Gravity Seven Gaps Metric Refinement Carrier Blocker Unit Decoration First Edge

The first edge's length

In the geometry of simplicial complexes, a tetrahedron is the four-triangle solid whose faces are triangles. The declaration unitDecoration_firstEdgeLength states, in the framework's machine-checked library of formal theorems, that for a particular tetrahedron given all edge lengths equal to 1, the length of its first edge is exactly 1. This is a definitional fact: the edge length is the square root of the squared edge length, and the squared edge length was set to 1.

The fact matters because it is half of a recognition obstruction, a proof that a certain kind of bookkeeping cannot work. The framework's library shows that two different metric decorations of the same tetrahedron, one with all edge lengths 1 and one with all edge lengths 2, both map to the same combinatorial class under the quotient that identifies shapes up to relabeling. Yet their first edge lengths differ, 1 versus 2, and their Cayley-Menger observables, a volume-like quantity computable from edge lengths, differ as well, 4 versus 256. No function on the combinatorial class alone can recover both edge length and observable for both decorations, because the class does not carry metric information.

This is a proved theorem in the framework's library, not a conjecture or a model choice. The theorem no_class_only_mesh_recovers_both states that no function from the quotient class to the real numbers can equal both 1 and 2 at the same class. The theorem no_class_only_cayleyMenger_recovers_both states the same for the observable. Together they establish that the forgetful map from metric-decorated complexes to the quotient is not injective: two distinct geometric objects collapse to the same combinatorial type.

The declaration does not claim that the framework has constructed a metric refinement family from its recognition substrate, nor that it has derived a measure or action for such a family, nor that the geometric continuum limit exists. Those remain open targets. What the declaration establishes is a precise, kernel-checked obstruction: the current quotient of combinatorial types is too coarse to carry metric geometry, and any proposed carrier for the geometric path sum must add metric data beyond the class. The first edge length being 1 is the concrete witness that makes the obstruction visible.

theorem unitDecoration_firstEdgeLength :
    firstEdgeLength unitDecoration = 1 := by
  norm_num [firstEdgeLength, unitDecoration]

theorem unit_double_toClass_eq :
    unitMetricOneTet.toClass = doubleMetricOneTet.toClass := rfl

/-- **P2.5 MESH BLOCKER.** No function of the current combinatorial quotient
class alone can recover the first-edge length of both admissible metric
decorations. -/
theorem no_class_only_mesh_recovers_both
    (mesh : TriangulationClass 6 → ℝ) :
    ¬ (mesh oneTetClass = firstEdgeLength unitDecoration ∧
      mesh oneTetClass = firstEdgeLength doubleDecoration) := by
  rw [unitDecoration_firstEdgeLength, doubleDecoration_firstEdgeLength]
  rintro ⟨h₁, h₂⟩
  linarith

/-- The same obstruction holds for an action-relevant Cayley-Menger
observable, not only for a chosen edge coordinate. -/
theorem no_class_only_cayleyMenger_recovers_both
    (observable : TriangulationClass 6 → ℝ) :
    ¬ (observable oneTetClass = cayleyMengerObservable unitDecoration ∧
      observable oneTetClass = cayleyMengerObservable doubleDecoration) := by
  rw [unitDecoration_cayleyMenger, doubleDecoration_cayleyMenger]
  rintro ⟨h₁, h₂⟩
  linarith

What this page does not claim

The framework has constructed a metric refinement family from its recognition substrate. The framework has derived a measure or action for such a family. The geometric continuum limit of the path sum has been proved to exist.

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/MetricRefinementCarrierBlocker.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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