Gravity Seven Gaps Metric Refinement Carrier Blocker No Class Only Cayley Menger

The metric obstruction

A recognition ledger, a discrete record of events, can be described in two ways: by its combinatorial type, which events connect to which, and by its metric geometry, the actual lengths and angles. In the framework's library of formal theorems, a machine-checked collection of proofs, a single combinatorial type can carry two different geometries. The declaration no_class_only_cayleyMenger_recovers_both proves that no function on the combinatorial type alone can recover the geometric observable for both decorations.

The proof is concrete. Take a single tetrahedron, the simplest simplicial complex. Decorate it with edge lengths all equal to 1, then decorate the same complex with edge lengths all equal to 2. Both decorations are valid, positive, nondegenerate geometries. The Cayley-Menger observable, a number computed from the edge lengths that determines the volume, is 4 for the unit decoration and 256 for the double decoration. The theorem states that any function from the combinatorial class to the real numbers must fail to match both values simultaneously, because 4 and 256 are not equal.

This obstruction is not a failure of the framework but a precise diagnosis. The existing quotient, TriangulationClass, identifies combinatorial types and therefore forgets metric information. The proof shows this forgetful map is not injective: two distinct metric-decorated complexes map to the same class. Consequently, the framework's library introduces a new carrier, MetricRefinementFamily, a structure with finite metric-decorated configuration spaces, a mesh tending to zero, and coarse projections, designed to carry the missing geometric data.

In Recognition Science, the framework models the path sum over geometries as a limit over this refinement family. The declaration establishes that the combinatorial quotient is insufficient for this purpose; the metric refinement family is proposed as the minimal carrier interface. What remains open is the construction of such a family from the recognition substrate itself, the derivation of its measure and action, and the proof of the continuum limit.

/-- 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

theorem unitDecoration_cayleyMenger :
    cayleyMengerObservable unitDecoration = 4 := by
  norm_num [cayleyMengerObservable, unitDecoration, cm3]

theorem doubleDecoration_cayleyMenger :
    cayleyMengerObservable doubleDecoration = 256 := by
  norm_num [cayleyMengerObservable, doubleDecoration, cm3]

/-- **CARRIER BLOCKER.** The current `TriangulationClass` quotient forgets
physical metric data: its forgetful map from decorated simplicial geometries
is not injective. -/
theorem metricForget_not_injective :
    ¬ Function.Injective
      (MetricDecoratedComplex.toClass :
        MetricDecoratedComplex 6 → TriangulationClass 6) := by
  intro hinj
  exact unitMetricOneTet_ne_doubleMetricOneTet
    (hinj unit_double_toClass_eq)

What this page does not claim

The declaration does not prove that the metric refinement family exists; it only proposes its structure as a model. The declaration does not claim that the continuum limit of the path sum converges. The declaration does not show that the combinatorial quotient is useless, only that it is insufficient for metric observables.

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)

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: