Masses Mass Genesis Admissible Mass Image Realized Excludes Nonrealized Sector R

The exclusion rule

The standard model of particle physics recognizes nine charged matter particles: the electron, muon, and tau leptons, and the up, charm, top, down, strange, and bottom quarks. Each carries a distinct mass and a distinct electric charge. Recognition Science, a framework that derives physical structure from a forced cost of recognition, models these nine particles as occupying specific slots on a generation ladder. The declaration realized_excludes_nonrealized_sector_rung is a formal theorem in the framework's machine-checked library of formal theorems. It states a simple exclusion rule: if a slot on the ladder is not occupied by any of the nine known particles, then no realized topology, no actual particle configuration, can carry the labels of that empty slot.

The theorem proves a consistency property. The framework first defines a list of the nine charged rows, one per particle, and assigns each a sector (lepton, up quark, or down quark) and a rung number on the ladder. It then defines a realized topology as any pattern that matches one of these nine rows. The exclusion rule takes a sector and a rung that appear in no realized row, and a topology that is realized; if that topology were labeled with the empty sector and rung, the theorem derives a contradiction. In plainer terms: the allowed slots are exactly the occupied ones. The proof is computational, checking all nine rows by direct evaluation, and the library records zero unproved assumptions and zero new axioms.

The declaration does not claim that the nine particles are the only possible stable configurations. That stronger statement, that stable admissible topology has exactly this image and no extra neighboring rows, is a separate obligation named ExactChargedTopologyImage in the library. The framework's own documentation marks it as a real physical theorem, not a table lookup. The exclusion rule is a consequence of the definition of realized topology, not a proof about which patterns physics allows. It also does not assign masses to the particles; the rung numbers are labels on a ladder, not measured mass values.

What the theorem changes is the shape of the framework's mass model. It certifies that the generation ladder, as defined, has no gaps: the nine known particles fill their assigned slots, and the labels derived from topology match the labels assigned by hand. The exclusion rule is the consistency backbone that lets the framework treat the particle list as a closed set. A reader can now see the nine rows as a complete, non-overlapping partition of the allowed sector-rung space, with the empty slots provably unreachable by any realized topology.

/-- The realized-image predicate excludes every sector/rung pair absent from the
candidate image: a concrete consequence of the instantiated exact image. -/
theorem realized_excludes_nonrealized_sector_rung
    {s : Anchor.Sector} {r : ℤ}
    (hnone : NoRealizedSectorRung s r)
    {T : PatternTopology}
    (hP : IsRealizedChargedTopology T)
    (hlabels : sectorFromTopology T = s ∧ rungFromTopology T = r) :
    False :=
  exact_image_excludes_nonrealized_sector_rung exactImage_realized hnone hP hlabels

/-- The realized-image predicate excludes every sector/rung pair absent from the
candidate image: a concrete consequence of the instantiated exact image. -/
theorem realized_excludes_nonrealized_sector_rung
    {s : Anchor.Sector} {r : ℤ}
    (hnone : NoRealizedSectorRung s r)
    {T : PatternTopology}
    (hP : IsRealizedChargedTopology T)
    (hlabels : sectorFromTopology T = s ∧ rungFromTopology T = r) :
    False :=
  exact_image_excludes_nonrealized_sector_rung exactImage_realized hnone hP hlabels

theorem chargedRows_nodup : chargedRows.Nodup := by
  decide

theorem chargedRows_length : chargedRows.length = 9 := by
  rfl

What this page does not claim

The theorem does not prove that the nine particles are the only stable configurations; that stronger statement remains an open obligation. The rung numbers are not measured masses, only labels on a generation ladder. The exclusion rule says nothing about neutral particles, which are not in the nine-row list.

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/MassGenesis/AdmissibleMassImage.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: