Holography Landauer Calorimeter Descent Six Channel Heat Eq Scaled Step Heat Cel

The six-channel heat identity

In the Recognition Science framework, a ledger is a discrete record of events, and recognition is the forced cost of recording one. The declaration `sixChannelHeat_eq_scaled_stepHeatCell` is a theorem in the framework's machine-checked library of formal theorems. It states that for any real scaling factor q, the six-channel heat function equals q times the cell's posted heat. In plain terms: the heat attributed to a cell by summing its six faces is the same as the heat the cell's own ledger entry posts, up to that scaling factor.

The proof is a chain of equalities. The framework defines six face channels, each a Clausius one-channel map on a cell configuration. A theorem, `stepHeatCell_eq_sum_face_channels`, proves that the cell's posted heat is exactly the sum of these six channel heats. The declaration in question then multiplies both sides by q. This is a type-lift: it shows that the six-channel construction satisfies the same carrier premise as the direct posted heat, but through aggregation rather than by definitional fiat.

This matters because the framework distinguishes between what is true by definition and what is true by proof. The six-channel heat is not defined as q times the cell heat; it is constructed from face atoms and then shown to equal that product. The theorem also records uniqueness: any heat function satisfying the carrier premise must equal the tautological selector pointwise. A five-channel truncation, omitting one face, provably fails the carrier premise on a concrete posting step, which sharpens the result.

In Recognition Science, this is a step in a descent from a whole cell to its faces. It establishes that the cell-level heat posting rule is consistent with a face-level aggregation rule. It does not, however, supply an independent external calorimeter beyond that rule. The honest residual, stated in the framework's own documentation, is that this is still the ledger posting rule, lifted from one-channel atoms to cell type. The sharper wall, that packaging unit face quantum with face additivity forces the tautological carrier, is out of scope for this aggregation module.

/-- The six-channel calorimeter equals scaled posted flux by channel
aggregation, not by definitional identification with `stepHeatCell`. -/
theorem sixChannelHeat_eq_scaled_stepHeatCell (q : ℝ) (c c' : CellCfg) :
    sixChannelHeat q c c' = q * (stepHeatCell c c' : ℝ) := by
  simp only [sixChannelHeat, stepHeatCell_eq_sum_face_channels c c']

/-- 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

/-- Any inhabitant of the carrier premise is pointwise the tautological
selector. The six-channel construction meets that unique value by the
aggregation theorem above. -/
theorem heat_carrier_unique (q : ℝ) (heat : PhysicalStepHeat)
    (h : HeatIsPostedRecordFlux q heat) :
    heat = tautologicalPostedHeat q := by
  funext c c'
  exact h c c'

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)

What this page does not claim

This theorem does not provide an independent external calorimeter; it is a type-lift of the existing posting rule. The proof does not establish that the six-channel construction is the only way to satisfy the carrier premise; it shows uniqueness of the value, not of the construction. The scaling factor q is not derived here; it is a free real parameter in the statement.

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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Derived articles

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