Foundation Recognition Time Delta Recognition Step Is Delta Succ

One tick forward

Think of a ledger, a discrete record of events, with a clock that counts how many entries have been written. In Recognition Science, a recognition step is one such entry, and its effect on the clock is the subject of a proved theorem. The declaration recognition_step_is_delta_succ states, in the framework's formal language, that after any single recognition step, the output tick equals the input tick plus one. It is a theorem, not a definition or a hope: the machine-checked library of formal theorems proves it from the prior fact that recognition steps iterate the successor operation.

The content is deliberately modest. The theorem does not say how long a recognition step takes in seconds, nor does it identify any physical device as the ledger's carrier. It fixes the abstract structure: recognition time behaves like the natural numbers, with a zero tick and a successor function that each step applies once. This makes recognition time a Peano model, a structure satisfying the usual induction axioms, which the library also proves. The practical payoff is that any theorem proved about such discrete counting structures automatically holds for recognition time as well.

The framework is careful about what a finite observer can see. A separate theorem, recognitionPrefix_agrees, certifies that for any finite bound n, the ticks from zero through n in the recognition model match the corresponding ticks in a bounded prefix exactly. The prefix has a last point with no successor inside it, so the finite picture is honest about its own edge. The unbounded Peano claim applies to recognition time as a mathematical object, not to any finite observation of it.

What the theorem does not claim is equally precise. It does not assert that a particular laboratory apparatus realizes the ledger or its tick readout; that identification remains an external premise. It does not claim that recognition steps are the only way time advances, only that each step advances the clock by one. And it does not promote the finite prefix to a full Peano model, since the prefix's last element lacks a successor. The theorem gives the framework a faithful clock, and leaves the wiring of that clock to the world outside the proof.

/-- 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

/-- 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

/-- 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 specific laboratory device is identified as the ledger or its tick readout. The theorem does not assign a duration in seconds to a recognition step. The finite prefix is not promoted to a full Peano model.

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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