Gravity Record Flux Boost Heat Exterior Step Heat Cast Eq Sum Channel Delta

The heat sum

A recognition cut is a discrete step in a ledger of events. The framework's gravity model assigns to each such step a real number called the posted heat, and it also assigns a change, called channelDelta, to each of the finitely many active channels that carry the step. The declaration exteriorStepHeat_cast_eq_sum_channelDelta proves that these two quantities are the same: the posted heat equals the sum of the channel changes. In plain terms, the heat recorded at the step is nothing more than the total of the individual channel contributions.

This is a theorem, not a definition. The equality is proved from the existing definitions of exterior step heat and channel delta, using a cast from the integer-valued heat to the real numbers. The proof is a direct algebraic consequence of the earlier integer-valued identity exteriorStepHeat_eq_sum_channelDeltaZ. The theorem does not introduce any new physical assumption; it merely states an exact relationship that holds by construction of the model.

The theorem is one link in a longer chain. A separate theorem, quadContr_cutEventStress_eq_sq_mul_heat, shows that when every channel covector has the same pairing q with a probe, the null contraction of the event-stress matrix equals q² times the posted heat. A further theorem, matchesPostedBoostHeat_of_attachment, combines that with an explicit calibration law to derive the local boost-heat equation. The exteriorStepHeat theorem supplies the heat-as-sum step that those later results rely on.

What the theorem does not claim is just as important. It does not claim that the channel covectors, the probe, or the uniform pairing are derived from the recognition geometry; those are explicit model inputs, stated in the definition UniformProbeAttachment. It does not claim that the physical heat scale is fixed; that comes from a separate model assumption, PostedBoostHeatNormalizationAssumption, which is not a theorem. And it does not claim that zero channel covectors can represent a nonzero heat; a separate theorem, zero_covectors_fail_nonzero_posted_heat, shows that they cannot. The theorem is a clean algebraic bridge, and the physical interpretation of its inputs remains open.

/--
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

/--
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

/--
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

What this page does not claim

The channel covectors, probe, and uniform pairing are derived from recognition geometry. The physical heat scale is fixed by the theorem. Zero channel covectors can represent a nonzero posted heat.

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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