Foundation Ledger Field Commit At Local

Locality of a field commit

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition field is an assignment of an independent, append-only ledger to each voxel of a spatial index. The declaration commitAt_local proves a locality property: a commit at one voxel does not change any other voxel. Formally, for a field F, a voxel v, an entry e, and any other voxel w with w ≠ v, the equation commitAt F v e w = F w holds. This means writes are strictly local; no spooky cross-voxel edits occur.

The proof is a direct consequence of the definition of commitAt, which uses Function.update to replace only the list at voxel v. The theorem is tagged THEOREM, meaning it is proved in the machine-checked library of formal theorems with no axioms beyond the standard three. It is part of a broader set of field-level theorems: past_immutable_at shows the committed past of a voxel cannot be altered, writeHeadAt_advances shows the per-voxel write-head advances by exactly one at the written voxel, and writeHeadAt_other shows it is unchanged elsewhere. Together these establish the field as an append-only, local, and addressable structure.

What commitAt_local does not claim is any physical identification. The theorem is about the abstract type LedgerField V E := V → List E, where V is a spatial index and E is the type of entries. The framework models physical voxels and recognition entries by choosing V and E, but that identification is a MODEL choice argued in a companion paper, not a theorem. The declaration does not assert that physical space is discrete, that recognition events exist, or that the field is causally connected to any conventional physics. It only establishes a formal property of a mathematical structure.

The consequence is that any theory built on this field can rely on locality without further proof. The hub content-emptiness lemma and the field-level widening cone build on this type, so the locality property is a keystone for those results. For a reader, the practical takeaway is that the framework's core data structure behaves like a distributed ledger with independent per-voxel histories, and the locality guarantee is not an assumption but a proved theorem.

/-- **Locality.** A commit at voxel `v` leaves every other voxel `w ≠ v` exactly as it was. -/
theorem commitAt_local (F : LedgerField V E) (v : V) (e : E) (w : V) (hw : w ≠ v) :
    commitAt F v e w = F w := by
  unfold commitAt
  rw [Function.update_of_ne hw]

/-- **Field past immutability.** Committing at `v` cannot alter the committed past of `v`. -/
theorem past_immutable_at (F : LedgerField V E) (v : V) (e : E) :
    (commitAt F v e v).take (F v).length = F v := by
  rw [commitAt_self]
  exact past_immutable (F v) e

/-- **Write-head advance (written voxel).** The present index at `v` moves forward by one. -/
theorem writeHeadAt_advances (F : LedgerField V E) (v : V) (e : E) :
    writeHeadAt (commitAt F v e) v = writeHeadAt F v + 1 := by
  unfold writeHeadAt
  rw [commitAt_self]
  exact writeHead_advances (F v) e

/-- **Write-head unchanged (other voxels).** No other voxel's present index moves. -/
theorem writeHeadAt_other (F : LedgerField V E) (v : V) (e : E) (w : V) (hw : w ≠ v) :
    writeHeadAt (commitAt F v e) w = writeHeadAt F w := by
  unfold writeHeadAt
  rw [commitAt_local F v e w hw]

What this page does not claim

The declaration does not claim that physical space is discrete or that recognition events exist. The identification of V with physical voxels and E with recognition entries is a model choice, not a theorem. The theorem does not assert any causal connection to conventional physics.

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

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