Foundation Ledger Field Cone Field Cone Card

The field cone count

A ledger, in the Recognition Science framework, is a discrete record of events. Each cell, called a voxel, holds a list of entries. The declaration fieldConeCard is a definition: it counts, over a chosen finite set of voxels, the total number of admissible continuations each voxel currently allows. Think of it as a census of possible next steps the ledger could take from its present state.

The key theorem attached to this definition, fieldCone_card_monotone, proves that this count never decreases when the ledger takes one step forward. If at a given moment the field permits, say, 1,000 possible continuations, then after one tick, the new count is at least 1,000. The future cone widens; it never contracts. This is a purely combinatorial statement about finite sets, and it is proved in the machine-checked library of formal theorems without extra axioms.

In Recognition Science, the declaration also supports a companion fact: the hub, the vantage point that can address every committed entry, is content-empty at the present frontier. Reading a voxel at its own write-head index returns none, meaning no value has been committed there yet. Together the two facts form a field time certificate: no information is retrieved from the future, and the space of possible futures only grows.

What the declaration does not claim is any physical identification. The theorem is about abstract finite sets and successor relations. The framework models voxels as physical locations and entries as recognition events, but that modeling step is a choice, not a proved consequence. The cone count itself says nothing about what the ledger represents, only that its admissible continuations are monotone.

/-- One step of the field-level admissible cone at a fixed voxel set, using a per-voxel
successor relation `next`. The cone over the field is the union of the per-voxel cones. -/
def fieldConeCard (next : E → Finset E) [DecidableEq E] (S : V → Finset E) (vs : Finset V) : ℕ :=
  ∑ v ∈ vs, (S v).card

/-- **Field-level widening cone.** The total admissible-continuation count over a finite
voxel set is nondecreasing under one cone step at every voxel: the field future cone never
shrinks. -/
theorem fieldCone_card_monotone (next : E → Finset E) [DecidableEq E]
    (S : V → Finset E) (vs : Finset V) :
    fieldConeCard next S vs ≤ fieldConeCard next (fun v => coneStep next (S v)) vs := by
  unfold fieldConeCard
  apply Finset.sum_le_sum
  intro v _
  exact cone_card_monotone next (S v)

/-- **Hub content-emptiness.** Reading a voxel at its own present write-head index returns
`none`: the frontier is unwritten. The hub is address-complete (every committed index is
readable) but content-empty at the present (no committed value to retrieve from the
future). -/
theorem hub_content_empty (F : LedgerField V E) (v : V) :
    (F v)[writeHeadAt F v]? = none := by
  unfold writeHeadAt writeHead
  exact List.getElem?_eq_none (le_refl _)

/-- **Field-level widening cone.** The total admissible-continuation count over a finite
voxel set is nondecreasing under one cone step at every voxel: the field future cone never
shrinks. -/
theorem fieldCone_card_monotone (next : E → Finset E) [DecidableEq E]
    (S : V → Finset E) (vs : Finset V) :
    fieldConeCard next S vs ≤ fieldConeCard next (fun v => coneStep next (S v)) vs := by
  unfold fieldConeCard
  apply Finset.sum_le_sum
  intro v _
  exact cone_card_monotone next (S v)

What this page does not claim

The declaration does not identify voxels with physical locations; that identification is a modeling choice. The monotonicity theorem does not claim the cone count is bounded or that it grows at any particular rate. The field time certificate does not claim that no information can ever be retrieved from a voxel at a future index.

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/LedgerFieldCone.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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