Foundation Pair Kernel Posting Boundary Record Posting Record Is Moving Cut Reco

The moving-cut record

The declaration postingRecordIsMovingCutRecord_uninhabited_on_bit_carrier proves, in the framework's machine-checked library, that no posting record can be represented as a moving cut carried by a single bit. In plain language: if you try to encode the identity of a posting event in one bit, the bit must be the same for every event, so it cannot distinguish one posting from another. The theorem states that the proposition PostingRecordIsMovingCutRecord, which asserts the existence of such a bit-based encoding, is uninhabited: it has no instances.

The proof works by examining what it means for a bit to recover a posting boundary record. A posting boundary record is the credit leg of a posting read as flux across the boundary of the debit region. The framework proves that any bit assignment that recovers this record must be constant: it always returns the value 1. Since a moving cut requires a non-constant bit, the two requirements conflict. The theorem recovers_implies_constant establishes the constancy, and the main theorem applies it to rule out the moving-cut encoding.

This result is a boundary on what a bit can do, not a claim about the physical world. It does not say that posting records cannot be represented in other ways, nor that moving cuts do not exist in general. It only says that a single bit, as defined in this framework, cannot serve as the carrier of a moving-cut occurrence identity. The theorem is proved with no sorry and no new axioms, meaning it relies only on the framework's existing definitions and theorems.

The practical consequence is that occurrence identity, the property that distinguishes one posting event from another, requires more structure than a single bit can provide. This is a formal result about the framework's model of postings, not a statement about information theory or physics. It clarifies what the framework's definitions can and cannot express, and it leaves open the question of what minimal structure would suffice for occurrence identity.

theorem postingRecordIsMovingCutRecord_uninhabited_on_bit_carrier :
    ¬ PostingRecordIsMovingCutRecord := by
  rintro ⟨attach, hrec, hnonconst⟩
  exact hnonconst (recovers_implies_constant attach hrec)

theorem recovers_implies_constant
    (attach : CutBitOfOrientedPosting)
    (hrec : RecoversPostingBoundaryRecord attach) :
    IsConstantCutBit attach :=
  ⟨(1 : ZMod 2), fun _ _ _ => recovers_forces_posted_one attach hrec _ _ _⟩

theorem postingRecordIsMovingCutRecord_uninhabited_on_bit_carrier :
    ¬ PostingRecordIsMovingCutRecord := by
  rintro ⟨attach, hrec, hnonconst⟩
  exact hnonconst (recovers_implies_constant attach hrec)

What this page does not claim

The theorem does not claim that posting records cannot be represented in any form, only that a single bit cannot serve as the carrier. The theorem does not claim that moving cuts do not exist in general, only that they cannot be encoded by a single bit. The theorem does not make any statement about physical information theory or the nature of bits in the physical world.

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

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