Foundation Ledger Time Cone Card Monotone

The growing future cone

In the Recognition Science framework, time is not a smooth river but a series of discrete events. The framework models each event as an entry in a ledger, a discrete record of committed events that can only be appended to and never edited. This is the append-only structure of recognition time: the past is fixed and readable, while the future remains open to an expanding set of possibilities.

The declaration cone_card_monotone establishes a simple but central fact about this structure. It proves that the number of admissible future states never decreases as you look further ahead. In plainer terms, the set of possible continuations from any point in the ledger can only grow or stay the same size; it can never shrink. This is the formal guarantee that the future cone, the collection of all admissible next steps, is always widening or holding steady, never closing in.

The proof itself is almost trivial once the definitions are in place. The cone at one step is defined as the current frontier plus all its admissible successors under a given transition rule. Since the frontier is always a subset of this expanded set, the cardinality of the cone must be at least as large as the frontier's. The theorem cone_card_monotone simply applies this subset relation to conclude that the count of admissible states is nondecreasing in horizon.

What this theorem does not claim is just as important. It does not say that the future is predetermined or that every possible state will actually be reached. It only guarantees the count of possibilities does not shrink. The identification of the ledger's entries with actual recognition events, and of the cone step with the set of J-admissible continuations, is a modeling choice argued in the companion paper, not a theorem proved here. The structural facts about lists and finite sets are theorems; the physical interpretation is a model.

This monotonicity is what gives the framework its sense of an open future. If the cone could shrink, the ledger would be closing off possibilities as time moved forward. The theorem ensures that the past, however fixed, never constrains the future to fewer options than existed before. It is a formal expression of the idea that time's arrow points toward more, not fewer, admissible paths.

/-- **Admissible count nondecreasing.** The number of admissible states is
nondecreasing in horizon.

Strategy: `exact Finset.card_le_card (cone_grows next S)`. -/
theorem cone_card_monotone (next : E → Finset E) (S : Finset E) :
    S.card ≤ (coneStep next S).card := by
  exact Finset.card_le_card (cone_grows next S)

/-- One step of the admissible future cone: the current frontier together with all
its admissible successors under `next`. -/
def coneStep (next : E → Finset E) (S : Finset E) : Finset E :=
  S ∪ S.biUnion next

What this page does not claim

The cone_card_monotone theorem does not prove that the future is predetermined or that every possible state will be reached. The identification of ledger entries with recognition events is a modeling choice, not a theorem. The theorem does not specify what the transition rule 'next' must be; it holds for any finite set-valued function.

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

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