Gravity Seven Gaps Path Sum Measure Proved Count Le Structural Bound

A proved structural bound

A path sum in physics is a way of adding up contributions from all possible histories of a system, each weighted by a phase. In discrete gravity, the histories are combinatorial triangulations: networks of vertices, edges, and tetrahedra that approximate a curved spacetime. The Recognition Science framework's machine-checked library of formal theorems contains a proof about a specific class of these triangulations, called BoundedComplex, which are configurations with a fixed maximum size and equilateral edges at a minimum lattice scale.

The proof establishes that this class of configurations is finite: there are only finitely many ways to arrange vertices, edges, and tetrahedra within the size bound. This finiteness is proved by constructing an explicit bijection between the configurations and a finite set of codes, and the theorem instFintypeBoundedComplex records the result. Because the class is finite, the path sum over it is a finite sum, and the proof provides modulus bounds: the absolute value of the sum is at most the number of configurations, and also at most the sum of the symmetry factors. These bounds are proved, not assumed.

The proof also handles the symmetry of relabeling. Two configurations that differ only by renaming vertices, edges, and tetrahedra are considered equivalent, and the measure assigns each configuration a weight of 1 divided by the size of its automorphism group, the number of relabelings that leave it unchanged. The proof shows this weight is always between 0 and 1, and that it is invariant under relabeling, so the path sum respects the equivalence. This is the standard discrete-gravity convention of a labeled sum with a symmetry factor.

In Recognition Science, this result discharges part of an assumption. A family of admissible triangulations was previously assumed to have a finite growth base; the proof now supplies a finite cardinal for the scoped class, replacing a postulated bound with a proved one. However, the sharper exponential-growth semantics of that field for exact simplicial classes remains open, and the continuum limit of the path sum as the size bound goes to infinity is also open.

What the proof does not do is derive the measure from the recognition substrate. A prior claim that the measure could be derived from ledger information was retracted: the ledger state is pinned to zero, and the counting is label indifference, equivalent to a Gibbs weight on labeled configurations. The proof that the recognition-ledger cost function cannot supply this measure is recorded in Gap2LedgerSiteBlindness, which shows site-symmetry counts are cost-blind on two sites. The premise that a richer substrate structure derives the measure remains open, named but not discharged.

/-- **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 (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]

/-- **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]

What this page does not claim

The proof does not derive the measure from recognition-ledger information; that derivation remains open. The continuum limit of the path sum is not proved here. The scoped class contains all bounded triangulations but also non-simplicial configurations, so the finiteness result is for the broader class.

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)

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