Gravity Record Flux Stress Quad Contr Event Stress Zero Of Covector Zero

The zero-covector stress theorem

In the Recognition Science framework, stress-like quantities are built from discrete records of events, not from continuous fields. The framework models each event as a point with a weight and an assigned covector, a vector-like object that maps directions to numbers. The ledger, a discrete record of events, supplies the weights; the covectors are an explicit modeling choice. The central object is a symmetric matrix called event stress, defined as the sum of weighted outer products of the assigned covectors.

The declaration quadContr_eventStress_zero_of_covector_zero proves a simple but load-bearing fact: if every assigned covector is the zero vector, then the event stress matrix is the zero matrix, and its quadratic contraction against any probe vector is zero. In plain language, if you assign no direction to any recorded event, the resulting stress-like object carries no information. The theorem is tagged THEOREM, meaning it is proved in the machine-checked library of formal theorems with no unproved assumptions. It holds for any finite collection of events and any real weights, so it is fully general within the framework's definitions.

The proof is a direct computation. The event stress matrix is defined componentwise as a sum over events of weight times covector components. Setting all covectors to zero makes every summand zero, so the matrix is zero. The quadratic contraction, which pairs the matrix with a probe vector, then also vanishes. The library also proves the converse direction with a witness: a single event with unit weight and a nonzero covector produces a nonzero stress matrix, so the zero result is not vacuous.

What this theorem does not claim is equally important. It does not claim that this stress-like matrix represents continuum stress-energy, the Unruh effect, Ricci curvature, or any general-relativity quantity. It does not claim that the covector assignment is derived from the framework; the assignment is an explicit MODEL interface, a definitional choice. It does not claim that all-null equality holds, that the Einstein field equations follow, or that any gap in the framework's closure is filled. The theorem is a precise algebraic fact about a defined object, not a physical law.

The theorem's role is to anchor the framework's internal consistency. It shows that the event stress construction behaves as expected under a degenerate input, and it rules out a trivial tautology: the stress is defined once from the weights and covectors, then contracted, so there is no hidden dependence on the probe. This makes the construction a fixed object before any probe appears, a property the framework relies on for further derivations.

/-- Zero covectors force zero quadratic contraction for every probe. -/
theorem quadContr_eventStress_zero_of_covector_zero {E : Type*} [Fintype E]
    (w : E → ℝ) (k : Fin 4 → ℝ) :
    quadContr (eventStress w (fun _ _ => (0 : ℝ))) k = 0 := by
  rw [quadContr_eventStress]
  simp

/--
Probe-independent event stress: the sum of weighted outer products of the
assigned covectors.  Defined componentwise so the matrix is fixed before any
probe appears.
-/
def eventStress {E : Type*} [Fintype E] (w : E → ℝ) (p : E → Fin 4 → ℝ) :
    Matrix (Fin 4) (Fin 4) ℝ :=
  fun a b => ∑ e : E, w e * p e a * p e b

/--
Non-tautology witness: a single event with unit weight and a nonzero covector
produces a nonzero stress matrix.
-/
theorem eventStress_ne_zero_of_unit_channel :
    eventStress (fun _ : Fin 1 => (1 : ℝ))
        (fun _ μ => if μ = (0 : Fin 4) then (1 : ℝ) else 0) ≠ 0 := by
  intro h
  have h00 := congrFun (congrFun h (0 : Fin 4)) (0 : Fin 4)
  simp [eventStress] at h00

What this page does not claim

The theorem does not claim the event stress represents continuum stress-energy or any general-relativity quantity. The covector assignment is a MODEL choice, not a derived consequence of the framework. The theorem does not imply the Einstein field equations or any all-null equality.

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

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