Encyclopedia Gravity Gravity Record Flux Stress

ARTICLE 5 claims 5 theorems

Gravity Record Flux Stress

A machine-checked definition builds a stress-like matrix from discrete cut records, proving only what the construction itself guarantees.

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Record flux stress

In Recognition Science, gravity record flux stress is a symmetric 4 by 4 matrix built from a ledger of discrete events. The ledger here is a pair of local cuts, each a finite configuration of boolean bits posted on channels. The matrix sums, over every exterior channel, the signed change in that channel's bit between the two cuts, weighted by an assigned covector. The covector assignment is an explicit model choice, not a derived quantity.

The framework's machine-checked library of formal theorems proves three things about this construction. First, the matrix is symmetric: swapping row and column indices leaves it unchanged. Second, its quadratic contraction against any probe vector equals the sum over channels of the signed weight times the squared inner product of the covector with the probe. Third, if every covector is zero, the matrix and its contraction vanish for every probe. A separate witness theorem shows the construction is not vacuous: a single event with unit weight and a nonzero covector produces a nonzero matrix.

The module also bridges the new channel weights to the committed cut heat. It proves that the posted exterior heat between two cuts equals the sum of signed exterior channel deltas. This links the stress construction back to the framework's existing heat accounting, so the new object is not floating free.

The module is deliberately narrow. It does not claim continuum stress-energy, Unruh effect, Ricci curvature, geodesic focusing, equality of all null contractions, Einstein field equations, or closure of the C-gap. The stress is defined once from the pair (w, p) and then contracted; there is no interface that picks a target equality per probe. The anti-tautology point is that the matrix is fixed before any probe appears.

What this establishes in plain language is a minimal, probe-independent stress-like object from discrete records, with its symmetries and contraction behavior proved. It is a building block, not a theory of gravity. The physical bridge from these discrete records to continuum gravity remains open.

THEOREM eventStress_symmetric · IndisputableMonolith/Gravity/RecordFluxStress.lean
eventStress_symmetric · IndisputableMonolith/Gravity/RecordFluxStress.lean:42 
/-- Outer-product summands are symmetric, hence so is `eventStress`. -/
theorem eventStress_symmetric {E : Type*} [Fintype E]
    (w : E → ℝ) (p : E → Fin 4 → ℝ) :
    Symmetric4 (eventStress w p) := by
  intro i j
  simp only [eventStress]
  refine Finset.sum_congr rfl fun e _ => by ring
THEOREM quadContr_eventStress · IndisputableMonolith/Gravity/RecordFluxStress.lean
quadContr_eventStress · IndisputableMonolith/Gravity/RecordFluxStress.lean:71 
/--
Quadratic contraction of the fixed event stress against an arbitrary probe.
The stress is constructed from `(w, p)` before `k` appears.
-/
theorem quadContr_eventStress {E : Type*} [Fintype E]
    (w : E → ℝ) (p : E → Fin 4 → ℝ) (k : Fin 4 → ℝ) :
    quadContr (eventStress w p) k =
      ∑ e : E, w e * (∑ μ, p e μ * k μ) ^ 2 := by
  unfold quadContr eventStress
  have hpull (i j : Fin 4) :
      (∑ e : E, w e * p e i * p e j) * k i * k j =
        ∑ e : E, w e * p e i * p e j * k i * k j := by
    rw [mul_assoc, Finset.sum_mul]
    exact Finset.sum_congr rfl fun e _ => by ring
  simp_rw [hpull]
  have hinner (i : Fin 4) :
      (∑ j, ∑ e : E, w e * p e i * p e j * k i * k j) =
        ∑ e : E, ∑ j, w e * p e i * p e j * k i * k j :=
    Finset.sum_comm
  simp_rw [hinner]
  rw [Finset.sum_comm]
  exact Finset.sum_congr rfl fun e _ => sum_mul_sq (p e) k (w e)
THEOREM eventStress_zero_of_covector_zero · quadContr_eventStress_zero_of_covector_zero · IndisputableMonolith/Gravity/RecordFluxStress.lean
eventStress_zero_of_covector_zero · IndisputableMonolith/Gravity/RecordFluxStress.lean:94 
/-- Zero covectors force zero event stress (load-bearing decoy). -/
theorem eventStress_zero_of_covector_zero {E : Type*} [Fintype E]
    (w : E → ℝ) :
    eventStress w (fun _ _ => (0 : ℝ)) = 0 := by
  ext i j
  simp [eventStress]
quadContr_eventStress_zero_of_covector_zero · IndisputableMonolith/Gravity/RecordFluxStress.lean:101 
/-- 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
THEOREM eventStress_ne_zero_of_unit_channel · IndisputableMonolith/Gravity/RecordFluxStress.lean
eventStress_ne_zero_of_unit_channel · IndisputableMonolith/Gravity/RecordFluxStress.lean:164 
/--
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
THEOREM exteriorStepHeat_eq_sum_channelDeltaZ · IndisputableMonolith/Gravity/RecordFluxStress.lean
exteriorStepHeat_eq_sum_channelDeltaZ · IndisputableMonolith/Gravity/RecordFluxStress.lean:205 
/--
Posted exterior heat equals the sum of signed exterior channel deltas.
This links the new channel weights to the committed cut heat.
-/
theorem exteriorStepHeat_eq_sum_channelDeltaZ {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H) :
    exteriorStepHeat c c' = ∑ ch : ExteriorCutChannel a s, channelDeltaZ c c' ch := by
  unfold exteriorStepHeat
  rw [recordFlux_eq_sum_bitDelta, exteriorRecord_as_channels c, exteriorRecord_as_channels c']
  set fA := fun i : Fin a => channelBitReadout c (Sum.inl i)
  set fS := fun j : Fin s => channelBitReadout c (Sum.inr j)
  set gA := fun i : Fin a => channelBitReadout c' (Sum.inl i)
  set gS := fun j : Fin s => channelBitReadout c' (Sum.inr j)
  have hlen : (List.ofFn fA).length = (List.ofFn gA).length := by
    simp [List.length_ofFn]
  rw [List.zipWith_append (f := bitDelta) hlen]
  rw [zipWith_bitDelta_ofFn fA gA, zipWith_bitDelta_ofFn fS gS, List.sum_append]
  rw [sum_ofFn_eq_sum, sum_ofFn_eq_sum, Fintype.sum_sum_type]
  refine congrArg₂ (· + ·) ?_ ?_
  · refine Finset.sum_congr rfl fun i _ => ?_
    simp only [channelDeltaZ, bitDelta, fA, gA, channelBitReadout]
  · refine Finset.sum_congr rfl fun j _ => ?_
    simp only [channelDeltaZ, bitDelta, fS, gS, channelBitReadout]

What this page does not claim

This module does not claim to derive the Einstein field equations. This module does not claim to model the Unruh effect or any horizon thermodynamics. This module does not claim that the stress matrix is unique or that its covector assignment is forced.

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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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Derived articles

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