Zero-State Axioms: Minimal Boundary Structure, Structural Consequences, and Semantic Models

The Zero-State Axioms: Minimal Coherence, Structural Consequences, and Semantic Models Lawrence Ip Independent Researcher 5/12/2025 Abstract We show that any coherent system that supports both unbounded extensibility and an internally governed notion of measurement must operate relative to a primitive boundary constraint. This is formalised by the Infinity–Measurement Boundary Theorem, which demonstrates that no system can coherently realise extensibility and measurement together without such a boundary, and that weakening this boundary leads to regress or inconsistency. The Zero-State Axioms (ZSA) provide an explicit axiomatization of this minimal boundary. We develop the semantic framework required to interpret ZSA, construct canonical and non-trivial models, and derive structural consequences including admissibility conditions, coherence valuations, and stability under extension. We also establish sharpness and independence results, showing that the axioms are minimal and non-redundant. Finally, we show that any system extending ZSA can coherently realise both infinity and measurement without circularity, and we characterise the resulting semantic and structural regimes. The framework thereby identifies the unique minimal boundary required for unifying infinity and measurement, and provides their first explicit axiomatisation. MSC Classification (2020): 03E99, 03B30, 18B99, 94A15 Keywords: coherence category; minimal coherence; admissible morphism; axiomatic emergence; mutual information; semantic domain 1 Introduction Coherent systems in logic, mathematics, computation, and physics frequently rely on two structural capacities: unbounded extensibility and internally governed measurement. Unbounded extensibility allows indefinitely generable configurations such as infinite sequences, spectra, chains, or refinement procedures. Measurement provides an internally meaningful 1 rule of determination, assigning values in a manner constrained by the system’s own coherence conditions. Though common, these capacities jointly generate a structural tension: extensibility cannot be freely combined with measurement without additional foundational constraints, while measurement cannot remain stable in the presence of unbounded extension without presupposing conditions not derivable from within the system. Section 6formalises this tension via the Infinity–Measurement Boundary Theorem (Theorem 6.1). The theorem shows that any coherent system satisfying both the Infinity Condition and the Measurement Condition (Definitions 3.2–3.3) must operate relative to a primitive boundary constraint that is not derivable from within the system. Attempts to internalise this boundary lead to regress, circularity, or instability under unbounded extension; weakening the boundary reintroduces inconsistency or loss of determinacy. Thus the coexistence of infinity and measurement forces the presence of a minimal constraint-surface, the zero-state boundary (Definition 3.4). The Zero-State Axioms (ZSA), introduced in Section 7, provide an explicit axiomatization of this minimal boundary. The axioms specify coherence valuations, admissible configurations, and the constraint-layer governing the interaction between determinate measurement and unbounded extension. ZSA therefore constitute not an imposed theoretical structure, but the least structural commitments necessary for any coherent system realising both infinity and measurement. Section 5develops the semantic framework for interpreting ZSA. Canonical and nontrivial models are constructed in Section 9, demonstrating that ZSA admits robust and diverse realisations. These models also exhibit the independence and minimality phenomena established in Section 10.4, showing that the axioms form a sharp, non-redundant boundary. Structural consequences—including admissibility criteria, coherence-preservation results, and the stability of determination across unbounded configurations—are presented in Section 8. The overall picture is that systems extending ZSA can coherently realise both infinity and measurement without circularity, regress, or undefinedness. The framework identifies the unique minimal boundary required for coherent systems unifying these two structural capacities, and it provides their first explicit axiomatisation. 2 Preliminaries This section introduces the basic mathematical vocabulary used throughout the paper. The objective is not to anticipate the semantic framework for the Zero-State Axioms (ZSA)— developed later in Section 5—but to establish the underlying structures, orderings, and conventions that will be assumed when formulating coherence conditions, admissibility, and measurement. All background notions appearing here are standard, and no substantive results are stated. 2.1 Domains, Configurations, and Orderings Adomain is any set or structure Dserving as the underlying space over which configurations are formed. A configuration on Dis any finite, countable, or otherwise specified construction 2 derived from D. In general, configurations belong to a class C ⊆ D∗, where D∗denotes the finite or infinite sequences, trees, chains, or admissible composites drawn from D. Configurations are partially ordered by an extension relation c⪯c′(“c′extends c”), which is always assumed reflexive and transitive. 2.2 Coherence Relations Acoherence relation is any relation Con ⊆ C × C that determines which configurations may coexist. We assume: 1. Reflexivity: Con(c, c) holds for all c∈ C. 2. Monotonicity: if Con(c, c′) and c′⪯d′, then Con(c, d′). 2.3 Valuations and Ordered Value Sets We assume a partially ordered set (V,≤), called a value set, which supports the interpretation of coherence valuations in Section 5. A valuation is any function v:C → V. 2.4 Measurement as Partial Determination Ameasurement operator is any partial function m:C⇀V, where Vis a set of possible values. No conditions beyond partiality are assumed at this stage. 2.5 Chains, Limits, and Extensibility Achain is a sequence c0⪯c1⪯c2⪯ · · · in C. A limit configuration of such a chain, when it exists, is a configuration csuch that cn⪯cfor all n, and cis minimal with this property. 3 2.6 Conventions •If m(c) is defined, we write m(c)↓. •If m(c) is undefined, we write m(c)↑. •c≺c′indicates strict extension. •Symmetry of Con is never assumed unless explicitly stated. •The symbol Zis reserved for the zero-state introduced in Section 7. This concludes the Preliminaries, which provide minimal vocabulary and notation for stating the Infinity–Measurement Boundary Theorem and defining the semantic interpretation of the Zero-State Axioms. 3 Definitions This section collects the core formal definitions used throughout the paper. Preliminaries (Section 2) supply the underlying mathematical context; the definitions below fix the structural and semantic notions required for the Zero-State Axioms and the subsequent metatheory. 3.1 Coherent Systems Definition 3.1 (Coherent system).Acoherent system Sconsists of: 1. a domain of discourse DS; 2. a set of admissible configurations CSover DS; 3. a coherence relation ConS⊆ CSdetermining internally consistent configurations. We write c⪯c′for the extension order, refer to (ci)i∈Iasachain, and use supicifor the supremum of a chain when it exists. 3.2 Infinity Condition Definition 3.2 (Infinity Condition).A coherent system Ssatisfies the Infinity Condition if its configuration space CSsupports unbounded extensibility: for every c∈ CSthere exists c′∈ CSwith c≺c′, and there exist arbitrarily long chains under ⪯. We write Inf(S) for this property. 4 3.3 Measurement Condition Definition 3.3 (Measurement Condition).A coherent system Ssatisfies the Measurement Condition if it admits a partial map MeasS:CS⇀VS such that: 1. MeasSis non-trivial; 2. the interpretation of MeasSis governed by the coherence relation ConS. We write Meas(S) for this property. 3.4 Zero-State Boundary Definition 3.4 (Zero-state boundary).Azero-state boundary is a minimal constraint surface Zsatisfying: 1. Non-derivability; 2. Coherence sufficiency; 3. Minimality. 3.5 Semantic Universe Definition 3.5 (Semantic universe).Asemantic universe for a coherent system Sis a tuple US= (DS,CS,ConS, ZS,ValS,MeasS), where ValS:CS→VSis a valuation function and MeasSis the partial measurement map from Definition 3.3. 3.6 ZSA-Models Definition 3.6 (ZSA-model).AZSA-model is a semantic universe USthat satisfies the ZeroState Axioms introduced in Section 7. Models may additionally satisfy further conditions such as valuation stability or measurement continuity (Section 9). 4 Notation We record here the notational conventions used throughout the paper. The underlying mathematical objects are introduced in Section 2; the purpose of this section is to fix the symbols and typographic conventions employed in later sections. 5 4.1 Domains, Configurations, and Ordering •D: the domain of discourse. •C: the set of configurations over D. •c, c′, ci: elements of C. •c⪯c′: the extension order. •Achain is a linearly ordered family (ci)i∈Iunder ⪯. •supici: the supremum of a chain (when it exists). 4.2 Coherence, Valuation, and Measurement •Con ⊆ C: the coherence condition. •Z: a zero-state boundary (formalised in Section 7). •Val :C → V: a valuation function. •Meas :C⇀V: a partial measurement map. We use sans-serif font for system components and allow system-specific subscripts (e.g. MeasS). 4.3 System-Level Predicates For any coherent system S, we use the following macros: Inf(S),Meas(S),Val(S), where: •Inf(S) denotes the Infinity Condition; •Meas(S) denotes the Measurement Condition; •Val(S) denotes the existence of a valuation compatible with coherence. These are instantiated in the preamble by: \newcommand{\Inf}{\mathsf{Inf}} \newcommand{\Meas}{\mathsf{Meas}} \newcommand{\Val}{\mathsf{Val}} 6 4.4 Semantic Universes A semantic universe for a system Sis denoted (DS,CS,ConS, ZS,ValS,MeasS). We write Ufor a generic semantic universe. 4.5 Axioms, Models, and Consequences •ZSA: the Zero-State Axioms (Section 7). •AZSA-model is any structure satisfying ZSA. •Model classes—canonical, valuation-extended, measurement-stable, and product models— are introduced in Section 9. 4.6 Theorems and Cross-References The Infinity–Measurement Boundary Theorem (Section 6) is labelled thm:IM-boundary. We use the following standard section labels: •sec:semantics: Semantics section; •sec:ZSA: Zero-State Axioms; •sec:structural: Structural Consequences; •sec:models: Models; •sec:main: Main Theorems. This notation inventory fixes all symbols used in the development that follows and ensures coherence between syntactic and semantic layers. 5 Canonical Semantics for the Zero-State Axioms This section develops the semantic framework required to interpret the Zero-State Axioms (ZSA). The aim is not to impose a particular mathematical setting, but to specify the minimal semantic data necessary for interpreting coherence relations, admissible configurations, and the zero-state boundary introduced in the axioms. The semantics presented here are compatible with the Infinity–Measurement Boundary Theorem (Theorem 6.1) and provide the foundation for the model constructions in Section 9. Asemantic universe is a triple U= (D,C,Con), where Dis the underlying domain, C ⊆ D∗is the class of admissible configurations, and Con ⊆ C × C is a reflexive, monotone coherence relation. A designated element Z∈ C serves 7 as the interpretation of the zero-state boundary and is assumed to be minimal with respect to coherence and to support stability of measurement under unbounded extension. Acoherence valuation is a function Val :C → V, where Vis a partially ordered set, satisfying: 1. Monotonicity: if c⪯c′, then Val(c)≤Val(c′); 2. Grounding: Val(Z) is least in V; 3. Compatibility: if Con(c, c′) holds, then the corresponding valuations are jointly coherent in the sense required by the axioms. Measurement is interpreted as a partial function Meas :C⇀V, for some value space V, subject to: 1. Non-triviality: Meas is not everywhere undefined; 2. Zero-state constraint: if Meas(c) is defined, then Con(Z, c); 3. Stability under extension: if (cn) is an increasing chain in Cwith limit c, and all Meas(cn) are defined and stable, then Meas(c) is defined. Unbounded extensibility is modelled by chains c0⪯c1⪯ · · · in Cthat admit coherence-preserving limits. A semantic universe satisfies a semantic infinity condition if such chains exist and remain coherent with the zero-state boundary Z. This provides the semantic counterpart of the Infinity Condition used in the Infinity–Measurement Boundary Theorem. A semantic universe U= (D,C,Con,Z,Val,Meas) satisfies ZSA, written U |= ZSA, if: 1. all coherence and admissibility axioms of ZSA hold relative to Con; 2. the valuation axioms hold relative to Val, with Zinterpreting the zero-state boundary; 3. the measurement axioms hold relative to Meas, with stability under the specified notion of extensibility. This sets the stage for the canonical and non-trivial models constructed in Section 9. 8 6 The Infinity–Measurement Boundary In this section we formalise the structural tension between unbounded extensibility and internally governed measurement. The aim is to show that any coherent system realising both properties must include a minimal boundary constraint. This result provides the logical motivation for the Zero-State Axioms introduced in Section 7. We recall the definitions from Section 5: the Infinity Condition Inf(S) and the Measurement Condition Meas(S). 6.1 Boundary Phenomenon Unbounded extensibility threatens the stability of partial measurement unless governed by a primitive admissibility boundary. The following theorem states this precisely. Theorem 6.1 (Infinity–Measurement Boundary Theorem).Let Sbe a coherent system with semantic universe (DS,CS,ConS, ZS,ValS,MeasS). Suppose: 1. Ssatisfies the Infinity Condition: Inf(S); 2. Ssatisfies the Measurement Condition: Meas(S). Then: 1. There exists a minimal boundary constraint Zsuch that Scan be coherently realised only as an extension of Z; 2. Any coherent system S′extending such a boundary Zcan jointly realise Inf(S′)and Meas(S′)without circularity or infinite regress. In particular, the coexistence of infinity and measurement forces the existence of a minimal zero-state boundary. Proof sketch. If Inf(S) holds but Meas(S) fails, unbounded structure lacks consistent determination unless a primitive boundary is imposed, contradicting internal coherence. If Meas(S) holds but Inf(S) fails, measurement stability collapses under unbounded extension unless new admissibility constraints are added. When both conditions hold, coherent realisation requires a minimal admissibility boundary governing extension, valuation grounding, and measurement stability. Such a constraint cannot be derived internally without regress; hence a primitive boundary Zmust be included. Weakening Zrevives one of the instability modes above. 6.2 Consequences Theorem 6.1 motivates the Zero-State Axioms: the axioms of Section 7axiomatise precisely the minimal boundary required for the coexistence of infinity and measurement. The structural consequences in Section 8confirm that ZSA realises the behaviour predicted by the theorem. 9 The existence of a minimal boundary Zarises from reconciling these pressures. Zfixes the admissible region within which extension and measurement can interact without regress. ZSA makes this boundary explicit and axiomatic. .2 Three Toy Scenarios We briefly sketch three schematic systems to illustrate the roles of Inf(S), Meas(S), and the boundary. (i) Finite systems with measurement but no infinity. Let Sfin have: •a finite configuration space Cfin; •a total measurement map Measfin :Cfin → Vfin; •no non-trivial chains of unbounded length. Here Meas(Sfin) holds but Inf(Sfin) fails. There is no tension: measurement can be defined by explicit case analysis, and no regress arises. The IMB Theorem does not apply; no zero-state boundary is forced. (ii) Systems with infinity but no measurement. Let Sinf have: •countably many configurations; •arbitrarily long chains under ⪯; •no measurement operator MeasSinf . Here Inf(Sinf ) holds but Meas(Sinf ) fails. Again, there is no internal conflict: the system simply never attempts to determine values. (iii) Na¨ıve combination of infinity and measurement. The critical case is when a system Sis required to satisfy both Inf(S) and Meas(S) without additional constraints. Suppose: •CSsupports arbitrarily long chains; •MeasSis defined on some base configurations and extended “as far as possible” by informal rules. Without a primitive admissibility boundary, one of two things typically occurs: 1. extending along a chain forces inconsistent value assignments, violating coherence; or 2. measurement becomes undefined on large parts of the configuration space, contradicting Meas(S). The IMB Theorem abstracts these failure modes and shows that they can only be globally avoided if there is a minimal constraint surface Zgoverning admissibility and measurement stability. ZSA gives this surface in axiom form. 16 .3 Role of Minimality The minimality clause in Definition 3.4 and in the Main Theorems isolates the exact amount of structure needed to stabilise the interaction of infinity and measurement. If one strengthens Zbeyond this minimal set, the system remains coherent but carries redundant structure; if one weakens it, the pathologies reappear. Thus the boundary is not merely a convenient parameter but a necessary structural feature of any system in which Inf(S) and Meas(S) coexist in the strong sense formalised in Section 6. .4 Relation to ZSA-Models ZSA-models (Definition 3.6) can be understood as semantic universes in which: •the boundary Zis made explicit and axiomatic; •the behaviour of Val and Meas is controlled at and above Z; •the existence and stability of chains is regulated by propagation and continuity principles. From this perspective, Theorem 6.1 explains why a boundary is forced, the Zero-State Axioms explain how this boundary is realised, and the model constructions in Section 9show that such systems are consistent and non-trivial. Appendix B: On the Non-Metaphysical Status of the Zero-State Axioms This appendix addresses a natural question that may arise when interpreting the role of the zero-state boundary: does ZSA commit the framework to a metaphysical thesis concerning the existence or nature of infinity, measurement, or boundary conditions? We clarify here that it does not. ZSA is a structural axiomatic schema, and its commitments are strictly mathematical. .5 What ZSA Does Not Assume The Zero-State Axioms do not presuppose: 1. ontological commitments concerning actual infinities; 2. epistemic or metaphysical accounts of measurement; 3. philosophical interpretations of coherence or determination; 4. any form of model-theoretic realism stronger than that used in ordinary logic. In particular, ZSA does not assert that infinity exists “in the world,” that measurement has metaphysical significance, or that the zero-state boundary corresponds to an external entity. 17 .6 What ZSA Actually Provides ZSA isolates the minimal structural constraints required for a coherent system to satisfy both the Infinity Condition Inf(S) and the Measurement Condition Meas(S). These principles are defined formally in Section 3. The Zero-State Axioms simply provide an explicit axiomatisation of the minimal boundary required to prevent regress and inconsistency when these principles interact. .7 Why the Boundary Is Not a Metaphysical Object Although the boundary terminology might suggest otherwise, the zero-state boundary is not interpreted ontologically. Formally, Zfunctions as: •a minimal admissibility constraint on configurations; •a regress-blocking condition for chains; •a stability principle for valuation and measurement. Its status is structural, comparable to axioms of extensionality, admissibility, or regularity in other formal systems. .8 Why No Circularity or Philosophical Load Is Introduced ZSA does not attempt to define the nature of infinity or measurement. Instead, it provides the structurally weakest axioms under which Inf(S) and Meas(S) can coexist. The philosophical interpretation of either condition is left open, as in standard model-theoretic practice. .9 Relation to the IMB Theorem Theorem 6.1 shows that a minimal boundary must exist whenever infinity and measurement coexist in a coherent system. This is a logical implication, not a metaphysical claim. ZSA gives an explicit axiomatisation of such a boundary, without committing to any philosophical interpretation of the underlying conditions. .10 Conclusion ZSA is a structural axiomatisation, not a metaphysical thesis. Its role is to capture the minimal boundary conditions necessary for the coexistence of unbounded extensibility and internally governed measurement. The framework operates entirely within standard mathematical logic and does not introduce metaphysical commitments. 18