Gravity Record Flux Boost Heat Record Flux Boost Heat Cert

The heat certificate

In classical physics, the connection between heat and gravity is a deep problem, not a simple bookkeeping identity. The declaration recordFluxBoostHeatCert addresses a narrow, formal slice of that problem. It establishes a conditional bridge: if every active channel covector has the same pairing q with one probe k, then the contraction of the event-stress matrix is exactly q squared times the posted record heat. An explicit calibration law then converts this into the local boost-heat equation.

The ledger, a discrete record of events, is the framework's central object. The recognition cost, the forced expense of noting an event, is what the ledger tracks. The certificate proves that under two named inputs, the sum of the real channel weights, the posted heat, equals a null stress flux scaled by a constant. The first input is the uniform attachment of the cut channels to one probe. The second is the calibration assumption that sets the heat scale to minus surface gravity times boost moment times q squared.

The certificate is a theorem in the framework's machine-checked library of formal theorems. It derives the equality from the two separately named inputs. It also includes a load-bearing decoy: zero channel covectors produce zero stress flux, so they cannot represent a nonzero calibrated posted heat. This guards against a trivial or degenerate representation.

What the certificate does not claim is as important as what it proves. It does not derive the channel covectors, the probe, the uniform-pairing attachment, or the physical heat calibration from recognition geometry. Those are model inputs, chosen by definition, not forced by the framework. The certificate is a bridge between two formal structures, not a derivation of gravity from first principles.

/--
With uniform channel attachment and the explicit calibration, calibrated
posted heat equals minus surface gravity times boost moment times null stress
flux.  The theorem derives the equality from two separately named inputs.
-/
theorem matchesPostedBoostHeat_of_attachment
    {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 surfaceGravity boostMoment heatScale : ℝ)
    (hattach : UniformProbeAttachment p k q)
    (hcal : PostedBoostHeatNormalizationAssumption
      surfaceGravity boostMoment heatScale q) :
    heatScale * (exteriorStepHeat c c' : ℝ) =
      -surfaceGravity * boostMoment *
        quadContr (cutEventStress c c' p) k := by
  rw [quadContr_cutEventStress_eq_sq_mul_heat c c' p k q hattach]
  unfold PostedBoostHeatNormalizationAssumption at hcal
  rw [hcal]
  ring

/--
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 certificate does not derive the channel covectors, probe, or calibration from recognition geometry. It does not claim that the heat-scale conversion is a theorem rather than a definitional choice. It does not establish any physical law of gravity, only a formal equality under two named model inputs.

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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