Foundation Ledger Field Cone Field Cone Card Monotone

The widening field cone

A ledger here is a discrete record of events, and a recognition is one event being recorded. In the Recognition Science framework, the ledger's future is not a single prediction but a cone: the set of all admissible continuations from the present state. The theorem fieldCone_card_monotone states that this cone never contracts. If you count the admissible continuations over any finite collection of ledger positions, the count at a longer horizon is always at least the count at a shorter one. The future widens; it never shrinks.

The proof is a direct lift of a single-position fact to the whole field. The framework's machine-checked library of formal theorems first proves cone_card_monotone, which says the admissible-continuation count at one position is nondecreasing under one step of the successor relation. The field-level theorem sums that count over a finite set of positions and applies the single-position result to each one. Since a sum of nondecreasing terms is nondecreasing, the field cone is nondecreasing in horizon. The statement is tagged THEOREM, meaning it is proved in the library with no special axioms beyond the ambient type theory.

What the theorem does not claim is just as important. It does not say the cone grows, only that it does not shrink; a cone that stays the same size still satisfies the inequality. It says nothing about the content of the futures, only their count. A companion theorem, hub_content_empty, supplies the content side: at the present write-head, the frontier is unwritten, so no committed value can be retrieved from the future. Together they form the field time certificate: addressing is complete, content at the frontier is empty, and the cone of possibilities widens monotonically.

The practical consequence is a consistency guarantee for the framework's model of time. If the cone could contract, some futures would become inadmissible as the horizon grows, which would mean the ledger's own rules erased possibilities. The theorem rules that out for any finite set of positions, uniformly. It is a structural fact about the ledger's time, not a physical prediction; the identification of these formal positions with physical voxels and recognition entries is a modeling choice, not part of the proof.

/-- **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)

/-- **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 _)

/-- 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

What this page does not claim

The theorem does not claim the cone grows, only that it does not shrink. It does not claim anything about the content of the futures, only their count. It does not claim that physical voxels and recognition entries are anything more than a modeling identification.

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)

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