Gravity Seven Gaps Measure Invariance No Go Measure Invariance No Go Status Grou

A proved limit on symmetry

In the Recognition Science framework, a central question is how to assign a numerical weight to each possible configuration in a path-sum. The natural guess is the symmetry-factor measure, written 1/|Aut|, which counts how many symmetries a configuration has and weights it inversely. This measure has a plausible pedigree: it is positive, it never exceeds one, it treats relabeled copies of the same configuration equally, and it gives weight one to the empty configuration. The question is whether these four properties, taken together, force that measure and no other.

The answer, proved in a machine-checked library of formal theorems, is no. The declaration measureInvarianceNoGoStatus_grounded establishes that the symmetry-factor measure, the uniform weight that assigns one to every configuration, and the squared symmetry factor all satisfy the same four axioms. In fact, a countably infinite family of distinct measures does. The separation is concrete: on a configuration with two vertices and no edges, the symmetry factor gives 1/2 while the uniform weight gives 1. Relabeling invariance alone underdetermines the path-sum measure.

This is a no-go theorem, not a failure of the framework. It kills a specific positive claim, that invariance fixes 1/|Aut|, and it does so with explicit witnesses. What it does not do is close the subject. The theorem is scoped to the four named axioms; a richer axiom set, for example one that includes a gluing or factorization rule, could in principle restore uniqueness. That possibility remains open. The declaration also records that no substrate derivation of the measure exists yet, and that the continuum limit is untouched. The framework's honesty here is structural: the status record carries flags for what is proved, what is open, and what is not even stateable with the current machinery.

For a reader, the consequence is a precise boundary. Symmetry arguments can justify a measure's elegance, but they cannot justify its necessity. Any future derivation of the path-sum measure must bring in additional structure, and the theorem names the kind of structure that would be needed. The page therefore marks a genuine advance: a clean, kernel-checked negative result that redirects the search rather than ending it.

/-- **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⟩

/-- **HEADLINE (KILL + WITNESS).**  The named invariance axioms do NOT
determine the path-sum measure: the symmetry-factor measure `1/|Aut|`
and the uniform weight 1 BOTH satisfy every named axiom, yet they are
unequal, with the pointwise strict inequality exhibited at the concrete
two-point witness (where `|Aut| = 2`).  This is the kernel refutation of
the killed "mu-from-invariance" determination claim; the substrate
derivation of a unique measure remains OPEN. -/
theorem mu_not_determined_by_invariance (B : ℕ) (hB : 2 ≤ B) :
    InvarianceAxioms B (muMeasure B) ∧
    InvarianceAxioms B (uniformMeasure B) ∧
    muMeasure B ≠ uniformMeasure B ∧
    muMeasure B (twoPointComplex B hB) <
      uniformMeasure B (twoPointComplex B hB) :=
  ⟨muMeasure_satisfies B, uniformMeasure_satisfies B,
    muMeasure_ne_uniformMeasure B hB,
    muMeasure_lt_uniform_at_witness B hB⟩

/-- **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.**  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

The theorem does not prove that the symmetry-factor measure is the unique measure satisfying the axioms. The theorem does not derive the 1/|Aut| measure from recognition-ledger substrate axioms. The theorem does not address the continuum limit of the path-sum measure.

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)

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.

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