Gravity Seven Gaps Cap Shell Bridge Phased Zq Eq Exact Complexity Cutoff

The bridge at the cap

In the Recognition Science framework, a ledger is a discrete record of events, and one of its central objects is a bounded complex: a finite collection of vertices, edges, and tetrahedra whose size is controlled by a number called the cap, B. Each such complex has a unique exact complexity, the largest of three counts (vertices, edges, tetrahedra), and this complexity is always at most B. The declaration phasedZq_eq_exactComplexityCutoff proves that a sum over all bounded complexes up to cap B, weighted by a phase, equals a sum over exact complexes in shells up to B, with the shell range running to B + 1. In plain terms, two different ways of organizing the same geometric objects give the same weighted total.

The proof works by constructing an explicit equivalence between the two collections. Every bounded complex maps to an exact complex in a shell, and every exact complex in a shell maps back to a bounded complex. These maps preserve the incidence data (which vertices belong to which edges and tetrahedra) and the relabeling witnesses, so they descend to the quotient carriers, the classes of complexes up to relabeling. The two maps are inverse to each other, giving a bijection between the two sets of classes. The bridge also preserves the automorphism cardinality, meaning the size of the symmetry group of each complex, and therefore preserves the 1 / |Aut| class measure used in the weighted sum.

The theorem is stated for an arbitrary phase, a real number assigned to each exact complex in a shell. The bridge transports any such phase to a phase model at every cap, and the equality then follows by reindexing the finite quotient sum along the carrier equivalence. The result is a compatibility statement: the capped-quotient construction and the exact-shell construction agree at every finite level, not just in some limit. This is what the declaration establishes, and it is a theorem in the machine-checked library of formal theorems, with no unproved assumptions beyond the standard axioms of the ambient type theory.

The declaration does not claim more than this. It does not assert any equality of infinite sums, any convergence statement, any substrate phase, or any physical continuum interpretation. It is a statement about finite combinatorial objects and their weighted sums, not about the physical meaning of those sums. The bridge is a structural fact about the framework's own constructions, and it is a prerequisite for further work, not a conclusion about the physical world.

/-- **HEADLINE FINITE-SUM REINDEXING.**  The phased capped quotient sum
transported from any exact-shell phase equals its exact-shell cutoff through
shell `B`. -/
theorem phasedZq_eq_exactComplexityCutoff
    (phase : ∀ n : ℕ, ExactPathClass n → ℝ) (B : ℕ) :
    phasedZqSequence (capPhaseFamily phase) B =
      exactComplexityCutoff phase B := by
  unfold phasedZqSequence
  change Zq B (phasedWeight (phaseModelAtCap phase B)) =
    exactComplexityCutoff phase B
  rw [Zq]
  calc
    (∑ q : TriangulationClass B,
      (mu (Quotient.out q) : ℂ) *
        phasedWeight (phaseModelAtCap phase B) q)
        = ∑ x : ShellsUpTo B,
            (classMu x.2 : ℂ) *
              Complex.exp (Complex.I * (phase x.1 x.2 : ℂ)) := by
            apply Fintype.sum_equiv (capShellEquiv B)
            intro q
            change (mu (Quotient.out q) : ℂ) *
                phasedWeight (phaseModelAtCap phase B) q =
              (classMu (capToShell q).2 : ℂ) *
                Complex.exp (Complex.I * (phase (capToShell q).1 (capToShell q).2 : ℂ))
            rw [classMu_capToShell]
            simp only [phasedWeight, classPhase_phaseModelAtCap]
    _ = exactComplexityCutoff phase B :=
      sum_shellsUpTo_eq_exactComplexityCutoff phase B

/-- **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)

/-- The transported phase model at every cap. -/
def capPhaseFamily
    (phase : ∀ n : ℕ, ExactPathClass n → ℝ) : CapPhaseFamily :=
  fun B => phaseModelAtCap phase B

What this page does not claim

No equality of infinite sums or convergence statement is asserted. No substrate phase or physical continuum interpretation is assumed. The declaration does not assign physical meaning to the phase or the weighted sum.

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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