Masses Mass Genesis Admissible Mass Image Exact Image Excludes Nonrealized Secto

The exclusion lemma

The declaration exact_image_excludes_nonrealized_sector_rung is a formal lemma in the Recognition Science framework's machine-checked library of formal theorems. It says: if a proposed set of particle states has exactly the nine known charged rows and no extras, then any state in that set whose sector and rung labels match a forbidden combination cannot exist. The proof is a direct contradiction: the exact-image assumption forces the state to be one of the nine rows, but the forbidden labels contradict every one of those rows' own labels. The library checks this argument with no unproved assumptions and no new axioms.

The lemma is the formal backbone of a larger separation. The framework's mass genesis module distinguishes three claims that are easy to mix: the labels that the nine realized charged rows carry, the finite candidate image that the generation ladder supplies, and the physical closure theorem that stable admissible topology has exactly this image and nothing more. The exclusion lemma proves the first two are consistent: the candidate image does not secretly include a forbidden neighboring row. It packages the candidate image as a certificate, and it names the remaining physical closure as an explicit obligation, ExactChargedTopologyImage, rather than pretending the table lookup settles it.

What the lemma does not claim is the physical closure itself. It takes as input an assumption that a predicate P exactly matches the realized topologies, and it derives the exclusion from that assumption. It does not prove that any actual physical stability predicate satisfies that assumption. The docstring is explicit: proving that no extra stable closed light patterns exist is a real physical theorem, not a table lookup. The library keeps that as a named obligation, ExactImageFromPhysicalStability, and the certificate's final field records it as such.

In plain terms: the lemma is a consistency check on the list of nine rows, and it is airtight. The step from that list to a physical law that forces exactly that list is a separate, unfinished theorem. The framework's own documentation says so in the same breath that it states the lemma. A reader who takes the exclusion lemma as proof that nature realizes exactly these nine rows would be overreading it; the lemma proves the exclusion only under the exact-image assumption, which remains open.

/-- If the exact image theorem holds, any sector/rung pair absent from the
candidate image is physically excluded. -/
theorem exact_image_excludes_nonrealized_sector_rung
    {P : PatternTopology → Prop}
    (H : ExactChargedTopologyImage P)
    {s : Anchor.Sector} {r : ℤ}
    (hnone : NoRealizedSectorRung s r)
    {T : PatternTopology}
    (hP : P T)
    (hlabels : sectorFromTopology T = s ∧ rungFromTopology T = r) :
    False := by
  obtain ⟨row, hrow, hT⟩ := H.no_extra T hP
  rw [hT, row_sector_eq row, row_rung_eq row] at hlabels
  exact hnone row hrow hlabels

/-- If the exact image theorem holds, any sector/rung pair absent from the
candidate image is physically excluded. -/
theorem exact_image_excludes_nonrealized_sector_rung
    {P : PatternTopology → Prop}
    (H : ExactChargedTopologyImage P)
    {s : Anchor.Sector} {r : ℤ}
    (hnone : NoRealizedSectorRung s r)
    {T : PatternTopology}
    (hP : P T)
    (hlabels : sectorFromTopology T = s ∧ rungFromTopology T = r) :
    False := by
  obtain ⟨row, hrow, hT⟩ := H.no_extra T hP
  rw [hT, row_sector_eq row, row_rung_eq row] at hlabels
  exact hnone row hrow hlabels

/-- If the exact image theorem holds, any sector/rung pair absent from the
candidate image is physically excluded. -/
theorem exact_image_excludes_nonrealized_sector_rung
    {P : PatternTopology → Prop}
    (H : ExactChargedTopologyImage P)
    {s : Anchor.Sector} {r : ℤ}
    (hnone : NoRealizedSectorRung s r)
    {T : PatternTopology}
    (hP : P T)
    (hlabels : sectorFromTopology T = s ∧ rungFromTopology T = r) :
    False := by
  obtain ⟨row, hrow, hT⟩ := H.no_extra T hP
  rw [hT, row_sector_eq row, row_rung_eq row] at hlabels
  exact hnone row hrow hlabels

/-- The genuine remaining physical obligation: a physical-stability predicate has the
exact charged image iff it coincides, pointwise, with the canonical realized image.
This is the precise bridge that the Mass Genesis stability theorem must supply. -/
def ExactImageFromPhysicalStability (P : PatternTopology → Prop) : Prop :=
  ∀ T : PatternTopology, P T ↔ IsRealizedChargedTopology T

What this page does not claim

The lemma does not prove that any actual physical stability predicate satisfies the exact-image assumption. The lemma does not prove that nature realizes exactly these nine charged rows and no others. The lemma does not derive the nine rows from first principles; it takes them as the given chargedRows 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.

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