Encyclopedia Gravity Gravity Track1 Bcorrected Quadratic Not Both Correspondences Of Quadratics Diffe

ARTICLE 3 claims 3 theorems

Gravity Track1 Bcorrected Quadratic Not Both Correspondences Of Quadratics Diffe

Two candidate formulas for gravity's local energy cannot both be right; the framework proves why, and names which one survives.

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The exclusivity theorem

In the Recognition Science framework, gravity's local behavior is studied through a discrete record of geometric data on a triangulated space, a structure the framework calls a ledger, a finite list of values attached to the vertices and edges of the mesh. The framework's library, a machine-checked collection of formal theorems, asks a precise question: when a small disturbance is applied to the geometry, which quadratic expression correctly describes the leading change in the gravitational action? Two candidate expressions had been proposed. The older one, the edge stencil, sums contributions along the edges of the mesh. The newer one, the axis stencil, sums along the coordinate axes of each tetrahedron. The declaration not_both_correspondences_of_quadratics_differ is a theorem stating that these two candidates cannot both be correct.

The proof is an exercise in rigidity. The library first proves a general uniqueness result: if two homogeneous quadratic expressions both match the gravitational action to third order near a flat configuration, then they must be identical everywhere. The theorem reggeLocalQuadraticCorrespondence_quadratic_unique states this for any triangulation. The exclusivity result applies this to the periodic torus used in the framework's calculations. It assumes that the two stencils differ at some configuration, which the framework's earlier audit at a specific mesh size confirms. From that difference and the uniqueness theorem, it follows that at most one of the two stencils can satisfy the local correspondence condition. The theorem does not itself say which one is correct; it only rules out the possibility that both are.

The framework's separate audit then chooses the axis stencil. At a mesh with five vertices along each side, the mixed quadratic evaluates to 12, while the seven-class edge stencil evaluates to 6 + 6√2 + 2√3. These scalars differ, so the two stencils are not the same quadratic. Since the uniqueness theorem forces any two valid candidates to be equal, the edge stencil cannot be the correct Taylor coefficient. The axis stencil is the surviving candidate, and the framework's library proves the corrected gate at that mesh size is closed. The all-cardinality generalization, for any mesh size, remains an open target.

What the theorem does not claim is as important as what it proves. It does not assert that the axis stencil is the true physical law of gravity; that would require the all-cardinality closure, which is not proved. It does not claim the edge stencil is useless, only that it cannot serve as the local quadratic correspondence. And it does not derive any new physics from the choice; it is a statement about which mathematical expression is consistent with the framework's axioms.

THEOREM not_both_correspondences_of_quadratics_differ · IndisputableMonolith/Gravity/Track1BCorrectedQuadratic.lean
not_both_correspondences_of_quadratics_differ · IndisputableMonolith/Gravity/Track1BCorrectedQuadratic.lean:300 
/-- **EXCLUSIVITY.**  Given the audit witness, the legacy seven-class endpoint
and the corrected axis endpoint are mutually exclusive: at most one of them is
the true cubic-Taylor statement for the Regge action. -/
theorem not_both_correspondences_of_quadratics_differ
    (Nx Ny Nz : ℕ) [NeZero Nx] [NeZero Ny] [NeZero Nz]
    (hx : 2 < Nx) (hy : 2 < Ny) (hz : 2 < Nz)
    (hdiff : AxisEdgeStencilQuadraticsDiffer Nx Ny Nz hx hy hz) :
    ¬(CanonicalPeriodicEdgeStencilLocalCorrespondence Nx Ny Nz hx hy hz ∧
      CanonicalPeriodicAxisStencilLocalCorrespondence Nx Ny Nz hx hy hz) := by
  rintro ⟨hLegacy, hCorrected⟩
  obtain ⟨ξ, hξ⟩ := hdiff
  exact hξ (both_correspondences_force_equal_quadratics
    Nx Ny Nz hx hy hz hLegacy hCorrected ξ)
THEOREM reggeLocalQuadraticCorrespondence_quadratic_unique · IndisputableMonolith/Gravity/Track1BCorrectedQuadratic.lean
reggeLocalQuadraticCorrespondence_quadratic_unique · IndisputableMonolith/Gravity/Track1BCorrectedQuadratic.lean:171 
/-- **RIGIDITY.**  If two quadratically homogeneous candidates both satisfy
the local correspondence on the same complex, they are pointwise equal.  The
quadratic coefficient of a cubic-Taylor expansion is unique, so at most one
stencil can be the true second-order content of the Regge action. -/
theorem reggeLocalQuadraticCorrespondence_quadratic_unique
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (Q₁ Q₂ : VertexPotential K → ℝ)
    (hQ₁ : ∀ (a : ℝ) (ξ : VertexPotential K), Q₁ (a • ξ) = a ^ (2 : ℕ) * Q₁ ξ)
    (hQ₂ : ∀ (a : ℝ) (ξ : VertexPotential K), Q₂ (a • ξ) = a ^ (2 : ℕ) * Q₂ ξ)
    (h₁ : ReggeLocalQuadraticCorrespondence K hK Q₁)
    (h₂ : ReggeLocalQuadraticCorrespondence K hK Q₂) :
    ∀ ξ : VertexPotential K, Q₁ ξ = Q₂ ξ := by
  obtain ⟨r₁, C₁, hr₁, hC₁, hb₁⟩ := h₁
  obtain ⟨r₂, C₂, hr₂, hC₂, hb₂⟩ := h₂
  intro ξ
  by_contra hne
  have hΔpos : 0 < |Q₁ ξ - Q₂ ξ| := abs_pos.mpr (sub_ne_zero.mpr hne)
  set Δ : ℝ := |Q₁ ξ - Q₂ ξ| with hΔdef
  -- Choose the probe scale `t`.
  have hA : (0 : ℝ) < 1 + ‖ξ‖ := by positivity
  have hB : (0 : ℝ) < 1 + 2 * (C₁ + C₂) * ‖ξ‖ ^ (3 : ℕ) := by positivity
  set t : ℝ :=
    min (min r₁ r₂ / (1 + ‖ξ‖)) (Δ / (1 + 2 * (C₁ + C₂) * ‖ξ‖ ^ (3 : ℕ)))
    with ht_def
  have ht_pos : 0 < t := by
    refine lt_min (div_pos (lt_min hr₁ hr₂) hA) (div_pos hΔpos hB)
  -- The scaled probe sits inside both radii.
  have ht_norm : ‖t • ξ‖ = t * ‖ξ‖ := by
    rw [norm_smul, Real.norm_eq_abs, abs_of_pos ht_pos]
  have hsmall : t * ‖ξ‖ < min r₁ r₂ := by
    have h1 : t ≤ min r₁ r₂ / (1 + ‖ξ‖) := min_le_left _ _
    have h2 : ‖ξ‖ < 1 + ‖ξ‖ := by linarith [norm_nonneg ξ]
    have hq_pos : 0 < min r₁ r₂ / (1 + ‖ξ‖) := div_pos (lt_min hr₁ hr₂) hA
    calc t * ‖ξ‖ ≤ (min r₁ r₂ / (1 + ‖ξ‖)) * ‖ξ‖ :=
          mul_le_mul_of_nonneg_right h1 (norm_nonneg ξ)
      _ < (min r₁ r₂ / (1 + ‖ξ‖)) * (1 + ‖ξ‖) :=
          mul_lt_mul_of_pos_left h2 hq_pos
      _ = min r₁ r₂ := div_mul_cancel₀ _ hA.ne'
  have hsmall₁ : ‖t • ξ‖ < r₁ := by
    rw [ht_norm]; exact lt_of_lt_of_le hsmall (min_le_left _ _)
  have hsmall₂ : ‖t • ξ‖ < r₂ := by
    rw [ht_norm]; exact lt_of_lt_of_le hsmall (min_le_right _ _)
  -- The two cubic bounds at the scaled probe.
  have hb₁' := hb₁ (t • ξ) hsmall₁
  have hb₂' := hb₂ (t • ξ) hsmall₂
  rw [hQ₁ t ξ, Real.norm_eq_abs, ht_norm] at hb₁'
  rw [hQ₂ t ξ, Real.norm_eq_abs, ht_norm] at hb₂'
  -- Triangle inequality forces the quadratic gap below a linear-in-`t` bound.
  have hdiff :
      (reggeAction K hK (t • ξ) - reggeAction K hK (zeroPotential K) -
          (1 / 2) * (t ^ (2 : ℕ) * Q₂ ξ)) -
        (reggeAction K hK (t • ξ) - reggeAction K hK (zeroPotential K) -
          (1 / 2) * (t ^ (2 : ℕ) * Q₁ ξ)) =
        (1 / 2) * t ^ (2 : ℕ) * (Q₁ ξ - Q₂ ξ) := by ring
  have hgap : (1 / 2) * t ^ (2 : ℕ) * Δ ≤ (C₁ + C₂) * (t * ‖ξ‖) ^ (3 : ℕ) := by
    have htri :
        |(1 / 2) * t ^ (2 : ℕ) * (Q₁ ξ - Q₂ ξ)| ≤
          C₂ * (t * ‖ξ‖) ^ (3 : ℕ) + C₁ * (t * ‖ξ‖) ^ (3 : ℕ) := by
      rw [← hdiff]
      exact le_trans (abs_sub _ _) (add_le_add hb₂' hb₁')
    have habs :
        |(1 / 2) * t ^ (2 : ℕ) * (Q₁ ξ - Q₂ ξ)| =
          (1 / 2) * t ^ (2 : ℕ) * Δ := by
      rw [abs_mul, abs_of_nonneg (by positivity : (0:ℝ) ≤ (1/2) * t ^ (2:ℕ))]
    rw [habs] at htri
    linarith
  -- Divide by `t²` and contradict the choice of `t`.
  have ht2_pos : (0 : ℝ) < t ^ (2 : ℕ) := by positivity
  have hΔle : Δ ≤ 2 * (C₁ + C₂) * t * ‖ξ‖ ^ (3 : ℕ) := by
    have hexp : (t * ‖ξ‖) ^ (3 : ℕ) = t ^ (2 : ℕ) * (t * ‖ξ‖ ^ (3 : ℕ)) := by
      ring
    rw [hexp] at hgap
    calc Δ = (1 / 2) * t ^ (2 : ℕ) * Δ * (2 / t ^ (2 : ℕ)) := by
          field_simp
      _ ≤ (C₁ + C₂) * (t ^ (2 : ℕ) * (t * ‖ξ‖ ^ (3 : ℕ))) * (2 / t ^ (2 : ℕ)) :=
          mul_le_mul_of_nonneg_right hgap (by positivity)
      _ = 2 * (C₁ + C₂) * t * ‖ξ‖ ^ (3 : ℕ) := by
          field_simp
  have ht_le : t ≤ Δ / (1 + 2 * (C₁ + C₂) * ‖ξ‖ ^ (3 : ℕ)) := min_le_right _ _
  have hfinal : Δ < Δ := by
    have hC12 : 0 ≤ C₁ + C₂ := by linarith
    have hfrac :
        2 * (C₁ + C₂) * ‖ξ‖ ^ (3 : ℕ) <
          1 + 2 * (C₁ + C₂) * ‖ξ‖ ^ (3 : ℕ) := by linarith
    calc Δ ≤ 2 * (C₁ + C₂) * t * ‖ξ‖ ^ (3 : ℕ) := hΔle
      _ = t * (2 * (C₁ + C₂) * ‖ξ‖ ^ (3 : ℕ)) := by ring
      _ ≤ (Δ / (1 + 2 * (C₁ + C₂) * ‖ξ‖ ^ (3 : ℕ))) *
            (2 * (C₁ + C₂) * ‖ξ‖ ^ (3 : ℕ)) := by
          refine mul_le_mul_of_nonneg_right ht_le ?_
          positivity
      _ < (Δ / (1 + 2 * (C₁ + C₂) * ‖ξ‖ ^ (3 : ℕ))) *
            (1 + 2 * (C₁ + C₂) * ‖ξ‖ ^ (3 : ℕ)) := by
          refine mul_lt_mul_of_pos_left hfrac ?_
          exact div_pos hΔpos hB
      _ = Δ := div_mul_cancel₀ _ hB.ne'
  exact absurd hfinal (lt_irrefl Δ)
THEOREM not_both_correspondences_of_quadratics_differ · IndisputableMonolith/Gravity/Track1BCorrectedQuadratic.lean
not_both_correspondences_of_quadratics_differ · IndisputableMonolith/Gravity/Track1BCorrectedQuadratic.lean:300 
/-- **EXCLUSIVITY.**  Given the audit witness, the legacy seven-class endpoint
and the corrected axis endpoint are mutually exclusive: at most one of them is
the true cubic-Taylor statement for the Regge action. -/
theorem not_both_correspondences_of_quadratics_differ
    (Nx Ny Nz : ℕ) [NeZero Nx] [NeZero Ny] [NeZero Nz]
    (hx : 2 < Nx) (hy : 2 < Ny) (hz : 2 < Nz)
    (hdiff : AxisEdgeStencilQuadraticsDiffer Nx Ny Nz hx hy hz) :
    ¬(CanonicalPeriodicEdgeStencilLocalCorrespondence Nx Ny Nz hx hy hz ∧
      CanonicalPeriodicAxisStencilLocalCorrespondence Nx Ny Nz hx hy hz) := by
  rintro ⟨hLegacy, hCorrected⟩
  obtain ⟨ξ, hξ⟩ := hdiff
  exact hξ (both_correspondences_force_equal_quadratics
    Nx Ny Nz hx hy hz hLegacy hCorrected ξ)

What this page does not claim

The theorem does not prove the axis stencil is the true physical law of gravity. The theorem does not claim the edge stencil is useless, only that it cannot serve as the local quadratic correspondence. The theorem does not derive any new physics from the choice of stencil.

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/Track1BCorrectedQuadratic.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

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