Gravity Seven Gaps Cap Shell Bridge

Cap and shell bridge

In the Recognition Science framework, the study of gravity proceeds by counting discrete geometric objects called triangulations, which are built from vertices, edges, and tetrahedra. The framework's library works with two ways of organizing these objects. A bounded complex is a triangulation whose size, measured by a number called its complexity, is capped at some limit B. An exact shell is a triangulation whose size is exactly n, for some n, with no cap. The cap shell bridge is a machine-checked theorem that these two organizations are equivalent: every capped object corresponds to exactly one shell object, and vice versa, in a way that preserves all the relevant data.

The equivalence is not merely a statement about counts. The bridge constructs explicit maps in both directions. Given a bounded complex, it produces the unique shell whose complexity is the bounded complex's complexity, and it records the exact triangulation data. Given a shell, it reattaches the three proofs that the shell's size is within the cap, producing a bounded complex. The two maps are inverse to each other on the nose, meaning that applying one and then the other returns the original object. This is the kind of precise, structural correspondence that the framework's library is built to establish.

The bridge also preserves a key quantity: the number of symmetries of a triangulation. This matters because the framework's measure on triangulation classes is built from the inverse of the symmetry count, written 1/|Aut|. The theorem shows that this measure is identical whether it is computed on the capped side or on the shell side. Because of this, any phase, which is a real number assigned to each exact shell, can be transported to a phase model on the capped side without changing the physics.

The final result is a single equality. The sum over all capped triangulation classes of the phase-weighted measure equals the sum over all exact shells up to the cap of the same phase-weighted measure. The theorem, named capShellCompatibility, proves this equality for any choice of phase function. The library states plainly what this result does not assume: no target sum equality, no convergence statement, no substrate phase, and no physical continuum interpretation. The bridge is a formal fact about the discrete combinatorics, and it is a step toward connecting the discrete sums of the framework to a continuum limit.

/-- **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)

/-- **P2.3 CLOSER.**  Every exact-shell phase has a canonically transported
capped phase family satisfying the previously missing
`ZqContinuumBlocker.CapShellCompatibility`. -/
theorem capShellCompatibility
    (phase : ∀ n : ℕ, ExactPathClass n → ℝ) :
    CapShellCompatibility (capPhaseFamily phase) phase :=
  ⟨phasedZq_eq_exactComplexityCutoff phase⟩

What this page does not claim

The bridge does not assume or prove any convergence statement for the sums involved. The bridge does not assign a physical interpretation to the phase function. The bridge does not establish any connection to a physical continuum of spacetime.

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)

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