Gravity Record Flux Boost Heat Quad Contr Cut Event Stress Eq Sq Mul Heat

The heat-stress bridge

In the Recognition Science framework, the declaration establishes a precise algebraic bridge between two quantities defined on a local cut: the posted heat, which is the sum of real channel weights, and the null contraction of the event-stress matrix. The theorem states that if every active channel covector has the same pairing q with a single probe vector k, then the stress contraction is exactly q squared times the posted record heat. This is a proved equality in the framework's machine-checked library of formal theorems.

The proof itself is a direct algebraic computation from the definitions. The posted heat is shown to equal the sum of the channel deltas, and the uniform attachment condition converts the stress contraction into that same sum multiplied by the common squared pairing. The theorem does not stand alone: it is one link in a chain that, with an additional explicit calibration assumption, yields the local boost-heat equation, where calibrated posted heat equals minus surface gravity times boost moment times null stress flux.

What the declaration does not claim is equally important. The uniform attachment of channels to a probe is an explicit geometric model input, not something inferred from the cut record. The calibration law that converts bit-valued posted heat into a physical heat scale is also a named model assumption, not a derived consequence. The framework itself marks the derivation of these model inputs from recognition geometry as an open target. A load-bearing decoy theorem shows why the model inputs matter: zero channel covectors produce zero stress flux, so they cannot represent a nonzero calibrated posted heat, ruling out a trivial reading of the bridge.

In plain terms, the declaration proves a conditional statement: if the channels attach uniformly to a probe and the calibration is chosen, then heat and stress flux are locked together by a square. The conditions are the price of the exactness. The framework's contribution is to make the algebra of that conditional statement airtight, while leaving the physical interpretation of the inputs as a separate, open modeling question.

/--
Uniform probe pairing converts the fixed event-stress contraction into the
posted cut heat times the common squared pairing.
-/
theorem quadContr_cutEventStress_eq_sq_mul_heat
    {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)
    (q : ℝ)
    (hattach : UniformProbeAttachment p k q) :
    quadContr (cutEventStress c c' p) k =
      q ^ 2 * (exteriorStepHeat c c' : ℝ) := by
  rw [quadContr_cutEventStress]
  unfold UniformProbeAttachment at hattach
  simp_rw [hattach]
  rw [← Finset.sum_mul]
  rw [← exteriorStepHeat_cast_eq_sum_channelDelta c c']
  ring

/--
The real-valued posted heat is the sum of the real channel weights.
-/
theorem exteriorStepHeat_cast_eq_sum_channelDelta
    {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H) :
    (exteriorStepHeat c c' : ℝ) =
      ∑ ch : ExteriorCutChannel a s, channelDelta c c' ch := by
  have h := exteriorStepHeat_eq_sum_channelDeltaZ c c'
  unfold channelDelta
  exact_mod_cast h

/--
Uniform attachment of the cut channels to one probe.  The common pairing is
an explicit geometric MODEL input; it is not inferred from the cut record.
-/
def UniformProbeAttachment
    {a s : ℕ}
    (p : ExteriorCutChannel a s → Fin 4 → ℝ)
    (k : Fin 4 → ℝ)
    (q : ℝ) : Prop :=
  ∀ ch, (∑ μ, p ch μ * k μ) = q

/--
Named MODEL normalization assumption required to convert bit-valued posted
heat into the local boost-energy normalization.  It supplies the whole
physical heat-scale conversion and is not derived from the record data.
-/
def PostedBoostHeatNormalizationAssumption
    (surfaceGravity boostMoment heatScale q : ℝ) : Prop :=
  heatScale = -surfaceGravity * boostMoment * q ^ 2

/--
Load-bearing decoy: zero channel covectors produce zero stress flux, so they
cannot represent a nonzero calibrated posted heat.
-/
theorem zero_covectors_fail_nonzero_posted_heat
    {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H)
    (k : Fin 4 → ℝ)
    (surfaceGravity boostMoment heatScale : ℝ)
    (hheat : heatScale * (exteriorStepHeat c c' : ℝ) ≠ 0) :
    ¬ heatScale * (exteriorStepHeat c c' : ℝ) =
      -surfaceGravity * boostMoment *
        quadContr
          (cutEventStress c c' (fun _ _ => (0 : ℝ))) k := by
  intro h
  rw [cutEventStress_zero_of_covector_zero] at h
  have hz :
      quadContr (0 : Matrix (Fin 4) (Fin 4) ℝ) k = 0 := by
    simp [quadContr]
  rw [hz, mul_zero] at h
  exact hheat h

What this page does not claim

The theorem does not claim the uniform attachment condition holds for any actual physical cut. The theorem does not claim the calibration assumption is derived from the record data. The theorem does not claim the model inputs are derivable from recognition geometry.

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/RecordFluxBoostHeat.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.

Derived articles

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