Holography Landauer Calorimeter Descent Five Channel Sum Ne Step Heat Cell

The five-channel decoy

The declaration five_channel_sum_ne_stepHeatCell is a machine-checked theorem in the Recognition Science framework's library of formal proofs. It states a precise inequality: if you add up the heat posted through five of the six faces of a cell, on a particular concrete step, the sum is not equal to the heat posted through the whole cell. The proof is computational; the library checks it by direct evaluation on the specific cell configuration named cell0 and the step named postingStepTarget.

In plain terms, the framework models a discrete ledger of recognition events, where each event posts a cost. A cell is a unit with six faces, and each face can be open or closed. The heat posted by a step is the sum over the closed faces. The theorem shows that if you drop one face, the sum changes: five faces do not give the same heat as six. This is not a numerical approximation; it is an exact inequality proved by computation.

The framework's library also proves the converse for six faces: the six-channel sum equals the posted cell heat exactly, by a theorem called stepHeatCell_eq_sum_face_channels. The five-channel result is a decoy: it shows that a truncated sum fails the carrier premise, meaning it cannot serve as a valid heat posting. The library records this failure in the theorem five_channel_fails_unit_carrier, which states that the five-channel heat does not inhabit the required type HeatIsPostedRecordFlux.

What the declaration does not claim: it does not establish that five faces are physically impossible, nor that a five-sided cell cannot exist. It only shows that, within this framework's posting rule, omitting one face breaks the equality between the face sum and the cell heat on the tested step. The framework's own documentation calls this an honest residual: the result is still the ledger posting rule, lifted from one-channel atoms to a cell type. It does not supply an independent external calorimeter beyond that rule.

The practical consequence is that the framework's heat accounting is exact only when all six faces are counted. A five-channel truncation is not a small error; it is a categorical failure. This matters for anyone building on the framework: the six-channel aggregation is the minimal correct unit, and any reduction to five channels invalidates the posting.

/-- On the banked single-vertex posting step, the five-channel truncation
(misses face 0) disagrees with posted cell flux. -/
theorem five_channel_sum_ne_stepHeatCell :
    (∑ i : Fin 5, stepHeat (faceBit i.succ) cell0 postingStepTarget) ≠
      stepHeatCell cell0 postingStepTarget := by
  native_decide

theorem five_channel_fails_unit_carrier :
    ¬ HeatIsPostedRecordFlux 1 (fiveChannelHeat 1) := by
  intro h
  have hstep := h cell0 postingStepTarget
  have hne := five_channel_sum_ne_stepHeatCell
  simp only [fiveChannelHeat] at hstep
  exact hne (by exact_mod_cast hstep)

/-- Channel aggregation: posted cell flux is exactly the sum of the six
Clausius face-channel heats. -/
theorem stepHeatCell_eq_sum_face_channels (c c' : CellCfg) :
    stepHeatCell c c' = sixChannelNativeHeat c c' := by
  unfold sixChannelNativeHeat stepHeatCell recordFlux
  rw [faceRecord_eq_faceBits c, faceRecord_eq_faceBits c', sum_fin6]
  simp [stepHeat, faceBit]
  ring

What this page does not claim

Five-faced cells are physically impossible. The theorem provides an independent external calorimeter. The result applies to all possible cell configurations beyond the tested step.

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/LandauerCalorimeterDescent.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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