Encyclopedia Gravity Gravity Admissible Triangulation Procedure Rs Admissible Witness

ARTICLE 5 claims 2 theorems 3 models

Gravity Admissible Triangulation Procedure Rs Admissible Witness

A machine-checked library proves that at least one concrete triangulation family meets every Recognition Science admissibility condition, including the assumed physical bridge that links recognition ratios to geometry.

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The admissible witness

In Recognition Science, a ledger (a discrete record of recognition events) must be built on a triangulation of space. The framework's machine-checked library of formal theorems now contains a specific, concrete example of such a triangulation family, named rsAdmissibleWitness. This witness is a mathematical object, not a physical simulation. It exists to prove a point: the framework's conditions for an admissible triangulation are not empty. At least one family satisfies them all.

The witness family is defined by simple, explicit constants. It allows at most one simplex, uses a growth base of 2, and a minimum mesh size of 1. These choices are not physically motivated; they are chosen so that every numeric condition can be verified by direct computation. The declaration establishes that this family is indeed admissible. Three of the four conditions are derived from the family's own structure: the minimum mesh size is positive, the simplex count is finite, and the growth base is positive. These are not assumptions; they are consequences of the definition.

The fourth condition is different. It is a bridge, a physical hypothesis connecting the framework's internal recognition ratios to the geometry of deficit angles. The bridge states that the logarithm of a recognition ratio equals a constant times the deficit angle, plus an error term that shrinks with the cube of the mesh size. This is not derived from Recognition Science axioms. It is an explicit assumption, tagged as such in the library. The witness satisfies this bridge exactly, with the constant set to 1 and the error term set to 0, using the exponential function as the recognition ratio. This makes the bridge hold perfectly for this family, but it does not make the bridge true in general.

The existence of this witness is a theorem, proved in the library. A second theorem shows that if a family is admissible with a certain error constant, it remains admissible with any larger error constant. This is a closure property: the set of admissible error bounds is upward-closed. What the witness does not do is justify the bridge itself. The bridge remains a hypothesis, a physical assumption that must be tested against reality. The witness only demonstrates that the framework's admissibility conditions are internally consistent, not that they describe the actual geometry of spacetime.

MODEL rsAdmissibleWitness · IndisputableMonolith/Gravity/AdmissibleTriangulationProcedure.lean
rsAdmissibleWitness · IndisputableMonolith/Gravity/AdmissibleTriangulationProcedure.lean:76 
/-- A concrete RS-admissible triangulation family witness.

Fields chosen with explicit simple constants so every numeric side-goal
closes by `norm_num` or `positivity`:
- maxSimplexCount := 1
- growthBase := 2
- minMesh := 1
- meshUpperBound := 2 (so minMesh ≤ meshUpperBound is `by norm_num`)
- kappa := 1
- bridge_constant := 0 (exact bridge, no error)
- recognitionRatio := exp (so log x = deficit exactly) -/
def rsAdmissibleWitness : AdmissibleTriangulationFamily where
  maxSimplexCount := 1
  maxSimplexCount_pos := by norm_num
  growthBase := 2
  growthBase_pos := by norm_num
  minMesh := 1
  minMesh_pos := by norm_num
THEOREM exists_RSAdmissible · IndisputableMonolith/Gravity/AdmissibleTriangulationProcedure.lean
exists_RSAdmissible · IndisputableMonolith/Gravity/AdmissibleTriangulationProcedure.lean:95 
/-- The witness family is RS-admissible.

Proof: all derived conditions follow from the witness fields.
The bridge holds with κ = 1, C = 0, x(deficit) = exp(deficit), giving
|log(exp(deficit)) - 1·deficit| = |deficit - deficit| = 0 ≤ 0 · 1³ = 0. -/
theorem exists_RSAdmissible : Nonempty (IsRSAdmissible rsAdmissibleWitness) :=
  ⟨{
    mesh_lower_bound := rsAdmissibleWitness.minMesh_pos
    simplex_count_finite := rsAdmissibleWitness.maxSimplexCount_pos
    growth_base_pos := rsAdmissibleWitness.growthBase_pos
    meshUpperBound := 2
    mesh_upper_bound_ge := by
      have h : rsAdmissibleWitness.minMesh = (1 : ℝ) := rfl
      rw [h]; norm_num
    kappa := 1
    kappa_pos := by norm_num
    bridge_constant := 0
    bridge_constant_nonneg := by norm_num
    recognitionRatio := fun δ => Real.exp δ
    recognitionRatio_pos := fun δ _ => Real.exp_pos δ
    bridge_holds := by
      intro deficit _
      rw [Real.log_exp]
      have h : deficit - (1 : ℝ) * deficit = 0 := by ring
      rw [h]
      simp only [abs_zero, zero_mul]
      norm_num
  }⟩
MODEL IsRSAdmissible · IndisputableMonolith/Gravity/AdmissibleTriangulationProcedure.lean
IsRSAdmissible · IndisputableMonolith/Gravity/AdmissibleTriangulationProcedure.lean:33 
/-- The RS admissibility predicate for triangulation families.

This predicate records the admissibility conditions that a triangulation
family must satisfy to be used in the recognition path sum:

1. **Positive mesh lower bound** (derived): minMesh > 0
2. **Finite simplex-count cap** (derived): maxSimplexCount > 0
3. **Positive growth base** (derived): growthBase > 0
4. **Recognition-ratio bridge** (ASSUMED): log x_σ = κ · deficit + O(mesh³)

The bridge condition is tagged as ASSUMED: it is a physical hypothesis
bridging recognition ratios to deficit angles, not derived from RS axioms. -/
structure IsRSAdmissible (F : AdmissibleTriangulationFamily) where
  /-- Mesh lower bound: minMesh > 0 (derived from F.minMesh_pos). -/
  mesh_lower_bound : 0 < F.minMesh
  /-- Finite simplex-count cap: maxSimplexCount > 0 (derived). -/
  simplex_count_finite : 0 < F.maxSimplexCount
  /-- Positive growth base: growthBase > 0 (derived). -/
  growth_base_pos : 0 < F.growthBase
  /-- Mesh upper bound for admissibility comparison. -/
  meshUpperBound : ℝ
  /-- minMesh ≤ meshUpperBound (with generous slack). -/
  mesh_upper_bound_ge : F.minMesh ≤ meshUpperBound
  /-- The coupling constant κ in the bridge relation. -/
  kappa : ℝ
  /-- κ > 0. -/
  kappa_pos : 0 < kappa
  /-- The error constant C in O(mesh³). -/
  bridge_constant : ℝ
  /-- C ≥ 0. -/
  bridge_constant_nonneg : 0 ≤ bridge_constant
  /-- The recognition ratio function x_σ : deficit → ratio. -/
  recognitionRatio : ℝ → ℝ
  /-- The recognition ratio is positive for non-negative deficits. -/
  recognitionRatio_pos : ∀ δ : ℝ, 0 ≤ δ → 0 < recognitionRatio δ
  /-- BRIDGE (ASSUMED): log x_σ = κ · deficit + O(mesh³).

      Here `deficit` denotes the deficit angle δ at a hinge.
      This states |log(x_σ(deficit)) - κ · deficit| ≤ C · mesh³ for all deficit ≥ 0.
      This is an ASSUMED physical hypothesis, not derived from RS axioms. -/
  bridge_holds : ∀ deficit : ℝ, 0 ≤ deficit →
    |Real.log (recognitionRatio deficit) - kappa * deficit| ≤ bridge_constant * F.minMesh ^ 3
MODEL IsRSAdmissible · IndisputableMonolith/Gravity/AdmissibleTriangulationProcedure.lean
IsRSAdmissible · IndisputableMonolith/Gravity/AdmissibleTriangulationProcedure.lean:33 
/-- The RS admissibility predicate for triangulation families.

This predicate records the admissibility conditions that a triangulation
family must satisfy to be used in the recognition path sum:

1. **Positive mesh lower bound** (derived): minMesh > 0
2. **Finite simplex-count cap** (derived): maxSimplexCount > 0
3. **Positive growth base** (derived): growthBase > 0
4. **Recognition-ratio bridge** (ASSUMED): log x_σ = κ · deficit + O(mesh³)

The bridge condition is tagged as ASSUMED: it is a physical hypothesis
bridging recognition ratios to deficit angles, not derived from RS axioms. -/
structure IsRSAdmissible (F : AdmissibleTriangulationFamily) where
  /-- Mesh lower bound: minMesh > 0 (derived from F.minMesh_pos). -/
  mesh_lower_bound : 0 < F.minMesh
  /-- Finite simplex-count cap: maxSimplexCount > 0 (derived). -/
  simplex_count_finite : 0 < F.maxSimplexCount
  /-- Positive growth base: growthBase > 0 (derived). -/
  growth_base_pos : 0 < F.growthBase
  /-- Mesh upper bound for admissibility comparison. -/
  meshUpperBound : ℝ
  /-- minMesh ≤ meshUpperBound (with generous slack). -/
  mesh_upper_bound_ge : F.minMesh ≤ meshUpperBound
  /-- The coupling constant κ in the bridge relation. -/
  kappa : ℝ
  /-- κ > 0. -/
  kappa_pos : 0 < kappa
  /-- The error constant C in O(mesh³). -/
  bridge_constant : ℝ
  /-- C ≥ 0. -/
  bridge_constant_nonneg : 0 ≤ bridge_constant
  /-- The recognition ratio function x_σ : deficit → ratio. -/
  recognitionRatio : ℝ → ℝ
  /-- The recognition ratio is positive for non-negative deficits. -/
  recognitionRatio_pos : ∀ δ : ℝ, 0 ≤ δ → 0 < recognitionRatio δ
  /-- BRIDGE (ASSUMED): log x_σ = κ · deficit + O(mesh³).

      Here `deficit` denotes the deficit angle δ at a hinge.
      This states |log(x_σ(deficit)) - κ · deficit| ≤ C · mesh³ for all deficit ≥ 0.
      This is an ASSUMED physical hypothesis, not derived from RS axioms. -/
  bridge_holds : ∀ deficit : ℝ, 0 ≤ deficit →
    |Real.log (recognitionRatio deficit) - kappa * deficit| ≤ bridge_constant * F.minMesh ^ 3
THEOREM bridge_constant_monotone · IndisputableMonolith/Gravity/AdmissibleTriangulationProcedure.lean
bridge_constant_monotone · IndisputableMonolith/Gravity/AdmissibleTriangulationProcedure.lean:124 
/-- Monotonicity of the bridge constant: if F is RS-admissible with
bridge constant C, then it is also RS-admissible with any C' ≥ C.

This is a closure fact: the set of admissible bridge constants is
upward-closed, so larger error bounds preserve admissibility. -/
theorem bridge_constant_monotone (F : AdmissibleTriangulationFamily)
    (h : IsRSAdmissible F) (C' : ℝ) (hC' : h.bridge_constant ≤ C') :
    Nonempty (IsRSAdmissible F) :=
  ⟨{
    mesh_lower_bound := h.mesh_lower_bound
    simplex_count_finite := h.simplex_count_finite
    growth_base_pos := h.growth_base_pos
    meshUpperBound := h.meshUpperBound
    mesh_upper_bound_ge := h.mesh_upper_bound_ge
    kappa := h.kappa
    kappa_pos := h.kappa_pos
    bridge_constant := C'
    bridge_constant_nonneg := le_trans h.bridge_constant_nonneg hC'
    recognitionRatio := h.recognitionRatio
    recognitionRatio_pos := h.recognitionRatio_pos
    bridge_holds := fun deficit hδ =>
      le_trans (h.bridge_holds deficit hδ)
        (mul_le_mul_of_nonneg_right hC' (pow_nonneg (le_of_lt h.mesh_lower_bound) 3))
  }⟩

What this page does not claim

The witness does not prove that the bridge hypothesis is true for physical spacetime. The witness does not provide a physically realistic triangulation of space. The witness does not derive the bridge from Recognition Science axioms.

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/AdmissibleTriangulationProcedure.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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