Remarks on Primitive Regulation

Milan Rosko

arxiv: 2605.18924 · v2 · pith:SJKS4IESnew · submitted 2026-05-18 · 🧮 math.LO

Remarks on Primitive Regulation

Milan Rosko This is my paper

Pith reviewed 2026-05-20 02:43 UTC · model grok-4.3

classification 🧮 math.LO

keywords closure predicatesobstruction theoremevaluation completenessexcluded-middle completenessimplication-falsity fragmentreflective fixed-pointconsistencymodus ponens

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The pith

The conjunction of evaluation and excluded-middle completeness for any closure predicate on the implication-falsity fragment generates a reflective fixed-point that forces acceptance of falsity, contradicting consistency.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that no consistent closure predicate C can satisfy both Eval(C) and LEM(C) at once. Eval(C) generates a code B such that C treats B as equivalent to its negation. LEM(C) then requires C to accept either B or its negation, and modus ponens turns this into acceptance of falsity. Consistency of the underlying fragment turns the internal collapse into an outright contradiction. The result separates Boolean decision, which strengthens LEM(C) and is therefore blocked, from mere refutation, which avoids the coverage demand and remains possible.

Core claim

For a closure predicate C over formulas built from ⊥ and implication, the joint assumption of Eval(C) and LEM(C) produces a reflective fixed-point B ≃_C ¬B; LEM(C) forces C to classify this point, each branch collapses to C(⊥) by modus ponens, and consistency converts the collapse into an external contradiction. A strengthened Boolean decision principle is therefore obstructed, while the always-false classifier satisfies refutation without coverage.

What carries the argument

The reflective fixed-point B ≃_C ¬B generated by Eval(C) and then classified by LEM(C), which collapses under modus ponens.

If this is right

Boolean decision strengthens LEM(C) and is therefore obstructed by the theorem.
Refutation imposes no coverage requirement and is inhabited by the always-false classifier.
The obstruction applies uniformly to any C that meets the two completeness conditions over the given fragment.
Consistency is required to turn the internal acceptance of ⊥ into an external contradiction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same pattern may block joint completeness conditions in richer fragments that still contain implication and falsity.
Refutation-only regulators remain viable candidates for primitive notions that avoid the fixed-point collapse.
Any attempt to regulate truth or provability by a single predicate must choose between generative and decisional completeness.

Load-bearing premise

The implication-falsity fragment is consistent and the closure predicate respects modus ponens when the fixed-point is classified.

What would settle it

Exhibiting a consistent closure predicate C that satisfies both Eval(C) and LEM(C) without deriving C(⊥).

read the original abstract

We prove, and mechanize in Rocq, an abstract obstruction theorem for closure predicates $C : \mathsf{Form} \to \mathsf{Prop}$ over the closed implication-falsity fragment $A,B ::= \bot \mid A \to B$. Evaluation completeness, $\mathsf{Eval}(C)$, says that every formula-valued behavior of codes is represented up to closure equivalence, where $A \simeq_C B$ abbreviates $C(A \to B) \land C(B \to A)$. For any $C$ closed under modus ponens and satisfying consistency, this completeness principle is incompatible with the internal excluded-middle schema $\mathsf{LEM}(C)$, namely $\forall A, C(A)\lor C(\neg A)$. Thus $\mathsf{Eval}(C), \mathsf{MP}(C), \mathsf{Cons}(C)$, and $\mathsf{LEM}(C)$ cannot hold jointly. The proof uses evaluation completeness to obtain a formula $B$ such that $B \simeq_C \neg B$. Applying $\mathsf{LEM}(C)$ to this $B$, either alternative gives $C(\bot)$ by detachment, contradicting consistency. Consequently, any Boolean decision procedure for $C$ induces the obstructed excluded-middle schema. Mere refutation behaves differently: its false branch carries no closure condition, and is therefore inhabited by the always-false classifier.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean obstruction showing Eval(C) plus LEM(C) forces a primitive closure predicate to accept falsehood, mechanized in Rocq, but the collapse step needs explicit justification that C respects modus ponens on the fixed-point instances.

read the letter

The core result is that no primitive closure predicate C can satisfy both evaluation completeness (every behavior has a representing code up to C-equivalence) and excluded-middle completeness (C decides every formula or its negation) without C(⊥) holding internally; consistency then yields the external contradiction. This pairing and the reflective fixed-point construction look new relative to earlier work on closure predicates and completeness principles in minimal logic fragments. The mechanization in Rocq is a genuine strength—it turns the derivation into something that can be checked rather than just asserted. The separation between generative Eval(C) and decisional LEM(C) is also handled cleanly, and the note that Boolean decision is obstructed while plain refutation is not is a useful distinction. The argument stays within the ⊥,→ fragment and avoids extra assumptions beyond consistency. The main soft spot is exactly the one flagged in the stress-test: the collapse from B ≃_C ¬B plus LEM(C) to C(⊥) relies on applying modus ponens inside C twice. The definition given is only C : Form → Prop, so the paper must derive or stipulate that C is closed under MP for the relevant implications generated by the fixed point. If that step is only implicit or left to the reader, the obstruction does not yet follow from the stated hypotheses alone. The consistency appeal inherits the same dependency. This is a fixable gap rather than a fatal one, but it is load-bearing. The work is aimed at logicians and formal-verification people who care about limits on self-referential completeness principles in weak fragments. A reader already comfortable with closure predicates and Rocq will get the most out of it. It is worth sending to peer review; the mechanization and the focused obstruction give it enough substance that referees can usefully engage even if they ask for more detail on the MP closure.

Referee Report

2 major / 1 minor

Summary. The paper proves and mechanizes in Rocq an abstract obstruction theorem for primitive closure predicates C : Form → Prop over the implication-falsity fragment. Eval(C) generates a reflective fixed-point B ≃_C ¬B (via representing codes up to closure equivalence), while LEM(C) forces C to decide B or ¬B; either case collapses to C(⊥) by repeated modus ponens, and consistency yields an external contradiction. Boolean decision is obstructed while refutation is not.

Significance. If the central derivation holds, the result cleanly separates generative and decisional completeness notions for closure predicates and isolates an obstruction that does not affect refutation. The Rocq mechanization is a concrete strength, supplying machine-checked verification of the fixed-point construction and collapse from the stated definitions of Eval(C), LEM(C), and consistency.

major comments (2)

[§3] §3 (obstruction argument): the collapse from B ≃_C ¬B together with LEM(C) to C(⊥) invokes modus ponens on the two directions of the equivalence and then again on the resulting pair C(B) ∧ C(¬B). The definition of a primitive closure predicate is given only as an arbitrary map Form → Prop; unless a preceding lemma explicitly shows that every such C is closed under MP for the ⊥,→ fragment (or that the particular B constructed by Eval(C) supplies the exact antecedents required), the internal collapse step does not follow from the hypotheses alone. This is load-bearing for the main theorem.
[Mechanization appendix] Mechanization appendix: the abstract asserts a Rocq development, yet the manuscript provides neither the code nor the key extracted statements for the fixed-point construction and the MP applications in the collapse. Without these, the soundness of the central steps cannot be directly inspected.

minor comments (1)

[§2] The notation A ≃_C B is introduced as C(A → B) ∧ C(B → A); a short remark clarifying whether this equivalence is intended to be an equivalence relation on the nose or only up to further closure would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. The points raised about the justification of the collapse step and the presentation of the mechanization are well-taken and will strengthen the paper. We address each major comment below.

read point-by-point responses

Referee: [§3] §3 (obstruction argument): the collapse from B ≃_C ¬B together with LEM(C) to C(⊥) invokes modus ponens on the two directions of the equivalence and then again on the resulting pair C(B) ∧ C(¬B). The definition of a primitive closure predicate is given only as an arbitrary map Form → Prop; unless a preceding lemma explicitly shows that every such C is closed under MP for the ⊥,→ fragment (or that the particular B constructed by Eval(C) supplies the exact antecedents required), the internal collapse step does not follow from the hypotheses alone. This is load-bearing for the main theorem.

Authors: We agree that the current presentation leaves the MP applications insufficiently justified. The definition of a primitive closure predicate is indeed an arbitrary C : Form → Prop, and no general closure under modus ponens is assumed for arbitrary predicates. The specific B arising from Eval(C) does supply the antecedents B → ¬B and ¬B → B via the equivalence ≃_C, but this requires explicit verification rather than being left implicit. We will therefore insert a short preceding lemma in §3 that derives the two required MP steps directly from the definition of Eval(C) and the construction of B. This lemma will be stated and proved before the main obstruction theorem, making the collapse fully rigorous from the given hypotheses. revision: yes
Referee: [Mechanization appendix] Mechanization appendix: the abstract asserts a Rocq development, yet the manuscript provides neither the code nor the key extracted statements for the fixed-point construction and the MP applications in the collapse. Without these, the soundness of the central steps cannot be directly inspected.

Authors: We accept that the mechanization appendix is currently too thin for independent verification. Although the full Rocq development exists separately, the manuscript itself does not display the key extracted statements. In the revised version we will expand the appendix to include the relevant Coq definitions and lemmas: the inductive construction of the reflective fixed-point B under ≃_C, the formal statements of Eval(C) and LEM(C), and the mechanized derivation of the collapse to C(⊥). These excerpts will be presented in a readable form so that the central steps can be inspected directly from the paper. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained with no circularity

full rationale

The paper derives its obstruction theorem directly from the stated definitions of primitive closure predicates C : Form → Prop, Eval(C) as a generative completeness principle, and LEM(C) as a decisional principle, together with consistency (¬C(⊥)). The fixed-point B ≃_C ¬B is generated from Eval(C), classified by LEM(C), and collapsed via modus ponens applications that the paper treats as following from the fragment's rules; the mechanization in Rocq supplies independent verification of the entire chain. No self-citations, fitted parameters renamed as predictions, or definitional reductions appear in the argument, so the result does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions of closure predicates, modus ponens, and consistency in the implication-falsity fragment; no free parameters or invented entities are introduced.

axioms (2)

standard math The implication-falsity fragment is closed under modus ponens.
Invoked in the collapse of the fixed-point to C(⊥).
domain assumption Consistency of the logic means C(⊥) is not derivable.
Converts internal collapse into external contradiction.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Eval(C) ∧ MP(C) ∧ Cons(C) ∧ LEM(C) =⇒ ⊥ (Theorem 3.2.4); negation fixed-point B ≃_C ¬B from goal-restricted evaluation frame (Theorem 3.1.8)

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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