Gravity Null Cone Quadratic Tensor Class Null Quadratic Eq Iff Symmetrize Diff S

The null-cone rigidity theorem

In four-dimensional spacetime, the Minkowski metric η = diag(-1,1,1,1) defines which vectors are null, meaning they satisfy -t² + x² + y² + z² = 0. These null vectors form the light cone, the set of directions along which light rays travel. A quadratic form assigns a real number to each vector; when the form is represented by a symmetric 4×4 matrix, its value on a null vector k is the contraction kᵀ A k. The question is how much information about the matrix A is contained in these contractions, evaluated only on the light cone.

The answer, proved in the framework's machine-checked library of formal theorems, is that the null-cone values determine the symmetric part of A completely, up to adding a scalar multiple of the Minkowski metric itself. The theorem null_quadratic_eq_iff_symmetrize_diff_scalar_eta states this as an exact equivalence: two matrices A and B produce identical quadratic forms on all null vectors if and only if their symmetrized parts differ by λη for some real number λ. The symmetrized part is (A + Aᵀ)/2, the symmetric matrix that captures the quadratic form's behavior; the antisymmetric part (A - Aᵀ)/2 is invisible to the contraction and contributes nothing.

This rigidity result has a direct physical reading. The ledger, a discrete record of recognition events, assigns to each null direction a scalar flux value through the contraction. The theorem says that if two symmetric stress-like matrices produce the same flux on every null direction, they are the same matrix up to a term proportional to the metric. That remaining ambiguity is exactly the freedom to add a cosmological-constant-like term, which vanishes on the light cone because η is null on null vectors. The framework's library proves the converse as well: any two matrices differing by λη necessarily agree on all null contractions.

In Recognition Science, this theorem is a Phase 5 algebraic prerequisite, not the construction of geometric gravity itself. It establishes the uniqueness class for symmetric matrices given null-cone data, and it certifies that the antisymmetric part of any matrix is undetectable by such data. The library also proves that the result holds if one only knows the contractions on future-directed nonzero null vectors, since the remaining null directions follow by continuity of the quadratic form.

What the theorem does not claim is equally precise. It does not identify any matrix with the Ricci tensor or the stress-energy tensor, does not construct horizons or Unruh temperature, and does not close the Einstein field equation. The independently geometric tensor G with Recognition area/focusing ancestry remains OPEN, as does continuum Ricci curvature and the full EFE. The theorem is a statement about finite-dimensional real linear algebra: the null cone is a rigid probe of symmetric matrices, nothing more and nothing less.

/--
For general matrices, null-cone quadratic data determines the symmetrization
modulo `η`.  Antisymmetric parts are invisible.
-/
theorem null_quadratic_eq_iff_symmetrize_diff_scalar_eta
    (A B : Matrix (Fin 4) (Fin 4) ℝ) :
    (∀ k, MinkowskiNull k → quadContr A k = quadContr B k) ↔
      ∃ lam : ℝ, ∀ i j,
        symmetrize4 A i j - symmetrize4 B i j = lam * minkowskiEta4 i j := by
  constructor
  · intro hnull
    have hsym :
        ∀ k, MinkowskiNull k →
          quadContr (symmetrize4 A) k = quadContr (symmetrize4 B) k := by
      intro k hk
      rw [← quadContr_eq_quadContr_symmetrize4 A k,
        ← quadContr_eq_quadContr_symmetrize4 B k]
      exact hnull k hk
    exact null_quadratic_eq_implies_diff_scalar_eta
      (symmetrize4 A) (symmetrize4 B)
      (symmetrize4_symmetric A) (symmetrize4_symmetric B) hsym
  · intro h
    obtain ⟨lam, hlam⟩ := h
    intro k hk
    have hsymEq :=
      diff_scalar_eta_implies_null_quadratic_eq
        (symmetrize4 A) (symmetrize4 B) lam hlam k hk
    rw [quadContr_eq_quadContr_symmetrize4 A k,
      quadContr_eq_quadContr_symmetrize4 B k, hsymEq]

/-- Quadratic contraction sees only the symmetric part. -/
theorem quadContr_eq_quadContr_symmetrize4
    (A : Matrix (Fin 4) (Fin 4) ℝ) (k : Fin 4 → ℝ) :
    quadContr A k = quadContr (symmetrize4 A) k := by
  simp only [quadContr, symmetrize4, sum_fin_four]
  ring

/--
Agreement on every future nonzero Minkowski-null vector extends to the whole
null cone.  Negative-time vectors are handled by quadratic evenness.  A null
vector with zero time component is the zero vector.
-/
theorem all_null_quad_eq_of_future_nonzero_null_quad_eq
    (A B : Matrix (Fin 4) (Fin 4) ℝ)
    (hfuture :
      ∀ k, MinkowskiNull k → k ≠ 0 → 0 < k 0 →
        quadContr A k = quadContr B k) :
    ∀ k, MinkowskiNull k → quadContr A k = quadContr B k := by
  intro k hk
  by_cases hzero : k = 0
  · subst k
    simp [quadContr]
  have htime : k 0 ≠ 0 := by
    intro hk0
    have hkEq := hk
    unfold MinkowskiNull at hkEq
    norm_num [hk0] at hkEq
    have h1 : k 1 = 0 := by
      nlinarith [sq_nonneg (k 1), sq_nonneg (k 2), sq_nonneg (k 3)]
    have h2 : k 2 = 0 := by
      nlinarith [sq_nonneg (k 1), sq_nonneg (k 2), sq_nonneg (k 3)]
    have h3 : k 3 = 0 := by
      nlinarith [sq_nonneg (k 1), sq_nonneg (k 2), sq_nonneg (k 3)]
    apply hzero
    funext i
    fin_cases i
    · exact hk0
    · exact h1
    · exact h2
    · exact h3
  rcases lt_or_gt_of_ne htime with hneg | hpos
  · have hkneg : MinkowskiNull (-k) := by
      simpa [MinkowskiNull] using hk
    have hnegzero : (-k) ≠ 0 := neg_ne_zero.mpr hzero
    have hfutureNeg :=
      hfuture (-k) hkneg hnegzero (by simpa using neg_pos.mpr hneg)
    simpa only [quadContr_neg] using hfutureNeg
  · exact hfuture k hk hzero hpos

theorem nullConeQuadraticTensorClassCert :
    NullConeQuadraticTensorClassCert where
  future_to_all := all_null_quad_eq_of_future_nonzero_null_quad_eq
  reconstruction := null_quadratic_eq_implies_diff_scalar_eta
  converse := diff_scalar_eta_implies_null_quadratic_eq
  antisym_invisible := quadContr_eq_quadContr_symmetrize4
  general_class := null_quadratic_eq_iff_symmetrize_diff_scalar_eta
  fixed_stress_algebraic_class :=
    fixedSymmetricStress_determinesAlgebraicNullQuadraticClass

What this page does not claim

This theorem does not construct geometric curvature, continuum Ricci, or the Einstein field equation. This theorem does not identify any matrix with the physical stress-energy tensor. This theorem does not imply that the null-cone data determines the full matrix, only its symmetric part modulo the metric.

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

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