Gravity Seven Gaps Exact Shell Gauge Preflight Status Counting Principle Open

The open counting principle

In the Recognition Science framework's machine-checked library of formal theorems, a status flag named status_counting_principle_open records a precise state of knowledge: the counting principle that assigns weights to triangulations in a discrete gravity path sum is still an assumption, not a theorem derived from the framework's core cost function. The flag's value is false for the boolean field counting_principle_derived_from_ledger, meaning the derivation has not been achieved. This is an honest bookkeeping device: the framework's own ledger of what has been proved and what remains open.

The counting principle in question is the rule that each labeled configuration in a discrete gravity sum carries equal weight, so that the physical mass of an equivalence class of triangulations is the number of labeled copies divided by the gauge volume. The module derives the standard symmetry factor 1/|Aut K| from this principle using orbit-stabilizer counting: the number of relabeling witnesses between equivalent complexes equals the size of the automorphism group, and the total pair count factorizes accordingly. These are proved theorems. What is not proved is the counting principle itself. The module explicitly identifies it as a MODEL premise, and a separate theorem proves that the ledger cost function cannot supply it.

The flag also records a retraction. A previous claim that the counting principle had been derived from the ledger was flipped back to false on 2026-07-26. The theorem that seemed to support the derivation is true and remains in the library, but scrutiny showed its history carrier was equivalent to the plain complex carrier, its enrichment parameter had no fields, and its ledger state was pinned to zero. The derivation claim was withdrawn; the counting principle remains the named premise of the module, exactly as the paragraph above says.

What the flag does not claim is equally important. It does not claim the counting principle is false, only that it is not derived from the ledger. It does not claim the orbit-stabilizer theorems are unsound; they are proved with zero axioms beyond the standard three. It does not claim the counting principle is unprovable in principle; it remains a target, not an impossibility. The flag is a precise statement of what the framework knows and what it does not, and that precision is what makes the rest of the derivation trustworthy.

/-- OPEN (retracted 2026-07-26): the counting principle remains this
module's named premise and is not derived from the ledger. -/
theorem status_counting_principle_open :
    gaugePreflightStatus.counting_principle_derived_from_ledger = false := rfl

/-- **THEOREM (the derivation).**  The counting-defined class mass equals
the postulated symmetry-factor measure: GIVEN the pair-counting principle,
`1/|Aut|` follows from orbit-stabilizer (`pairCount = orbitCard * |Aut|`)
rather than being written into the definition. -/
theorem gaugeOrbitMass_eq_mu (K : BoundedComplex B) :
    gaugeOrbitMass (Quotient.mk (relabelSetoid B) K) = mu K := by
  have ho : (gaugeOrbitCard K : ℝ) ≠ 0 := by
    exact_mod_cast (gaugeOrbitCard_pos K).ne'
  unfold gaugeOrbitMass mu
  rw [orbitCardClass_mk, pairCountClass_mk, pairCount_eq_orbitCard_mul_autCard,
    Nat.cast_mul, div_mul_eq_div_div, div_self ho]

/-- **Grounding theorem.**  The status flags are backed by the actual
theorems: orbit-stabilizer, pair-count factorization, the derivation
`gaugeOrbitMass = mu`, and uniqueness. -/
theorem gaugePreflight_grounded (B : ℕ) :
    (∀ K K' : BoundedComplex B, Equivalent K K' →
        relabelingCount K K' = Nat.card (Aut K)) ∧
    (∀ K : BoundedComplex B,
        pairCount K = gaugeOrbitCard K * Nat.card (Aut K)) ∧
    (∀ K : BoundedComplex B,
        gaugeOrbitMass (Quotient.mk (relabelSetoid B) K) = mu K) ∧
    (∀ ν : TriangulationClass B → ℝ,
        (∀ c, ν c * (pairCountClass c : ℝ) = (orbitCardClass c : ℝ)) →
        ∀ c, ν c = gaugeOrbitMass c) :=
  ⟨fun _ _ h => relabelingCount_eq_autCard h,
   pairCount_eq_orbitCard_mul_autCard,
   gaugeOrbitMass_eq_mu,
   gaugeCountingMass_unique⟩

/-- OPEN (retracted 2026-07-26): the counting principle remains this
module's named premise and is not derived from the ledger. -/
theorem status_counting_principle_open :
    gaugePreflightStatus.counting_principle_derived_from_ledger = false := rfl

What this page does not claim

The counting principle is false or unsound. The orbit-stabilizer theorems are unproved or withdrawn. The counting principle is unprovable in principle from the ledger.

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/ExactShellGaugePreflight.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: