Foundation Ledger Field Write Head At Advances

The advancing write-head

A recognition field is a spatial arrangement of independent histories. The framework models it as an assignment of an append-only ledger to each voxel, where a ledger is a discrete record of events that can only grow by appending. A commit is the act of appending one event to one voxel's history. The declaration writeHeadAt_advances proves that this commit moves the written voxel's write-head, the index of the next unwritten position, forward by exactly one.

The theorem is local. It states that for any field, voxel, and event, the write-head at the written voxel after the commit equals its previous value plus one. This is the multi-voxel generalization of the single-voxel rule that appending advances the present index by one. The proof is a direct lift of that single-voxel theorem, and the framework's machine-checked library of formal theorems certifies it as an axiom-clean result, meaning it relies only on the standard logical axioms and no framework-specific postulates.

In Recognition Science, this locality is the keystone for a larger claim. Together with the companion theorems that a commit leaves every other voxel untouched and that the committed past is immutable, it certifies that a recognition field behaves as a set of independent, append-only ledgers. The framework identifies the spatial index V with physical voxels and the entries E with recognition events, but that identification is a modeling choice, not a proved theorem. The theorem itself is about the abstract type: it proves what happens to the write-head when a commit occurs, not that physical space is such a field.

What this changes is the picture of time in the framework. Time is not a single global stream but a per-voxel, append-only record. The write-head advancing by exactly one at the written voxel, and nowhere else, is what makes each voxel's history a clean, addressable sequence. The past is read-only, the present is a single advancing index, and a commit never edits another voxel's record. This is the structural basis on which the framework builds its field-level widening cone and content-emptiness results, though those later developments are not part of this theorem.

/-- **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 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

/-- **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 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 theorem does not claim that physical space is a recognition field; that identification is a modeling choice. The theorem does not claim anything about the content of the events, only about the structure of the write-head. The theorem does not claim that the write-head advances by more than one, or that it can be reset.

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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