung table

The rung table is a discrete ladder of integers that the Recognition Science framework assigns to each fermion species: a rung is a whole number, and the framework's mass predictions hang off these rungs. The table has twelve entries, one for each of the three generations of neutrinos, charged leptons, and quarks. The theorem rung_eq_predicted_of_ne_nu3 states that for every fermion except the third neutrino, the rung equals a predicted value built from two ingredients: a base that counts how many gauge channels the species couples to, and a generation-dependent torsion term. In plain words: the framework proves that eleven of the twelve rungs are not free numbers; they are forced by a rule that counts coupling channels and adds a known twist.

The rule itself is simple. A species that couples to no gauge channel gets a base of zero; one that couples to one channel gets two; one that couples to both gets four. The theorem rungBase_values fixes these bases for the neutrino, electron, up, and down. The prediction then adds the generation torsion, which is already derived elsewhere. So the first neutrino gets rung 0, the electron gets 2, the up and down get 4, and the pattern continues through the table. The theorem rung_eq_predicted_of_ne_nu3 is the machine-checked statement that this rule reproduces the live table for all but one entry. It is a theorem in the framework's library: a machine-checked collection of formal theorems, checked by a computer against the axioms.

The one exception is the third neutrino. The theorem nu3_departs_by_two states that its rung is exactly two higher than the rule predicts. The departure lands it on the same rung as the tau lepton, its charged partner in the third generation. The framework records this as a fact, not as an explanation: the docstring says it is either a clue or a coincidence, and the framework does not decide. The theorem nu3_shares_the_tau_rung proves the rung equality, but nothing in the pack claims to explain why the third neutrino breaks the pattern.

What the theorem does not claim is as important as what it proves. It does not claim that the rung rule is physically fundamental; the rule is a definitional choice, not a derived law. It does not claim that the third neutrino's departure is predicted or understood; the departure is an open anomaly. And it does not claim that the rung table by itself predicts any mass; the rungs are an input to mass predictions, not the masses themselves. The framework's own honesty is explicit: the neutrino base is not evidence for the rule, because a species coupling to nothing returns zero whatever the parameters are. The strength of the fit is not overcounted.

The practical consequence is that the framework has a precise, machine-checked statement of where its mass ladder stands: eleven rungs forced, one unexplained. That boundary is what a later session needs. The table is not twelve independent integers; it is one rule plus one anomaly. Knowing exactly which entry is unexplained, and by how much, is the difference between a vague gap and a sharp target.

/-- **Eleven of the twelve rungs are sector base plus generation torsion.** This is the live
table used by the mass predictions, not the parallel per-sector definitions in
`Masses.Anchor`. -/
theorem rung_eq_predicted_of_ne_nu3 (f : Fermion) (h : f ≠ .nu3) :
    rung f = rungPredicted f := by
  cases f <;>
    simp_all [rung, rungPredicted, rungBase, activeChannelClasses, couplesToCharge,
      couplesToColor, tildeQ, sectorOf, genOf, Integers.tau, Anchor.E_passive, Anchor.W,
      passive_field_edges, cube_edges, active_edges_per_tick, D, wallpaper_groups]

/-- **The twelfth departs by exactly two.** The only entry the structure does not explain. -/
theorem nu3_departs_by_two : rung .nu3 = rungPredicted .nu3 + 2 := by
  simp [rung, rungPredicted, rungBase, activeChannelClasses, couplesToCharge, couplesToColor,
    tildeQ, sectorOf, genOf, Integers.tau, Anchor.E_passive, Anchor.W, passive_field_edges,
    cube_edges, active_edges_per_tick, D, wallpaper_groups]

/-- And the departure lands it on the charged-lepton rung of the same generation, which is
either a clue or a coincidence and is recorded as neither. -/
theorem nu3_shares_the_tau_rung : rung .nu3 = rung .tau := by
  simp [rung]

/-- The neutrino base, by contrast, is NOT evidence for the rule: the form returns zero for a
species coupling to nothing whatever the parameters are. Stated so the strength of the fit is
not overcounted. -/
theorem neutrino_base_is_forced (a b : ℤ) :
    a * 0 + b * 0 = rungBase .nu1 := by
  have := rungBase_values
  omega

What this page does not claim

The rung rule is a derived law of physics; it is a definitional choice. The third neutrino's departure is predicted or explained by the framework. The rung table by itself predicts any particle mass. The neutrino base provides evidence for the rung rule.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Masses/RungTableStructure.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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