Foundation Recognition Time Delta Recognition Prefix Agrees

The finite prefix certificate

Recognition time is the discrete sequence of steps a recognition process takes. The framework's library proves that this sequence forms a Peano model, meaning it behaves like the natural numbers 0, 1, 2, and so on, with a zero and a successor operation. That unbounded theorem, however, is about the mathematical object itself. A real observation is always finite: it can only see ticks 0 through some bound n, and it cannot see what comes after its last tick.

The declaration recognitionPrefix_agrees (a theorem in the machine-checked library) certifies exactly what a finite observation can know. For any chosen bound n, it proves that the first n ticks of recognition time agree with the standard counting numbers: zero maps to zero, each successor maps to the next number, distinct ticks stay distinct, and no tick is its own successor. It also proves exact coverage: every tick up to index n appears in the prefix, and the last tick of the prefix has no successor inside it. That terminal boundary is the honest limit of a finite view.

What the theorem does not claim is just as important. It does not promote the finite prefix to a full Peano model. The last tick in the prefix has no successor within the prefix, so the successor operation is only partial there. The unbounded induction law, which needs a successor for every element, does not hold on the finite carrier. The certificate is a bounded agreement, not a finite copy of the infinite structure.

This distinction matters for the framework's physicality seam. A laboratory device can be identified with the recognition ledger only up to a finite bound; the theorem guarantees that the first n ticks of the device, if it is a faithful carrier, match the recognition-time arithmetic exactly. It does not, and cannot, certify that the device realizes the unbounded Peano structure. That empirical identification remains an external premise, not a proved theorem.

/-- 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

/-- 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

/-- The terminal point has no in-prefix successor. This is the formal boundary
that blocks any claim that the finite carrier is a full Peano realization. -/
theorem prefixLast_not_in_succ_domain (n : Nat) :
    ¬ (prefixLast n).val < n := by
  simp [prefixLast]

What this page does not claim

The finite prefix is not a full Peano model; its last tick has no successor inside the prefix. No specific laboratory device is identified with the recognition ledger; that remains an external premise. The unbounded induction law does not hold on the finite prefix.

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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