Gravity Seven Gaps Exact Shell Gauge Preflight Pair Count Eq Orbit Card Mul Aut

The pair count identity

In the discrete gravity models studied here, space is built from labeled complexes: collections of vertices, edges, and triangles with explicit labels on each piece. Two complexes are considered equivalent if one can be relabeled to become the other. A key quantity is the pair count, the number of pairs consisting of a complex in a given equivalence class and a specific relabeling that connects it to a chosen representative. The theorem pairCount_eq_orbitCard_mul_autCard states that this pair count equals the size of the equivalence class (the orbit) multiplied by the number of symmetries of the representative complex (its automorphism group).

This identity is a direct consequence of the orbit-stabilizer theorem from group theory, applied to the relabeling action. For any two equivalent complexes, the set of relabelings between them forms a torsor over the automorphism group of the first: pick one relabeling, and any other is obtained by composing with a unique symmetry. Therefore, the number of relabelings between any two equivalent complexes equals the number of symmetries of either one. Summing this over all complexes in the orbit gives the factorization. The proof is formalized in the machine-checked library of formal theorems, with no axioms beyond the standard logical ones.

The identity becomes physically meaningful when combined with a counting principle: that each labeled configuration carries equal weight in the path sum. Under that assumption, the statistical weight of an entire equivalence class is the number of labeled copies divided by the pair count. The theorem shows this ratio equals the familiar symmetry factor 1 divided by the size of the automorphism group, the same factor used in standard discrete gravity. The counting principle itself is a model choice, not a theorem; it is the premise that uniform gauge density on labeled representatives is the correct physical measure.

What the identity does not do is derive that counting principle from deeper physics. The principle that each labeled configuration is equally likely remains an explicit premise, flagged as such in the framework. A different principle, such as weighting each labeled copy equally, would give a different measure. The theorem also does not compute the pair count in closed form for any specific complex; that requires additional structure about the size of the relabeling group, which is a separate result. What the identity provides is a clean bridge from a symmetry count to a statistical weight, making the standard 1 over automorphism factor a consequence of orbit-stabilizer rather than an independent postulate.

/-- **THEOREM (pair-count factorization).**  The gauge volume of the orbit
is the orbit size times the automorphism count: every fiber of the
projection `(K', r) ↦ K'` is a torsor over `Aut K`. -/
theorem pairCount_eq_orbitCard_mul_autCard (K : BoundedComplex B) :
    pairCount K = gaugeOrbitCard K * Nat.card (Aut K) := by
  have e : (Σ K' : {K' : BoundedComplex B // Equivalent K K'}, Relabel K K'.val)
      ≃ {K' : BoundedComplex B // Equivalent K K'} × Aut K :=
    Equiv.sigmaEquivProdOfEquiv fun K' =>
      (torsorEquiv (Classical.choice K'.property)).symm
  unfold pairCount gaugeOrbitCard
  rw [Nat.card_congr e, Nat.card_prod]

/-- **THEOREM (torsor).**  Fixing one witness `r0 : Relabel K K'`, the map
`a ↦ a.trans r0` is a bijection `Aut K ≃ Relabel K K'`: the witnesses
between equivalent complexes are a torsor over the automorphism group. -/
def torsorEquiv {K K' : BoundedComplex B} (r0 : Relabel K K') :
    Aut K ≃ Relabel K K' where
  toFun a := a.trans r0
  invFun r := r.trans r0.symm
  left_inv a := by
    apply Relabel.ext <;>
      · apply Equiv.ext
        intro x
        simp only [Relabel.trans_vEquiv, Relabel.trans_eEquiv, Relabel.trans_tEquiv,
          Relabel.symm_vEquiv, Relabel.symm_eEquiv, Relabel.symm_tEquiv,
          Equiv.trans_apply, Equiv.symm_apply_apply]
  right_inv r := by
    apply Relabel.ext <;>
      · apply Equiv.ext
        intro x
        simp only [Relabel.trans_vEquiv, Relabel.trans_eEquiv, Relabel.trans_tEquiv,
          Relabel.symm_vEquiv, Relabel.symm_eEquiv, Relabel.symm_tEquiv,
          Equiv.trans_apply, Equiv.apply_symm_apply]

/-- **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]

What this page does not claim

This identity does not derive the counting principle of uniform gauge density from the ledger cost function or any other deeper principle. It does not provide a closed-form numerical value for the pair count of any specific complex. It does not establish that the measure is unique without assuming the counting principle.

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)

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