Foundation Ledger Field Past Immutable At

A past that cannot be rewritten

In the Recognition Science framework, reality is modeled as a ledger: a discrete record of events, where each new event is appended to the end of a list. The declaration past_immutable_at is a theorem about this ledger when it is extended to a field, meaning an assignment of an independent ledger to every point, or voxel, of a spatial index. It states a precise property: when a new entry is committed at a particular voxel, the committed past of that voxel remains exactly as it was. Truncating the new ledger back to its previous length returns the original ledger, unchanged.

This property is not a design choice but a proved consequence of how the field is built. In the machine-checked library of formal theorems, a field commit is defined as updating only the specified voxel, leaving all others untouched. The theorem past_immutable_at then follows directly from the single-voxel theorem of the same name. The proof is axiom-clean, meaning it relies only on the standard logical axioms of the ambient type theory, with no framework-specific assumptions.

What the theorem establishes, in plain terms, is that the past is read-only. Once an entry is committed, it can be read back at any later time and will always return the same value. This is what makes the ledger a reliable record: the history is not just stored, it is guaranteed to be immutable. The theorem also guarantees that the write-head, the present index at a voxel, advances by exactly one per commit, and that no other voxel's write-head moves. Together these properties form a certificate that the field behaves as a proper append-only ledger.

What past_immutable_at does not claim is equally important. It does not claim that the physical world is actually a ledger, nor that the spatial index V corresponds to real physical voxels. That identification is a modeling choice, argued in a companion paper, not a proved theorem. The theorem operates entirely within the abstract type: given a field of lists and a commit operation, the past is immutable. Whether reality instantiates this structure is a separate question that the declaration leaves open.

/-- **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

/-- **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

What this page does not claim

The theorem does not claim that physical reality is actually a ledger, only that the abstract type has this property. The theorem does not claim that the spatial index V corresponds to real physical voxels; that identification is a modeling choice. The theorem does not claim anything about the content of the entries, only the structure of the record.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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