Holography Landauer Calorimeter Descent Face Record Eq Face Bits

The face record theorem

In Recognition Science, a cell is a discrete unit of space with eight corners. The theorem faceRecord_eq_faceBits states that the cell's face record, the list of which of its six faces are closed, is exactly the list of six binary bits computed from which corners are closed. Each bit asks a simple yes or no question about a group of four corners: are all four of them closed? The theorem proves, by direct computation, that the face record and the bit list are the same object. There is no hidden information in the face record beyond what the corners already determine.

The proof is a definitional equality, meaning the two sides are the same by definition once the terms are unfolded. This is the strongest kind of equality in the machine-checked library of formal theorems: it requires no argument, only the expansion of definitions. The theorem is tagged THEOREM because it is proved in the library with no axioms beyond the standard three. It is a structural fact about the cell, not a physical law.

In Recognition Science, this theorem is a load-bearing step in a larger construction. The library uses it to show that the heat posted by a cell, the discrete record of energy change, equals the sum of six separate face-channel heats. Each face channel is a Clausius one-channel map, a term for a single face's contribution. The aggregation theorem stepHeatCell_eq_sum_face_channels proves this equality. The six-channel calorimeter built from face atoms then satisfies the posting rule, and the library proves that any heat carrier satisfying the rule is unique.

The theorem does not claim that the face record is physically meaningful on its own. It does not provide an independent external calorimeter; the six-channel construction is still the ledger posting rule, lifted from one-channel atoms to cell type. The docstring is explicit: aggregation is not an independent calorimeter. The sharper independence wall, where packaging unit face quantum with face additivity would force the tautological carrier, is out of scope. That remains an OPEN target. The theorem also does not say that the corners cause the faces; it says the face record and the bit list are the same list, nothing more.

theorem faceRecord_eq_faceBits (c : CellCfg) :
    faceRecord c =
      [faceBit 0 c, faceBit 1 c, faceBit 2 c, faceBit 3 c, faceBit 4 c,
        faceBit 5 c] :=
  rfl

theorem faceRecord_eq_faceBits (c : CellCfg) :
    faceRecord c =
      [faceBit 0 c, faceBit 1 c, faceBit 2 c, faceBit 3 c, faceBit 4 c,
        faceBit 5 c] :=
  rfl

/-- 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

/-- The six-channel Clausius lift inhabits `HeatIsPostedRecordFlux` via
channel aggregation (not by defining heat as `q * stepHeatCell`). This is a
type-lift of the banked posting rule, not an independent external calorimeter. -/
theorem sixChannelHeat_is_posted_record_flux (q : ℝ) :
    HeatIsPostedRecordFlux q (sixChannelHeat q) := by
  intro c c'
  exact sixChannelHeat_eq_scaled_stepHeatCell q c c'

/-- 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'

What this page does not claim

The theorem does not provide an independent external calorimeter. The theorem does not say that corners cause faces, only that the face record and bit list are the same list. The theorem does not establish a physical recognition-to-linking bridge.

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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