Holography Moving Recognition Horizon Cut Moving Step Heat Eq Exterior Step Heat

The moving cut

A moving boundary in this framework is a discrete record of events, a ledger that gains one new slot for a bit while preserving every old slot and its reading intact. The framework's library of machine-checked theorems proves that when such a boundary moves, the heat it registers is exactly the change in its active posted bits. This is the theorem named movingStepHeat_eq_exteriorStepHeat, and it holds for a single step: the new heat equals the old heat plus the value of the newly exposed bit, which is either 0 or 1.

The proof is short and exact. The library defines moving heat as the difference between active exterior bits on two cuts, and it proves that this difference equals the previously established exterior step heat when the context is fixed. It also proves that extending a cut by one bit adds exactly that bit's value to the active count. So the theorem is not an approximation or a limit; it is an identity for discrete steps, with no smooth interpolation involved.

What the theorem does not claim is just as important. It does not provide a smooth interpolation between steps, nor does it introduce null covectors, Raychaudhuri's equation, Ricci curvature, or a sourced field equation. It is a discrete moving-boundary theorem, not a statement about continuous spacetime geometry. The framework's own documentation says so explicitly, and the proof relies only on the definitions and the previously proved exterior step heat, not on any additional physical postulate.

In plain terms, the declaration establishes that for a single discrete move of the recognition boundary, the heat is the change in active bits. It is a consistency check: the moving case reduces to the fixed case, and the newly exposed bit is the only source of heat. This is a small but load-bearing step in the framework's account of how a boundary that records events behaves when it moves.

/--
On a fixed context, varying-context heat reduces to the already proved
`exteriorStepHeat`.
-/
theorem movingStepHeat_eq_exteriorStepHeat
    {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H) :
    movingStepHeat c c' = exteriorStepHeat c c' := by
  unfold movingStepHeat
  rw [activeExteriorBits_eq_exteriorPotential,
    activeExteriorBits_eq_exteriorPotential,
    exteriorStepHeat_eq_potential]

/-- Exact moving-cut heat: only the newly exposed bit is posted. -/
theorem movingStepHeat_extendCut {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c : LocalCut H) (x : ZMod 2) :
    movingStepHeat c (extendCut c x) = postedBitValue x := by
  unfold movingStepHeat
  rw [activeExteriorBits_extendCut]
  ring

/--
On a fixed context, varying-context heat reduces to the already proved
`exteriorStepHeat`.
-/
theorem movingStepHeat_eq_exteriorStepHeat
    {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H) :
    movingStepHeat c c' = exteriorStepHeat c c' := by
  unfold movingStepHeat
  rw [activeExteriorBits_eq_exteriorPotential,
    activeExteriorBits_eq_exteriorPotential,
    exteriorStepHeat_eq_potential]

What this page does not claim

The theorem does not claim to describe a smooth or continuous moving boundary. It does not claim to derive any curvature or field equation from the moving cut. It does not claim that the moving cut is the same as a fixed cut in all respects.

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/MovingRecognitionHorizonCut.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: