Gravity Record Flux Stress Cut Event Stress Symmetric

A symmetric stress from cut events

In classical physics, a stress tensor is a symmetric matrix that describes how forces act across a surface. Its symmetry is a physical fact: the stress across one face equals the stress across the adjacent face, otherwise a small cube would spin on its own. The Recognition Science framework builds a mathematical object with a similar shape from a very different source: a discrete record of events, called a ledger, which is a list of signed changes in bit states at a boundary.

The framework's library, a machine-checked collection of formal theorems, defines a matrix called cutEventStress. It is constructed by taking each exterior channel of a cut, computing the signed change in its bit value between two records, and multiplying that weight by an assigned covector. The covector assignment is an explicit modeling choice, not a derived quantity. The theorem cutEventStress_symmetric proves that this matrix is symmetric: the entry at row i, column j equals the entry at row j, column i. The proof is a direct calculation from the definition, since each summand is a symmetric outer product w * p_i * p_j.

The construction also proves a contraction property: for any probe vector k, the quadratic contraction of the stress equals the sum over channels of the signed weight times the square of the covector-probe inner product. This identity holds for every probe, but the stress matrix is fixed before any probe appears. There is no interface that chooses a different stress for each probe, which would make the result tautological. A separate theorem shows that if all covectors are zero, the stress is zero, and a witness theorem shows that a single event with unit weight and a nonzero covector produces a nonzero matrix.

In Recognition Science, the framework uses this object as a probe-independent stress-like matrix. The word "probe-independent" means the matrix is built once from the ledger and the covector assignment, then contracted against any probe. The framework does not claim this matrix is the continuum stress-energy tensor of general relativity. It does not claim any connection to Unruh radiation, Ricci curvature, geodesic focusing, or the Einstein field equations. It does not claim that all null directions give equal stress. The theorem is purely algebraic: given a finite set of channels, signed weights, and assigned covectors, the resulting matrix is symmetric and has the stated contraction identity.

What this establishes is a bridge between the discrete ledger and a familiar structural property of physics. Symmetry is the first step toward anything that could behave like a stress tensor. The framework has shown that a symmetric matrix can be built from record-flux events without any continuum assumptions. Whether that matrix can be promoted to a physical stress-energy tensor remains an open target, not a proved result.

theorem cutEventStress_symmetric {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H) (p : ExteriorCutChannel a s → Fin 4 → ℝ) :
    Symmetric4 (cutEventStress c c' p) :=
  eventStress_symmetric _ _

theorem quadContr_cutEventStress {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H) (p : ExteriorCutChannel a s → Fin 4 → ℝ)
    (k : Fin 4 → ℝ) :
    quadContr (cutEventStress c c' p) k =
      ∑ ch : ExteriorCutChannel a s,
        channelDelta c c' ch * (∑ μ, p ch μ * k μ) ^ 2 :=
  quadContr_eventStress _ _ _

/-- Zero cut-channel covectors force zero cut event stress. -/
theorem cutEventStress_zero_of_covector_zero {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H) :
    cutEventStress c c' (fun _ _ => (0 : ℝ)) = 0 :=
  eventStress_zero_of_covector_zero _

/--
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

/--
Cut event stress: one fixed symmetric matrix from signed exterior channel
weights and an explicit MODEL covector assignment.
-/
def cutEventStress {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H) (p : ExteriorCutChannel a s → Fin 4 → ℝ) :
    Matrix (Fin 4) (Fin 4) ℝ :=
  eventStress (channelDelta c c') p

What this page does not claim

This theorem does not claim the matrix is the stress-energy tensor of general relativity. This theorem does not claim any connection to Unruh radiation, Ricci curvature, or the Einstein field equations. This theorem does not claim that all null directions give equal stress.

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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