Foundation Recognition Time Delta Forced True In Recognition Time

Recognition time

Recognition Science (RS) builds its physical models on a discrete record of events, which it calls a ledger. Time in this framework is not a continuous flow but a sequence of ticks, each tick marking one recognition step. The declaration forced_true_in_recognition_time is a theorem in the framework's machine-checked library of formal theorems. It states that any formula forced by the empty ledger, meaning any statement that follows from the framework's basic rules without any prior assumptions, is satisfied in this model of recognition time.

The theorem is a transport result. The framework has a general theory of formulas that are forced by an empty ledger. This declaration proves that this general theory applies to the specific model of recognition time. The proof works by showing that recognition time is a Peano model, a structure satisfying the same induction and successor axioms as the natural numbers. This means that the framework's general theorems about forced formulas automatically hold in recognition time.

One consequence is that a statement like the commutativity of addition, which is forced in the general theory, is also true in recognition time. The declaration does not claim that any particular physical device, such as a clock or a computer, is a realization of this model. That identification remains an external premise. The theorem is about the mathematical structure of recognition time as the framework defines it.

The library also contains a separate, bounded result for finite observations. A finite observation of a ledger can only see a finite prefix of ticks. This prefix has a last tick, which has no successor within the prefix. The bounded agreement theorem states that this finite prefix agrees with the full model on zero, successor, and equality, but it is not promoted to a full Peano model. The unbounded theorem and the bounded certificate together show how the framework's infinite mathematical time relates to what a finite observer can actually see.

/-- Every empty-ledger δ theorem is true in RS recognition time. Transport adds
no premise and does not alter the checker ledger. -/
theorem forced_true_in_recognition_time {d : Deriv} {φ : DFormula}
    (h : Forced [] d φ) (ρ : Env) :
    msat recognitionTimeAlgebra ρ φ :=
  transport_forced h recognitionTimeAlgebra recognitionTime_isPeano 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

/-- Concrete regression instance: addition commutativity holds in recognition
time by the same empty-ledger derivation that proves it in the canonical
δ-model. -/
theorem addComm_true_in_recognition_time (ρ : Env) :
    msat recognitionTimeAlgebra ρ
      (.all (.all GodelTest.commFormula)) :=
  addComm_transported recognitionTimeAlgebra recognitionTime_isPeano ρ

/-- 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

The theorem does not claim that any specific laboratory device realizes the recognition ledger. The bounded finite prefix is not a full Peano model because its last point has no successor. No empirical premise is manufactured by the theorem.

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)

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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