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

# A Unified Framework via Closure, Comparability, and Joint Admissibility
Source: [https://arxiv.org/html/2608.07476](https://arxiv.org/html/2608.07476)
## Determinization in Structure Theories: A Unified Framework via Closure, Comparability, and Joint Admissibility

\(Version v2\.16\.4 April 2026\)

###### Abstract

Motivated by structural failures in LLM\-assisted reasoning and decision systems, we develop a formal framework for constructing canonical interpretations from plural structure theories\. A*structure theory*is a tripleT=\(Σ,A,I\)T=\(\\Sigma,A,I\)of a signature, axioms, and inference policy\. Its*admissible interpretation family*𝒫T\\mathcal\{P\}\_\{T\}collects all globally consistent assignments of structural conclusions to positions\. In LLM\-assisted reasoning, hallucination can be viewed as unsupported canonicalization—the system emits a determinate answer when the underlying admissible set has not collapsed to a singleton\. Our framework formalizes when canonicalization is licensed\.

Not all structure theories are deterministic\. We classify non\-determinism into*structural plurality*\(Type S\) and*epistemic plurality*\(Type E\), instantiated by Wyckoff and ICT respectively, relative to the intrinsically deterministic Chan theory\.

Our framework distinguishes three increasingly strong notions of “canonical”:*closure stabilization*\(per\-seed convergence to some fixed point, property \(4’\)\);*global completion*\(all seeds converge to a single fixed point, property \(4\)\); and*determinization*in the AC\-6 sense \(a unique admissible interpretation per input\)\. The construction lemma of §5\.3 \(Theorem 6b\) records sufficient structural conditions under which a canonicalization mechanism \(completion operatorDcompD\_\{\\mathrm\{comp\}\}for Type E, selector PhaseClassify for Type S\-strong\) can be constructed at closure\-stabilization strength\. Theorem 6b is a construction lemma; its substantive content is unpacked in Corollary 6b’ \(the Class C classification of pureIcoreI^\{\\mathrm\{core\}\}\-computable completion under R1\+R2\+R3 plus rule Soundness, proved via Lemma C and the Monotonic Exhaustion Lemma in §8\.2\)\. For Type E theories \(ICT\), upgrading from \(4’\) to \(4\) and AC\-6 requires an additional global confluence result; for ICT this is the open question OQ\-GC\-1, and we are explicit that*the present paper does not establish ICT determinization at AC\-6 strength*\. For Type S\-strong theories \(Wyckoff\), AC\-6 is achieved directly via canonical selection \(PhaseClassify\)\.

Two canonicalization forms exist:*operator\-based completion*and*selector\-based construction*\. These are not the same kind of mathematical object: completion is operator\-theoretic; selection is selector\-based\. Completion requires closure, comparability, and a compatible\-extension condition \(C5\-E\); selection requires theory\-intrinsic comparability \(N4\) and global compatibility of selected choices \(C5\-S\)\. A minimal counterexampleTc​eT\_\{ce\}shows that closure alone does not suffice: theory\-intrinsic comparability \(N4\) is necessary\.

We further show that completion extends to multi\-timeframe hierarchies via a staged operatorDcompHTF=DL∘DHD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}=D\_\{L\}\\circ D\_\{H\}, and establish that this operator isstructurally non\-commutative: within the specified two\-operator raw staged architecture, HTF\-first ordering is the only admissibility\-preserving linearization\.

Section 8 presents a classification of canonicalization mechanisms within the MST framework\. We define theory\-intrinsic operations independently of the mechanism taxonomy \(definable fromTTand automorphism\-invariant\); Theorem U3, conditional on the realization\-coverage assumption tracked as OQ\-Realization and on per\-realization branch hypotheses \(R1\+R2\+R3 \+ Soundness \+ elementaryFF\-step for \(3a\); theory\-intrinsic⪯sel\\preceq\_\{\\mathrm\{sel\}\}\+ C5\-S for \(3b\); R1\+R2\+R3 \+ \(G2\) \+ \(G3\) \+ Soundness for \(3c\)\), classifies every theory\-intrinsic mechanism realized within the \(3a\)/\(3b\)/\(3c\) grammar into Class C \(closure\) or Class S \(selection\)\. The classification is at closure\-stabilization strength \(4’\); whether a Class C operator additionally achieves property \(4\) or AC\-6 is instance\-specific\. The section also introduces canonicalization as a partial pseudofunctor \(Det\)\.

For Type S, the relationship to selection is more nuanced than a strict dichotomy\. We isolateType S\-strongas the subclass where𝒫T\\mathcal\{P\}\_\{T\}contains two admissible interpretations with no common upper bound under⪯spec\\preceq\_\{\\mathrm\{spec\}\}\(§4\.2\)\. Only for Type S\-strong instances is property \(4\) \(global completion\) provably impossible; the weaker \(4’\) remains attainable\. Wyckoff is Type S\-strong, so the selection construction is justified forTW​yT\_\{Wy\}; the generic Type S→\\toselection inference is conditional on this strengthening and is left open \(OQ\-TypeS\-Imp\)\.

Phase 2 \(§10\) characterizes canonicalization under multiple operators with precedence constraints \(summary statement; full proofs in companion artifacts\)\. Feasibility is equivalent to the vanishing of the safety gapΦ∗\\Phi^\{\*\}\. Under pairwise local compatibility \(PLC𝒮\\mathrm\{PLC\}\_\{\\mathcal\{S\}\}\), the existence conditions \(C2\) and \(C2’\) collapse\. The compatibility\-condition strength ordering isGCC⇒RUC𝒮⇒PLC𝒮\\mathrm\{GCC\}\\Rightarrow\\mathrm\{RUC\}\_\{\\mathcal\{S\}\}\\Rightarrow\\mathrm\{PLC\}\_\{\\mathcal\{S\}\}\(strict in the seed\-dependent setting, collapsing toGCC⇒RUC≡PLC\\mathrm\{GCC\}\\Rightarrow\\mathrm\{RUC\}\\equiv\\mathrm\{PLC\}seed\-independently\); detailed proofs are recorded in companion artifacts and not reproduced in this paper\.

Core principle\.*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\.*

###### Contents

1. [1Introduction](https://arxiv.org/html/2608.07476#S1)
2. [2Framework](https://arxiv.org/html/2608.07476#S2)1. [2\.1Structure Theories](https://arxiv.org/html/2608.07476#S2.SS1)
3. [3Taxonomy of Non\-Determinism](https://arxiv.org/html/2608.07476#S3)1. [3\.1Formal Predicates](https://arxiv.org/html/2608.07476#S3.SS1) 2. [3\.2Examples](https://arxiv.org/html/2608.07476#S3.SS2)
4. [4Determinization Mechanisms](https://arxiv.org/html/2608.07476#S4)1. [4\.1Operator\-Based Completion \(Type E\)](https://arxiv.org/html/2608.07476#S4.SS1) 2. [4\.2Selector\-Based Construction \(Type S\-strong\)](https://arxiv.org/html/2608.07476#S4.SS2)
5. [5Sufficient Conditions for Canonicalization \(Theorem 6b\)](https://arxiv.org/html/2608.07476#S5)1. [5\.1Conditions](https://arxiv.org/html/2608.07476#S5.SS1) 2. [5\.2Counterexample](https://arxiv.org/html/2608.07476#S5.SS2) 3. [5\.3Theorem 6b \(Construction Lemma\) and Corollary 6b’](https://arxiv.org/html/2608.07476#S5.SS3) 4. [5\.4Summary](https://arxiv.org/html/2608.07476#S5.SS4)
6. [6Examples](https://arxiv.org/html/2608.07476#S6)1. [6\.1Wyckoff \(Type S\-strong, Complete\)](https://arxiv.org/html/2608.07476#S6.SS1) 2. [6\.2ICT \(Type E\) — Single\-Timeframe and Multi\-HTF](https://arxiv.org/html/2608.07476#S6.SS2)
7. [7Non\-Commutativity of Staged Completion](https://arxiv.org/html/2608.07476#S7)
8. [8Mechanism Classification and Categorical Structure](https://arxiv.org/html/2608.07476#S8)1. [8\.1Definitions](https://arxiv.org/html/2608.07476#S8.SS1) 2. [8\.2Asymmetric Reduction \(Lemma R2\*\)](https://arxiv.org/html/2608.07476#S8.SS2) 3. [8\.3Conditional Classification within the MST Realization Grammar \(Theorem U3\)](https://arxiv.org/html/2608.07476#S8.SS3) 4. [8\.4Comparability Structures \(Refined N4\)](https://arxiv.org/html/2608.07476#S8.SS4) 5. [8\.5Determinization as a Pseudofunctor](https://arxiv.org/html/2608.07476#S8.SS5) 6. [8\.6Status of §8 Results](https://arxiv.org/html/2608.07476#S8.SS6)
9. [9Discussion](https://arxiv.org/html/2608.07476#S9)
10. [10Phase 2: Multi\-Operator Canonicalization](https://arxiv.org/html/2608.07476#S10)1. [10\.1Setting and Definitions](https://arxiv.org/html/2608.07476#S10.SS1) 2. [10\.2Phase 2 Summary Theorem](https://arxiv.org/html/2608.07476#S10.SS2) 3. [10\.3Strictness Witness](https://arxiv.org/html/2608.07476#S10.SS3)
11. [11Changelog](https://arxiv.org/html/2608.07476#S11)1. [11\.1v2\.16\.3→\\tov2\.16\.4 \(mathematical correctness fixes\)](https://arxiv.org/html/2608.07476#S11.SS1) 2. [11\.2v2\.16\.2→\\tov2\.16\.3 \(claim\-strength tightening\)](https://arxiv.org/html/2608.07476#S11.SS2) 3. [11\.3v2\.16→\\tov2\.16\.2 \(M1 fix and minor follow\-up\)](https://arxiv.org/html/2608.07476#S11.SS3) 4. [11\.4v2\.15\.1→\\tov2\.16 \(errata revision, 13 issues\)](https://arxiv.org/html/2608.07476#S11.SS4)
12. [AConflict Matrices](https://arxiv.org/html/2608.07476#A1)1. [A\.1M2 — ICT Single\-Timeframe Closure Domain](https://arxiv.org/html/2608.07476#A1.SS1) 2. [A\.2M2\-HTF — Cross\-Timeframe Forbidden Pairs](https://arxiv.org/html/2608.07476#A1.SS2)

## 1Introduction

This 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\. Canonicalization is the problem of deciding when the system is licensed to output a unique interpretation rather than a set of admissible alternatives\. Hallucination, in this view, is unsupported canonicalization—the system emits a determinate answer when the underlying admissible set has not collapsed to a singleton\. We formalize the structural conditions under which canonicalization is licensed\.

Most market structure frameworks are*not*deterministic: multiple valid interpretations coexist at the same position\. We formalize this question and characterize sufficient structural conditions under which a canonical reading can be constructed\.

We formalize using structure theoriesT=\(Σ,A,I\)T=\(\\Sigma,A,I\), and introduce*canonicalization*as the construction of a canonical extensionTcanonT^\{\\mathrm\{canon\}\}from a plural baseTT\. Our framework operates at three strengths:

- •Closure stabilization\(property \(4’\), §4\.1\): each admissible seed converges under iteration of an operator to*some*fixed point\. This is the strength established by Theorem 6b’s completion branch and by Corollary 6b’\.
- •Global completion\(property \(4\)\): all seeds converge to a*single*fixed point\. This requires global confluence of the operator’s ARS, which is theory\-specific and not implied by \(4’\)\. For ICT this is open \(OQ\-GC\-1\)\.
- •Determinization\(AC\-6\): the unique fixed point is the sole admissible interpretation\. This is the strongest claim and is achieved directly by canonical selection \(e\.g\., PhaseClassify for Wyckoff\)\. For ICT, AC\-6 is conditional on OQ\-GC\-1\.

We are explicit throughout the paper about which claim applies at each strength, and in particular about the fact thatthe present paper establishes ICT closure stabilization but does not establish ICT determinization at AC\-6 strength\.

Our main contributions are:

1. 1\.Taxonomy\(§3, §4\.2\): two\-type classification of non\-determinism \(Type S vs\. Type E\), with the*Type S\-strong*refinement \(no common upper bound\) for which selection is provably the unique mechanism reaching AC\-6\.
2. 2\.Construction lemma\(§5\.3, Theorem 6b\): sufficient structural conditions \(N0–N4 \+ C5, with mechanism\-specific variants C5\-E and C5\-S\) under which the completion operatorDcompD\_\{\\mathrm\{comp\}\}or the selector PhaseClassify can be constructed\. Theorem 6b is a construction lemma: given the conditions, the mechanism is built\. Its substantive depth is unpacked by Corollary 6b’ \(§5\.3\)\.
3. 3\.Classification ofIcoreI^\{\\mathrm\{core\}\}\-computable completion\(Corollary 6b’, §5\.3 \+ Lemma C and Monotonic Exhaustion Lemma, §8\.2\): under R1\+R2\+R3*plus rule Soundness relative toAA*, every \(3a\)\-computable completion operator \(interpreted as the saturated closureFωF^\{\\omega\}under elementary one\-rule firing\) is automatically monotone, extensive, admissibility\-preserving, and converges per seed to a fixed point—hence is Class C at closure\-stabilization strength\.
4. 4\.Dual mechanism\(§4\.1, §4\.2\): two fundamentally different canonicalization forms—operator\-based completion \(Type E\) vs\. selector\-based construction \(Type S\-strong, axiom\-induced order\)\. The two constructions yield different mathematical objects\.
5. 5\.CounterexampleTc​eT\_\{ce\}\(§5\.2\): a minimal theory satisfying N0\+N1\+N3 but failing both N4\-sel and N4\-comp; demonstrates that theory\-intrinsic comparability \(N4\) is necessary\.
6. 6\.Multi\-timeframe extension\(§6\.2\): staged operatorDcompHTF=DL∘DHD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}=D\_\{L\}\\circ D\_\{H\}on𝒫I​C​Tmulti\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}, with explicit verification of §4\.1 properties at \(4’\) strength\.
7. 7\.Structural non\-commutativity\(§7\): a concrete witnessPPsuch that\(DL∘DH\)​\(P\)∈𝒫I​C​Tmulti\(D\_\{L\}\\circ D\_\{H\}\)\(P\)\\in\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}while\(DH∘DL\)​\(P\)∉𝒫I​C​Tmulti\(D\_\{H\}\\circ D\_\{L\}\)\(P\)\\notin\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}, with ordering uniqueness scoped to the two\-operator raw staged architecture\.
8. 8\.Mechanism classification\(§8\): asymmetric reduction \(R2\*, parts i and ii\); independent definition of “theory\-intrinsic” not predicated on \(3a\)/\(3b\)/\(3c\); Theorem U3, conditional on realization\-coverage \(OQ\-Realization\) and on per\-realization branch hypotheses, classifies every theory\-intrinsic mechanism realized within \(3a\)/\(3b\)/\(3c\) into Class C or Class S; Det as a partial pseudofunctor \(§8\.4\)\.
9. 9\.Phase 2 \(multi\-operator canonicalization\)\(§10\): summary of existence conditions \(Φ∗=0⇔\\Phi^\{\*\}=0\\Leftrightarrowadmissible linearization\), collapse underPLC𝒮\\mathrm\{PLC\}\_\{\\mathcal\{S\}\}\(C2≡\\equivC2’≡\\equivexistence\), and the compatibility\-strength hierarchyGCC⇒RUC𝒮⇒PLC𝒮\\mathrm\{GCC\}\\Rightarrow\\mathrm\{RUC\}\_\{\\mathcal\{S\}\}\\Rightarrow\\mathrm\{PLC\}\_\{\\mathcal\{S\}\}\(summary statement; full proofs are recorded in companion artifacts and not reproduced here\)\.

Scope note\.This paper unifies several lines of MST development under a single framework\. Each of contributions 1–3, 6–7, 8, and 9 could be developed in a focused follow\-up paper, and we anticipate such future splits as the framework matures\. The §11 changelog tracks the v2\.15\.1→\\tov2\.16→\\tov2\.16\.2→\\tov2\.16\.3→\\tov2\.16\.4 errata revisions\.

### 2\.1Structure Theories

###### Definition 2\.1\.

A*structure theory*isT=\(Σ,A,I\)T=\(\\Sigma,A,I\)whereΣ\\Sigmais a signature,AAaxioms, andI=Icore∪IextI=I^\{\\mathrm\{core\}\}\\cup I^\{\\mathrm\{ext\}\}an inference policy \(this paper concerns onlyIcoreI^\{\\mathrm\{core\}\}\)\.

Input tuple\.Theories are evaluated on inputs\(N,Θ,Π\)\(N,\\Theta,\\Pi\)whereNNis an observed market realization,Θ\\Thetais the set of theory\-relevant parameters, andΠ\\Piis a structural evidence assignment\. Domain objects:RR= set of regions/instruments,𝖳𝗂𝗆𝖾\\mathsf\{Time\}= temporal index set,𝒞\\mathcal\{C\}= structural conclusion vocabulary\.

###### Definition 2\.2\(Finite evaluation domain\)\.

Throughout we assume: \(a\)𝒞\\mathcal\{C\}is finite; \(b\)Dom⁡\(N,Θ\)⊆R×𝖳𝗂𝗆𝖾\\operatorname\{Dom\}\(N,\\Theta\)\\subseteq R\\times\\mathsf\{Time\}is finite for every\(N,Θ\)\(N,\\Theta\)\. Hence\|𝒫T​\(N,Θ,Π\)\|≤\(2\|𝒞\|\)\|Dom⁡\(N,Θ\)\|<∞\|\\mathcal\{P\}\_\{T\}\(N,\\Theta,\\Pi\)\|\\leq\(2^\{\|\\mathcal\{C\}\|\}\)^\{\|\\operatorname\{Dom\}\(N,\\Theta\)\|\}<\\infty\.

Definitional constraints onIcoreI^\{\\mathrm\{core\}\}\.IcoreI^\{\\mathrm\{core\}\}is restricted to*positive, non\-retractive*inference:

- •\(R1\) Positive rule form: every rule only adds structural conclusions; no rule removes conclusions\.
- •\(R2\) No negation\-as\-failure inIcoreI^\{\\mathrm\{core\}\}: every rule has a premise that is a positive conjunction of conditions—each condition asserts the*presence*of a pattern or comparison on observed data, or the*presence*of an already\-derived predicate, but never the*absence*of a derived predicate\. *Scope*: R2 constrainsIcoreI^\{\\mathrm\{core\}\}\. The determinization layer \(§4\) may employ conflict\-resolution mechanisms \(e\.g\., M2/M2\-HTF guards, §6\.2\) that are notIcoreI^\{\\mathrm\{core\}\}rules and are not constrained by R2\.
- •\(R3\) No retraction: no rule withdraws or invalidates previously derived conclusions\.

These constraints are definitional and*syntactic*—they constrain the shape of rules\. They are independent of*semantic*soundness \(consistency of rule conclusions with the axiomsAA\); see §5\.3 and the soundness Remark following Corollary 6b’\. Conflict resolution among admissible conclusions is not the responsibility ofIcoreI^\{\\mathrm\{core\}\}; it is handled by the determinization mechanism \(Class C completion or Class S selection\)\. This separation is the architectural basis for the mechanism classification in §8\.

###### Definition 2\.3\.

The*admissible interpretation family*𝒫T​\(N,Θ,Π\)\\mathcal\{P\}\_\{T\}\(N,\\Theta,\\Pi\)is the set of all assignmentsP:Dom⁡\(N,Θ\)→2𝒞P:\\operatorname\{Dom\}\(N,\\Theta\)\\to 2^\{\\mathcal\{C\}\}satisfying all axiomsAAand consistent withIcoreI^\{\\mathrm\{core\}\}\.

###### Definition 2\.4\(Specification order\)\.

ForP,Q∈𝒫TP,Q\\in\\mathcal\{P\}\_\{T\}, writeP⪯specQP\\preceq\_\{\\mathrm\{spec\}\}QifP​\(r,t\)⊆Q​\(r,t\)P\(r,t\)\\subseteq Q\(r,t\)for all\(r,t\)∈Dom⁡\(N,Θ\)\(r,t\)\\in\\operatorname\{Dom\}\(N,\\Theta\)\. This is a partial order on𝒫T\\mathcal\{P\}\_\{T\}\.

###### Definition 2\.5\(AC\-6 and AC\-6’\)\.

TTis*deterministic*\(AC\-6\) if\|𝒫T​\(N,Θ,Π\)\|=1\|\\mathcal\{P\}\_\{T\}\(N,\\Theta,\\Pi\)\|=1for all inputs\.TTis*plural*\(AC\-6’\) if\|𝒫T​\(N,Θ,Π\)\|≥2\|\\mathcal\{P\}\_\{T\}\(N,\\Theta,\\Pi\)\|\\geq 2for some input\.

## 3Taxonomy of Non\-Determinism

### 3\.1Formal Predicates

###### Definition 3\.1\(Type E — Epistemic Plurality\)\.

TTis*Type E*if its plurality vanishes under structural evidence refinement: for every input with\|𝒫T\|\>1\|\\mathcal\{P\}\_\{T\}\|\>1, there exists a refinedΠ′⊇Π\\Pi^\{\\prime\}\\supseteq\\Pi\(withNN,Θ\\Thetafixed\) such that\|𝒫T​\(N,Θ,Π′\)\|=1\|\\mathcal\{P\}\_\{T\}\(N,\\Theta,\\Pi^\{\\prime\}\)\|=1\.

###### Definition 3\.2\(Πfull\\Pi\_\{\\mathrm\{full\}\}\)\.

A*maximally informative evidence assignment*assigns a definite truth value to every theory\-relevant structural predicate in𝒞\\mathcal\{C\}\.

###### Definition 3\.3\(Type S — Structural Plurality\)\.

TTis*Type S*if its plurality persists atΠfull\\Pi\_\{\\mathrm\{full\}\}: there exists\(N,Θ,Πfull\)\(N,\\Theta,\\Pi\_\{\\mathrm\{full\}\}\)with\|𝒫T​\(N,Θ,Πfull\)\|\>1\|\\mathcal\{P\}\_\{T\}\(N,\\Theta,\\Pi\_\{\\mathrm\{full\}\}\)\|\>1\.

###### Definition 3\.4\(Type S\-strong — No Common Upper Bound\)\.

TTis*Type S\-strong*ifTTis Type S and there exists an input\(N,Θ,Πfull\)\(N,\\Theta,\\Pi\_\{\\mathrm\{full\}\}\)at which two distinct admissible interpretationsP1,P2∈𝒫T​\(N,Θ,Πfull\)P\_\{1\},P\_\{2\}\\in\\mathcal\{P\}\_\{T\}\(N,\\Theta,\\Pi\_\{\\mathrm\{full\}\}\)have*no common upper bound*in\(𝒫T,⪯spec\)\(\\mathcal\{P\}\_\{T\},\\preceq\_\{\\mathrm\{spec\}\}\)\.

### 3\.2Examples

Type S\-strong: Wyckoff \(TW​yT\_\{Wy\}\)\.Axiom \[Wy\-Sem\-1\] makesin\_phaserelational; admissible phases atΠfull\\Pi\_\{\\mathrm\{full\}\}are axiomatically incomparable, with no element of𝒫W​y\\mathcal\{P\}\_\{Wy\}extending two distinct phase assignments\. Hence Type S\-strong\.

Type E: ICT \(TI​C​TT\_\{ICT\}\)\.Plurality is epistemic: once additional bars resolve the structure, exactly one completion survives\.

Intrinsically Deterministic: Chan \(TChanT\_\{\\mathrm\{Chan\}\}\)\.Chan is deterministic by the unique\-normal\-form theorem for its term rewrite systemℛChan\\mathcal\{R\}\_\{\\mathrm\{Chan\}\}\.

Chan rewrite presentation \(sketch\)\.ℛChan\\mathcal\{R\}\_\{\\mathrm\{Chan\}\}encodes the bi\-directional constituency relations of Chan theory, a hierarchical market structure framework with multiple structural levels \(elementary strokes, line segments, pivot zones\)\. The TRS is asserted to satisfy termination \(rules strictly decrease structural nesting depth\) and local confluence \(critical pairs are joinable\)\. By Newman’s Lemma,ℛChan\\mathcal\{R\}\_\{\\mathrm\{Chan\}\}is confluent: every expression has a unique normal form, establishing AC\-6 without external construction\.

Open question \(OQ\-Chan\-TRS\)\.Termination and local confluence are stated here as the standard informal account in the Chan literature\. A self\-contained formal proof is left as an open question\. The framework does not depend on resolving OQ\-Chan\-TRS for Type E or Type S\-strong results; Chan is used as a comparator, not a proof dependency\.

## 4Determinization Mechanisms

### 4\.1Operator\-Based Completion \(Type E\)

###### Definition 4\.1\(Determinization operator\)\.

Let𝒟T⊆𝒫T\\mathcal\{D\}\_\{T\}\\subseteq\\mathcal\{P\}\_\{T\}be a canonical closure domain\. A*determinization operator*is a mapD:𝒟T→𝒟TD:\\mathcal\{D\}\_\{T\}\\to\\mathcal\{D\}\_\{T\}satisfying:

1. 1\.Extensivity:P⪯specD​\(P\)P\\preceq\_\{\\mathrm\{spec\}\}D\(P\)for allP∈𝒟TP\\in\\mathcal\{D\}\_\{T\}\.
2. 2\.Idempotence:D​\(D​\(P\)\)=D​\(P\)D\(D\(P\)\)=D\(P\)for allP∈𝒟TP\\in\\mathcal\{D\}\_\{T\}\.
3. 3\.Admissibility preservation:D​\(P\)∈𝒟TD\(P\)\\in\\mathcal\{D\}\_\{T\}for allP∈𝒟TP\\in\\mathcal\{D\}\_\{T\}\.
4. 4\.Per\-seed convergence \(Property \(4’\)\): for every\(N,Θ,Π\)\(N,\\Theta,\\Pi\)and every seedP∈𝒟T​\(N,Θ,Π\)P\\in\\mathcal\{D\}\_\{T\}\(N,\\Theta,\\Pi\),∃k≤\|𝒟T​\(N,Θ,Π\)\|\\exists\\,k\\leq\|\\mathcal\{D\}\_\{T\}\(N,\\Theta,\\Pi\)\|and a fixed pointP∗​\(P\)∈𝒟TP^\{\*\}\(P\)\\in\\mathcal\{D\}\_\{T\}such thatDk​\(P\)=P∗​\(P\)D^\{k\}\(P\)=P^\{\*\}\(P\)andD​\(P∗​\(P\)\)=P∗​\(P\)D\(P^\{\*\}\(P\)\)=P^\{\*\}\(P\)\.

The strongerProperty \(4\)additionally requires all seeds converge to the same fixed point\.AC​\-​6⇒\(4\)⇒\(4′\)\\mathrm\{AC\\text\{\-\}6\}\\Rightarrow\(4\)\\Rightarrow\(4^\{\\prime\}\)\.

###### Definition 4\.3\(Uniqueness\-inducing operator\)\.

D:𝒟T→𝒟TD:\\mathcal\{D\}\_\{T\}\\to\\mathcal\{D\}\_\{T\}is*uniqueness\-inducing*if it satisfies property \(4\): there exists, for each instance\(N,Θ,Π\)\(N,\\Theta,\\Pi\), a singleP∗​\(N,Θ,Π\)∈𝒟T​\(N,Θ,Π\)P^\{\*\}\(N,\\Theta,\\Pi\)\\in\\mathcal\{D\}\_\{T\}\(N,\\Theta,\\Pi\)such thatDk​\(P\)=P∗​\(N,Θ,Π\)D^\{k\}\(P\)=P^\{\*\}\(N,\\Theta,\\Pi\)for all seedsPPand somekkuniformly bounded across seeds\.

An operator satisfying only \(4’\) \(per\-seed convergence\) is*not*uniqueness\-inducing in this sense—different seeds may stabilize at distinct fixed pointsP∗​\(P\)P^\{\*\}\(P\)\.

ICT instance\.Dcomp​\(P\)=ClAI​C​T,Π​\(P\)D\_\{\\mathrm\{comp\}\}\(P\)=\\mathrm\{Cl\}\_\{A\_\{ICT\},\\Pi\}\(P\)\. Under M1\+M2\+M3,DcompD\_\{\\mathrm\{comp\}\}satisfies all four properties on𝒟T=𝒫I​C​Tclosure\\mathcal\{D\}\_\{T\}=\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{closure\}\}, with property \(4\) at strength \(4’\) only—the upgrade to global \(4\) is OQ\-GC\-1\.

### 4\.2Selector\-Based Construction \(Type S\-strong\)

For Type E theories, completion naturally applies\. For Type S\-strong theories—those with incomparable admissible interpretations atΠfull\\Pi\_\{\\mathrm\{full\}\}—completion at property \(4\) strength is provably impossible, and canonicalization at AC\-6 strength must use a canonical selector\. The weaker property \(4’\) is not blocked\.

###### Proposition 4\.4\(Completion at \(4\)\-strength impossible for Type S\-strong\)\.

LetTTbe Type S\-strong, witnessed atΠfull\\Pi\_\{\\mathrm\{full\}\}byP1,P2∈𝒫T​\(N,Θ,Πfull\)P\_\{1\},P\_\{2\}\\in\\mathcal\{P\}\_\{T\}\(N,\\Theta,\\Pi\_\{\\mathrm\{full\}\}\)with no common upper bound in\(𝒫T,⪯spec\)\(\\mathcal\{P\}\_\{T\},\\preceq\_\{\\mathrm\{spec\}\}\)\. Then for any extensive operatorD:𝒟T→𝒟TD:\\mathcal\{D\}\_\{T\}\\to\\mathcal\{D\}\_\{T\}withP1,P2∈𝒟TP\_\{1\},P\_\{2\}\\in\\mathcal\{D\}\_\{T\}, property \(4\) fails on this instance\.

###### Proof\.

Suppose for contradiction thatDDis extensive and property \(4\) holds with unique convergence pointP∗P^\{\*\}\.

*Step 1 \(ascending chain\)\.*Extensivity givesP1⪯specD​\(P1\)⪯specD2​\(P1\)⪯spec⋯P\_\{1\}\\preceq\_\{\\mathrm\{spec\}\}D\(P\_\{1\}\)\\preceq\_\{\\mathrm\{spec\}\}D^\{2\}\(P\_\{1\}\)\\preceq\_\{\\mathrm\{spec\}\}\\cdots\. Since𝒟T⊆𝒫T\\mathcal\{D\}\_\{T\}\\subseteq\\mathcal\{P\}\_\{T\}is finite, the chain stabilizes:∃k1≤\|𝒟T\|\\exists\\,k\_\{1\}\\leq\|\\mathcal\{D\}\_\{T\}\|withDk1​\(P1\)=Dk1\+1​\(P1\)D^\{k\_\{1\}\}\(P\_\{1\}\)=D^\{k\_\{1\}\+1\}\(P\_\{1\}\)\.

*Step 2 \(stabilization equalsP∗P^\{\*\}\)\.*Dk1​\(P1\)D^\{k\_\{1\}\}\(P\_\{1\}\)is a fixed point ofDDin𝒟T\\mathcal\{D\}\_\{T\}, hence a valid seed\. By \(4\), every seed converges toP∗P^\{\*\}; the constant orbit fromDk1​\(P1\)D^\{k\_\{1\}\}\(P\_\{1\}\)converges to itself, soDk1​\(P1\)=P∗D^\{k\_\{1\}\}\(P\_\{1\}\)=P^\{\*\}\. By transitivity,P1⪯specP∗P\_\{1\}\\preceq\_\{\\mathrm\{spec\}\}P^\{\*\}\.

*Step 3 \(symmetric\)\.*P2⪯specP∗P\_\{2\}\\preceq\_\{\\mathrm\{spec\}\}P^\{\*\}\.

*Step 4 \(contradiction\)\.*P∗P^\{\*\}is a common upper bound of\{P1,P2\}\\\{P\_\{1\},P\_\{2\}\\\}, contradicting Type S\-strong\.

The weaker property \(4’\) is not ruled out:P1P\_\{1\}andP2P\_\{2\}may individually converge to distinct fixed pointsP1∗,P2∗P\_\{1\}^\{\*\},P\_\{2\}^\{\*\}\. ∎∎

###### Definition 4\.8\(Canonical selector\)\.

A functionϕ:\(r,t,Θ,Π\)↦c∗∈Ar,t\\phi:\(r,t,\\Theta,\\Pi\)\\mapsto c^\{\*\}\\in A\_\{r,t\}, theory\-intrinsically defined and globally compatible\.

Wyckoff instance\.PhaseClassify​\(r,t;Θ,Π\)=max⪯sel⁡Ar,t\\mathrm\{PhaseClassify\}\(r,t;\\Theta,\\Pi\)=\\max\_\{\\preceq\_\{\\mathrm\{sel\}\}\}A\_\{r,t\}, where⪯sel\\preceq\_\{\\mathrm\{sel\}\}is induced by event dominance \(C4\-Q2\) and MEP ensures global compatibility\.

## 5Sufficient Conditions for Canonicalization \(Theorem 6b\)

### 5\.1Conditions

- •N0\(universal, non\-emptiness\):𝒫T​\(N,Θ,Π\)≠∅\\mathcal\{P\}\_\{T\}\(N,\\Theta,\\Pi\)\\neq\\varnothingfor every input\.
- •N1\(selector\-specific, finiteness\):\|Ar,t\|<∞\|A\_\{r,t\}\|<\\infty\.
- •N3\(operator\-specific, bounded closure\):∃k:Clk​\(P\)=Clk\+1​\(P\)\\exists\\,k:\\mathrm\{Cl\}^\{k\}\(P\)=\\mathrm\{Cl\}^\{k\+1\}\(P\)on the canonical seed domain \(M3 ensures this; bound\|𝒞\|⋅\|Dom⁡\(N,Θ\)\|\|\\mathcal\{C\}\|\\cdot\|\\operatorname\{Dom\}\(N,\\Theta\)\|\)\.
- •N4\(unified comparability\):TTadmits a theory\-intrinsic comparability\-inducing structure𝒦\\mathcal\{K\}providing local comparability, unique selection or unique closure fixed point, derivability, and global compatibility\. *Refined treatment\.*The structural definition—order tuple𝒦=\(D,⪯𝒦,𝒜,𝒞\)\\mathcal\{K\}=\(D,\\preceq\_\{\\mathcal\{K\}\},\\mathcal\{A\},\\mathcal\{C\}\)with selector\-/closure\-/piecewise\-compatible specializations—is given in §8\.3\. Theorem 6b’s branches use N4\-comp \(closure\-compatible\) and N4\-sel \(selector\-compatible\)\.
- •C5\(joint admissibility, mechanism\-specific\)\. C5 is expressed as two distinct conditions according to mechanism type: C5\-S\(selection variant,*global compatibility of selected choices*\): letcr,t∗:=max⪯sel⁡Ar,tc^\{\*\}\_\{r,t\}:=\\max\_\{\\preceq\_\{\\mathrm\{sel\}\}\}A\_\{r,t\}be the selector output at\(r,t\)\(r,t\)\. Then the selected assignment P∗:\(r,t\)↦\{cr,t∗\}P^\{\*\}:\(r,t\)\\mapsto\\\{c^\{\*\}\_\{r,t\}\\\}extends to an element of𝒫T\\mathcal\{P\}\_\{T\}\. In multi\-label form,\{cr,t∗:\(r,t\)∈Dom⁡\(N,Θ\)\}∈𝒫T\\\{c^\{\*\}\_\{r,t\}:\(r,t\)\\in\\operatorname\{Dom\}\(N,\\Theta\)\\\}\\in\\mathcal\{P\}\_\{T\}\. C5\-S asserts global admissibility of the selector’s output, not joint admissibility of local alternatives—selection in Type S\-strong precisely operates over alternatives that lack a common upper bound, so requiring local alternatives to be jointly admissible would conflict with the Type S\-strong hypothesis\. C5\-E\(completion variant,*compatible extension within the closure domain*\): for every input and every pairc1,c2∈𝒞c\_\{1\},c\_\{2\}\\in\\mathcal\{C\}such that∃P1,P2∈𝒫Tclosure\\exists\\,P\_\{1\},P\_\{2\}\\in\\mathcal\{P\}\_\{T\}^\{\\mathrm\{closure\}\}and shared\(r,t\)\(r,t\)withc1∈P1​\(r,t\)c\_\{1\}\\in P\_\{1\}\(r,t\),c2∈P2​\(r,t\)c\_\{2\}\\in P\_\{2\}\(r,t\): if the closure\-domain conflict relation does not mark\{c1,c2\}\\\{c\_\{1\},c\_\{2\}\\\}as forbidden at\(r,t\)\(r,t\), then∃P\+∈𝒫Tclosure\\exists\\,P^\{\+\}\\in\\mathcal\{P\}\_\{T\}^\{\\mathrm\{closure\}\}with\{c1,c2\}⊆P\+​\(r,t\)\\\{c\_\{1\},c\_\{2\}\\\}\\subseteq P^\{\+\}\(r,t\)\. The*closure\-domain conflict relation*is the set of forbidden predicate pairs whose exclusion defines𝒫Tclosure\\mathcal\{P\}\_\{T\}^\{\\mathrm\{closure\}\}\. ForTI​C​TT\_\{ICT\}single\-timeframe, this is M2 \(Appendix A\.1\); forTI​C​TmultiT\_\{ICT\}^\{\\mathrm\{multi\}\}, it isM2∪M2​\-​HTF\\mathrm\{M2\}\\cup\\mathrm\{M2\\text\{\-\}HTF\}\(Appendix A\.2\)\. C5\-E asserts that non\-forbidden co\-generating predicate pairs have a common closure\-domain witness\. *Asymmetry between C5\-S and C5\-E\.*The two variants are not parallel: C5\-S is a forward condition on the selector’s output \(the selected assignment is admissible\); C5\-E is a closure\-domain extension property \(non\-conflicting predicates jointly extend\)\. The two share the structural intuition that canonicalization output must be globally consistent, but operate on different mathematical objects \(a selector function vs\. a set\-valued closure operator\)\.

*C5 and N4 are independent conditions\.*T′T^\{\\prime\}witnesses N4 without C5\-S \(𝒫T′=\{\{a\},\{b\}\}\\mathcal\{P\}\_\{T^\{\\prime\}\}=\\\{\\\{a\\\},\\\{b\\\}\\\}witha≺selba\\prec\_\{\\mathrm\{sel\}\}b, soc∗=bc^\{\*\}=bat every point, but if selectingbbglobally violates some other axiom inA′A^\{\\prime\}, the global admissibility of the selected assignment fails\)\.T′′T^\{\\prime\\prime\}witnesses C5\-E without N4 \(𝒫T′′=\{\{a\},\{b\},\{a,b\}\}\\mathcal\{P\}\_\{T^\{\\prime\\prime\}\}=\\\{\\\{a\\\},\\\{b\\\},\\\{a,b\\\}\\\}, with\{a,b\}\\\{a,b\\\}jointly realizable but no theory\-intrinsic order or closure fixed point\)\.

### 5\.2Counterexample

###### Proposition 5\.1\(N0\+N1\+N3 do not suffice\)\.

LetTc​eT\_\{ce\}have two conclusionsc1,c2c\_\{1\},c\_\{2\}: individually admissible, axiomatically incompatible \(so\{c1,c2\}∉𝒫Tc​e\\\{c\_\{1\},c\_\{2\}\\\}\\notin\\mathcal\{P\}\_\{T\_\{ce\}\}\), unrelated by any ordering axiom\. Then𝒫Tc​e=\{\{c1\},\{c2\}\}\\mathcal\{P\}\_\{T\_\{ce\}\}=\\\{\\\{c\_\{1\}\\\},\\\{c\_\{2\}\\\}\\\}\.Tc​eT\_\{ce\}satisfies N0, N1, N3, but fails N4 in both forms \(N4\-sel: no theory\-intrinsic order; N4\-comp: trivial closure has multiple fixed points\)\. Hence no canonicalization mechanism exists, showingtheory\-intrinsic comparability \(N4\) is necessary\.

### 5\.3Theorem 6b \(Construction Lemma\) and Corollary 6b’

###### Theorem 5\.2\(Construction lemma for canonicalization mechanisms\)\.

LetTTsatisfy AC\-6’\. Then:

*Completion branch:*

N0\+N3\+N4​\-​comp\+C5​\-​E⇒Dcomp​exists and satisfies \(4’\)\\mathrm\{N0\}\+\\mathrm\{N3\}\+\\mathrm\{N4\\text\{\-\}comp\}\+\\mathrm\{C5\\text\{\-\}E\}\\;\\Rightarrow\\;D\_\{\\mathrm\{comp\}\}\\text\{ exists and satisfies \(4'\)\}
*Selection branch:*

N0\+N1\+N4​\-​sel\+C5​\-​S⇒PhaseClassify​exists\\mathrm\{N0\}\+\\mathrm\{N1\}\+\\mathrm\{N4\\text\{\-\}sel\}\+\\mathrm\{C5\\text\{\-\}S\}\\;\\Rightarrow\\;\\mathrm\{PhaseClassify\}\\text\{ exists\}

###### Proof\.

*Type E construction\.*Given the conditions,DcompD\_\{\\mathrm\{comp\}\}is constructed from M1\+M2\+M3 on the closure domain\. N0 ensures non\-emptiness; N3 ensures termination; N4\-comp provides the closure structure; C5\-E ensures co\-generating predicates have common closure\-domain extensions\.

*Type S construction\.*Given the conditions, PhaseClassify is constructed from event dominance \(C4\-Q2\) and MEP\. N0 ensures non\-emptiness; N1 ensures finiteness of local sets; N4\-sel provides the total order⪯sel\\preceq\_\{\\mathrm\{sel\}\}definingcr,t∗c^\{\*\}\_\{r,t\}; C5\-S ensures the selected assignmentP∗P^\{\*\}is globally admissible\. ∎∎

###### Corollary 5\.5\(Classification of pureIcoreI^\{\\mathrm\{core\}\}\-computable completion\)\.

LetFFbe the*elementary*one\-stepIcoreI^\{\\mathrm\{core\}\}\-rule operator on the state space2𝒞↑Dom⁡\(N,Θ\)2^\{\\mathcal\{C\}\\,\\uparrow\\,\\operatorname\{Dom\}\(N,\\Theta\)\}—i\.e\., a one\-stepFF\-move applies exactly one rule occurrence under a fixed fair scheduling convention, not simultaneous firing of all enabled rules—and letf:=Fωf:=F^\{\\omega\}denote its saturated closure \(applyFFuntil no further rule fires\)\. LetIcoreI^\{\\mathrm\{core\}\}satisfy R1\+R2\+R3, and𝒫T\\mathcal\{P\}\_\{T\}be finite\. Assume additionally:

- \(Soundness\)EveryIcoreI^\{\\mathrm\{core\}\}\-rule is*sound relative toAA*: for everyP∈𝒫TP\\in\\mathcal\{P\}\_\{T\}and every ruleρ∈Icore\\rho\\in I^\{\\mathrm\{core\}\}whose premise is satisfied byPPat some\(r,t\)∈Dom⁡\(N,Θ\)\(r,t\)\\in\\operatorname\{Dom\}\(N,\\Theta\), the stateP′P^\{\\prime\}obtained by addingcρc\_\{\\rho\}toP​\(r,t\)P\(r,t\)is again in𝒫T\\mathcal\{P\}\_\{T\}\.

Thenf=Fωf=F^\{\\omega\}restricts to a map𝒫T→𝒫T\\mathcal\{P\}\_\{T\}\\to\\mathcal\{P\}\_\{T\}, satisfies properties \(1\)–\(4’\) of §4\.1, and is therefore Class C\.

###### Proof\.

*Saturation well\-definedness\.*We take a one\-stepFF\-move to mean one elementaryIcoreI^\{\\mathrm\{core\}\}\-rule firing, not simultaneous firing of all enabled rules\. By R1, every nontrivialFF\-move adds at least one conclusion and removes none\. Since\|𝒞\|⋅\|Dom⁡\(N,Θ\)\|<∞\|\\mathcal\{C\}\|\\cdot\|\\operatorname\{Dom\}\(N,\\Theta\)\|<\\infty, no strictly increasingFF\-trajectory has length exceeding this bound\. HenceFω​\(P\)F^\{\\omega\}\(P\)exists as a finite\-step saturation for every statePP, under the chosen scheduling convention\.

*Order independence \(under R1\+R2\+R3 \+ finite \+ fair scheduling\)\.*In the pure \(3a\) positive\-rule setting, under fair elementary scheduling, saturation is order\-independent: every conclusion whose positive premises eventually become true is eventually added \(R1 \+ fair scheduling\), no rule is conditioned on the*absence*of a derived predicate \(R2, no negation\-as\-failure\), and no rule retracts conclusions \(R3\)\. HenceFω​\(P\)F^\{\\omega\}\(P\)is independent of the fair scheduling convention\.

*Restriction to𝒫T\\mathcal\{P\}\_\{T\}\.*By Soundness, each elementaryFF\-step from a𝒫T\\mathcal\{P\}\_\{T\}\-state yields a𝒫T\\mathcal\{P\}\_\{T\}\-state\. Finite composition along the saturation trajectory givesFω​\(P\)∈𝒫TF^\{\\omega\}\(P\)\\in\\mathcal\{P\}\_\{T\}for everyP∈𝒫TP\\in\\mathcal\{P\}\_\{T\}\. Henceffrestricts to𝒫T→𝒫T\\mathcal\{P\}\_\{T\}\\to\\mathcal\{P\}\_\{T\}\.

*Monotonicity and extensivity\.*By Lemma C \(§8\.2\), R1\+R2\+R3 implyffis monotone and extensive on the finite poset\(𝒫T,⪯spec\)\(\\mathcal\{P\}\_\{T\},\\preceq\_\{\\mathrm\{spec\}\}\)\. \(Lemma C does not require Soundness—it is a syntactic consequence of R1\+R2\+R3\.\)

*Per\-seed convergence \(4’\)\.*By the Monotonic Exhaustion Lemma \(§8\.2\), an extensive operator on a finite poset reaches a fixed point in at most\|𝒫T\|\|\\mathcal\{P\}\_\{T\}\|steps\. Combined with the saturation interpretation,f​\(P\)=Fω​\(P\)f\(P\)=F^\{\\omega\}\(P\)is itself a fixed point:f​\(f​\(P\)\)=Fω​\(Fω​\(P\)\)=Fω​\(P\)=f​\(P\)f\(f\(P\)\)=F^\{\\omega\}\(F^\{\\omega\}\(P\)\)=F^\{\\omega\}\(P\)=f\(P\)\.

*Idempotence\.*From the previous line,f∘f=ff\\circ f=fas a saturation tautology\.

All four Class C properties \(1\)–\(4’\) hold\. ∎∎

### 5\.4Summary

Canonicalization==Closure\+\+Comparability\+\+Joint AdmissibilityTheorem 6b \(construction lemma\):Type E:N0\+N3\+N4​\-​comp\+C5​\-​E⇒Dcomp\\mathrm\{N0\}\+\\mathrm\{N3\}\+\\mathrm\{N4\\text\{\-\}comp\}\+\\mathrm\{C5\\text\{\-\}E\}\\Rightarrow D\_\{\\mathrm\{comp\}\}exists at \(4’\)\.Type S\-strong:N0\+N1\+N4​\-​sel\+C5​\-​S⇒\\mathrm\{N0\}\+\\mathrm\{N1\}\+\\mathrm\{N4\\text\{\-\}sel\}\+\\mathrm\{C5\\text\{\-\}S\}\\RightarrowPhaseClassify exists at AC\-6\.Corollary 6b’ \(substantive content\):R1\+R2\+R3\+\\mathrm\{R1\}\{\+\}\\mathrm\{R2\}\{\+\}\\mathrm\{R3\}\+Soundness\+\+\(3a\)\-computability \(saturated closureFωF^\{\\omega\}under elementary one\-rule firing\)⇒\\RightarrowClass C at \(4’\) via Lemma C and Monotonic Exhaustion\.N0 universal; N1/N3 mechanism\-specific\. C5\-E and C5\-S are mechanism\-specific variants of joint admissibility, not parallel restatements: C5\-S is global admissibility of the selector’s output; C5\-E is compatible extension within the closure domain\. Completion branch yields \(4’\); upgrade to AC\-6 requires \(4\) \(OQ\-GC\-1 for ICT\)\.Type E→\\tocompletion \(at \(4’\)\); Type S\-strong→\\toselection \(at AC\-6\)\. Generic Type S→\\toselection: not proven \(OQ\-TypeS\-Imp\)\.*Not all canonicalization is operator\-theoretic\.*

## 6Examples

### 6\.1Wyckoff \(Type S\-strong, Complete\)

Event dominance \(C4\-Q2\) givesPhase​\_​A≺sel⋯≺selPhase​\_​E\\mathrm\{Phase\\\_A\}\\prec\_\{\\mathrm\{sel\}\}\\cdots\\prec\_\{\\mathrm\{sel\}\}\\mathrm\{Phase\\\_E\}\. MEP ensures the selected assignment is globally admissible\.TW​ycanon=TW​yv​2\+PhaseClassify∈AC​\-​6T\_\{Wy\}^\{\\mathrm\{canon\}\}=T\_\{Wy\}^\{v2\}\+\\mathrm\{PhaseClassify\}\\in\\mathrm\{AC\\text\{\-\}6\}\(full domain\)\.

### 6\.2ICT \(Type E\) — Single\-Timeframe and Multi\-HTF

Single\-timeframe\.M1\+M2 define𝒫I​C​Tclosure\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{closure\}\}; M3 ensuresDcompD\_\{\\mathrm\{comp\}\}terminates\. Forbidden pairs \(M2\):

\{LTF​\_​BOSup,LTF​\_​BOSdown\},\{LTF​\_​bullish​\_​cont,LTF​\_​bearish​\_​cont\},\\displaystyle\\\{\\mathrm\{LTF\\\_BOS\_\{up\}\},\\mathrm\{LTF\\\_BOS\_\{down\}\}\\\},\\quad\\\{\\mathrm\{LTF\\\_bullish\\\_cont\},\\mathrm\{LTF\\\_bearish\\\_cont\}\\\},\{LTF​\_​BOSup,LTF​\_​bearish​\_​cont\},\{LTF​\_​BOSdown,LTF​\_​bullish​\_​cont\}\.\\displaystyle\\\{\\mathrm\{LTF\\\_BOS\_\{up\}\},\\mathrm\{LTF\\\_bearish\\\_cont\}\\\},\\quad\\\{\\mathrm\{LTF\\\_BOS\_\{down\}\},\\mathrm\{LTF\\\_bullish\\\_cont\}\\\}\.TI​C​TcanonT\_\{ICT\}^\{\\mathrm\{canon\}\}achieves \(4’\) on𝒫I​C​Tclosure\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{closure\}\}\. Whether it achieves \(4\) and hence AC\-6 is conditional on OQ\-GC\-1\.

Multi\-timeframe extension\.DcompHTF:=DL∘DH:𝒫I​C​T→𝒫I​C​THTF→𝒫I​C​TmultiD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}:=D\_\{L\}\\circ D\_\{H\}:\\mathcal\{P\}\_\{ICT\}\\to\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{HTF\}\}\\to\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}\.

LTF inference rules \(positive premises only\):

Premise mutual exclusion\.Within each priority tier, rule premises are pairwise mutually exclusive on observed data:

- •BOS tier:close\\mathrm\{close\}cannot simultaneously satisfy\>swing​\_​high\>\\mathrm\{swing\\\_high\}and<swing​\_​low<\\mathrm\{swing\\\_low\}\(assumingswing​\_​low≤swing​\_​high\\mathrm\{swing\\\_low\}\\leq\\mathrm\{swing\\\_high\}\)\.
- •Continuation tier:close\\mathrm\{close\}cannot simultaneously satisfy\>prev​\_​close\>\\mathrm\{prev\\\_close\}and<prev​\_​close<\\mathrm\{prev\\\_close\}\. The boundary caseclose=prev​\_​close\\mathrm\{close\}=\\mathrm\{prev\\\_close\}triggers neither rule \(non\-event by convention\)\.

Cross\-timeframe forbidden pairs \(M2\-HTF\):

\{HTF​\_​BOSdown,LTF​\_​BOSup\},\{HTF​\_​BOSup,LTF​\_​BOSdown\},\\displaystyle\\\{\\mathrm\{HTF\\\_BOS\_\{down\}\},\\mathrm\{LTF\\\_BOS\_\{up\}\}\\\},\\quad\\\{\\mathrm\{HTF\\\_BOS\_\{up\}\},\\mathrm\{LTF\\\_BOS\_\{down\}\}\\\},\{HTF​\_​bearish​\_​bias,LTF​\_​bullish​\_​cont\},\{HTF​\_​bullish​\_​bias,LTF​\_​bearish​\_​cont\}\.\\displaystyle\\\{\\mathrm\{HTF\\\_bearish\\\_bias\},\\mathrm\{LTF\\\_bullish\\\_cont\}\\\},\\quad\\\{\\mathrm\{HTF\\\_bullish\\\_bias\},\\mathrm\{LTF\\\_bearish\\\_cont\}\\\}\.
###### Definition 6\.1\(Guard mechanism\)\.

For stateSS, an LTF ruleρ\\rhowith conclusioncρc\_\{\\rho\}is*guard\-suppressed*iff: \(1\)∃c′∈πL​\(S\)\\exists\\,c^\{\\prime\}\\in\\pi\_\{L\}\(S\)with\{c′,cρ\}∈M2\\\{c^\{\\prime\},c\_\{\\rho\}\\\}\\in\\mathrm\{M2\}, or \(2\)∃h∈πH​\(S\)\\exists\\,h\\in\\pi\_\{H\}\(S\)with\{h,cρ\}∈M2​\-​HTF\\\{h,c\_\{\\rho\}\\\}\\in\\mathrm\{M2\\text\{\-\}HTF\}\.

###### Definition 6\.2\(Rule priority forDLD\_\{L\}\)\.

ρ1,ρ2\\rho\_\{1\},\\rho\_\{2\}\(BOS\) have higher priority thanρ3,ρ4\\rho\_\{3\},\\rho\_\{4\}\(continuation\)\.

###### Definition 6\.3\(Reduction relation forDLD\_\{L\}\)\.

On admissibleS∈𝒫I​C​THTFS\\in\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{HTF\}\},S→𝜌S′S\\xrightarrow\{\\rho\}S^\{\\prime\}iff: \(a\)ρ∈AI​C​TL​T​F\\rho\\in A\_\{ICT\}^\{LTF\}; \(b\) premise\(ρ\)\(\\rho\)satisfied inSS; \(c\)ρ\\rhonot guard\-suppressed onSS; \(d\)ρ\\rhohas highest priority among rules satisfying \(a\)–\(c\); \(e\)S′=S∪\{cρ\}S^\{\\prime\}=S\\cup\\\{c\_\{\\rho\}\\\}\.

###### Lemma 6\.4\(Functionality of→\\to\)\.

For every admissibleS∈𝒫I​C​THTFS\\in\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{HTF\}\}, there is at most oneS′S^\{\\prime\}withS→𝜌S′S\\xrightarrow\{\\rho\}S^\{\\prime\}\.

###### Proof\.

Two\-part argument:*\(1\) Across tiers:*\(d\) selects only the highest\-priority tier among applicable rules\. If any BOS rule satisfies \(a\)–\(c\), no continuation rule passes \(d\)\.*\(2\) Within a tier:*Premise mutual exclusion gives at most one rule per tier with premise \(b\) satisfied for given\(N,Θ\)\(N,\\Theta\)\. Combined, at most one rule satisfies \(a\)–\(d\)\. ∎∎

###### Lemma 6\.6\(Single\-step admissibility preservation under guard\)\.

IfSSis admissible andS→𝜌S′S\\xrightarrow\{\\rho\}S^\{\\prime\}, thenS′S^\{\\prime\}is admissible\.

###### Proof\.

*Same\-timeframe\.*If\{c′,cρ\}∈M2\\\{c^\{\\prime\},c\_\{\\rho\}\\\}\\in\\mathrm\{M2\}for somec′∈πL​\(S\)c^\{\\prime\}\\in\\pi\_\{L\}\(S\), thenρ\\rhois guard\-suppressed—contradicting \(c\)\.*Cross\-timeframe\.*cρ∈𝒞L​T​Fc\_\{\\rho\}\\in\\mathcal\{C\}\_\{LTF\}\(sinceρ∈AI​C​TL​T​F\\rho\\in A\_\{ICT\}^\{LTF\}\), soπH​\(S′\)=πH​\(S\)\\pi\_\{H\}\(S^\{\\prime\}\)=\\pi\_\{H\}\(S\)\. If\{cH,cρ\}∈M2​\-​HTF\\\{c\_\{H\},c\_\{\\rho\}\\\}\\in\\mathrm\{M2\\text\{\-\}HTF\}withcH∈πH​\(S\)c\_\{H\}\\in\\pi\_\{H\}\(S\), thenρ\\rhois guard\-suppressed—contradicting \(c\)\. OtherwisecH∉πH​\(S′\)c\_\{H\}\\notin\\pi\_\{H\}\(S^\{\\prime\}\), so the forbidden pair is not inS′S^\{\\prime\}\. ∎∎

Verification of §4\.1 properties forDcompHTFD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}\.DcompHTFD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}is the saturated closure\(DL∘DH\)ω\(D\_\{L\}\\circ D\_\{H\}\)^\{\\omega\}of its underlying one\-step guarded operator; the verification below is for this saturated form\. We verify \(1\)–\(4’\) by direct argument from the reduction relation, without appeal to external lemmas\.

1. 1\.*Extensivity\.*P⪯specDH​\(P\)⪯specDL​\(DH​\(P\)\)P\\preceq\_\{\\mathrm\{spec\}\}D\_\{H\}\(P\)\\preceq\_\{\\mathrm\{spec\}\}D\_\{L\}\(D\_\{H\}\(P\)\)by additive updates \(DHD\_\{H\}adds HTF predicates;DLD\_\{L\}adds LTF predicates viaS′=S∪\{cρ\}S^\{\\prime\}=S\\cup\\\{c\_\{\\rho\}\\\}\)\.
2. 2\.*Admissibility preservation\.*Induction on reduction length using Lemma[6\.6](https://arxiv.org/html/2608.07476#S6.Thmtheorem6)for eachDLD\_\{L\}step\.DHD\_\{H\}admissibility is direct \(no guard layer;DHD\_\{H\}does not modify LTF predicates\)\.
3. 3\.*Idempotence\.*FixPP, letP∗:=DcompHTF​\(P\)=DL​\(DH​\(P\)\)P^\{\*\}:=D\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}\(P\)=D\_\{L\}\(D\_\{H\}\(P\)\)\. We showDcompHTF​\(P∗\)=P∗D\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}\(P^\{\*\}\)=P^\{\*\}\. *Step 1 \(DH​\(P∗\)=P∗D\_\{H\}\(P^\{\*\}\)=P^\{\*\}\)\.*DHD\_\{H\}’s rules reference external HTF evidence, not derived predicates\. SinceπH​\(P∗\)=πH​\(DH​\(P\)\)\\pi\_\{H\}\(P^\{\*\}\)=\\pi\_\{H\}\(D\_\{H\}\(P\)\)\(DLD\_\{L\}does not add HTF predicates\), everyDHD\_\{H\}\-applicable rule has already fired inP∗P^\{\*\}\. *Step 2 \(DL​\(P∗\)=P∗D\_\{L\}\(P^\{\*\}\)=P^\{\*\}\)\.*By construction,DLD\_\{L\}runs to fixed point, so noρ\\rhosatisfies \(a\)–\(d\) onP∗P^\{\*\}\. *Combining\.*DcompHTF​\(P∗\)=DL​\(DH​\(P∗\)\)=DL​\(P∗\)=P∗D\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}\(P^\{\*\}\)=D\_\{L\}\(D\_\{H\}\(P^\{\*\}\)\)=D\_\{L\}\(P^\{\*\}\)=P^\{\*\}\.
4. 4\.*Per\-seed convergence \(4’\)\.*DHD\_\{H\}terminates: HTF rules are finitely many on𝒞H​T​F\\mathcal\{C\}\_\{HTF\}\.DLD\_\{L\}onDH​\(P\)D\_\{H\}\(P\)terminates: each step strictly increases\|S∩𝒞L​T​F\|\|S\\cap\\mathcal\{C\}\_\{LTF\}\|\(extensive addition of new predicates, bound\|𝒞L​T​F\|⋅\|Dom⁡\(N,Θ\)\|\|\\mathcal\{C\}\_\{LTF\}\|\\cdot\|\\operatorname\{Dom\}\(N,\\Theta\)\|\); the relation is functional \(Lemma[6\.4](https://arxiv.org/html/2608.07476#S6.Thmtheorem4)\)\. Each seed converges to a fixed point ofDcompHTFD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}in𝒫I​C​Tmulti\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}within\|𝒞H​T​F\|\+\|𝒞L​T​F\|⋅\|Dom⁡\(N,Θ\)\|\|\\mathcal\{C\}\_\{HTF\}\|\+\|\\mathcal\{C\}\_\{LTF\}\|\\cdot\|\\operatorname\{Dom\}\(N,\\Theta\)\|iterations\. The fixed point is seed\-dependent; whether all seeds converge to the same fixed point is OQ\-GC\-1\.

###### Definition 6\.8\(DHD\_\{H\}architectural scope\)\.

DHD\_\{H\}applies HTF inference rules whose premises reference external HTF evidence and do not depend on derived LTF predicates\.DHD\_\{H\}does not employ an M2\-HTF guard—cross\-timeframe conflict resolution is the responsibility ofDLD\_\{L\}via the Guard Mechanism\. Staged ordering \(§7\) resolves the resulting non\-commutativity\.

TI​C​Tmulti=TI​C​Tv​2\+DcompHTFT\_\{ICT\}^\{\\mathrm\{multi\}\}=T\_\{ICT\}^\{v2\}\+D\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}achieves closure stabilization \(4’\) on𝒫I​C​Tmulti\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}\. Full determinization \(AC\-6, requiring \(4\)\) is conditional on OQ\-GC\-1\.

## 7Non\-Commutativity of Staged Completion

###### Proposition 7\.1\(Structural non\-commutativity\)\.

There existsP∈𝒫I​C​TP\\in\\mathcal\{P\}\_\{ICT\}such that\(DL∘DH\)​\(P\)∈𝒫I​C​Tmulti\(D\_\{L\}\\circ D\_\{H\}\)\(P\)\\in\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}while\(DH∘DL\)​\(P\)∉𝒫I​C​Tmulti\(D\_\{H\}\\circ D\_\{L\}\)\(P\)\\notin\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}\.

###### Proof\.

LetP=∅P=\\varnothing; assumeclose\>prev​\_​close\\mathrm\{close\}\>\\mathrm\{prev\\\_close\}, no BOS condition holds, HTF bearish BOS evidence confirmed\.

Path A\(DL∘DHD\_\{L\}\\circ D\_\{H\}, canonical\): \(1\)DH​\(P\)=\{HTF​\_​bearish​\_​bias\}D\_\{H\}\(P\)=\\\{\\mathrm\{HTF\\\_bearish\\\_bias\}\\\}; \(2\)DLD\_\{L\}: BOS rules do not fire;ρ3\\rho\_\{3\}is guard\-suppressed \(since\{HTF​\_​bearish​\_​bias,LTF​\_​bullish​\_​cont\}∈M2​\-​HTF\\\{\\mathrm\{HTF\\\_bearish\\\_bias\},\\mathrm\{LTF\\\_bullish\\\_cont\}\\\}\\in\\mathrm\{M2\\text\{\-\}HTF\}\)\.RA=\{HTF​\_​bearish​\_​bias\}∈𝒫I​C​TmultiR\_\{A\}=\\\{\\mathrm\{HTF\\\_bearish\\\_bias\}\\\}\\in\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}\.

Path B\(DH∘DLD\_\{H\}\\circ D\_\{L\}, non\-canonical\): \(1\)DL​\(∅\)=\{LTF​\_​bullish​\_​cont\}D\_\{L\}\(\\varnothing\)=\\\{\\mathrm\{LTF\\\_bullish\\\_cont\}\\\}\(ρ3\\rho\_\{3\}fires, no active guard\); \(2\)DHD\_\{H\}fires \(premise external; no M2\-HTF guard\)\.RB=\{HTF​\_​bearish​\_​bias,LTF​\_​bullish​\_​cont\}R\_\{B\}=\\\{\\mathrm\{HTF\\\_bearish\\\_bias\},\\mathrm\{LTF\\\_bullish\\\_cont\}\\\}violates M2\-HTF, soRB∉𝒫I​C​TmultiR\_\{B\}\\notin\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}\. ∎∎

###### Corollary 7\.2\(Canonical ordering uniqueness within the staged architecture\)\.

Within the two\-operator raw staged architecture\{DL∘DH,DH∘DL\}\\\{D\_\{L\}\\circ D\_\{H\},D\_\{H\}\\circ D\_\{L\}\\\}, with no repair operator, no M2\-HTF guard added toDHD\_\{H\}, and no joint fixed\-point construction, HTF\-firstDL∘DHD\_\{L\}\\circ D\_\{H\}is the only admissibility\-preserving linearization\.

## 8Mechanism Classification and Categorical Structure

*This section classifies, conditional on a realization\-coverage assumption \(OQ\-Realization\) and per\-realization branch hypotheses, every theory\-intrinsic canonicalization mechanism realized within the \(3a\)/\(3b\)/\(3c\) grammar into Class C \(closure\) or Class S \(selection\)\. The classification is at closure\-stabilization strength \(4’\); whether Class C operators additionally achieve property \(4\) or AC\-6 is instance\-specific\.*

### 8\.1Definitions

###### Definition 8\.1\(Theory\-intrinsic operation\)\.

f:𝒫T→𝒫Tf:\\mathcal\{P\}\_\{T\}\\to\\mathcal\{P\}\_\{T\}is*theory\-intrinsic*if:

- \(I1\)ffis definable fromT=\(Σ,A,I\)T=\(\\Sigma,A,I\)—i\.e\.,ffis given by a fixed expression in the signature, axioms, and inference policy ofTT, with no external parameters beyond input\(N,Θ,Π\)\(N,\\Theta,\\Pi\);
- \(I2\)ffis invariant under automorphisms ofTT: ifσ\\sigmais an automorphism of\(Σ,A\)\(\\Sigma,A\), thenf​\(σ⋅P\)=σ⋅f​\(P\)f\(\\sigma\\cdot P\)=\\sigma\\cdot f\(P\)for everyP∈𝒫TP\\in\\mathcal\{P\}\_\{T\}\.

This definition is independent of any mechanism taxonomy\. Conditions \(I1\)–\(I2\) specify*what*makes an operation theory\-intrinsic; the trichotomy below specifies*how*such operations are realized, used as the case partition for Theorem U3\.

Examples\.DcompD\_\{\\mathrm\{comp\}\}\(ICT closure\) and PhaseClassify \(Wyckoff event dominance\) are both theory\-intrinsic—the former by \(3a\) realization \(saturated closure under elementary firing\), the latter by \(3b\)\.DcompHTFD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}is theory\-intrinsic by \(3c\) realization \(saturated guardedIcoreI^\{\\mathrm\{core\}\}\)\.

Realizations\(case partition for Theorem U3\)\.

- •\(3a\)ffis*IcoreI^\{\\mathrm\{core\}\}\-computable*as the saturated closuref=Fωf=F^\{\\omega\}under elementary one\-rule firing:f​\(P\)f\(P\)is the result of applyingIcoreI^\{\\mathrm\{core\}\}rules toPPone rule at a time under a fair scheduling convention until no further rule fires, with no guard or scheduling discipline beyond fairness\. The one\-step operatorFFis not itself the \(3a\) realization; saturation is built in\.
- •\(3b\)ffis*selector\-computable*from axiom\-induced comparability:f​\(P\)​\(r,t\):=max⪯sel⁡Ar,tf\(P\)\(r,t\):=\\max\_\{\\preceq\_\{\\mathrm\{sel\}\}\}A\_\{r,t\}where⪯sel\\preceq\_\{\\mathrm\{sel\}\}is theory\-intrinsic\.
- •\(3c\)ffis*guardedIcoreI^\{\\mathrm\{core\}\}\(saturated form\)*: there is a one\-step guarded operatorGGsuch thatf:=Gωf:=G^\{\\omega\}, whereGGappliesIcoreI^\{\\mathrm\{core\}\}rules under a guard mechanism and/or scheduling discipline that: \(i\) suppresses or ordersIcoreI^\{\\mathrm\{core\}\}rule applications based on state and axiom\-derived conflict/priority relations; \(ii\) adds no conclusions beyond those derivable byIcoreI^\{\\mathrm\{core\}\}rules—every predicate inG​\(P\)∖PG\(P\)\\setminus Pis the conclusion of someρ∈Icore\\rho\\in I^\{\\mathrm\{core\}\}; andf​\(P\)=Gω​\(P\)f\(P\)=G^\{\\omega\}\(P\)is obtained by repeated application ofGGuntil no guarded step fires\.

\(3a\)⊂\\subset\(3c\) \(pureIcoreI^\{\\mathrm\{core\}\}saturation is the special caseG=FG=Fwith trivial guard\)\.

Realization\-coverage assumption \(OQ\-Realization\)\.Theorem U3 below is conditional on the assumption that every valid theory\-intrinsic canonicalization mechanism is realized by exactly one of \(3a\), \(3b\), or \(3c\)\. This is an open question, tracked as OQ\-Realization in §8\.5; whether the trichotomy is provably exhaustive for all theory\-intrinsic operations definable in MST is not established in this paper\. Theorem U3’s conclusion is therefore explicitly scoped to mechanisms realized within this grammar\.

###### Definition 8\.2\(Class C — Completion\)\.

ffis*Class C*ifffsatisfies properties \(1\)–\(3\) and \(4’\) of §4\.1\. Class C is a mechanism class; it does not imply \(4\) or AC\-6\.

###### Definition 8\.3\(Class S — Selection\)\.

ffis*Class S*ifffis induced by a canonical selector—i\.e\.,ffis the \(3b\) realization with respect to some⪯sel\\preceq\_\{\\mathrm\{sel\}\}\. Class S directly achieves AC\-6 and need not satisfy extensivity in⪯spec\\preceq\_\{\\mathrm\{spec\}\}\.

### 8\.2Asymmetric Reduction \(Lemma R2\*\)

###### Lemma 8\.4\(Asymmetric reduction\)\.

Let𝒫T\\mathcal\{P\}\_\{T\}be a dcpo \(which holds under M3\) and letf:𝒫T→𝒫Tf:\\mathcal\{P\}\_\{T\}\\to\\mathcal\{P\}\_\{T\}be Scott\-continuous, theory\-intrinsic, and uniqueness\-inducing in the sense of §4\.1\.

*\(i\) No outward drift belowP∗P^\{\*\}\.*If∃P≺P∗\\exists\\,P\\prec P^\{\*\}withP≺f​\(P\)P\\prec f\(P\), then\{fn​\(P\)\}n\\\{f^\{n\}\(P\)\\\}\_\{n\}is a directed increasing chain; Scott\-continuity givesf​\(P∞\)=P∞f\(P\_\{\\infty\}\)=P\_\{\\infty\}, and uniqueness forcesP∞=P∗P\_\{\\infty\}=P^\{\*\}\. Hence uniqueness\-inducing upward motion is closure/completion\-like towardP∗P^\{\*\}\.

*\(ii\) Asymmetry aboveP∗P^\{\*\}\.*ForP≻P∗P\\succ P^\{\*\}, descending\-limit arguments do not apply in a general dcpo \(infima of decreasing chains are not guaranteed\)\. Uniqueness from above cannot be derived via Scott\-continuous downward convergence alone\.

### 8\.3Conditional Classification within the MST Realization Grammar \(Theorem U3\)

###### Theorem 8\.6\(Conditional primitive mechanism classification under R1\+R2\+R3\)\.

LetTThaveIcoreI^\{\\mathrm\{core\}\}satisfying R1\+R2\+R3 and finite𝒞\\mathcal\{C\}\. Letf:𝒫T→𝒫Tf:\\mathcal\{P\}\_\{T\}\\to\\mathcal\{P\}\_\{T\}be theory\-intrinsic \(Definition[8\.1](https://arxiv.org/html/2608.07476#S8.Thmtheorem1), conditions \(I1\)–\(I2\)\)\.

*Assume the realization\-coverage hypothesis \(OQ\-Realization\):ffis 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 \(§5\.3 Corollary[5\.5](https://arxiv.org/html/2608.07476#S5.Thmtheorem5)\) \+ finite𝒫T\\mathcal\{P\}\_\{T\};FF\-step is elementary one\-rule firing under fair scheduling\.
- •\(3b\)\-branch: total theory\-intrinsic selector⪯sel\\preceq\_\{\\mathrm\{sel\}\}\+ global compatibility of selected output \(C5\-S, §5\.1\)\.
- •\(3c\)\-branch: R1\+R2\+R3 \+ \(G2\) functionality \+ \(G3\) guard\-stability \+ Soundness \(§8\.2 Lemma[8\.13](https://arxiv.org/html/2608.07476#S8.Thmtheorem13)\)\.

Then:

- •\(3a\) and \(3c\) realizations yieldf∈f\\inClass C;
- •\(3b\) realization yieldsf∈f\\inClass S\.

Within the \(3a\)/\(3b\)/\(3c\) realization grammar, every theory\-intrinsicffsatisfying the corresponding per\-branch hypotheses is Class C or Class S\.

###### Proof\.

*Case partition\.*By \(3a\)⊂\\subset\(3c\), pureIcoreI^\{\\mathrm\{core\}\}\-computableffsatisfies both \(3a\) and \(3c\); we treat suchffunder Case 1 to exploit the stronger conclusions via Lemma C and Monotonic Exhaustion\. Case 3 handles \(3c\)\-but\-not\-\(3a\) mechanisms\.

*Case 1 \(\(3a\)\-branch\)\.*Lemma[8\.8](https://arxiv.org/html/2608.07476#S8.Thmtheorem8)gives monotonicity and extensivity of the saturated closure under R1\+R2\+R3\. Single\-rule Soundness combined with elementary\-stepFFgives admissibility preservation along the saturation trajectory \(finite composition of admissibility\-preserving steps\)\. By Lemma[8\.10](https://arxiv.org/html/2608.07476#S8.Thmtheorem10)\(Monotonic Exhaustion\), finite extensivity gives \(4’\)\. Saturation gives idempotence \(Fω∘Fω=FωF^\{\\omega\}\\circ F^\{\\omega\}=F^\{\\omega\}\)\. Henceffis Class C\.

*Case 2 \(\(3b\)\-branch\)\.*By the Class S definition \(§8\.0\), anffrealized as \(3b\) with theory\-intrinsic⪯sel\\preceq\_\{\\mathrm\{sel\}\}is Class S; C5\-S gives global admissibility of the selector’s output\.

*Case 3 \(\(3c\)\-but\-not\-\(3a\)\)\.*By Lemma[8\.13](https://arxiv.org/html/2608.07476#S8.Thmtheorem13)\(Guard Preservation\) applied toGωG^\{\\omega\}under \(G2\)\+\(G3\)\+Soundness,ffis extensive, admissibility\-preserving, idempotent \(by saturation\), and satisfies \(4’\)\. Henceffis Class C by the property\-based definition\. The realization is Class C, not Class S, since by \(3c\)\(ii\) every added predicate is anIcoreI^\{\\mathrm\{core\}\}conclusion, not a selector output\. ∎∎

###### Lemma 8\.8\(Operator monotonicity from R1\+R2\+R3, \(3a\)\-scope\)\.

Letffbe the saturated closureFωF^\{\\omega\}of anIcoreI^\{\\mathrm\{core\}\}\-computable operator in the sense of \(3a\) \(elementaryFF\-step, fair scheduling\)\. Under R1\+R2\+R3,ffis monotone and extensive on finite𝒫T\\mathcal\{P\}\_\{T\}\.

###### Proof\.

SupposeP⪯specQP\\preceq\_\{\\mathrm\{spec\}\}Q\. To showf​\(P\)⪯specf​\(Q\)f\(P\)\\preceq\_\{\\mathrm\{spec\}\}f\(Q\), we show every conclusion inf​\(P\)f\(P\)is inf​\(Q\)f\(Q\)\.

A ruleρ∈Icore\\rho\\in I^\{\\mathrm\{core\}\}has a premise that, by R2, is a positive conjunction of two classes:

\(Class A\) Conditions on observed data: comparisons or pattern matches on\(N,Θ\)\(N,\\Theta\)alone, e\.g\.,close\>swing​\_​high\\mathrm\{close\}\>\\mathrm\{swing\\\_high\}\. These depend only on\(N,Θ\)\(N,\\Theta\), not onPPorQQ\. Equivalent across allP,Q∈𝒫TP,Q\\in\\mathcal\{P\}\_\{T\}with the same\(N,Θ\)\(N,\\Theta\)\.

\(Class B\) Conditions on derived predicates: positive presence conditions “c∈P​\(r,t\)c\\in P\(r,t\)”\. By R2, no Class B condition is of the form “c∉P​\(r,t\)c\\notin P\(r,t\)”\.

For Class B: if “c∈P​\(r,t\)c\\in P\(r,t\)” holds, thenP⪯specQP\\preceq\_\{\\mathrm\{spec\}\}QgivesP​\(r,t\)⊆Q​\(r,t\)P\(r,t\)\\subseteq Q\(r,t\), so “c∈Q​\(r,t\)c\\in Q\(r,t\)” also holds\. Hence positive presence conditions monotone\-preserve under⪯spec\\preceq\_\{\\mathrm\{spec\}\}\.

Combining: every premise triggeringρ\\rhoonPPalso triggersρ\\rhoonQQ\. By R1,ρ\\rhoaddscρc\_\{\\rho\}\. By R3, no rule retracts\. Hence every conclusion inFω​\(P\)=f​\(P\)F^\{\\omega\}\(P\)=f\(P\)is inf​\(Q\)f\(Q\)\.

Extensivity: eachIcoreI^\{\\mathrm\{core\}\}rule application adds zero or more conclusions, never removing any, soP⪯specf​\(P\)P\\preceq\_\{\\mathrm\{spec\}\}f\(P\)\. ∎∎

###### Lemma 8\.10\(Monotonic exhaustion \(per\-seed\)\)\.

Letf:𝒫T→𝒫Tf:\\mathcal\{P\}\_\{T\}\\to\\mathcal\{P\}\_\{T\}be extensive, with𝒫T\\mathcal\{P\}\_\{T\}finite\. Then for every seedP0∈𝒫TP\_\{0\}\\in\\mathcal\{P\}\_\{T\}there existk​\(P0\)≤\|𝒫T\|k\(P\_\{0\}\)\\leq\|\\mathcal\{P\}\_\{T\}\|andP∗​\(P0\)∈𝒫TP^\{\*\}\(P\_\{0\}\)\\in\\mathcal\{P\}\_\{T\}with:

P0⪯specf​\(P0\)⪯spec⋯⪯specfk​\(P0\)​\(P0\)=P∗​\(P0\)P\_\{0\}\\preceq\_\{\\mathrm\{spec\}\}f\(P\_\{0\}\)\\preceq\_\{\\mathrm\{spec\}\}\\cdots\\preceq\_\{\\mathrm\{spec\}\}f^\{k\(P\_\{0\}\)\}\(P\_\{0\}\)=P^\{\*\}\(P\_\{0\}\)andf​\(P∗​\(P0\)\)=P∗​\(P0\)f\(P^\{\*\}\(P\_\{0\}\)\)=P^\{\*\}\(P\_\{0\}\)\. That is,ffsatisfies property \(4’\)\.

###### Proof\.

Extensivity gives the ascending chain in the finite poset\(𝒫T,⪯spec\)\(\\mathcal\{P\}\_\{T\},\\preceq\_\{\\mathrm\{spec\}\}\)\. Any ascending chain in a finite poset stabilizes within\|𝒫T\|\|\\mathcal\{P\}\_\{T\}\|steps, yielding a fixed point\. Monotonicity is not needed for this conclusion; it is recorded as a companion structural property where it holds \(Lemma[8\.8](https://arxiv.org/html/2608.07476#S8.Thmtheorem8), \(3a\)\-scope\)\. ∎∎

###### Example 8\.12\(Witnessing \(4’\)⇏\\not\\Rightarrow\(4\)\)\.

𝒫T=\{a,b\}\\mathcal\{P\}\_\{T\}=\\\{a,b\\\}incomparable under⪯spec\\preceq\_\{\\mathrm\{spec\}\},f=id𝒫Tf=\\mathrm\{id\}\_\{\\mathcal\{P\}\_\{T\}\}\. Thenffis extensive, idempotent, with two distinct fixed points\. Hence extensive \+ finite does not imply uniqueness\.

###### Lemma 8\.13\(Guard preservation under R1\+R2\+R3, \(3c\)\-scope\)\.

LetGGbe a one\-step guarded operator satisfying §8\.0 conditions \(3c\)\(i\)–\(ii\), and letf:=Gωf:=G^\{\\omega\}denote its saturated closure\. Suppose:

- \(G2\)*Functionality:*at each state, at most one rule \(highest\-priority unsuppressed rule with satisfied premise\) fires next underGG\.
- \(G3\)*Additive\-stable guard:*along anyGG\-trajectory, predicates added byGGat stateSSdo not subsequently un\-suppress rules that weregg\-suppressed at earlier states \(the conflict relation definingggis monotone under⊆\\subseteqat the witness level\)\.
- \(Soundness\)for everyP∈𝒫TP\\in\\mathcal\{P\}\_\{T\}and everyGG\-step fromPPyieldingP′P^\{\\prime\},P′∈𝒫TP^\{\\prime\}\\in\\mathcal\{P\}\_\{T\}\.

Thenf=Gωf=G^\{\\omega\}is well\-defined on𝒫T\\mathcal\{P\}\_\{T\}, restricts to a map𝒫T→𝒫T\\mathcal\{P\}\_\{T\}\\to\\mathcal\{P\}\_\{T\}, and satisfies extensivity, admissibility preservation, idempotence, and per\-seed convergence \(4’\) on finite𝒫T\\mathcal\{P\}\_\{T\}\.

###### Proof\.

*Step 1 —GG\-trajectory terminates\.*LetP0=PP\_\{0\}=PandPn\+1=G​\(Pn\)P\_\{n\+1\}=G\(P\_\{n\}\)whenever aGG\-step is available atPnP\_\{n\}\. By R1 and \(3c\)\(ii\), each non\-trivialGG\-step adds at least oneIcoreI^\{\\mathrm\{core\}\}conclusion and removes none, so\|Pn∩𝒞⋅Dom⁡\(N,Θ\)\|\|P\_\{n\}\\cap\\mathcal\{C\}\\cdot\\operatorname\{Dom\}\(N,\\Theta\)\|strictly increases along the trajectory\. Since\|𝒞\|⋅\|Dom⁡\(N,Θ\)\|<∞\|\\mathcal\{C\}\|\\cdot\|\\operatorname\{Dom\}\(N,\\Theta\)\|<\\infty, the trajectory has length at most\|𝒞\|⋅\|Dom⁡\(N,Θ\)\|\|\\mathcal\{C\}\|\\cdot\|\\operatorname\{Dom\}\(N,\\Theta\)\|and reaches a stateP∗=PNP^\{\*\}=P\_\{N\}at which noGG\-step is available\. By \(G2\), the trajectory is functional and hence unique\.

*Step 2 —f:=Gωf:=G^\{\\omega\}is well\-defined\.*By Step 1 theGG\-trajectory terminates, and by \(G2\) it is functional; henceGω​\(P\)G^\{\\omega\}\(P\)is well\-defined as the terminal state of the uniqueGG\-trajectory fromPP\. By \(G3\), guard suppression is stable along the trajectory: predicates added later do not un\-suppress rules that were suppressed earlier by positive conflict witnesses\. \(\(G3\) is not needed for well\-definedness itself—that follows from finite additive termination plus \(G2\)—but it records the additional stability property satisfied by the ICT guard\.\)

*Step 3 — Restriction to𝒫T\\mathcal\{P\}\_\{T\}and admissibility preservation\.*By Soundness, eachGG\-step from a𝒫T\\mathcal\{P\}\_\{T\}\-state yields a𝒫T\\mathcal\{P\}\_\{T\}\-state\. Finite composition along the trajectory givesf​\(P\)∈𝒫Tf\(P\)\\in\\mathcal\{P\}\_\{T\}for everyP∈𝒫TP\\in\\mathcal\{P\}\_\{T\}\.

*Step 4 — Extensivity\.*P0⪯specP1⪯spec⋯⪯specPN=f​\(P\)P\_\{0\}\\preceq\_\{\\mathrm\{spec\}\}P\_\{1\}\\preceq\_\{\\mathrm\{spec\}\}\\cdots\\preceq\_\{\\mathrm\{spec\}\}P\_\{N\}=f\(P\)by \(3c\)\(ii\) and R1, soP⪯specf​\(P\)P\\preceq\_\{\\mathrm\{spec\}\}f\(P\)\.

*Step 5 — Idempotence\.*By construction, noGG\-step is available atf​\(P\)=P∗f\(P\)=P^\{\*\}, so applyingGωG^\{\\omega\}again yieldsf​\(P\)f\(P\)itself\. HenceGω​\(Gω​\(P\)\)=Gω​\(P\)G^\{\\omega\}\(G^\{\\omega\}\(P\)\)=G^\{\\omega\}\(P\), i\.e\.f∘f=ff\\circ f=f\. \(Idempotence is a saturation tautology and does not require G3\.\)

*Step 6 — Per\-seed convergence \(4’\)\.*Forf=Gωf=G^\{\\omega\}, everyPPreaches its fixed pointf​\(P\)f\(P\)ink=1k=1step offf\-iteration:f1​\(P\)=f​\(P\)f^\{1\}\(P\)=f\(P\)andf2​\(P\)=f​\(f​\(P\)\)=f​\(P\)f^\{2\}\(P\)=f\(f\(P\)\)=f\(P\)\. The deeper content of \(4’\)—that the underlyingGG\-iteration terminates—is Step 1\. ∎∎

### 8\.4Comparability Structures \(Refined N4\)

###### Definition 8\.18\(Comparability structure\)\.

𝒦=\(D,⪯𝒦,𝒜,𝒞\)\\mathcal\{K\}=\(D,\\preceq\_\{\\mathcal\{K\}\},\\mathcal\{A\},\\mathcal\{C\}\)whereDDis a domain of admissible candidates,⪯𝒦\\preceq\_\{\\mathcal\{K\}\}an order onDD,𝒜\\mathcal\{A\}a local admissibility predicate,𝒞\\mathcal\{C\}a global compatibility condition\.

###### Definition 8\.19\(N4 — refined\)\.

TTsatisfies N4 if there exists a theory\-intrinsic𝒦\\mathcal\{K\}such that:

1. 1\.local setsAr,t⊆DA\_\{r,t\}\\subseteq Dare well\-defined;
2. 2\.⪯𝒦\\preceq\_\{\\mathcal\{K\}\}induces uniqueness on eachAr,tA\_\{r,t\}via:*selector\-compatible*\(N4\-sel\): unique maximum;*closure\-compatible*\(N4\-comp\): extensive, idempotent, admissibility\-preserving map with at least one fixed point;*piecewise\-compatible*: domain decomposes into subdomains, each admitting one of the above;
3. 3\.the induced mechanism extends to globally admissible assignments under𝒞\\mathcal\{C\}\.

Relationship to §5\.1\.The §5\.1 statement is a working summary; §8\.3 is the structural form\. Theorem 6b’s branches use N4\-comp \(≡\\equivclosure\-compatible\) and N4\-sel \(≡\\equivselector\-compatible\)\.

### 8\.5Determinization as a Pseudofunctor

###### Definition 8\.21\(Det\)\.

Det​\(T,𝒦\)=T𝒦canon\\mathrm\{Det\}\(T,\\mathcal\{K\}\)=T^\{\\mathrm\{canon\}\}\_\{\\mathcal\{K\}\}, obtained by adjoining the uniqueness mechanism induced by𝒦\\mathcal\{K\}\. Signature and axioms preserved; inference policy extended:Icanon=Icore∪Idet​\(𝒦\)I^\{\\mathrm\{canon\}\}=I^\{\\mathrm\{core\}\}\\cup I^\{\\mathrm\{det\}\}\(\\mathcal\{K\}\)\. This construction is mechanism\-relative: closure\-based in Class C \(N4\-comp\), selector\-based in Class S \(N4\-sel\)\.

Category𝐌𝐒𝐓𝒦\\mathbf\{MST\}\_\{\\mathcal\{K\}\}: objects are pairs\(T,𝒦\)\(T,\\mathcal\{K\}\)whereTTis determinizable\. Morphisms\(φ,η\)\(\\varphi,\\eta\):φ:T1⇀T2\\varphi:T\_\{1\}\\rightharpoonup T\_\{2\}a partial morphism,η\\etapreserves𝒦\\mathcal\{K\}\. Det is a partial pseudofunctor𝐌𝐒𝐓𝒦→𝐌𝐒𝐓\\mathbf\{MST\}\_\{\\mathcal\{K\}\}\\to\\mathbf\{MST\}: composition and identities hold up to 2\-morphism \(domain inclusion\)\.

### 8\.6Status of §8 Results

Theorem 6b \(Construction Lemma\):ESTABLISHED by direct construction; substantive content is Corollary 6b’\.Corollary 6b’ \(\(3a\)\-scope\):ESTABLISHED at \(4’\) strength via Lemma C \+ Monotonic Exhaustion \(per\-seed\) \+ saturation,under the Soundness hypothesis \(§5\.3\) and elementary one\-rule firing interpretation ofFF\. Does not establish \(4\) or AC\-6\. \(3a\) interpreted as saturated closureFωF^\{\\omega\}; one\-step operatorFFalone is generally not idempotent\.Lemma R2\*:ESTABLISHED \(parts i, ii\); characterizes asymmetry, not mechanism reduction \(see U3\)\.N4 \(refined\):ESTABLISHED \(§8\.3\)\. Type E: N4\-comp; Type S\-strong: N4\-sel\.Theorem U3 \(conditional classification\):ESTABLISHED as classification*within*\(3a\)/\(3b\)/\(3c\) realization grammar, conditional on OQ\-Realizationand on per\-realization branch hypotheses now lifted into the U3 statement\(R1\+R2\+R3 \+ Soundness \+ elementaryFFfor \(3a\)/\(3c\);⪯sel\\preceq\_\{\\mathrm\{sel\}\}\+ C5\-S for \(3b\); \(G2\) \+ \(G3\) for \(3c\)\)\. Without those hypotheses, U3 does not establish global no\-third\-class\. Theory\-intrinsic via \(I1\)–\(I2\)\. U3 is no longer a tautology inClass​C∪Class​S\\mathrm\{Class\\ C\}\\cup\\mathrm\{Class\\ S\}, but a derivation of class membership from per\-realization assumptions\.Lemma G \(\(3c\)\-scope\):ESTABLISHED forf=Gωf=G^\{\\omega\}under \(G2\) functionality, \(G3\) guard\-stability \(suppression persists along the trajectory\), and Soundness\. Provides extensivity \+ admissibility preservation \+ idempotence \(by saturation\) \+ \(4’\) \(withk=1k=1forff\)\. Well\-definedness ofGωG^\{\\omega\}does not require G3 \(only R1 \+ finite \+ G2\)\. NOT monotone in⪯spec\\preceq\_\{\\mathrm\{spec\}\}\(counterexample §6\.2, Remark[6\.7](https://arxiv.org/html/2608.07476#S6.Thmtheorem7)\)\.DcompHTFD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}classification:\(3c\)\-mechanism \(saturated form\)\. Class C by direct §4\.1 verification in §6\.2 and by Lemma G with \(G2\)\+\(G3\)\+Soundness\. NOT monotone in⪯spec\\preceq\_\{\\mathrm\{spec\}\}\(concrete witness in §6\.2\)\.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\.External dependencies \(non\-OQ\):•Phase 2 results \(§10\.2\): summary statement; full proofs in companion artifactsphase2bandphase2c, not reproduced in this paper\.ICT canonicalization status:•Closure stabilization \(4’\): ESTABLISHED \(§6\.2\)\.•Global completion \(4\): CONDITIONAL on OQ\-GC\-1\.•Determinization \(AC\-6\): CONDITIONAL on OQ\-GC\-1\.Type S\-strong→\\toselection \(AC\-6\):ESTABLISHED \(Proposition[4\.4](https://arxiv.org/html/2608.07476#S4.Thmtheorem4), §4\.2\)\. Generic Type S→\\toselection: NOT proven \(OQ\-TypeS\-Imp\)\.Non\-commutativity:Witness ESTABLISHED \(§7 Proposition[7\.1](https://arxiv.org/html/2608.07476#S7.Thmtheorem1)\)\. Ordering uniqueness scoped to two\-operator raw staged architecture \(Corollary[7\.2](https://arxiv.org/html/2608.07476#S7.Thmtheorem2)\); uniqueness across all multi\-stage architectures is NOT claimed\.

## 9Discussion

Stabilization vs\. determinization\.A key structural distinction is between closure stabilization \(per\-seed convergence, property \(4’\)\) and full determinization \(global unique fixed point \+ AC\-6\)\. The current framework fully characterizes closure stabilization \(Class C\) and selection\-based canonicalization \(Class S\); the upgrade from stabilization to determinization is governed by additional global confluence properties that remain to be characterized\. For ICT, this is OQ\-GC\-1\.

Mechanism classification\.§8 establishes that, conditional on the realization\-coverage assumption \(OQ\-Realization\) and on per\-branch hypotheses, no third primitive canonicalization mechanism arises within the \(3a\)/\(3b\)/\(3c\) realization grammar\. The proof works in two layers: theory\-intrinsic operations are defined independently via \(I1\)–\(I2\); the trichotomy \(3a\)/\(3b\)/\(3c\) is the realization case partition; Theorem U3 then classifies each case using Lemma C \(\(3a\)\-scope monotonicity from R1\+R2\+R3\) and Lemma G \(\(3c\)\-scope extensivity \+ admissibility preservation \+ \(4’\) \+ idempotence\)\. This two\-layer structure replaces the v2\.15\.1 formulation in which the trichotomy was embedded in the theory\-intrinsic definition \(definitional circularity\)\. Realization\-coverage is tracked as OQ\-Realization, and U3 is explicitly conditional on it\. \(3a\)\-scope Class C derivation requires both syntactic R1\+R2\+R3 and the semantic Soundness hypothesis; the two are independent\.

Monotonicity in⪯spec\\preceq\_\{\\mathrm\{spec\}\}is mechanism\-specific\.PureIcoreI^\{\\mathrm\{core\}\}\-computable mechanisms \(3a, in saturated form under elementary firing\) are automatically monotone in⪯spec\\preceq\_\{\\mathrm\{spec\}\}under R1\+R2\+R3 \(Lemma C\)\. Guarded mechanisms \(3c\) are generically*not*monotone: adding predicates may activate previously\-inactive conflict guards, blocking derivations that fire in the smaller state\. The §6\.2 explicit witness forDcompHTFD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}\(Remark[6\.7](https://arxiv.org/html/2608.07476#S6.Thmtheorem7)\) makes this concrete\. This is not a defect: Lemma G’s \(4’\) conclusion does not require monotonicity, and the Monotonic Exhaustion Lemma uses only finite \+ extensive\. The asymmetry between \(3a\)\- and \(3c\)\-scope monotonicity is a structural feature of guard\-based determinization, reflecting the architectural separation betweenIcoreI^\{\\mathrm\{core\}\}derivation \(monotone by R1\+R2\+R3\) and conflict resolution \(not monotone in general\)\.

Type S, Type S\-strong, and the scope of selection\.The dichotomy “Type E↔\\leftrightarrowcompletion, Type S↔\\leftrightarrowselection” is sound only with the Type S\-strong qualifier\. Type S theories without the no\-common\-upper\-bound strengthening may admit completion at \(4\) strength depending on upper\-bound structure \(OQ\-TypeS\-Imp\)\.TW​yT\_\{Wy\}is Type S\-strong by axiomatic phase incomparability, justifying the selection construction\. The framework’s substantive claim is therefore narrower than a generic Type S→\\toselection inference: Type S\-strong provably blocks \(4\)\-strength completion, while leaving the broader question open\. Crucially, the selection mechanism in Type S\-strong precisely operates over alternatives that lack a common upper bound, so the C5\-S condition asserts global admissibility of the selector’s output, not joint admissibility of local alternatives—these are distinct claims\.

Obstacle to global completion in ICT\.The obstruction to upgrading ICT from \(4’\) to \(4\) is temporal: directional alternatives \(e\.g\., BOS direction\) are not yet decided at a given input but become decided under further evidence refinement\. Different seeds may stabilize at different fixed points precisely because the evidence has not yet forced unique directional commitment\. Hence OQ\-GC\-1 remains open\.

The role of C5\.C5 captures the requirement that canonicalization output is globally consistent\. The two variants operate on different mathematical objects: C5\-E asserts compatible extension within the closure domain \(a set\-valued condition\); C5\-S asserts global admissibility of the selector’s output \(a forward condition on the selected assignment\)\. The variants share the intuition of canonicalization output coherence but apply to operators versus selectors respectively\.

Non\-commutativity and hierarchy\.§7 establishes that, within the two\-operator raw staged architecture, multi\-level canonicalization is not reducible to iterated single\-level canonicalization with arbitrary ordering\. The HTF\-first ordering is a structural property within this declared architecture; alternative architectures \(with repair operators, joint fixed\-point constructions, or symmetrically guardedDHD\_\{H\}\) are not excluded but lie outside the present scope\.

Toward recursive theories\.Elliott Wave and similar hierarchical theories require an ordered family\{Dℓ\}ℓ∈L\\\{D\_\{\\ell\}\\\}\_\{\\ell\\in L\}with level\-indexed M2\-HTF guards and a compatibility condition across levels\. The Det pseudofunctor of §8\.4 provides the categorical template\.

Connection to LLM\-assisted reasoning\.The framework’s conceptual payoff for AI systems is the formalization of when an LLM\-assisted system is licensed to commit to a unique structural interpretation\. In this view, 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\.

## 10Phase 2: Multi\-Operator Canonicalization

*All definitions in §10\.1 are self\-contained\. The existence and hierarchy results in §10\.2 are summary statements; full proofs are recorded in the companion artifactsphase2bandphase2cand are not reproduced in this paper\. The present section serves as a unified reference point and as scaffolding for the Det pseudofunctor of §8\.4\.*

### 10\.1Setting and Definitions

###### Definition 10\.1\(DQA — Determinization Quasi\-Algebra\)\.

𝒬=\(𝔇,∘,𝒜,≺,𝒪\)\\mathcal\{Q\}=\(\\mathfrak\{D\},\\circ,\\mathcal\{A\},\\prec,\\mathcal\{O\}\): finite family𝔇=\{Dℓ\}ℓ∈L\\mathfrak\{D\}=\\\{D\_\{\\ell\}\\\}\_\{\\ell\\in L\}, partial composition∘\\circ, admissible sequences𝒜\\mathcal\{A\}, precedence poset≺\\prec, obstructed sequences𝒪\\mathcal\{O\}\.

EachDℓD\_\{\\ell\}satisfies:

\(Q1\) Local closureExtensivity, idempotence, admissibility preservation, per\-seed convergence \(4’\) on its standard domain\.

\(Q2\) Partial compositionDefined only on admissible sequences\.

\(Q3\) Obstructionσ∈𝒪⇔∃P∈𝒫T:σ​\(P\)∉𝒫T\\sigma\\in\\mathcal\{O\}\\iff\\exists\\,P\\in\\mathcal\{P\}\_\{T\}:\\sigma\(P\)\\notin\\mathcal\{P\}\_\{T\}\.

\(Q4\) Idempotent stability \(partial\-function form\)Dℓ∘DℓD\_\{\\ell\}\\circ D\_\{\\ell\}andDℓD\_\{\\ell\}agree as partial functions: for allP∈𝒫TP\\in\\mathcal\{P\}\_\{T\}on which both sides are defined,\(Dℓ∘Dℓ\)​\(P\)=Dℓ​\(P\)\(D\_\{\\ell\}\\circ D\_\{\\ell\}\)\(P\)=D\_\{\\ell\}\(P\)\.

###### Definition 10\.3\(Standard domain\)\.

The*standard domain*ofDℓD\_\{\\ell\}is the canonical seed domain on whichDℓD\_\{\\ell\}is verified in isolation to satisfy Q1’s four properties\. For ICT, the standard domain ofDcompD\_\{\\mathrm\{comp\}\}is𝒫I​C​Tclosure\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{closure\}\}; forDcompHTFD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}it is𝒫I​C​Tmulti\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}\.

###### Definition 10\.4\(Safe application domain\)\.

Safe⁡\(Dℓ\):=\{P∈𝒫T∣Dℓ​\(P\)∈𝒫T\}\\operatorname\{Safe\}\(D\_\{\\ell\}\):=\\\{P\\in\\mathcal\{P\}\_\{T\}\\mid D\_\{\\ell\}\(P\)\\in\\mathcal\{P\}\_\{T\}\\\}\.

###### Definition 10\.6\(Prefix\-reachable states\)\.

For topological orderingℓ1,…,ℓn\\ell\_\{1\},\\ldots,\\ell\_\{n\}and seed set𝒮\\mathcal\{S\}:Reach0⁡\(𝒮\):=𝒮\\operatorname\{Reach\}\_\{0\}\(\\mathcal\{S\}\):=\\mathcal\{S\};Reachk\+1⁡\(𝒮\):=\{Dℓk\+1​\(Q\)∣Q∈Reachk⁡\(𝒮\)\}\\operatorname\{Reach\}\_\{k\+1\}\(\\mathcal\{S\}\):=\\\{D\_\{\\ell\_\{k\+1\}\}\(Q\)\\mid Q\\in\\operatorname\{Reach\}\_\{k\}\(\\mathcal\{S\}\)\\\}\.

###### Definition 10\.7\(Admissible linearization\)\.

A topological orderingℓ1,…,ℓn\\ell\_\{1\},\\ldots,\\ell\_\{n\}of\(L,≺\)\(L,\\prec\)is*admissible on𝒮\\mathcal\{S\}*if the composed operator\(Dℓn∘⋯∘Dℓ1\)​\(P\)∈𝒫T\(D\_\{\\ell\_\{n\}\}\\circ\\cdots\\circ D\_\{\\ell\_\{1\}\}\)\(P\)\\in\\mathcal\{P\}\_\{T\}for allP∈𝒮P\\in\\mathcal\{S\}\. Equivalently,Reachk⁡\(𝒮\)⊆Safe⁡\(Dℓk\+1\)\\operatorname\{Reach\}\_\{k\}\(\\mathcal\{S\}\)\\subseteq\\operatorname\{Safe\}\(D\_\{\\ell\_\{k\+1\}\}\)for allk<nk<n\.

###### Definition 10\.8\(Safety gap\)\.

Φ​\(ℓ1,…,ℓn;𝒮\):=∑k=0n−1𝟏​\{Reachk⁡\(𝒮\)⊈Safe⁡\(Dℓk\+1\)\}\\Phi\(\\ell\_\{1\},\\ldots,\\ell\_\{n\};\\mathcal\{S\}\):=\\sum\_\{k=0\}^\{n\-1\}\\mathbf\{1\}\\\{\\operatorname\{Reach\}\_\{k\}\(\\mathcal\{S\}\)\\nsubseteq\\operatorname\{Safe\}\(D\_\{\\ell\_\{k\+1\}\}\)\\\}Φ∗​\(𝒮\):=mintopological⁡Φ​\(⋅;𝒮\)\\Phi^\{\*\}\(\\mathcal\{S\}\):=\\min\_\{\\text\{topological\}\}\\Phi\(\\cdot;\\mathcal\{S\}\)\.Φ∗​\(𝒮\)=0⇔∃\\Phi^\{\*\}\(\\mathcal\{S\}\)=0\\iff\\existsadmissible linearization on𝒮\\mathcal\{S\}\.

Compatibility conditions\.

\(GCC\) Graph\-Compatible Guard Condition\(seed\-independent\):Im​\(Dℓk\)∩𝒫T⊆Safe⁡\(Dℓk\+1\)\\mathrm\{Im\}\(D\_\{\\ell\_\{k\}\}\)\\cap\\mathcal\{P\}\_\{T\}\\subseteq\\operatorname\{Safe\}\(D\_\{\\ell\_\{k\+1\}\}\)for all consecutiveℓk,ℓk\+1\\ell\_\{k\},\\ell\_\{k\+1\}\.

\(RUC𝒮\\mathrm\{RUC\}\_\{\\mathcal\{S\}\}\) Reach\-Uniform Compatibility\(seed\-dependent\): for every extendable prefixUkU\_\{k\}and everyℓ,ℓ′∈Next⁡\(Uk\)\\ell,\\ell^\{\\prime\}\\in\\operatorname\{Next\}\(U\_\{k\}\):Safe⁡\(Dℓ\)∩Reachk⁡\(𝒮\)=Safe⁡\(Dℓ′\)∩Reachk⁡\(𝒮\)\\operatorname\{Safe\}\(D\_\{\\ell\}\)\\cap\\operatorname\{Reach\}\_\{k\}\(\\mathcal\{S\}\)=\\operatorname\{Safe\}\(D\_\{\\ell^\{\\prime\}\}\)\\cap\\operatorname\{Reach\}\_\{k\}\(\\mathcal\{S\}\)\.

\(PLC𝒮\\mathrm\{PLC\}\_\{\\mathcal\{S\}\}\) Pairwise Local Compatibility\(seed\-dependent\): for every extendable prefixUkU\_\{k\}and everyℓ,ℓ′∈Next⁡\(Uk\)\\ell,\\ell^\{\\prime\}\\in\\operatorname\{Next\}\(U\_\{k\}\):Reachk⁡\(𝒮\)⊆Safe⁡\(Dℓ\)⇔Reachk⁡\(𝒮\)⊆Safe⁡\(Dℓ′\)\\operatorname\{Reach\}\_\{k\}\(\\mathcal\{S\}\)\\subseteq\\operatorname\{Safe\}\(D\_\{\\ell\}\)\\Leftrightarrow\\operatorname\{Reach\}\_\{k\}\(\\mathcal\{S\}\)\\subseteq\\operatorname\{Safe\}\(D\_\{\\ell^\{\\prime\}\}\)\.

Extension conditions\.\(C2\)∃ℓ∈Next⁡\(Uk\)\\exists\\,\\ell\\in\\operatorname\{Next\}\(U\_\{k\}\)withReachk⁡\(𝒮\)⊆Safe⁡\(Dℓ\)\\operatorname\{Reach\}\_\{k\}\(\\mathcal\{S\}\)\\subseteq\\operatorname\{Safe\}\(D\_\{\\ell\}\)at each prefix\.\(C2’\)∀ℓ∈Next⁡\(Uk\)\\forall\\,\\ell\\in\\operatorname\{Next\}\(U\_\{k\}\),Reachk⁡\(𝒮\)⊆Safe⁡\(Dℓ\)\\operatorname\{Reach\}\_\{k\}\(\\mathcal\{S\}\)\\subseteq\\operatorname\{Safe\}\(D\_\{\\ell\}\)at each prefix\.

### 10\.2Phase 2 Summary Theorem

###### Theorem 10\.9\(Phase 2 summary; proofs in companion artifacts\)\.

\(Statement; full proofs are not reproduced in this paper\. They are recorded in the companion artifactsphase2bandphase2c\. The present theorem serves as a unified reference point and as scaffolding for the Det pseudofunctor of §8\.4\.\)

Let𝒬\\mathcal\{Q\}be a DQA with finiteLLand𝒮⊆𝒫T\\mathcal\{S\}\\subseteq\\mathcal\{P\}\_\{T\}\.

*\(I\) Existence\.*\(Requires\(L,≺\)\(L,\\prec\)acyclic\.\)

Φ∗​\(𝒮\)=0⟺∃admissible linearization on​𝒮\.\\Phi^\{\*\}\(\\mathcal\{S\}\)=0\\Longleftrightarrow\\exists\\,\\text\{admissible linearization on \}\\mathcal\{S\}\.UnderPLC𝒮\\mathrm\{PLC\}\_\{\\mathcal\{S\}\}:Φ∗=0⇔\(C2\)⇔\(C2′\)⇔∃\\Phi^\{\*\}=0\\Leftrightarrow\\mathrm\{\(C2\)\}\\Leftrightarrow\\mathrm\{\(C2^\{\\prime\}\)\}\\Leftrightarrow\\existsadmissible linearization\.

*\(II\) Compatibility\-strength hierarchy\.*The compatibility conditions are ordered by logical implication strength \(not by set inclusion\):

GCC⇒RUC𝒮⇒PLC𝒮\(seed\-dependent, both implications strict\)\\mathrm\{GCC\}\\Rightarrow\\mathrm\{RUC\}\_\{\\mathcal\{S\}\}\\Rightarrow\\mathrm\{PLC\}\_\{\\mathcal\{S\}\}\\quad\\text\{\(seed\-dependent, both implications strict\)\}GCC⇒RUC≡PLC\(seed\-independent\)\\mathrm\{GCC\}\\Rightarrow\\mathrm\{RUC\}\\equiv\\mathrm\{PLC\}\\quad\\text\{\(seed\-independent\)\}

###### Proof sketch\.

\(I\): Theorems 2b\-14/2b\-15 of phase2b artifact; collapse underPLC𝒮\\mathrm\{PLC\}\_\{\\mathcal\{S\}\}by Theorem 2c\-1 of phase2c artifact\. \(II\): Propositions 2c\-2 through 2c\-6 of phase2c artifact\. Detailed proofs are not reproduced here; the present section is a summary\. ∎∎

### 10\.3Strictness Witness

###### Example 10\.11\(PLC𝒮\\mathrm\{PLC\}\_\{\\mathcal\{S\}\}withoutRUC𝒮\\mathrm\{RUC\}\_\{\\mathcal\{S\}\}\)\.

L=\{a,b\}L=\\\{a,b\\\},≺=∅\\prec=\\varnothing,𝒫T=\{Q1,Q2,Q3\}\\mathcal\{P\}\_\{T\}=\\\{Q\_\{1\},Q\_\{2\},Q\_\{3\}\\\},𝒮=𝒫T\\mathcal\{S\}=\\mathcal\{P\}\_\{T\}\.Safe⁡\(Da\)∩Reach0=\{Q1,Q2\}\\operatorname\{Safe\}\(D\_\{a\}\)\\cap\\operatorname\{Reach\}\_\{0\}=\\\{Q\_\{1\},Q\_\{2\}\\\};Safe⁡\(Db\)∩Reach0=\{Q1,Q3\}\\operatorname\{Safe\}\(D\_\{b\}\)\\cap\\operatorname\{Reach\}\_\{0\}=\\\{Q\_\{1\},Q\_\{3\}\\\}\. Bothχa=χb=0\\chi\_\{a\}=\\chi\_\{b\}=0\(PLC𝒮\\mathrm\{PLC\}\_\{\\mathcal\{S\}\}holds\); restricted safe sets differ \(RUC𝒮\\mathrm\{RUC\}\_\{\\mathcal\{S\}\}fails\)\. PLC checks the binary all\-or\-nothing judgment; RUC checks set identity\.

## 11Changelog

This section records changes across the v2\.15\.1→\\tov2\.16→\\tov2\.16\.2→\\tov2\.16\.3→\\tov2\.16\.4 revision sequence\. v2\.16 was an errata revision of v2\.15\.1 addressing 13 issues; v2\.16\.2 was an incremental fix addressing one identified gap \(M1, Lemma G monotonicity\) plus minor follow\-up; v2\.16\.3 tightens four claim\-strength items identified in post\-v2\.16\.2 review; v2\.16\.4 is a mathematical\-correctness patch addressing four findings from a third\-round adversary review \(S1 Soundness, S3 Lemma G saturated form with G3 reframed as guard\-stability, S4 Phase 2 summary qualifier, U3 statement\-level de\-tautology\)\.

### 11\.1v2\.16\.3→\\tov2\.16\.4 \(mathematical correctness fixes\)

S1 \(Soundness assumption added to Corollary 6b’\)\.v2\.16\.3 §5\.3 Corollary 6b’ implicitly assumed admissibility preservation under the phrase “by construction ofIcoreI^\{\\mathrm\{core\}\}\-computability” but did not declare it as a hypothesis\. The reviewer’s reflexive counterexample \(rule “a⇒ba\\Rightarrow b” plus axiom forbidding\{a,b\}\\\{a,b\\\}satisfies R1\+R2\+R3 syntactically while saturated closure exits𝒫T\\mathcal\{P\}\_\{T\}\) shows the corollary as stated is false\. v2\.16\.4 \(i\) reintroducesffasFωF^\{\\omega\}on the unrestricted state space and uses Soundness to derive the restriction𝒫T→𝒫T\\mathcal\{P\}\_\{T\}\\to\\mathcal\{P\}\_\{T\}\(avoiding type\-circularity of writing “f:𝒫T→𝒫Tf:\\mathcal\{P\}\_\{T\}\\to\\mathcal\{P\}\_\{T\}” and then proving it\), \(ii\) declares Soundness as an explicit additional hypothesis, \(iii\) updates the proof body to invoke Soundness for admissibility preservation, \(iv\) adds Remark[5\.6](https://arxiv.org/html/2608.07476#S5.Thmtheorem6)distinguishing the syntactic constraints R1\+R2\+R3 from the semantic Soundness condition, and \(v\) makes the elementary one\-rule\-firing interpretation ofFFexplicit at statement level, so single\-rule Soundness suffices \(a separate counterexample exists for simultaneous\-firing semantics: rules “a⇒ba\\Rightarrow b” and “a⇒ca\\Rightarrow c” plus axiom forbidding\{b,c\}\\\{b,c\\\}are individually sound but jointly unsound\)\. The proof also adds an order\-independence sentence noting that, in the pure \(3a\) positive\-rule setting under fair elementary scheduling, saturation is order\-independent \(every conclusion whose positive premises eventually become true is eventually added; no rule retracts\)\. ForTI​C​TT\_\{ICT\}the soundness role is filled by M1\.

S3 \(Lemma G saturated form, mirror of T4; G3 reframed as guard\-stability\)\.v2\.16\.3 T4 fixed the \(3a\) realization as the saturated closuref=Fωf=F^\{\\omega\}\. v2\.16\.3 Lemma G \(\(3c\)\-scope\) retained the ambiguity between one\-step operator and saturation: the proof showed iteration reaches a fixed pointP∗P^\{\*\}and concludedf​\(P∗\)=P∗f\(P^\{\*\}\)=P^\{\*\}, but did not establishf∘f=ff\\circ f=ffor arbitraryPP\. v2\.16\.4 \(i\) declaresf:=Gωf:=G^\{\\omega\}explicitly \(one\-step guarded operatorGG, saturated closureGωG^\{\\omega\}\), making idempotence a saturation tautology, \(ii\) rewrites the proof body as a 6\-step argument \(GG\-trajectory termination→\\toGωG^\{\\omega\}well\-defined→\\torestriction by Soundness→\\toextensivity→\\toidempotence by saturation→\\to\(4’\) withk=1k=1forff\), \(iii\)reframes G3 as a guard\-stability property of the ICT\-style guard, not as the source of saturation well\-definedness — well\-definedness ofGωG^\{\\omega\}comes from R1 \+ finite \+ \(G2\) functionality, while \(G3\) ensures that suppression persists along the trajectory, \(iv\) synchronizes the §8\.0 \(3c\) definition with the saturated form \(f=Gωf=G^\{\\omega\}explicit\), and \(v\) adds the same Soundness hypothesis as S1, applied toGG\. Internal consistency with the \(3a\) treatment under T4\.

S4 \(Phase 2 theorem qualifier\)\.v2\.16\.3 §10\.2 stated a theorem whose detailed proofs reside in companion artifactsphase2b/phase2c\. v2\.16\.4 \(i\) extends the theorem name with “Phase 2 summary; proofs in companion artifacts”, \(ii\) prepends a bold parenthetical clarifying that proofs are not reproduced in the paper and recording the companion artifacts, and \(iii\) extends the §10 opener accordingly\. Wording is restrained \(no claim that the artifacts constitute “established external results”\), reflecting that the artifacts are not necessarily publicly accessible\.

U3 statement\-level de\-tautology\.v2\.16\.3 Theorem U3 had a hypothesis of the form “ffis a valid canonicalization mechanism, i\.e\. Class C or Class S”—putting the conclusion inside the hypothesis\. v2\.16\.4 deletes this clause and lifts the per\-realization assumptions into the statement: R1\+R2\+R3 \+ Soundness \+ elementaryFFfor \(3a\); theory\-intrinsic⪯sel\\preceq\_\{\\mathrm\{sel\}\}\+ C5\-S for \(3b\); R1\+R2\+R3 \+ \(G2\) \+ \(G3\) \+ Soundness for \(3c\)\. The Class C / Class S conclusion is now derived per\-case from per\-branch hypotheses, not pre\-assumed\.

No statement of any other theorem, lemma, or corollary changes; no classification result changes scope\. v2\.16\.4 is a mathematical\-correctness patch, not a content revision\.

### 11\.2v2\.16\.2→\\tov2\.16\.3 \(claim\-strength tightening\)

T1 \(C5\-S redefined: global compatibility of selected choices\)\.v2\.16\.2 §5\.1 defined C5\-S as “C5 applies over all locally admissible pairsc1,c2∈Ar,tc\_\{1\},c\_\{2\}\\in A\_\{r,t\}”—i\.e\., requiring joint admissibility of local alternatives\. This conflicts with Type S\-strong, which is defined precisely by alternatives lacking a common upper bound\. v2\.16\.3 redefines C5\-S as global admissibility of the selector’s output: the assignmentP∗:\(r,t\)↦\{cr,t∗\}P^\{\*\}:\(r,t\)\\mapsto\\\{c^\{\*\}\_\{r,t\}\\\}extends to an element of𝒫T\\mathcal\{P\}\_\{T\}, wherecr,t∗=max⪯sel⁡Ar,tc^\{\*\}\_\{r,t\}=\\max\_\{\\preceq\_\{\\mathrm\{sel\}\}\}A\_\{r,t\}\. Selection in Type S\-strong now operates correctly: the selector picks one alternative per locus, and C5\-S asserts that the resulting global assignment is admissible\. C5\-E \(completion variant\) is unchanged\.

T2 \(Theorem U3 explicitly conditional on OQ\-Realization\)\.v2\.16\.2 stated U3 as “no third primitive class arises in MST” with realization\-coverage tracked as informal\. v2\.16\.3 restates U3 as conditional on OQ\-Realization: “*Assume the realization\-coverage hypothesis \(OQ\-Realization\)\.*Then within the \(3a\)/\(3b\)/\(3c\) realization grammar, every valid theory\-intrinsic canonicalization mechanism is Class C or Class S\.” Theorem title changed to “Conditional Classification within the MST Realization Grammar\.” Realization\-coverage examples expanded in Remark[8\.16](https://arxiv.org/html/2608.07476#S8.Thmtheorem16)to flag mechanisms outside the declared grammar \(cardinality\-minimization, orbit selectors, priority\-valued canonicalizers, hybrid selector/closure operators\)\.

T3 \(Non\-commutativity ordering uniqueness scoped\)\.v2\.16\.2 stated “HTF\-first ordering is the unique admissibility\-preserving staged completion order” without scope qualifier\. v2\.16\.3 §7 separates the witness \(Proposition, unchanged\) from a scoped ordering\-uniqueness corollary \(Corollary[7\.2](https://arxiv.org/html/2608.07476#S7.Thmtheorem2)\): “*Within the two\-operator raw staged architecture\{DL∘DH,DH∘DL\}\\\{D\_\{L\}\\circ D\_\{H\},D\_\{H\}\\circ D\_\{L\}\\\}, with no repair operator, no M2\-HTF guard added toDHD\_\{H\}, and no joint fixed\-point construction*, HTF\-first is the only admissibility\-preserving linearization\.” A scope remark explicitly notes that alternative architectures \(with repair operators, symmetrically guardedDHD\_\{H\}, or joint fixed\-point constructions\) lie outside the present scope\.

T4 \(\(3a\) saturated form made explicit\)\.v2\.16\.2 referred to “IcoreI^\{\\mathrm\{core\}\}\-computable in the pure sense \(3a\)” without distinguishing one\-step rule application from saturated closure\. v2\.16\.3 makes the saturated interpretationf=Fωf=F^\{\\omega\}explicit in §8\.0 \(3a\) definition, Corollary 6b’ statement and proof, Lemma C statement, U3 Case 1, and §8\.5 status board\. The one\-step operatorFFalone is generally not idempotent; saturation is required for idempotence and is built into the \(3a\) realization throughout\.

T5 \(LLM\-assisted reasoning hook restored\)\.v2\.16\.2 abstract removed the LLM\-motivation framing present in v2\.15\.1\. v2\.16\.3 restores an opening LLM/cs\.AI hook in the abstract and adds an explicit §1 introduction paragraph and a closing §9 paragraph connecting canonicalization to hallucination as “unsupported canonicalization\.”

T6 \(Phase 2 hierarchy notation: implication, not inclusion\)\.v2\.16\.2 wroteGCC⊋RUC𝒮⊋PLC𝒮\\mathrm\{GCC\}\\supsetneq\\mathrm\{RUC\}\_\{\\mathcal\{S\}\}\\supsetneq\\mathrm\{PLC\}\_\{\\mathcal\{S\}\}, which is set\-inclusion notation but in the wrong direction \(GCC is the strongest condition, so the set of GCC\-satisfying DQAs is the*smallest*\)\. v2\.16\.3 changes this to implication notation:GCC⇒RUC𝒮⇒PLC𝒮\\mathrm\{GCC\}\\Rightarrow\\mathrm\{RUC\}\_\{\\mathcal\{S\}\}\\Rightarrow\\mathrm\{PLC\}\_\{\\mathcal\{S\}\}\. A notational convention remark is added to §10\.2\.

T7 \(Core principle restated to align with C5\-S/C5\-E asymmetry\)\.v2\.16\.2 abstract closed with “Canonicalization is not a consequence of closure alone, but of closure together with comparability and joint admissibility\.” This treated C5\-S and C5\-E as parallel, conflicting with T1\. v2\.16\.3 restates: “Completion\-based canonicalization requires closure, comparability, and compatible extension; selection\-based canonicalization requires theory\-intrinsic comparability and global compatibility of selected choices\.”

The \(4’\) closure\-stabilization conclusion is unchanged in scope and strength\. All Class C/Class S classifications, the Type S\-strong impossibility result, and ICT/Wyckoff/Chan instance assignments are unchanged\.

### 11\.3v2\.16→\\tov2\.16\.2 \(M1 fix and minor follow\-up\)

M1 \(Lemma G monotonicity claim removed\)\.v2\.16 §8\.2 Lemma G asserted that \(3c\)\-realized mechanisms inherit monotonicity in⪯spec\\preceq\_\{\\mathrm\{spec\}\}fromIcoreI^\{\\mathrm\{core\}\}rules under hypotheses \(G1\) state\-monotone guard, \(G2\) functionality, \(G3\) additive\-stable guard\. Audit identified that \(G1\) does not hold in the asserted direction for ICTDcompHTFD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}: guard suppression onQQdoes not imply guard suppression onPPwhenP⪯specQP\\preceq\_\{\\mathrm\{spec\}\}Q—the opposite direction is what holds \(more predicates means more witnesses, hence more suppression\)\. This made the v2\.16 monotonicity claim incorrect in its application to ICT\.

A concrete witness showsDcompHTFD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}is not monotone in⪯spec\\preceq\_\{\\mathrm\{spec\}\}: takeP=∅P=\\emptysetandQ=\{HTF​\_​bearish​\_​bias\}Q=\\\{\\mathrm\{HTF\\\_bearish\\\_bias\}\\\}at an input withclose\>prev​\_​close\\mathrm\{close\}\>\\mathrm\{prev\\\_close\}\. Thenρ3\\rho\_\{3\}fires onPP\(yieldingLTF​\_​bullish​\_​cont∈DcompHTF​\(P\)\\mathrm\{LTF\\\_bullish\\\_cont\}\\in D\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}\(P\)\) but is M2\-HTF\-suppressed onQQ\(soLTF​\_​bullish​\_​cont∉DcompHTF​\(Q\)\\mathrm\{LTF\\\_bullish\\\_cont\}\\notin D\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}\(Q\)\)\.

v2\.16\.2 fixes this as follows:

- •§8\.2 Lemma G: removed monotonicity claim and \(G1\) hypothesis\. Lemma G now establishes only extensivity, per\-seed \(4’\), and idempotence under \(G2\)\+\(G3\)\. Sufficient because Monotonic Exhaustion already requires only finite \+ extensive \(monotonicity was redundant\)\.
- •§8\.2 Monotonic Exhaustion Lemma: hypothesis tightened to “extensive \+ finite”\.
- •§8\.2 Lemma C: added Remark[8\.9](https://arxiv.org/html/2608.07476#S8.Thmtheorem9)“Lemma C does not extend to \(3c\) under naive⪯spec\\preceq\_\{\\mathrm\{spec\}\}”\.
- •§6\.2: added Remark[6\.7](https://arxiv.org/html/2608.07476#S6.Thmtheorem7)with the explicit ICT counterexample\.
- •§8\.5 status board: updated Lemma G entry andDcompHTFD\_\{\\mathrm\{comp\}\}^\{\\mathrm\{HTF\}\}classification entry\.
- •§9 Discussion: added paragraph “Monotonicity in⪯spec\\preceq\_\{\\mathrm\{spec\}\}is mechanism\-specific”\.

m4 \(Realization\-coverage marked informal; OQ\-Realization added\)\.v2\.16 §8\.2 Realization\-coverage remark argued informally that every theory\-intrinsic operation is realized by \(3a\)/\(3b\)/\(3c\)\. v2\.16\.2 explicitly tagged this as informal and added OQ\-Realization to §8\.5 status board\. \(v2\.16\.3 further escalates this to a conditional hypothesis on Theorem U3; see T2\.\)

c2 \(§9 Discussion reordered\)\.v2\.16\.2 moved “Stabilization vs\. determinization” to the first paragraph of §9, followed by “Mechanism classification” and “Monotonicity in⪯spec\\preceq\_\{\\mathrm\{spec\}\}is mechanism\-specific”\.

### 11\.4v2\.15\.1→\\tov2\.16 \(errata revision, 13 issues\)

This subsection records the v2\.15\.1→\\tov2\.16 changes\. Changes are grouped by issue\.

P1 \(Theorem U3 Case 3 tautology, framing\)\.§8\.2 Theorem U3 renamed “Primitive Mechanism Classification” \(was “Primitive Mechanism Exhaustion”\)\. Hypothesis “uniqueness\-inducing” removed from U3 statement\. Added scope note: U3 does not derive \(4\) or AC\-6\. Added Lemma G \(Guard Preservation\) in §8\.2 to give substantive proof for Case 3 mechanisms; previous tautology replaced by explicit \(G2\)\+\(G3\) hypothesis \(v2\.16\.2: Lemma G scope restricted; \(G1\) removed—see M1; v2\.16\.3: U3 made explicitly conditional—see T2; v2\.16\.4: per\-branch hypotheses lifted into U3 statement, G3 reframed as guard\-stability\)\. Added Realization\-coverage remark \(v2\.16\.2: marked informal, OQ\-Realization added\)\.

P2 \(R2\*\(iii\) restatement of U3\)\.§8\.1 Lemma R2\* part \(iii\) deleted\. R2\* now consists only of parts \(i\) and \(ii\)\. Mechanism reduction is the content of Theorem U3, not R2\*\.

P3 \(C5\-E vague definition\)\.§5\.1 C5\-E rewritten as framework\-level explicit definition with the closure\-domain conflict relation made abstract\. ICT instance \(M2 /M2∪M2​\-​HTF\\mathrm\{M2\}\\cup\\mathrm\{M2\\text\{\-\}HTF\}\) given as example, not embedded in definition\.

P4 \(Type S→\\toselection dichotomy overclaim\)\.Introduced Type S\-strong as named subclass in §3\.1\. §4\.2 Proposition renamed and re\-hypothesized for Type S\-strong\. Wyckoff explicitly classified as Type S\-strong\. Abstract, §1 contributions list, §3 examples table, §4\.2 architecture remark, §8\.5 status board, §9 Discussion all updated\. OQ\-TypeS\-Imp tracks the open generic\-Type\-S case\.

P5 \(uniqueness\-inducing terminology \+ Monotonic Exhaustion scope\)\.§4\.1: added Definition \(Uniqueness\-inducing operator\) as named alias for property \(4\)\. §8\.2 Monotonic Exhaustion Lemma renamed “\(per\-seed\)”, proof shrunk to use only finite \+ extensive→\\to\(4’\); counterexample \(\{a,b\}\\\{a,b\\\}\+ identity\) inserted showing \(4’\)⇏\\not\\Rightarrow\(4\)\. §5\.3 Corollary 6b’ hypothesis dropped “uniqueness\-inducing”\. §8\.0 Class C definition cleaned\. §8\.1 R2\* hypothesis explicitly references §4\.1\. §8\.2 U3 hypothesis dropped “uniqueness\-inducing”\.

P6 \(Q4 domain ambiguity\)\.§10\.1 Q4 reformulated as partial\-function form: agreement onPPwhere both sides are defined\.

P7 \(§6\.2 wording — “unique normal form for that seed”\)\.Replaced by “each seed converges to a fixed point within𝒫I​C​Tmulti\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}”\.

P10 \(§5\.3 Theorem 6b title\)\.Title changed to “Construction Lemma for Canonicalization Mechanisms”\.

R1 \(Theorem 6b vs\. Corollary 6b’ position\)\.Theorem 6b explicitly labeled “\(Construction Lemma\)”; remark added explaining “forward” arrows are constructions\. Corollary 6b’ explicitly labeled as “the substantive structural result”\.

R2 \(theory\-intrinsic definition circularity\)\.§8\.0 theory\-intrinsic definition rewritten using independent conditions \(I1\)–\(I2\)\. The \(3a\)/\(3b\)/\(3c\) trichotomy is now a separate “Realizations” structure, the case partition for Theorem U3\.

R3 \(U3 Case 3 still circular\)\.Addressed via Lemma G \(§8\.2\): substantive proof that \(3c\)\-realized mechanisms inherit \(1\), \(2\), \(4’\) properties under \(G2\)\+\(G3\) \(v2\.16\.2 scope; see M1; v2\.16\.4: G3 reframed as guard\-stability, Soundness added—see S3\)\.

R4 \(BOS premise mutual exclusion in §6\.2 Functionality Lemma\)\.§6\.2: added “Premise mutual exclusion” subsection\. Functionality Lemma proof restructured\.

R5 \(missing Lemmas 1–3 in §6\.2 idempotence\)\.§6\.2 idempotence verification rewritten as direct argument\.

R6 \(§5\.1 N4 vs\. §8\.3 refined N4\)\.§5\.1 N4 amended with “Refined treatment” subsection that explicitly references §8\.3\.

R7 \(abstract honesty re ICT determinization\)\.Abstract revised to explicitly state “the present paper does not establish ICT determinization at AC\-6 strength”\. ICT AC\-6 conditional on OQ\-GC\-1; Wyckoff AC\-6 directly via PhaseClassify\.

R8 \(Lemma C two\-class premise reasoning\)\.§8\.2 Lemma C proof expanded with explicit two\-class analysis\. R2 also clarified\.

R11 / OQ\-Chan\-TRS, OQ\-Det\-Coh tracking\.§3\.2 Chan rewrite presentation tagged OQ\-Chan\-TRS\. §8\.4 Det pseudofunctor coherence tagged OQ\-Det\-Coh\. §8\.5 status board updated with all OQs\.

## Appendix AConflict Matrices

### A\.1M2 — ICT Single\-Timeframe Closure Domain

*Scope*:𝒫I​C​Tclosure\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{closure\}\}\.×\\times= forbidden pair;✓\\checkmark= compatible\.

\(Full predicate names: LTF\_BOSup/down, LTF\_bullish/bearish\_cont\.\)

### A\.2M2\-HTF — Cross\-Timeframe Forbidden Pairs

*Scope*:𝒫I​C​Tmulti\\mathcal\{P\}\_\{ICT\}^\{\\mathrm\{multi\}\}\. HTF predicates dominate opposing LTF predicates\.