Holography Owner Indexed Horizon Cut Heat Eq Boundary Profile Ne Debit Credit

The owner-indexed horizon cut

The ledger, a discrete record of posted events, has a boundary: the set of channels through which it can be read. The Recognition Science framework's machine-checked library of formal theorems proves a structural fact about that boundary. When the channels are indexed by owner, meaning by account and by side (debit or credit), the boundary profile, the count of channels per owner, separates debit from credit completely. Two nonnegative ledger states with the same boundary profile have the same debit and credit entries. The proof is theorem-tier, meaning it is verified by the Lean kernel with no framework-specific axioms.

The same library proves how a posting, a single committed event, changes the boundary. A legal tick, one committed unit, adds exactly one new channel, and that channel's address is the posting itself: the pair (k, side) where k is the account and side is debit or credit. The old channels keep their identities and readings. This is the owner-indexed horizon cut: the boundary of the ledger after a posting encodes the posting's address, so the boundary shape recovers the posting from the two states alone. The heat of a legal tick, the change in active bits, is exactly 1, matching the corpus's moving step heat.

The theorem does not claim a physical bridge. The library leaves open the connection from this owner-indexed carrier to pair-kernel source normalization and the remaining physical equality. The owner reading itself is a model encoding, a definitional choice, not a derived fact: every committed unit is an active posted bit. The proof also does not involve the framework's constants: no source scale, no hbar, no G, no pi in the forcing path. The axioms are limited to the standard three: propext, Classical.choice, and Quot.sound.

What this changes: the boundary of a ledger is not a passive surface. It carries enough structure to identify the actor and the side of each posting. The boundary profile is not a free assignment; a theorem shows that no constant profile can hold for all ledgers. The boundary is a fingerprint of the ledger's history, and the framework has a machine-checked proof of that fact.

theorem boundaryProfile_determines_nonneg_state {d : Nat} {L L2 : LedgerState d}
    (hn : NonnegLedger (d := d) L) (hn2 : NonnegLedger (d := d) L2)
    (h : boundaryProfile L = boundaryProfile L2) :
    L.debit = L2.debit ∧ L.credit = L2.credit := by
  have hcount : ∀ k side, ownerCount L k side = ownerCount L2 k side :=
    (boundaryProfile_eq_iff_ownerCounts L L2).1 h
  constructor
  · funext k
    simp [(nonneg_entries_of_ownerCount hn k).1,
      (nonneg_entries_of_ownerCount hn2 k).1, hcount k Side.debit]
  · funext k
    simp [(nonneg_entries_of_ownerCount hn k).2,
      (nonneg_entries_of_ownerCount hn2 k).2, hcount k Side.credit]

theorem newChannel_owner {d : Nat} {L : LedgerState d}
    (hn : NonnegLedger (d := d) L) (k : Fin d) (side : Side) :
    (newChannel hn k side).1 = (k, side) :=
  rfl

theorem ownerStepHeat_legalAtomicTick {d : Nat} {L L' : LedgerState d}
    (hn : NonnegLedger (d := d) L)
    (h : LegalAtomicTick (d := d) L L') :
    ownerStepHeat L L' = 1 := by
  obtain ⟨k, side, rfl⟩ := legalAtomicTick_implies_PostingStep (d := d) h
  exact ownerStepHeat_post hn k side

theorem postingOfProfiles_of_post {d : Nat} [NeZero d] {L : LedgerState d}
    (hn : NonnegLedger (d := d) L) (k : Fin d) (side : Side) :
    postingOfProfiles L (post L k side) = (k, side) := by
  classical
  have hprop :
      boundaryProfile (post L k side) (k, side) =
          boundaryProfile L (k, side) + 1 ∧
        (∀ q : Fin d × Side, q ≠ (k, side) →
          boundaryProfile (post L k side) q = boundaryProfile L q) := by
    constructor
    · simp [boundaryProfile, ownerCount_post hn k side k side]
    · intro q hne
      simp [boundaryProfile, ownerCount_post hn k side q.1 q.2, hne]
  have hε :=
    Classical.epsilon_spec
      (p := fun p : Fin d × Side =>
        boundaryProfile (post L k side) p = boundaryProfile L p + 1 ∧
          (∀ q, q ≠ p → boundaryProfile (post L k side) q = boundaryProfile L q))
      ⟨(k, side), hprop⟩
  set p := postingOfProfiles L (post L k side)
  have hform := ownerCount_post hn k side p.1 p.2
  by_cases heq : p = (k, side)
  · exact heq
  · have h1 : boundaryProfile (post L k side) p = boundaryProfile L p := by
      simpa [boundaryProfile, heq] using hform
    have h2 : boundaryProfile (post L k side) p = boundaryProfile L p + 1 := hε.1
    omega

What this page does not claim

The physical bridge from this owner-indexed carrier to pair-kernel source normalization is not proved. The owner reading encoding is a model choice, not a derived fact. The theorem does not involve the framework's constants hbar, G, or pi.

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

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