Gravity Seven Gaps Measure Invariance No Go Mu Measure Lt Uniform At Witness

The witness that ends a claim

The declaration muMeasure_lt_uniform_at_witness is a machine-checked theorem in the framework's library of formal theorems. It states that on a specific configuration, the symmetry-factor measure assigns a strictly smaller weight than the uniform measure does. The configuration in question has two vertices and no edges or tetrahedra. Its automorphism group, the set of relabelings that preserve the structure, has exactly two elements: the identity and the swap of the two vertices. The symmetry-factor measure, which assigns to a configuration the reciprocal of the size of its automorphism group, therefore gives this two-point configuration weight 1/2. The uniform measure, which assigns weight 1 to every configuration, gives it weight 1. The theorem records the inequality 1/2 < 1 at this concrete witness.

The theorem is the sharp edge of a broader no-go result. A prior attempt in the framework had claimed that relabeling invariance, together with positivity and normalization requirements, singles out the symmetry-factor measure as the unique weight on the scoped path-sum configuration class. That positive claim is dead. The library proves that at least three genuinely different weight functions satisfy all the named axioms: the symmetry-factor measure, the uniform measure, and the squared symmetry factor. In fact, a countably infinite injective family of such measures exists. The two-point configuration is the explicit witness that separates the symmetry-factor measure from the uniform one. The theorem mu_not_determined_by_invariance packages this: both measures satisfy the axioms, they are unequal, and the inequality is exhibited pointwise.

The no-go is scoped to the axioms actually named. The named set includes relabeling invariance, strict positivity, per-configuration normalization, and unit weight on the canonical empty configuration. It does not include a disjoint-union or gluing factorization axiom, because the existing machinery carries no disjoint-union operation to state it against. A strictly richer axiom set could in principle restore uniqueness; that possibility is the open substrate-derivation frontier. The library's status record flags this: substrate_measure_derived is false, and the continuum limit is also open.

What the theorem does not claim is as important as what it proves. It does not resurrect the positive claim that invariance fixes the symmetry-factor measure; it refutes exactly that determination claim. It does not derive the symmetry-factor measure from recognition-ledger substrate axioms; no named substrate axiom set in the existing library forces a unique measure. The theorem is a no-go, not a derivation. It closes one door and leaves the search for a richer set of axioms open.

/-- **THEOREM.**  The symmetry-factor measure of the two-point witness is
exactly `1/2`. -/
theorem mu_twoPointComplex (B : ℕ) (hB : 2 ≤ B) :
    mu (twoPointComplex B hB) = 1 / 2 := by
  unfold mu
  rw [autCard_twoPointComplex B hB]
  norm_num

/-- **Pointwise strict separation.**  At the two-point witness the
symmetry-factor measure is strictly below the uniform weight:
`1/2 < 1`. -/
theorem muMeasure_lt_uniform_at_witness (B : ℕ) (hB : 2 ≤ B) :
    muMeasure B (twoPointComplex B hB) <
      uniformMeasure B (twoPointComplex B hB) := by
  show mu (twoPointComplex B hB) < 1
  rw [mu_twoPointComplex B hB]
  norm_num

/-- **THEOREM.**  `1/|Aut|` satisfies every named invariance axiom. -/
theorem muMeasure_satisfies (B : ℕ) : InvarianceAxioms B (muMeasure B) :=
  ⟨fun _ _ h => mu_congr h, fun K => mu_pos K, fun K => mu_le_one K,
    mu_emptyComplex B⟩

/-- **THEOREM.**  The uniform weight 1 satisfies every named invariance
axiom. -/
theorem uniformMeasure_satisfies (B : ℕ) :
    InvarianceAxioms B (uniformMeasure B) :=
  ⟨fun _ _ _ => rfl, fun _ => one_pos, fun _ => le_refl 1, rfl⟩

/-- **THEOREM.**  `1/|Aut|^2` satisfies every named invariance axiom. -/
theorem muSqMeasure_satisfies (B : ℕ) : InvarianceAxioms B (muSqMeasure B) :=
  ⟨fun K K' h => by
      show mu K ^ 2 = mu K' ^ 2
      rw [mu_congr h],
    fun K => pow_pos (mu_pos K) 2,
    fun K => pow_le_one₀ (mu_pos K).le (mu_le_one K),
    by
      show mu (emptyComplex B) ^ 2 = 1
      rw [mu_emptyComplex B]
      norm_num⟩

/-- **Grounding theorem.**  Every `true` status flag is tied to a kernel
statement; the RED flags remain false. -/
theorem measureInvarianceNoGoStatus_grounded :
    (measureInvarianceNoGoStatus.named_axioms_stated = true ∧
      ∀ B : ℕ, ∃ w : BoundedComplex B → ℝ, InvarianceAxioms B w) ∧
    (measureInvarianceNoGoStatus.mu_satisfies_axioms = true ∧
      ∀ B : ℕ, InvarianceAxioms B (muMeasure B)) ∧
    (measureInvarianceNoGoStatus.uniform_satisfies_axioms = true ∧
      ∀ B : ℕ, InvarianceAxioms B (uniformMeasure B)) ∧
    (measureInvarianceNoGoStatus.measures_separated = true ∧
      ∀ B : ℕ, 2 ≤ B → muMeasure B ≠ uniformMeasure B) ∧
    (measureInvarianceNoGoStatus.infinite_family_exhibited = true ∧
      (∀ B n : ℕ, InvarianceAxioms B (muPowMeasure B n)) ∧
      ∀ B : ℕ, 2 ≤ B → Function.Injective (muPowMeasure B)) ∧
    measureInvarianceNoGoStatus.factorization_axiom_stateable = false ∧
    measureInvarianceNoGoStatus.substrate_measure_derived = false ∧
    measureInvarianceNoGoStatus.Z_RS_continuum_limit = false :=
  ⟨⟨rfl, fun B => ⟨uniformMeasure B, uniformMeasure_satisfies B⟩⟩,
    ⟨rfl, muMeasure_satisfies⟩,
    ⟨rfl, uniformMeasure_satisfies⟩,
    ⟨rfl, muMeasure_ne_uniformMeasure⟩,
    ⟨rfl, fun B n => muPowMeasure_satisfies B n, muPowMeasure_injective⟩,
    rfl, rfl, rfl⟩

What this page does not claim

This theorem does not derive the symmetry-factor measure from recognition-ledger substrate axioms. This theorem does not claim that invariance alone fixes a unique measure; it refutes that claim. This theorem does not address the continuum limit, which 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/MeasureInvarianceNoGo.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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