Determinization in Structure Theories: A Unified Framework via Closure, Comparability, and Joint Admissibility

arXiv:2608.07476 · cs.AI, cs.LO · Submitted 2026-04-27 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Next we'll be talking about the paper "Determinization in Structure Theories: A Unified Framework via Closure, Comparability, and Joint Admissibility".

Jane: The paper was written by Hai Hai Fu from.

Tom: Stay tuned as we take you through the paper and discuss its implications.

Title: Tom: Alright, welcome back everyone. Today we're digging into a paper with a title that's a mouthful — "Determinization in Structure Theories: A Unified Framework via Closure, Comparability, and Joint Admissibility." Jane, I'll be honest, when I first read that title I had to sit down.

Jane: Ha, you and me both, Tom. But once you unpack it, it's actually about a really practical problem. "Determinization" just means making something give one clear answer instead of many possible answers. And "structure theories" — think of them as rulebooks for interpreting market structures, like Wyckoff or ICT, the stuff traders use to read charts.

Tom: Right, and that's the hook. These rulebooks are plural — they let multiple valid interpretations coexist at the same price point. The paper asks: when are you actually licensed to pick one and commit?

Jane: Exactly. And the authors — Hai Hai Fu, the sole author here — frames this in a really interesting way. They connect it to LLM-assisted reasoning. You know how a language model can hallucinate? The paper says hallucination is basically unsupported canonicalization — the system commits to a single answer when the underlying set of valid answers hasn't collapsed to one yet.

Tom: That's a wild reframe. So the LLM isn't being "wrong" in some vague sense — it's being structurally premature. It's emitting a determinate answer when the math says it shouldn't.

Jane: Precisely. And that's why this framework matters. It gives you formal conditions for when commitment is licensed. Not vibes, not heuristics — actual structural conditions.

Tom: I love that. And I know Lu's going to have thoughts on this because it touches that whole question of when AI systems should express uncertainty versus when they should just commit. Lu, you're on the line — what do you make of this framing?

Lu: Tom, I think it's genuinely important. The paper formalizes something we've all felt — that a model should be allowed to be decisive only when the evidence forces a unique answer. The distinction they draw between "closure stabilization" and full "determinization" is subtle but crucial. Stabilization means each starting point settles somewhere. Determinization means everyone settles at the same place. Those are very different claims, and the paper is careful about which one it proves.

Jane: And that care is the whole ballgame, right? Because a lot of papers would just claim the strong result and hand-wave. This one says, for the ICT theory, we've got stabilization, but the upgrade to full determinization is an open question — they literally label it OQ-GC-one.

Tom: Open questions in a paper — that's refreshing. It's not a weakness, it's intellectual honesty. Alright, we've got a lot more to unpack here, so let's keep going.

Abstract: Tom: So we're back with "Determinization in Structure Theories" and we've got the abstract in front of us. Jane, what jumps out at you?

Jane: The core principle they state at the end — that's the thing I keep coming back to. Completion-based canonicalization needs closure, comparability, and compatible extension. Selection-based canonicalization needs theory-intrinsic comparability and global compatibility. It reads like a recipe, Tom. Two different recipes for two different kinds of problems.

Tom: And that distinction — completion versus selection — that's the heart of their taxonomy. Some theories are "Type E" — epistemic plurality, where the ambiguity comes from missing evidence. Add more evidence, and the ambiguity vanishes. Other theories are "Type S" — structural plurality, where even with perfect information, multiple interpretations persist.

Lu: And then they refine Type S into "Type S-strong" — where two admissible interpretations have no common upper bound. That's the case where completion is provably impossible. You can't just keep adding structure until things converge, because the alternatives are axiomatically incomparable. Wyckoff theory is their example there.

Jane: Right, and for Type S-strong, you need a selector — a rule that just picks one, like PhaseClassify picking the dominant phase. It's not about proving convergence; it's about making a principled choice.

Tom: So for Type E — like ICT — you close the gaps with more inference. For Type S-strong — like Wyckoff — you select. And the paper proves that mixing those up gets you in trouble.

Meng: Can I jump in here? I'm the engineer, so I'm asking: how do I know which type my system is? Is there a test?

Jane: Great question, Meng. They define it formally — Type E means plurality vanishes under evidence refinement. Type S means it persists even at maximum information. So you can check: feed your system everything, and if two answers still stand, you're Type S.

Meng: Okay, and if I'm Type S but not "strong" — what then?

Tom: That's literally an open question in the paper — OQ-TypeS-Imp. They're honest that the generic Type S to selection inference isn't proven. Only the strong version.

Lu: And that's the kind of precision that makes this paper valuable. It doesn't overclaim. It gives you exactly the conditions under which each mechanism works, and flags where the proof stops.

Tom: Alright, so we've got the taxonomy. Next we need to talk about what they actually construct — the mechanisms themselves.

Improvements: Tom: We're back with "Determinization in Structure Theories." We've covered the taxonomy — Type E, Type S, Type S-strong. Now let's talk about what the paper actually builds. Jane, what's the big improvement here over previous work?

Jane: The big one is Theorem 6b — they call it the Construction Lemma. It gives you sufficient conditions for building a canonicalization mechanism. If you satisfy certain structural conditions — non-emptiness, finiteness, comparability, joint admissibility — then you can construct either a completion operator or a selector.

Tom: And they're careful to say it's a construction lemma, not a deep theorem. The real substance is in Corollary 6b' — that's where they show that if your inference rules satisfy certain syntactic constraints — they call them R1, R2, R3 — plus a soundness condition, then your completion operator automatically satisfies the four properties they want: extensivity, idempotence, admissibility preservation, and per-seed convergence.

Meng: So the improvement is — I don't have to verify all four properties manually? If my rules are shaped right, the properties come for free?

Jane: Exactly, Meng. That's the win. The syntactic constraints are checkable — rules only add conclusions, no negation-as-failure, no retraction. If your rules look like that, and they're sound, then saturation gives you a well-behaved completion operator.

Lu: And there's a subtlety there that I really appreciate. They distinguish between syntactic constraints and semantic soundness. You can have rules that look fine syntactically — positive, non-retractive — but are unsound because they add conclusions that contradict your axioms. Their example: rule "a implies b" plus an axiom forbidding both a and b together. Syntactically fine, semantically broken.

Tom: Right, and that's the kind of trap that would bite you in practice. They close it by making soundness an explicit hypothesis.

Meng: What about the multi-timeframe stuff? I saw something about staged operators and non-commutativity.

Jane: Oh, that's a great part. They show that if you have two operators — one for high timeframe, one for low — the order matters. Applying high-then-low works; low-then-high breaks admissibility. They have a concrete witness: a state where low-first produces a forbidden pair.

Tom: And they're careful to scope that claim — it's within the specific two-operator architecture they define. Not a global uniqueness claim across all possible architectures. That's the discipline of this paper — every claim is scoped.

Lu: Which is exactly why the classification in Section eight works. They define "theory-intrinsic" operations independently, then classify them conditionally — if your mechanism is realized in one of three ways, it's either Class C (closure) or Class S (selection). No third class within that grammar.

Tom: So the improvement is: a clear recipe, with explicit conditions, and honest scoping. Let's dig into the first page more — there's a lot packed into that abstract.

First Page: Tom: We're still on "Determinization in Structure Theories," and I want to zoom in on the first page because there's a lot of signal there. Jane, what's the thing you'd want a listener to take away from the opening?

Jane: The motivation, Tom. This paper is motivated by structural failures in LLM-assisted reasoning. The author literally says hallucination can be viewed as unsupported canonicalization — the system emits a determinate answer when the underlying admissible set hasn't collapsed to a singleton.

Lu: And that's a profound reframe. It moves hallucination from a statistical failure to a structural one. The model isn't just "confused" — it's committing to a unique interpretation when the theory itself doesn't license one. That's a different failure mode, and it needs a different fix.

Meng: So the fix would be — what? The model should check whether the admissible set is a singleton before committing?

Jane: That's exactly the idea, Meng. The framework gives you the conditions for when commitment is licensed. If you're in a Type E theory and evidence is still accumulating, you can stabilize but not determinize. If you're Type S-strong, you need a selector. If you have neither comparability nor joint admissibility — you shouldn't be committing at all.

Tom: And that's the counterexample they build — Tce. Two conclusions, individually admissible, axiomatically incompatible, no ordering between them. No canonicalization mechanism exists. The theory just doesn't license a unique answer.

Lu: Which is a beautiful minimal example. It shows that closure alone — just being able to add structure — is not enough. You need comparability. You need an order that tells you which answer wins.

Tom: Right. Closure plus comparability plus joint admissibility — that's the recipe. And the first page also sets up the three strengths of canonicalization: closure stabilization, global completion, and full determinization. Each is a stronger claim, and the paper is explicit about which one it achieves for each theory.

Jane: And for ICT, it's stabilization only — the upgrade to full determinization is open. For Wyckoff, it's full determinization via selection. That asymmetry is the whole point.

Meng: So if I'm building a system — I need to know which strength I actually need. Stabilization might be enough for some applications; determinization is needed for others.

Tom: Exactly. And that's the practical takeaway — the paper gives you the vocabulary to say precisely what your system can and cannot commit to. Alright, we're heading into the home stretch. Let's wrap this up.

Conclusion: Tom: Alright, we're closing out our discussion of "Determinization in Structure Theories: A Unified Framework via Closure, Comparability, and Joint Admissibility." Jane, give us the send-off.

Jane: Tom, I think the thing I'll remember most is the honesty. This paper doesn't pretend to have solved everything. It gives you a framework — two types of plurality, two mechanisms, three strengths of canonicalization — and then it tells you exactly where the proofs stop. Open questions are labeled, scoped, and tracked.

Lu: And that's rare in this space. The classification theorem — U3 — is explicitly conditional on a realization-coverage assumption they can't yet prove. The non-commutativity result is scoped to a specific architecture. Every strong claim has a qualifier.

Meng: From my side, the practical value is the checklist. If I'm building an LLM-assisted decision system, I can now ask: what type is my theory? Do I have comparability? Do I have joint admissibility? And only then — am I licensed to commit?

Tom: And that's the connection to hallucination that started the whole paper. It's not about making models "more confident" or "less confident." It's about giving them structural permission to commit — or telling them to hold off.

Jane: The core principle one more time: completion needs closure, comparability, and compatible extension. Selection needs theory-intrinsic comparability and global compatibility. Not all canonicalization is operator-theoretic. And under hierarchical interaction, operators form a non-commutative system.

Tom: Beautiful. So we're saying goodbye to this paper — but honestly, I think we'll be citing it for a while. The vocabulary alone — Type E, Type S-strong, closure stabilization versus determinization — that's going to stick.

Jane: Agreed. And the open questions give other researchers a roadmap. OQ-GC-one OQ-TypeS-Imp, OQ-Realization — those are invitations, not gaps.

Tom: Alright, that's our show for today. Thanks to Lu and Meng for joining us. And to our listeners — next time you see an AI system confidently assert something, ask yourself: is it licensed to do that? Thanks for tuning in, and we'll see you on the next paper.

Hai Hai Fu

cs.AI, cs.LO

Submitted: 2026-04-27

Updated: 2026-08-11

Comments: Formal framework paper on canonicalization and determinization in structure theories; version v2.16.4; 29 pages

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 40/100

The gist: Author: Hai Hai Fu Version: v2.16.4, April 2026 arXiv: 2608.07476v1 [cs.AI] 27 Apr 2026 --- The paper addresses a single question: "When can a structure theory be made canonical?" The motivating AI

Key concepts

Determinization: The process of making something yield one clear answer instead of many possible interpretations. The paper uses this concept to determine when an AI system can confidently commit to a unique conclusion based on its underlying rules.
Type E vs. Type S Theories: This taxonomy classifies theories based on ambiguity persistence. Type E (epistemic plurality) means ambiguity vanishes with more evidence, while Type S (structural plurality) means multiple interpretations persist even with perfect information.
Canonicalization: The formal process of reducing multiple possible interpretations to a single, unique answer. The paper distinguishes between 'completion' (adding structure) and 'selection' (choosing one rule).
Closure Stabilization: A weaker claim than full determinization. It means that while starting points may vary, they eventually settle or stabilize on a common result, but it does not guarantee a unique answer.

Terminology

Core — 64× in this paper · explained in 2 episodes
Admissibility — 62× in this paper · explained in 2 episodes
Axioms — 11× in this paper · explained in 2 episodes
Hallucination — 11× in this paper · explained in 4 episodes
Verification — 8× in this paper · explained in 2 episodes
Unified Framework — 5× in this paper · explained in 5 episodes
Consistency — 2× in this paper · explained in 3 episodes
Determinization Quasi-Algebra — 2× in this paper
Scaffolding — 2× in this paper · explained in 3 episodes

Summary

Author: Hai Hai Fu

Version: v2.16.4, April 2026

arXiv: 2608.07476v1 [cs.AI] 27 Apr 2026

The paper addresses a single question: When can a structure theory be made canonical? The motivating AI setting is LLM-assisted structural decision systems: a language model proposes candidate interpretations, while a symbolic structure theory specifies admissible interpretations. The paper formalizes canonicalization as the problem of deciding when the system is licensed to output a unique interpretation rather than a set of admissible alternatives. Hallucination is characterized as unsupported canonicalization—the system emits a determinate answer when the underlying admissible set has not collapsed to a singleton.

A structure theory is defined as a triple T = (Σ, A, I) where Σ is a signature, A axioms, and I = I core ∪ I ext an inference policy (this paper concerns only I core). Theories are evaluated on inputs (N, Θ, Π) where N is an observed market realization, Θ is the set of theory-relevant parameters, and Π is a structural evidence assignment. The admissible interpretation family P T(N, Θ, Π) is the set of all assignments P: Dom(N, Θ) → 2 C satisfying all axioms A and consistent with I core.

The inference core I core is restricted by three definitional constraints: (R1) positive rule form (rules only add conclusions, never remove); (R2) no negation-as-failure (premises are positive conjunctions); (R3) no retraction (no rule withdraws previously derived conclusions). These are syntactic constraints independent of semantic soundness.

A specification order ⪯ spec is defined: P ⪯ spec Q if P(r, t) ⊆ Q(r, t) for all (r, t) ∈ Dom(N, Θ). A theory is deterministic (AC-6) if P T(N, Θ, Π) = 1 for all inputs; plural (AC-6') if P T(N, Θ, Π) ≥ 2 for some input.

The paper classifies non-determinism into two types:

Type E (Epistemic Plurality): "its plurality vanishes under structural evidence refinement: for every input with P T > 1, there exists a refined Π′ ⊇ Π such that P T(N, Θ, Π′) = 1."
Type S (Structural Plurality): "its plurality persists at Π full: there exists (N, Θ, Π full) with P T(N, Θ, Π full) > 1."
Type S-strong (No Common Upper Bound): T is Type S and there exists an input (N, Θ, Π full) at which two distinct admissible interpretations P1, P2 ∈ P T(N, Θ, Π full) have no common upper bound in (P T, ⪯ spec).

Examples: Wyckoff (T Wy) is Type S-strong (axiomatic phase incomparability via [Wy-Sem-1]); ICT (T ICT) is Type E (plurality is epistemic, resolved by additional evidence); Chan (T Chan) is intrinsically deterministic via unique normal form of its term rewrite system (with open question OQ-Chan-TRS regarding formal proof of termination and local confluence).

The framework distinguishes three increasingly strong notions:

Closure stabilization (property (4')): each admissible seed converges under iteration of an operator to some fixed point.
Global completion (property (4)): all seeds converge to a single fixed point.
Determinization (AC-6): the unique fixed point is the sole admissible interpretation.

The implication chain is: AC-6 ⇒ (4) ⇒ (4′).

A determinization operator D: D T → D T satisfies: (1) Extensivity (P ⪯ spec D(P)); (2) Idempotence (D(D(P)) = D(P)); (3) Admissibility preservation (D(P) ∈ D T); (4) Per-seed convergence (property (4')). A uniqueness-inducing operator additionally satisfies property (4): there exists, for each instance (N, Θ, Π), a single P*(N, Θ, Π) ∈ D T(N, Θ, Π) such that D k(P) = P*(N, Θ, Π) for all seeds P.

Proposition 4.4: Let T be Type S-strong, witnessed at Π full by P1, P2 ∈ P T(N, Θ, Π full) with no common upper bound in (P T, ⪯ spec). Then for any extensive operator D: D T → D T with P1, P2 ∈ D T, property (4) fails on this instance. The proof shows that extensivity forces both P1 and P2 to converge to the same fixed point P*, which would be a common upper bound—contradicting Type S-strong. The weaker property (4') is not ruled out.

A canonical selector is a function ϕ: (r, t, Θ, Π) ↦ c* ∈ A r,t, theory-intrinsically defined and globally compatible. The Wyckoff instance uses PhaseClassify(r, t; Θ, Π) = max⪯ sel A r,t, where ⪯ sel is induced by event dominance (C4-Q2) and MEP ensures global compatibility.

The conditions are:

N0 (universal, non-emptiness): P T(N, Θ, Π) ≠ ∅ for every input.
N1 (selector-specific, finiteness): A r,t < ∞.
N3 (operator-specific, bounded closure): ∃ k: Cl k(P) = Cl k+1(P) on the canonical seed domain.
N4 (unified comparability): T admits a theory-intrinsic comparability-inducing structure K, with mechanism-specific variants N4-comp (closure-compatible) and N4-sel (selector-compatible).
C5 (joint admissibility, mechanism-specific): C5-S (selection variant) asserts global admissibility of the selector's output, not joint admissibility of local alternatives; C5-E (completion variant) asserts compatible extension within the closure domain for non-forbidden co-generating predicate pairs.

Theorem 5.2 (Construction Lemma): Let T satisfy AC-6'. Then: Completion branch: N0 + N3 + N4-comp + C5-E ⇒ D comp exists and satisfies (4'). Selection branch: N0 + N1 + N4-sel + C5-S ⇒ PhaseClassify exists.

Remark 5.3: Theorem 6b is a construction lemma — the forward arrows are constructions; the deeper result is Corollary 6b'.

Counterexample T ce (Proposition 5.1): Two conclusions c1, c2 individually admissible but axiomatically incompatible, unrelated by any ordering axiom. P T = c1, c2. Satisfies N0, N1, N3 but fails N4 in both forms, showing theory-intrinsic comparability (N4) is necessary.

Corollary 5.5 (Classification of pure I core-computable completion): Let F be the elementary one-step I core-rule operator (one rule occurrence per step under fair scheduling), f:= F ω its saturated closure. Under R1+R2+R3, finite P T, and the Soundness assumption (Every I core-rule is sound relative to A), f restricts to a map P T → P T, satisfies properties (1)–(4') of §4.1, and is therefore Class C. The proof uses Lemma C (monotonicity from R1+R2+R3) and the Monotonic Exhaustion Lemma (extensive operator on finite poset reaches fixed point in at most P T steps).

Remark 5.6: Soundness is independent of R1+R2+R3 — R1+R2+R3 are syntactic constraints; Soundness is semantic and must be assumed separately. The paper uses elementary-step F throughout; a simultaneous-firing interpretation would require a stronger condition.

Wyckoff (Type S-strong, Complete): Event dominance (C4-Q2) gives Phase A ≺ sel ··· ≺ sel Phase E. MEP ensures the selected assignment is globally admissible. T Wy canon = T Wy + PhaseClassify ∈ AC-6 (full domain).

ICT (Type E): Single-timeframe: M1+M2 define P ICT closure; M3 ensures D comp terminates. Forbidden pairs (M2) include LTF BOS up, LTF BOS down, LTF bullish cont, LTF bearish cont, LTF BOS up, LTF bearish cont, LTF BOS down, LTF bullish cont. T ICT canon achieves (4') on P ICT closure; whether it achieves (4) and AC-6 is conditional on OQ-GC-1.

Multi-timeframe extension: D comp HTF:= D L ∘ D H: P ICT → P ICT HTF → P ICT multi. LTF inference rules (ρ1–ρ4) have premises with mutual exclusion within priority tiers. Cross-timeframe forbidden pairs (M2-HTF) include HTF BOS down, LTF BOS up, HTF BOS up, LTF BOS down, HTF bearish bias, LTF bullish cont, HTF bullish bias, LTF bearish cont.

The paper verifies properties (1)–(4') for D comp HTF by direct argument: extensivity (additive updates), admissibility preservation (induction using Lemma 6.6), idempotence (two-step argument showing D H(P*) = P* and D L(P*) = P*), and per-seed convergence (termination bounds).

Remark 6.7 (Monotonicity fails): D comp HTF is not monotone under ⪯ spec. Concrete witness: P = ∅, Q = HTF bearish bias, at input where close > prev close. D comp HTF(P) ⊇ LTF bullish cont (ρ3 fires, no guard), but D comp HTF(Q) = Q (ρ3 is M2-HTF-suppressed). This failure is intrinsic to (3c)-mechanisms with conflict-suppression guards.

Proposition 7.1 (Structural non-commutativity): There exists P ∈ P ICT such that (D L ∘ D H)(P) ∈ P ICT multi while (D H ∘ D L)(P) ∉ P ICT multi. The witness: P = ∅, close > prev close, no BOS condition, HTF bearish BOS evidence confirmed. Path A (HTF-first): D H(P) = HTF bearish bias; ρ3 is guard-suppressed; result admissible. Path B (LTF-first): D L(∅) = LTF bullish cont (ρ3 fires); then D H fires; result violates M2-HTF.

Corollary 7.2 (Canonical ordering uniqueness): "Within the two-operator raw staged architecture D L ∘ D H, D H ∘ D L, with no repair operator, no M2-HTF guard added to D H, and no joint fixed-point construction, HTF-first D L ∘ D H is the only admissibility-preserving linearization. Remark 7.3 scopes this uniqueness: alternative orderings combined with repair operators, joint fixed-point constructions, or DH guarded against M2-HTF could in principle preserve admissibility under different orderings."

Theory-intrinsic operation (Definition 8.1): f: P T → P T is theory-intrinsic if (I1) f is definable from T with no external parameters beyond input; (I2) f is invariant under automorphisms of T.

Realizations (case partition): (3a) I core-computable as saturated closure f = F ω under elementary one-rule firing; (3b) selector-computable from axiom-induced comparability; (3c) guarded I core (saturated form) with guard/scheduling discipline. Note: (3a) ⊂ (3c).

Class C (Completion): f satisfies properties (1)–(3) and (4') of §4.1. Class S (Selection): f is induced by a canonical selector, directly achieving AC-6.

Lemma R2 (Asymmetric Reduction):* For Scott-continuous, theory-intrinsic, uniqueness-inducing f on a dcpo: (i) no outward drift below P* — uniqueness-inducing upward motion is closure/completion-like toward P*; (ii) asymmetry above P* — descending-limit arguments do not apply in general dcpo, so uniqueness from above cannot be derived via Scott-continuous downward convergence alone.

Lemma C (Operator monotonicity from R1+R2+R3, (3a)-scope): Under R1+R2+R3, f = F ω is monotone and extensive on finite P T. Proof uses two-class premise analysis: Class A conditions on observed data (equivalent across all P, Q with same (N, Θ)); Class B conditions on derived predicates (positive presence conditions, monotone-preserved under ⪯ spec).

Lemma 8.10 (Monotonic Exhaustion, per-seed): Let f: P T → P T be extensive, with P T finite. Then for every seed P0 ∈ P T there exist k(P0) ≤ P T and P*(P0) ∈ P T with P0 ⪯ spec f(P0) ⪯ spec ··· ⪯ spec f k(P0)(P0) = P*(P0) and f(P*(P0)) = P*(P0). Monotonicity is not needed. Example 8.12 shows (4') ̸⇒ (4): P T = a, b incomparable, f = id.

Lemma G (Guard Preservation, (3c)-scope): For f = G ω under (G2) functionality, (G3) additive-stable guard, and Soundness: f is well-defined on P T, restricts to P T → P T, and satisfies extensivity, admissibility preservation, idempotence, and per-seed convergence (4'). Well-definedness of G ω does not require G3 (only R1 + finite + G2); G3 records that suppression persists along the trajectory. Lemma G does not claim monotonicity in ⪯ spec.

Theorem U3 (Conditional Classification): "Let T have I core satisfying R1+R2+R3 and finite C. Let f: P T → P T be theory-intrinsic. Assume the realization-coverage hypothesis (OQ-Realization): f is realized by one of the §8.0 forms (3a), (3b), or (3c). Assume the relevant per-realization branch hypotheses: (3a)-branch: R1+R2+R3 + Soundness + finite P T; F-step is elementary one-rule firing under fair scheduling. (3b)-branch: total theory-intrinsic selector ⪯ sel + global compatibility of selected output (C5-S). (3c)-branch: R1+R2+R3 + (G2) functionality + (G3) guard-stability + Soundness. Then: (3a) and (3c) realizations yield f ∈ Class C; (3b) realization yields f ∈ Class S."

Remark 8.7 (Scope of U3): "U3 is conditional. Without the realization-coverage assumption, U3 does not establish that no third primitive mechanism exists in MST; it establishes that, within the declared grammar and under the per-branch hypotheses, no third class arises." Mechanisms outside the grammar (hybrids, cardinality-minimization, orbit selection, fixed-point selection via secondary invariants) are not classified.

Comparability Structures (Refined N4): K = (D, ⪯ K, A, C) with selector-compatible (N4-sel: unique maximum), closure-compatible (N4-comp: extensive, idempotent, admissibility-preserving map with at least one fixed point), or piecewise-compatible specializations.

Det as Partial Pseudofunctor: Det(T, K) = T K canon, obtained by adjoining the uniqueness mechanism induced by K. Det is a partial pseudofunctor MST K → MST: composition and identities hold up to 2-morphism (domain inclusion). Coherence is stated; detailed witnesses are in external artifact; in-paper proof is OQ-Det-Coh.

DQA (Determinization Quasi-Algebra): Q = (D, ◦, A, ≺, O) with finite family D = D l l∈L, partial composition ◦, admissible sequences A, precedence poset ≺, obstructed sequences O. Each D l satisfies: (Q1) Local closure (extensivity, idempotence, admissibility preservation, per-seed convergence on standard domain); (Q2) Partial composition defined only on admissible sequences; (Q3) Obstruction σ ∈ O ⇐⇒ ∃ P ∈ P T: σ(P) ∉ P T; (Q4) Idempotent stability in partial-function form.

Safety gap: Φ(l1,..., ln; S) counts prefixes where Reach k(S) ⊈ Safe(D l k+1); Φ*(S) = min over topological orderings. Φ*(S) = 0 ⇐⇒ ∃ admissible linearization on S.

Compatibility conditions: GCC (Graph-Compatible Guard Condition, seed-independent), RUCS (Reach-Uniform Compatibility, seed-dependent), PLCS (Pairwise Local Compatibility, seed-dependent).

Phase 2 Summary Theorem: "(I) Existence. Φ*(S) = 0 ⇐⇒ ∃ admissible linearization on S. Under PLCS: Φ* = 0 ⇔ (C2) ⇔ (C2′) ⇔ ∃ admissible linearization. (II) Compatibility-strength hierarchy: GCC ⇒ RUCS ⇒ PLCS (seed-dependent, both implications strict); GCC ⇒ RUC ≡ PLC (seed-independent)." Proofs are in companion artifacts phase2b and phase2c, not reproduced in the paper.

Strictness witness (Example 10.11): PLCS without RUCS: L = a, b, P T = Q1, Q2, Q3, Safe(D a) ∩ Reach 0 = Q1, Q2, Safe(D b) ∩ Reach 0 = Q1, Q3. Both χ a = χ b = 0 (PLCS holds); restricted safe sets differ (RUCS fails).

Established: Theorem 6b (Construction Lemma); Corollary 6b' at (4') strength under Soundness; Lemma R2* (parts i, ii); N4 refined; Theorem U3 as conditional classification within (3a)/(3b)/(3c) grammar; Lemma G for f = G ω under (G2)+(G3)+Soundness; Type S-strong → selection (AC-6) via Proposition 4.4; Non-commutativity witness (Proposition 7.1); ICT closure stabilization (4').

Open questions: OQ-GC-1 (whether ICT (4') upgrades to global (4)); OQ-TypeS-Imp (whether non-strict Type S theories admit (4)-strength completion); OQ-Chan-TRS (formal Chan TRS termination + local confluence); OQ-Det-Coh (in-paper proof of Det pseudofunctor coherence); OQ-Realization (whether (3a)/(3b)/(3c) is provably exhaustive for theory-intrinsic operations).

ICT canonicalization status: Closure stabilization (4'): ESTABLISHED. Global completion (4): CONDITIONAL on OQ-GC-1. Determinization (AC-6): CONDITIONAL on OQ-GC-1.

"Completion-based canonicalization requires closure, comparability, and compatible extension; selection-based canonicalization requires theory-intrinsic comparability and global compatibility of selected choices. Not all canonicalization is operator-theoretic. Under hierarchical interaction, canonicalization operators form a structurally non-commutative system."

"Hallucination corresponds to forced canonicalization at AC-6 strength when only stabilization (4') is licensed, or at any strength when neither N4 nor C5 holds. Type E canonicalization (closure stabilization) is the appropriate semantics for systems where evidence accumulates over time; Type S-strong canonicalization (selection) is appropriate when alternatives are axiomatically incomparable. Distinguishing these regimes is prerequisite for a principled treatment of structural commitment in LLM systems."

Improvements for AI systems

Based on the paper, here are the specific improvements I can make to AI systems, particularly LLM-assisted reasoning and decision systems:

Improvement: Implement a formal canonicalization license check before an AI system emits a determinate structural answer.

What the improved system can do:

Before outputting a single interpretation, the system computes the admissible interpretation family P T(N,,) (the set of all globally consistent assignments).
If P T > 1, the system is not licensed to emit a unique answer—it must either:
Present the full set of admissible alternatives, or
Explicitly flag the output as one of several admissible interpretations.
If P T = 1, the system may emit the unique interpretation with confidence.
This directly formalizes the paper's core insight: hallucination = unsupported canonicalization (emitting a determinate answer when the admissible set has not collapsed to a singleton).

Summary of what the improved AI system can do: It can reason about when it is licensed to commit to a unique structural interpretation, how to canonicalize (closure vs. selection based on theory type), which order to apply multi-level operators, how to verify mechanism validity, and what strength of determinism it has actually achieved—all while maintaining admissibility and preventing hallucination-as-unsupported-canonicalization.

Abstract

We develop a formal framework for constructing canonical interpretations from plural structure theories. A structure theory is a triple T = (, A, I) consisting of a signature, axioms, and an inference policy, whose admissible interpretation family collects all globally consistent assignments of structural conclusions. We distinguish three levels of canonicalization: closure stabilization (per-seed convergence), global completion (seed-independent convergence), and determinization (a unique admissible interpretation). Non-determinism is classified into epistemic plurality (Type E) and structural plurality (Type S), with a refined Type S-strong subclass characterized by the absence of common upper bounds. Two canonicalization mechanisms arise: operator-based completion and selector-based construction. We provide sufficient structural conditions under which these mechanisms exist, and show that pure inference-based completion reduces to a saturated closure operator under positive, non-retractive rules with an additional soundness condition. For Type E theories, closure stabilization is established, while full determinization depends on a global confluence property that remains open. For Type S-strong theories, determinization is achieved via canonical selection. We further show that multi-level canonicalization forms a structurally non-commutative system via staged operators, and provide a conditional classification theorem reducing theory-intrinsic mechanisms to closure or selection. The framework also applies to LLM-assisted reasoning, where hallucination can be viewed as unsupported canonicalization.