Masses Mass Genesis Admissible Mass Image Exact Image Of Physical Stability

The exact image theorem

The declaration exactImage_of_physicalStability is a theorem in the framework's machine-checked library of formal theorems. It states a conditional result: if a property P, which is intended to represent physical stability, is exactly equivalent to the property of being one of the nine realized charged rows (electron, muon, tau, up, charm, top, down, strange, bottom), then P has no other instances. In plainer terms, if stability is defined so that it picks out precisely those nine particles, then the theorem guarantees that stability cannot also pick out any additional, unobserved particle pattern.

The theorem's content splits into two directions. The first, called complete, says that every one of the nine realized rows satisfies P. The second, called no_extra, says that if any pattern T satisfies P, then T must be identical to one of those nine rows. Together these two halves form the claim that the image of P is exactly the nine-row list. The theorem does not by itself prove that any particular P is the correct stability predicate; that remains a separate obligation named ExactChargedTopologyImage in the framework's development.

What the theorem does establish is a bridge. It shows that the framework's candidate image, the nine rows supplied by the forced generation ladder, is the unique image compatible with any stability predicate that exactly matches it. The proof is short: it uses the equivalence between P and the realized-topology predicate, then applies the already-proved fact that the realized rows are exactly the nine listed. The library checks the whole argument with zero unproved assumptions and zero new axioms.

The theorem does not claim that physical stability has been derived from first principles. It does not say which property P is the true stability condition, nor does it prove that no other stable patterns exist outside the nine. Those remain open targets. What it does is make the conditional precise: if stability is exactly the realized set, then the nine-row list is closed under it, and no neighboring row can sneak in.

/-- **Reduction.** If a physical-stability predicate coincides with the canonical
realized image, then it has the exact charged image. This reduces the M2 closure to
proving the coincidence; the structural `complete` / `no_extra` content is already
discharged by `exactImage_realized`. -/
theorem exactImage_of_physicalStability
    {P : PatternTopology → Prop}
    (h : ExactImageFromPhysicalStability P) :
    ExactChargedTopologyImage P where
  complete := fun row hrow => (h (rowTopology row)).mpr ⟨row, hrow, rfl⟩
  no_extra := fun T hP => (h T).mp hP

/-- A physical admissibility predicate has the exact charged image if it contains
all candidate rows and contains no other charged topologies. This is the true
remaining M2 theorem, parameterized by the eventual primitive `P`. -/
structure ExactChargedTopologyImage
    (P : PatternTopology → Prop) : Prop where
  complete :
    ∀ row : ChargedMassRow, row ∈ chargedRows → P (rowTopology row)
  no_extra :
    ∀ T : PatternTopology, P T →
      ∃ row : ChargedMassRow, row ∈ chargedRows ∧ T = rowTopology row

/-- **Reduction.** If a physical-stability predicate coincides with the canonical
realized image, then it has the exact charged image. This reduces the M2 closure to
proving the coincidence; the structural `complete` / `no_extra` content is already
discharged by `exactImage_realized`. -/
theorem exactImage_of_physicalStability
    {P : PatternTopology → Prop}
    (h : ExactImageFromPhysicalStability P) :
    ExactChargedTopologyImage P where
  complete := fun row hrow => (h (rowTopology row)).mpr ⟨row, hrow, rfl⟩
  no_extra := fun T hP => (h T).mp hP

What this page does not claim

The theorem does not identify which property P is the correct stability predicate. The theorem does not prove that no other stable patterns exist; it only proves the conditional for a P that exactly matches the nine rows. The theorem does not derive the nine-row list itself; that list is supplied as a definition in the framework.

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

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