Gravity Seven Gaps Path Sum Measure Class Count Le Labeled Count

Counting configurations

In the discrete approach to gravity, spacetime is assembled from simple building blocks: triangles glued edge to edge, or tetrahedra glued face to face. A central question is how many distinct shapes can be built from a given number of blocks. In the Recognition Science framework, a machine-checked library of formal theorems proves a precise answer for the scoped configuration class BoundedComplex, a finite set of combinatorial triangulations at a fixed lattice scale. The declaration classCount_le_labeledCount establishes that the number of equivalence classes under relabeling is at most the number of labeled configurations.

Relabeling is the discrete analogue of renaming the vertices, edges, and tetrahedra of a complex. Two configurations that differ only by such a renaming are physically the same shape. The framework proves that relabeling is a genuine equivalence relation, with reflexivity, symmetry, and transitivity all verified, and that the quotient set of equivalence classes is finite. The labeled count is the raw number of configurations before any identification, and the class count is the number of distinct shapes after identifying relabelings. The inequality is a simple consequence of the fact that a quotient set cannot have more elements than the set it partitions.

The proof proceeds by showing that the scoped configuration class is finite via an explicit bijection to a code type, a tuple of natural numbers and functions that encodes all incidence data. This finiteness result discharges a count-finiteness assumption that was previously postulated in the framework's growth base. The sharper exponential-growth semantics of that field for exact simplicial classes remains open, but the finiteness content is now a proved theorem, not an assumed one.

In Recognition Science, this count bound is a stepping stone for the path-sum measure. The framework defines a measure on the labeled class by weighting each configuration by the reciprocal of the size of its automorphism group, the number of relabelings that map the configuration to itself. This symmetry factor is the standard discrete-gravity convention, and the framework proves it is positive, at most one, and invariant under relabeling. The path sum itself is a finite sum over the labeled class, and the count bound gives a modulus bound on that sum.

What the declaration does not claim is as important as what it proves. It does not establish that the quotient sum equals the weighted labeled sum; that orbit-counting identity is not proved here. It does not derive the continuum limit of the path sum as the size bound goes to infinity; that limit remains open. And it does not derive the measure from the recognition substrate itself; a named premise, the substrate measure derivation, remains open after a retraction. The count bound is a discrete, finite, combinatorial fact, and it is proved.

/-- **THEOREM (quotient finiteness).**  The set of combinatorially distinct
bounded triangulations is finite. -/
theorem triangulationClass_finite (B : ℕ) : Finite (TriangulationClass B) :=
  Quotient.finite _

/-- **THEOREM (count-finiteness).**  The scoped class of bounded
combinatorial triangulations is a finite type.  This is the proved
replacement for the assumed `growthBase` field. -/
instance instFintypeBoundedComplex (B : ℕ) : Fintype (BoundedComplex B) :=
  Fintype.ofEquiv (CodeType B) (codeEquiv B).symm

/-- **THEOREM (measure respects the equivalence).**  μ is a relabeling
invariant: equivalent complexes have equal measure. -/
theorem mu_congr {B : ℕ} {K K' : BoundedComplex B} (h : Equivalent K K') :
    mu K = mu K' := by
  obtain ⟨r⟩ := h
  unfold mu
  rw [Nat.card_congr r.autCongr]

/-- **THEOREM (UV-finiteness bound).**  For any weight of modulus at most 1,
`‖Z‖ ≤ card (BoundedComplex B)`: the path sum is finite with an explicit
proved bound, replacing the assumed `growthBase^N` count. -/
theorem Z_norm_le_card (B : ℕ) (w : BoundedComplex B → ℂ)
    (hw : ∀ K, ‖w K‖ ≤ 1) :
    ‖Z B w‖ ≤ (Fintype.card (BoundedComplex B) : ℝ) := by
  unfold Z
  calc ‖∑ K : BoundedComplex B, (mu K : ℂ) * w K‖
      ≤ ∑ K : BoundedComplex B, ‖(mu K : ℂ) * w K‖ := norm_sum_le _ _
    _ ≤ ∑ _K : BoundedComplex B, (1 : ℝ) := by
        refine Finset.sum_le_sum fun K _ => ?_
        rw [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos (mu_pos K)]
        calc mu K * ‖w K‖
            ≤ 1 * 1 := mul_le_mul (mu_le_one K) (hw K) (norm_nonneg _) zero_le_one
          _ = 1 := one_mul 1
    _ = (Fintype.card (BoundedComplex B) : ℝ) := by
        rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, mul_one]

What this page does not claim

The orbit-counting identity equating the weighted labeled sum with a quotient sum is not proved. The continuum limit of the path sum as B goes to infinity is not derived. The measure is not derived from the recognition substrate; the substrate measure derivation remains open.

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/PathSumMeasure.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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