Gravity Null Cone Quadratic Tensor Class

The null cone rigidity

In the four-dimensional spacetime of special relativity, the null cone is the set of directions along which light travels: vectors k whose squared length, computed with the Minkowski metric η = diag(-1,1,1,1), is zero. A symmetric quadratic form, such as a stress-energy tensor, assigns a number to each such direction by contraction, k·T·k. The question is how much of the tensor this collection of numbers reveals.

The answer, proved in the framework's machine-checked library of formal theorems, is that for a symmetric 4×4 matrix T, knowing the value of the quadratic form on every null direction determines T up to adding a scalar multiple of the metric η itself. In symbols: if two symmetric matrices T and B give the same values on all null covectors, then T - B = λη for some real number λ. The theorem is proved in Lean 4 (null_quadratic_eq_implies_diff_scalar_eta, in IndisputableMonolith/Gravity/NullConeQuadraticTensorClass.lean), with no axioms beyond the standard three. This is a rigidity result: the null cone is a sharp probe, but it cannot see the part of the tensor that is proportional to the metric, because the metric itself vanishes on all null vectors.

The framework also proves the converse: if two symmetric matrices differ by a scalar multiple of η, they agree on all null directions. Together these two directions give an exact characterization, an if-and-only-if statement. A related theorem shows that the antisymmetric part of any matrix is invisible to this probe: contracting with a null vector gives zero for the antisymmetric part, so only the symmetrized matrix matters. The general class of matrices that produce the same null-cone values is therefore exactly the symmetrization of a matrix plus a scalar multiple of η.

In Recognition Science, this result is a Phase 5 algebraic prerequisite. It is not the construction of the geometric tensor G that the phase ultimately needs. The module explicitly states that it does not identify any matrix with Ricci or stress-energy, does not construct horizons or Unruh temperature, and does not close the Einstein field equation. The construction of a symmetric G from Recognition area and focusing ancestry, independently of a fixed stress tensor, remains open. What the module does establish is a clean algebraic fact: the null cone determines a symmetric tensor up to the metric ambiguity, and this is the platform on which the later, geometric step must stand.

/--
Null-cone agreement of two symmetric quadratic forms determines their
difference as a scalar multiple of the Minkowski metric.
-/
theorem null_quadratic_eq_implies_diff_scalar_eta
    (A B : Matrix (Fin 4) (Fin 4) ℝ)
    (hA : Symmetric4 A)
    (hB : Symmetric4 B)
    (hnull : ∀ k, MinkowskiNull k → quadContr A k = quadContr B k) :
    ∃ lam : ℝ, ∀ i j, A i j - B i j = lam * minkowskiEta4 i j := by
  obtain ⟨lam, hlam⟩ := null_quadratic_eq_of_diff_scalar_eta A B hA hB hnull
  refine ⟨lam, ?_⟩
  intro i j
  have hij := congrFun (congrFun hlam i) j
  -- `A = B + lam • η` at `(i,j)`.
  change A i j = B i j + (lam • minkowskiEta4) i j at hij
  simp only [Matrix.smul_apply, smul_eq_mul] at hij
  linarith

/-- Converse: a pure metric difference is invisible on the null cone. -/
theorem diff_scalar_eta_implies_null_quadratic_eq
    (A B : Matrix (Fin 4) (Fin 4) ℝ) (lam : ℝ)
    (hlam : ∀ i j, A i j - B i j = lam * minkowskiEta4 i j) :
    ∀ k, MinkowskiNull k → quadContr A k = quadContr B k := by
  intro k hk
  have hAB : A = B + lam • minkowskiEta4 := by
    ext i j
    have hij := hlam i j
    simp only [Matrix.add_apply, Matrix.smul_apply, smul_eq_mul]
    linarith
  have hcontr := congrArg (fun M => quadContr M k) hAB
  change quadContr A k = quadContr (B + lam • minkowskiEta4) k at hcontr
  have hadd :
      quadContr (B + lam • minkowskiEta4) k =
        quadContr B k + quadContr (lam • minkowskiEta4) k := by
    simp only [quadContr, Matrix.add_apply, add_mul, Finset.sum_add_distrib]
  rw [hadd, quadContr_smul_eta_of_null lam k hk, add_zero] at hcontr
  exact hcontr

theorem quadContr_antisymmetrize4_eq_zero
    (A : Matrix (Fin 4) (Fin 4) ℝ) (k : Fin 4 → ℝ) :
    quadContr (antisymmetrize4 A) k = 0 := by
  have hA :
      A = symmetrize4 A + antisymmetrize4 A := by
    ext i j
    simp only [symmetrize4, antisymmetrize4, Matrix.add_apply]
    ring
  have hadd :
      quadContr A k =
        quadContr (symmetrize4 A) k + quadContr (antisymmetrize4 A) k := by
    have hcongr := congrArg (fun M => quadContr M k) hA
    change quadContr A k =
      quadContr (symmetrize4 A + antisymmetrize4 A) k at hcongr
    have hsum :
        quadContr (symmetrize4 A + antisymmetrize4 A) k =
          quadContr (symmetrize4 A) k +
            quadContr (antisymmetrize4 A) k := by
      simp only [quadContr, Matrix.add_apply, add_mul, Finset.sum_add_distrib]
    exact hcongr.trans hsum
  linarith [quadContr_eq_quadContr_symmetrize4 A k, hadd]

What this page does not claim

This module does not identify a matrix with Ricci or stress-energy. This module does not construct horizons or Unruh temperature. This module does not close the Einstein field equation.

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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