Gravity Seven Gaps Cap Shell Bridge Class Phase Phase Model At Cap

The carrier bridge

In the Recognition Science framework, a triangulation is a way of building a space from vertices, edges, and tetrahedra. The framework's machine-checked library of formal theorems works with two ways of organizing these triangulations. One way, the cap, limits the total complexity of a triangulation by a single number B. The other way, the shell, groups triangulations by their exact complexity n, where n is at most B. The declaration classPhase_phaseModelAtCap proves that these two organizations agree on the phases they assign.

A phase is a real number attached to a triangulation class, a way of treating triangulations that differ only by relabeling as the same object. The theorem states that the phase assigned by the capped model at cap B equals the phase assigned by the exact shell model, for every triangulation class q. The proof works by showing the two carrier sets, the capped quotients and the exact-shell carriers, are equivalent as sets, and that this equivalence preserves the phase values and the automorphism counts that define the class measure.

This agreement is not a physical claim. The theorem does not assert that any particular phase value is correct, that any sum converges, or that the phases correspond to anything in the physical continuum. It establishes a structural identity between two bookkeeping schemes inside the framework's formal system. The result matters because it lets the framework move between its cap-based and shell-based descriptions without changing the underlying assignments, a consistency property that later work can build on.

/-- The transported phase descends to exactly the original phase after the
carrier equivalence. -/
theorem classPhase_phaseModelAtCap
    (phase : ∀ n : ℕ, ExactPathClass n → ℝ) {B : ℕ}
    (q : TriangulationClass B) :
    classPhase (phaseModelAtCap phase B) q =
      phase (capToShell q).1 (capToShell q).2 := by
  refine Quotient.inductionOn q ?_
  intro K
  rfl

/-- **HEADLINE CARRIER EQUIVALENCE.**  Bounded quotient classes at cap `B`
are exactly the disjoint union of exact quotient shells `0, ..., B`. -/
def capShellEquiv (B : ℕ) : TriangulationClass B ≃ ShellsUpTo B where
  toFun := capToShell
  invFun := shellToCap
  left_inv := shellToCap_capToShell
  right_inv := capToShell_shellToCap

/-- **HEADLINE CARRIER EQUIVALENCE.**  Bounded quotient classes at cap `B`
are exactly the disjoint union of exact quotient shells `0, ..., B`. -/
def capShellEquiv (B : ℕ) : TriangulationClass B ≃ ShellsUpTo B where
  toFun := capToShell
  invFun := shellToCap
  left_inv := shellToCap_capToShell
  right_inv := capToShell_shellToCap

/-- **AUTOMORPHISM-CARDINALITY PRESERVATION.** -/
theorem autCard_toExact {B : ℕ} (K : BoundedComplex B) :
    Nat.card (Aut K) = Nat.card (ExactAut (toExact K)) :=
  Nat.card_congr (autEquivToExact K)

What this page does not claim

No claim that any specific phase value is physically correct. No claim that the sums involved converge or have a physical continuum interpretation. No claim that the theorem applies outside the framework's formal system.

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/Gravity/SevenGaps/CapShellBridge.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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