Foundation Recognition Time Delta Recognition Time Satisfaction Iff

The satisfaction bridge

The natural numbers, 0, 1, 2, and so on, are the counting numbers that arithmetic studies. Recognition Science works with a different object: a ledger, a discrete record of events, where each recognition step advances a tick. The declaration recognitionTime_satisfaction_iff establishes a precise equivalence: a formula is true in the recognition-time structure if and only if it is true in the standard natural numbers. This is a THEOREM, proved in the machine-checked library of formal theorems, and it means the two structures are indistinguishable from the standpoint of what can be expressed in the language of arithmetic.

The proof rests on a prior result, recognitionTime_isPeano, which shows that recognition time satisfies the Peano axioms, the defining properties of the natural numbers. Because it does, any statement provable about the natural numbers is automatically true about recognition time, and vice versa. The theorem is not a mere analogy; it is a formal identification. The library also proves that a single recognition step advances the tick exactly as the successor function advances a natural number, and that committing an event to the ledger does the same.

The equivalence has a deliberate limit. A finite observer, one who can only inspect a bounded prefix of ticks, does not see the full structure. The library proves that for any finite bound, the zero, successor, and equality relations agree on that prefix, but the last tick in the prefix has no successor within it. The finite certificate is a partial model, never promoted to a full Peano model. The unbounded theorem concerns recognition time as a mathematical object, not any particular physical device.

What the theorem does not claim is equally sharp. It does not assert that any laboratory instrument realizes the recognition ledger or its tick readout; that identification remains an external premise. It does not claim that the finite prefix can serve as a model of arithmetic, only that it agrees on the operations it can express. The bridge is exact where it is unbounded and bounded where it is finite, and the boundary is part of the theorem, not an omission.

/-- Satisfaction in recognition time is exactly canonical δ satisfaction.
Representation changes; truth does not. -/
theorem recognitionTime_satisfaction_iff (φ : DFormula) (ρ : Env) :
    msat recognitionTimeAlgebra ρ φ ↔ DFormula.sat ρ φ :=
  msat_iff_sat recognitionTimeAlgebra recognitionTime_isPeano φ ρ

/-- Recognition time satisfies the Peano laws required by the δ transport
surface: successor is injective, zero is not a successor, and every tick is
generated from zero by repeated successor. -/
theorem recognitionTime_isPeano : IsPeanoModel recognitionTimeAlgebra where
  succ_injective := by
    intro a b h
    apply tickEquivNat.injective
    have hi : a.index + 1 = b.index + 1 := congrArg Tick.index h
    exact Nat.add_right_cancel hi
  zero_not_succ := by
    intro x h
    have hi := congrArg Tick.index h
    simp [recognitionTimeAlgebra, tickZero, tickSucc] at hi
  induction := by
    intro P hzero hsucc x
    cases x with
    | mk n =>
        induction n with
        | zero => exact hzero
        | succ n ih =>
            simpa [recognitionTimeAlgebra, tickSucc] using hsucc ⟨n⟩ ih

/-- A recognition transition is literally the successor map of the
recognition-time δ-algebra. -/
theorem recognition_step_is_delta_succ (step : RecognitionStep) :
    step.output.tick = recognitionTimeAlgebra.succ step.input.tick :=
  recognitionStep_iterates_succ step

/-- Every finite initial segment of recognition time has exact bounded
δ-agreement with the unbounded recognition-time realization. -/
theorem recognitionPrefix_agrees (n : Nat) : BoundedDeltaAgreement n where
  zero_preserved := rfl
  successor_preserved := prefixToTick_succ
  identity_reflected := prefixToTick_injective n
  successor_injective := by
    intro i j hi hj h
    exact prefixSucc_injective (i := i) (j := j) hi hj h
  zero_not_successor := prefixZero_not_succ
  exact_coverage := tick_in_prefix_iff
  terminal_is_boundary := prefixLast_not_in_succ_domain n

What this page does not claim

No physical device is claimed to realize the recognition ledger or its tick readout. The finite prefix is not claimed to be a full Peano model, only an agreement on bounded operations. The theorem does not claim that recognition time is the only structure satisfying the equivalence.

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

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