Gravity Seven Gaps Measure Invariance No Go Invariance Admits Infinite Measure F

The measure remains underdetermined

In the framework's account of gravity, a path-sum assigns a number to each possible configuration of space, and that number is meant to contribute to a total. A natural guess is that the weight of a configuration should depend only on its symmetries: two configurations that are relabelings of each other should get the same weight. The question is whether this relabeling invariance, together with basic positivity and normalization rules, forces a unique choice of weights.

The answer is no. A machine-checked theorem in the framework's library proves that relabeling invariance, strict positivity, per-configuration normalization, and unit weight on the empty configuration are satisfied by a countably infinite family of distinct weight functions. The family is built from powers of the symmetry-factor measure: for any positive integer n, the weight 1/|Aut|^(n+1) obeys all four rules, and different n values give different weights. The proof exhibits the separation concretely on a two-vertex configuration with no edges, whose automorphism group has size exactly 2, so the n=0 weight is 1/2 while the uniform weight is 1.

This is a no-go result, not a constructive one. It does not derive the symmetry-factor measure from deeper substrate axioms; that derivation remains open. It also does not include a gluing or factorization axiom, because the existing configuration machinery has no disjoint-union operation to state one against. A strictly richer axiom set could in principle restore uniqueness, and that possibility is the open frontier the theorem deliberately leaves untouched.

The practical consequence is that any claim that relabeling invariance alone fixes the path-sum measure is false within this framework. The measure is underdetermined by symmetry, and the theorem gives a concrete, checkable witness for that failure. The search for a unique measure must look beyond invariance, toward additional structure that the current axioms do not capture.

/-- **HEADLINE (strengthened form).**  The named invariance axioms admit a
countably INFINITE injective family of measures `1/|Aut|^(n+1)`: the
underdetermination is not a two-point accident. -/
theorem invariance_admits_infinite_measure_family (B : ℕ) (hB : 2 ≤ B) :
    (∀ n : ℕ, InvarianceAxioms B (muPowMeasure B n)) ∧
      Function.Injective (muPowMeasure B) :=
  ⟨muPowMeasure_satisfies B, muPowMeasure_injective B hB⟩

/-- **THEOREM.**  Every member of the countable family satisfies every
named invariance axiom. -/
theorem muPowMeasure_satisfies (B n : ℕ) :
    InvarianceAxioms B (muPowMeasure B n) :=
  ⟨fun K K' h => by
      show mu K ^ (n + 1) = mu K' ^ (n + 1)
      rw [mu_congr h],
    fun K => pow_pos (mu_pos K) (n + 1),
    fun K => pow_le_one₀ (mu_pos K).le (mu_le_one K),
    by
      show mu (emptyComplex B) ^ (n + 1) = 1
      rw [mu_emptyComplex B]
      norm_num⟩

/-- The countable family is injective: distinct exponents give distinct
measures (separated at the two-point witness where `mu = 1/2`). -/
theorem muPowMeasure_injective (B : ℕ) (hB : 2 ≤ B) :
    Function.Injective (muPowMeasure B) := by
  have hval : ∀ n : ℕ,
      muPowMeasure B n (twoPointComplex B hB) = (1 / 2 : ℝ) ^ (n + 1) := by
    intro n
    show mu (twoPointComplex B hB) ^ (n + 1) = (1 / 2 : ℝ) ^ (n + 1)
    rw [mu_twoPointComplex B hB]
  have hanti : StrictAnti (fun n : ℕ => ((1 : ℝ) / 2) ^ (n + 1)) := by
    intro a b hab
    exact pow_lt_pow_right_of_lt_one₀ (by norm_num) (by norm_num)
      (Nat.succ_lt_succ hab)
  intro n m h
  have h2 : ((1 : ℝ) / 2) ^ (n + 1) = ((1 : ℝ) / 2) ^ (m + 1) := by
    rw [← hval n, ← hval m, h]
  exact hanti.injective h2

/-- **THEOREM.**  `|Aut(twoPoint)| = 2` exactly (the identity and the
vertex swap). -/
theorem autCard_twoPointComplex (B : ℕ) (hB : 2 ≤ B) :
    Nat.card (Aut (twoPointComplex B hB)) = 2 := by
  rw [Nat.card_congr (twoPointAutEquiv B hB), Nat.card_eq_fintype_card,
    Fintype.card_prod, Fintype.card_prod,
    Fintype.card_equiv (Equiv.refl (Fin 2)),
    Fintype.card_equiv (Equiv.refl (Fin 0)),
    Fintype.card_fin 2, Fintype.card_fin 0]
  norm_num [Nat.factorial]

/-- **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

What this page does not claim

This does not derive the symmetry-factor measure from substrate axioms. This does not include a gluing or factorization axiom in the no-go scoping. This does not claim the continuum limit is affected by the underdetermination.

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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