Holography Local Recognition Horizon Cut Exterior Step Heat Zero Of Same Project

The exterior record's blind spot

A ledger, a discrete record of events, can only report what it contains. The declaration exteriorStepHeat_zero_of_same_projection is a theorem in the framework's machine-checked library of formal theorems. It states that if two horizon cut configurations have the same exterior projection, then the one-step exterior heat between them is exactly zero. The exterior projection is the part of the record visible from outside the horizon: the exterior-private entries followed by the seam entries. Interior-private and rest-of-universe data are traced out, meaning they are not present in the exterior record at all.

The theorem is a direct consequence of how exterior heat is defined. Exterior heat is the change in the exterior record potential, which is a weighted count of the bits in that record. If two configurations share the same exterior projection, their exterior records are identical, so the potential is identical, and the heat is zero. This is a statement about the bookkeeping of the discrete record, not about any physical flux across a horizon. The framework's library proves this as a theorem, meaning it is a logical consequence of the definitions and axioms, with no additional assumptions.

In Recognition Science, this result is load-bearing for the record thermodynamics. It guarantees that the exterior ledger balances along every finite trajectory of closed local cuts, a theorem called exterior_books_balance. It also means that any change entirely hidden behind the same exterior projection is invisible to the exterior heat accounting. This is the precise sense in which the exterior record is one-sided: it cannot see what happens in the interior-private or rest-of-universe parts of the configuration.

What the theorem does not claim is equally important. It does not claim that no physical change occurs when the exterior projection is the same. It does not involve a stress tensor, Ricci tensor, focusing law, curvature match, Unruh effect, or Einstein equation. The theorem is about the discrete record and its bookkeeping, not about the continuum spacetime that the record might be thought to describe. The physical identification of the posted flux with horizon heat is a separate modeling choice, not part of this theorem.

/--
Exterior heat ignores every change hidden behind the same exterior projection.
This is the exact one-sidedness statement used by the record bookkeeping.
-/
theorem exteriorStepHeat_zero_of_same_projection
    {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H)
    (h : projA c.cfg = projA c'.cfg) :
    exteriorStepHeat c c' = 0 := by
  have hA : c.cfg.1 = c'.cfg.1 := by
    simpa [projA] using congrArg Prod.fst h
  have hS : c.cfg.2.1 = c'.cfg.2.1 := by
    simpa [projA] using congrArg Prod.snd h
  have hRecord : exteriorRecord c = exteriorRecord c' := by
    simp only [exteriorRecord]
    rw [hA, hS]
  unfold exteriorStepHeat
  rw [hRecord]
  exact recordFlux_self _

/--
Exterior heat ignores every change hidden behind the same exterior projection.
This is the exact one-sidedness statement used by the record bookkeeping.
-/
theorem exteriorStepHeat_zero_of_same_projection
    {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H)
    (h : projA c.cfg = projA c'.cfg) :
    exteriorStepHeat c c' = 0 := by
  have hA : c.cfg.1 = c'.cfg.1 := by
    simpa [projA] using congrArg Prod.fst h
  have hS : c.cfg.2.1 = c'.cfg.2.1 := by
    simpa [projA] using congrArg Prod.snd h
  have hRecord : exteriorRecord c = exteriorRecord c' := by
    simp only [exteriorRecord]
    rw [hA, hS]
  unfold exteriorStepHeat
  rw [hRecord]
  exact recordFlux_self _

/-- Posted exterior heat is exactly the change of record potential. -/
theorem exteriorStepHeat_eq_potential {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H) :
    exteriorStepHeat c c' = exteriorPotential c' - exteriorPotential c := by
  exact recordFlux_eq_weight_sub _ _ (by
    rw [exteriorRecord_length, exteriorRecord_length])

What this page does not claim

The theorem does not claim that no physical change occurs when the exterior projection is the same. The theorem does not involve any stress tensor, Ricci tensor, focusing law, curvature match, Unruh effect, or Einstein equation. The physical identification of the posted flux with horizon heat is a separate modeling choice, not part of this theorem.

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/Holography/LocalRecognitionHorizonCut.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.

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