The Oscillator as an Element: Static and Dynamical Analyses of the Imaginary Boolean Value in the Algebra of Eventually Periodic Truth Sequences

In the paper's words

Revision-theoretic treatments of the liar paradox produce an oscillator ι = ⟨ U,P,U,P,ldots⟩ that alternates between two states forever, but they leave ι a process rather than a value. This paper tests the conjecture that ι can be promoted to a static element of a genuine Boolean algebra of eventually periodic truth sequences. We define the algebra B_EP of eventually periodic binary sequences under pointwise negation, conjunction, and disjunction, and prove it is a Boolean algebra.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The minimal de-risking step is a mechanical check, not an experiment: encode the Boolean axioms and the collapse argument of Theorem 4.5 in a proof assistant, and separately implement a depth-$d$ liar evaluator that emits $\tau_{d}(\iota)$ and verify that successive outputs are related by the one-coordinate shift for $d$ up to a bound (say $d = 64$).

   from 7. Practical Implications and Engineering Pathways

2. For any implementation, the first binding constraint is representational: $\iota$ is an infinite object, so an implementation must carry truncations, and the depth $d$ is a memory/precision parameter with the exact cost of $d$ stored bits per tracked oscillator (Section 4.4).

   from 7. Practical Implications and Engineering Pathways

3. For the illustrative parameters $n = 10^{3}$ and $d = 10^{2}$, this is $10^{5}$ bits $\approx 12.5$ kilobytes of state — negligible by current standards, but the bound is a projection from the truncation model of Section 3.4, not a measurement.

   from 7. Practical Implications and Engineering Pathways

How it is done

1. We define the algebra $\mathcal{B}_{\mathrm{EP}}$ of eventually periodic binary sequences under pointwise negation, conjunction, and disjunction, and prove it is a Boolean algebra.

   from Abstract

2. Any truth-evaluation engine — a theorem prover with a reflective truth predicate, a database of self-referential definitions, an implementation of a distinction-based calculus — that admits the sentence $L$ with $L \leftrightarrow \neg L$ either halts with no value, diverges in an evaluation loop, or must be walled off by a syntactic hierarchy (Tarskian levels) that forbids the very reflexivity the application wants.

   from 1. Introduction

3. This motivates our search for a non-classical Boolean algebra based on sequence objects, and it cautions us: dualities that preserve Boolean behavior will not tolerate a self-negating element, which is exactly what our collapse theorem confirms. [2] extends the eliminant construction for commuting elements in the $q$-deformed Heisenberg algebra, providing a method for generating algebraic identities in non-commutative settings.

   from 2. Background and Related Work

4. For $s, t \in \mathcal{B}_{\mathrm{EP}}$ we define: $$ (s \wedge t)_{k} = s_{k} \cdot t_{k}, \qquad (s \vee t)_{k} = s_{k} + t_{k} - s_{k} \cdot t_{k}, \qquad (\neg s)_{k} = 1 - s_{k}, $$ where multiplication and addition are taken in $\{0,1\}$ with ordinary arithmetic, so that each coordinate value remains in $\{0,1\}$.

   from 3.2. Pointwise Boolean operations

What would falsify it

1. The oscillator is thus a computably approximable element of a Boolean algebra whose self-negation holds dynamically, not algebraically — a discrimination that turns the promotion conjecture into a precise, partially refuted, partially confirmed theorem.

   from Abstract

2. Refutation of the static self-negation claim: $\iota \neq \neg\iota$ in $\mathcal{B}_{\mathrm{EP}}$, plus a collapse theorem showing that no quotient of any nontrivial Boolean algebra can contain a self-negating element (Section 4.2).

   from 1. Introduction

3. What revision theory produces is a computably approximable element of $\mathcal{B}_{\mathrm{EP}}$ whose self-negation is realized by the dynamics ($\sigma(\iota) = \neg\iota$) and refuted by the algebra ($\iota \neq \neg\iota$).

   from 4.4. Dynamical characterization of $\iota$

Where the claim stops

1. Its dynamical half is fully right, and stronger than conjectured: the unfolding dynamics is not merely convergent but isomorphic to the shift, so the "approximation" is an exact orbit.

   from 6. Discussion

2. The negative and positive results rest on the following assumptions, with sensitivity noted for each.

   from 6. Discussion

3. Section 3.4 assumes the depth-$d$ unfolding's revision profile equals $\tau_{d}(\iota)$.

   from 6. Discussion

Where to read

1. Abstract240 words · 1 min
2. 1. Introduction655 words · 3 min
3. 2. Background and Related Work484 words · 2 min
4. 3. Methods403 words · 2 min
5. 4. Analysis1,125 words · 5 min
6. 5. Results356 words · 2 min
7. 6. Discussion586 words · 3 min
8. 7. Practical Implications and Engineering Pathways632 words · 3 min
9. 8. Conclusion227 words · 1 min
10. Provenance and AI Involvement225 words · 1 min