Gravity Rsnull Field Equation Quad Contr Metric Term Eq Zero

The vanishing metric term

In general relativity, the Einstein field equations relate the curvature of spacetime, encoded in the Ricci tensor, to the distribution of matter and energy. A common way to extract information from these tensor equations is to contract them with a null vector, a direction along which light travels. This produces a scalar equation, a single number, that is often easier to analyze. A key algebraic fact makes this contraction particularly clean: a term proportional to the metric tensor, which can appear in the equations, contributes nothing when contracted with a null vector.

The declaration quadContr_metric_term_eq_zero is a formal, machine-checked proof of this fact. It states that for any real number f and any Minkowski-null vector k, the quadratic contraction of the matrix f * η (the metric tensor scaled by f) with k is exactly zero. In plainer terms, if you take any multiple of the spacetime metric and feed it a light-like direction, the result is zero. This is not an approximation; it is an exact algebraic identity that follows from the definition of a null vector, which satisfies the condition that its squared length is zero.

This lemma is the engine behind a larger reduction. The framework's library proves that if you start with an equation shaped like Einstein's, namely κ T = Ric + f η, then contracting both sides with a null vector k eliminates the metric term and leaves the simpler scalar relation Ric(k,k) = κ T(k,k). The proof of this reduction relies directly on the vanishing of the metric term. The framework also proves that this vanishing is not reversible: there exist nonzero matrices, such as the metric itself, that are invisible to all null contractions. This means the scalar equation cannot be used to recover the original matrix equation; information is genuinely lost in the contraction.

In Recognition Science, this result is a stepping stone, not a destination. The framework models gravity by starting from a ledger of recognition events, and this theorem provides a rigorous bridge from an assumed matrix equation to its scalar form. It does not, however, derive that matrix equation from the ledger. The existence of an EinsteinShapedSource, the input equation, is taken as an external assumption, not a proven consequence. The theorem also does not construct a horizon patch, identify cut channels with spacetime covectors, or claim to have derived a sourced field equation from first principles. It is a precise algebraic tool, ready for use, but it does not by itself tell you where the source equation comes from.

What this means for a reader is that the framework has a verified handle on a standard piece of general-relativity algebra. The vanishing of the metric term on null directions is a classical fact, and here it is given a formal, machine-checked proof. The value is in the certainty and the clean interface it provides for further work, not in a new physical prediction. The framework's own documentation is explicit about this: the module proves the algebraic step, and the physical input remains an open question.

/-- Every scalar Minkowski metric term vanishes on a null probe. -/
theorem quadContr_metric_term_eq_zero
    (f : ℝ) (k : Fin 4 → ℝ)
    (hk : MinkowskiNull k) :
    quadContr (f • minkowskiEta4) k = 0 := by
  rw [quadContr_smul, quadContr_minkowskiEta4]
  rw [show -(k 0) ^ 2 + (k 1) ^ 2 + (k 2) ^ 2 + (k 3) ^ 2 = 0 by
    simpa [MinkowskiNull] using hk]
  ring

/--
Null reduction of one Einstein-shaped equation.  The undetermined scalar
metric term disappears on every Minkowski-null direction.
-/
theorem null_scalar_of_einstein_shaped
    (coupling : ℝ)
    (T Ric : Matrix (Fin 4) (Fin 4) ℝ)
    (f : ℝ)
    (h : coupling • T = Ric + f • minkowskiEta4)
    (k : Fin 4 → ℝ)
    (hk : MinkowskiNull k) :
    quadContr Ric k = coupling * quadContr T k := by
  have hcontract := congrArg (fun A => quadContr A k) h
  change quadContr (coupling • T) k =
    quadContr (Ric + f • minkowskiEta4) k at hcontract
  rw [quadContr_smul, quadContr_add,
    quadContr_metric_term_eq_zero f k hk, add_zero] at hcontract
  exact hcontract.symm

/--
The null equation does not recover the scalar metric term: adding a nonzero
multiple of `η` changes the matrix while preserving every null contraction.
-/
theorem scalar_metric_term_is_null_invisible :
    ∃ D : Matrix (Fin 4) (Fin 4) ℝ,
      D ≠ 0 ∧
      ∀ k, MinkowskiNull k → quadContr D k = 0 := by
  refine ⟨minkowskiEta4, ?_, ?_⟩
  · intro h
    have h00 := congrFun (congrFun h (0 : Fin 4)) (0 : Fin 4)
    norm_num [minkowskiEta4] at h00
  · intro k hk
    simpa using quadContr_metric_term_eq_zero 1 k hk

What this page does not claim

This theorem does not derive the Einstein field equations from the Recognition Science ledger. This theorem does not construct a horizon patch or identify cut channels with spacetime covectors. This theorem does not claim to recover the full matrix equation from its null-contracted scalar form.

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/RSNullFieldEquation.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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