Gravity Seven Gaps Path Sum Measure Proved Family Growth Base Derived

A proved growth base

A growth base is a number that bounds how many configurations can appear at a given size. In a discrete model of gravity, configurations are combinatorial triangulations, shapes built from vertices, edges, and tetrahedra. The framework's library, a machine-checked collection of formal theorems, had assumed such a bound existed. The declaration provedFamily_growthBase_derived replaces that assumption with a proved equality: the growth base for the scoped family equals the finite cardinality of the configuration class at each bound B.

The proof works by showing the configuration class is finite. The class BoundedComplex carries arbitrary bounded incidence data, so it contains all bounded triangulations but also non-simplicial configurations. Finiteness of this superclass gives finiteness of every subclass. The theorem is a ledger, a discrete record of events, of what was previously a postulate: a proved finite cardinal where a bound was assumed.

What the theorem does not claim is the sharper exponential-growth semantics of the growth base for exact simplicial classes. That remains open. The theorem also does not derive the continuum limit of the path sum as the bound B tends to infinity; that limit is recorded as open. Nor does it derive the substrate measure, which was retracted after a witness theorem turned out to be a presentation of the same ratio, not a derivation.

The practical consequence is that the scoped path-sum measure is now built on a proved finite foundation, not a postulated one. This matters because the measure, a sum over configurations with a symmetry factor, is only well-defined if the sum is finite. The proof supplies that finiteness, and the bounds on the sum's modulus follow. What a reader can now rely on is a finite, well-defined sum for the scoped class, with the sharper growth and continuum questions honestly marked as open.

/-- The growth base of the proved family is the derived cardinal. -/
theorem provedFamily_growthBase_derived (B : ℕ) :
    (provedFamily B).growthBase = (Fintype.card (BoundedComplex B) : ℝ) := rfl

/-- **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

What this page does not claim

The theorem does not prove exponential growth of simplicial classes. The theorem does not derive the continuum limit of the path sum. The theorem does not derive the substrate measure from ledger structure.

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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Derived articles

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