A Set-Theoretic Derivation of Physobser Axiom Irreducibility, Minimality, and Essential Uniqueness of the Physical–Observation Dual-Axis Structure

1. Introduction

The ordinary observation law $y=h\left(x\right)$ is exact when the conditions that form the record remain fixed. The difficulty begins when such a condition becomes a variable of the declared task. At one and the same physical state, a change of reference phase, sensor configuration, array weight, protocol state, or experimental setting may index a different certified record. Writing all variables inside a larger coordinate is always possible; proving that one typed role can be deleted without loss is not.

This paper isolates that representational question at the set-theoretic foundation of Physobser Theory, before probability, geometry, or dynamics is introduced. Its primitive datum is not an unlabelled set of outcomes but an operationally typed table: one label records the object or preparation role, and another records the operation or condition through which the observation is formed. The typing is empirical metadata supplied by the task. Set theory cannot infer it from outcome values alone; it determines what structural consequences follow once the typing and the task boundary are fixed.

The argument lies adjacent to several established theories without duplicating their aims. Measurement theory studies representations of antecedently specified empirical relations [13]. Automata and realization theory identify states through complete external behavior [14,15,16,17,18]. Experimental design and state-dependent channels make operating conditions explicit [9,10,11,12]. Identifiability and observability theory ask when states can be distinguished from data [23,24,25,26,27]. Here the narrower problem is to determine the minimal typed state responsibility required by a complete observation table, and to separate that responsibility from the physical-state identity that a prior physical theory may require the model to preserve.

The derivation has two layers. First, row and column behaviors produce a canonical typed reduction. Its first coordinate is merely an object-side behavioral state unless an external physical semantics has been supplied. Second, a compatible physical interface $r:\mathfrak{P}\twoheadrightarrow {\mathcal{X}}_{\mathrm{phys}}$ anchors the first coordinate in physical identity. The behavior-minimal second coordinate then carries the state of observation formation. This anchored structure is the physical–observation dual axis named by Physobser Axiom.

The main results are as follows. A fixed-label output difference obstructs every first-state-only exact representation. The row–column quotient is terminal among surjective typed exact factorizations and invariant under typed isomorphisms and redundant exact covers. A compatible physical interface descends the primitive table to ${\mathcal{X}}_{\mathrm{phys}}$ and factors canonically onto the object-side behavioral quotient, thereby separating physical identity from observational resolution. The formation quotient is terminal among physical-state-preserving exact realizations and is unique up to a unique behavior-preserving isomorphism when redundant states are removed. Finally, the same set-theoretic interface yields exact criteria for conditional, label-free, and joint identifiability, and identifies the fiber product over the physical projection as the natural domain for later same-source transport questions.

All statements are relative to a declared task, output resolution, test family, and exact equality criterion. The proofs establish a canonical interface within that mathematical class; they do not infer physical ontology from observational equivalence, nor do they prove that every physical experiment admits the required closed-world presentation. Probability, topology, smoothness, group actions, connections, and transport laws enter only after this set-theoretic foundation has been fixed.

2. Primitive Observation Records and the Limits of Single-Axis Representation

2.1. From Formed Information Back to the Conditions of Observation Formation

Classical information theory, theoretical statistics, and statistical decision theory generally begin with random variables, probability laws, channels, or statistical experiments that have already been formed. Shannon studied the limits of reliable communication under specified source and channel statistics [1]; Fisher treated likelihood, sufficiency, and estimation accuracy within an explicit parametric model [2]; Wald organized statistical inference through decision rules and risk [3]; and Blackwell compared the informational value of statistical experiments across all decision problems [4,5]. What these theories share is not a neglect of observation, but the convention that the observed result and its generating law are already given when the theory begins.

The standard statistical representation is

$$P_{Y\mid X}(\mathrm{d}y\mid x),$$

(2.1)

while the deterministic case is written as

$$y=h\left(x\right),h:\mathcal{X}⟶\mathcal{Y}.$$

(2.2)

Here, x denotes the state of the object under study, and y is the result entering recording, estimation, testing, coding, or decision. As long as the measurement rule, reference conditions, sampling procedure, and processing configuration remain fixed over the domain of inquiry, they may be absorbed into the definition of the map h or the probability kernel $P_{Y\mid X}$. Modern information theory starts from a given joint distribution and channel law [6], while the theory of statistical experiments begins with a parameter-indexed family of probability distributions and studies sufficiency, comparison, and reduction [7,8]. Within their respective domains of validity, these starting points are complete and effective.

We ask a preceding question. Suppose that conditions previously held fixed inside the observation law are included among the task variables, and distinct condition labels index distinct certified entries. What state structure must then be retained in a complete account of the observational facts? This question does not reject (2.1) or (2.2). It asks only when further compression remains legitimate and when a formation condition with an observable distinction must be made explicit again.

2.2. Exact Reduction Under Fixed Formation Conditions

Let $\mathcal{X}$, $\mathfrak{A}$, and $\mathcal{Y}$ be sets, let $a_0\in \mathfrak{A}$, and let a deterministic observation process be specified by a typed map

$$F_0:\mathcal{X}\times \mathfrak{A}⟶\mathcal{Y}.$$

If the domain of inquiry is restricted throughout to $a=a_0$, define

$$h_{a_0}\left(x\right):=F_0(x,a_0).$$

(2.3)

This gives the ordinary map $h_{a_0}:\mathcal{X}\to \mathcal{Y}$. If measurable structures and a Markov kernel are supplied in a later statistical model, the analogous fixed-condition notation is

$$P_{Y\mid X}^{\left(a_0\right)}(\mathrm{d}y\mid x).$$

(2.4)

Proposition 2.1

(Exact slice at a fixed formation condition). Let $F_0:\mathcal{X}\times \mathfrak{A}\to \mathcal{Y}$ and $a_0\in \mathfrak{A}$. With ${\iota }_{a_0}:\mathcal{X}\to \mathcal{X}\times \mathfrak{A}$ defined by ${\iota }_{a_0}\left(x\right)=(x,a_0)$, the fixed-condition map $h_{a_0}:\mathcal{X}\to \mathcal{Y}$ is exactly the restriction

$$h_{a_0}=F_0\circ {\iota }_{a_0}.$$

(2.5)

Consequently, $h_{a_0}\left(x\right)=F_0(x,a_0)$ for every $x\in \mathcal{X}$.

Proof. 

For every $x\in \mathcal{X}$,

$$(F_0\circ {\iota }_{a_0})\left(x\right)=F_0(x,a_0)=h_{a_0}\left(x\right),$$

which proves equality of the two typed maps. □

This gives the first boundary for what follows. The fact that a physical system contains many components does not require a mathematical model to record each of them explicitly. A separately typed state position is needed only for a condition that varies within the declared problem and indexes a distinguishable observational fact. A fixed observation condition is not a failure of the broader theory; it is an exact slice of the joint structure developed below. The displayed stochastic notation is only an explanatory analogue here; a formal stochastic theorem would additionally specify measurable spaces and a kernel.

2.3. Primitive Facts Under Variable Formation Conditions

Let p denote a primitive object label and let $a_0,a_1$ denote two admissible formation operations. If the same object label satisfies

$${\mu }_0(p,a_0)\ne {\mu }_0(p,a_1),$$

(2.6)

then the two entries in (2.6) share the declared object/preparation label p; hence their inequality is not indexed by a difference in that label. Continuing to conceal all formation conditions inside a single h would force the same argument to take two different values.

Allowing formation conditions to enter a model is not a new mathematical practice. Experimental design treats experimental conditions as selectable objects and studies their effects on estimation and discrimination [9,10]; state-dependent channel theory permits the channel law to vary with a state [11,12]; and the theory of statistical experiments compares directly the decision value of different data-generating mechanisms [4,5,7]. This foundational section invokes none of their probabilistic, decision-theoretic, or coding structures. It retains only the elementary fact that one declared object/preparation label may be paired with different record values under different formation-operation labels.

2.4. An Empirically Typed Closed-World Table

Definition 2.2

(Empirically typed closed-world presentation). Fix a declared observational task. An empirically typed closed-world presentation of that task is a tuple

$$\mathfrak{E}=(\mathfrak{P},\mathfrak{A},\mathcal{Y},\perp ,{\mu }_0)$$

with the following properties.

1.

$\mathfrak{P}$ and $\mathfrak{A}$ are nonempty typed sets. Equality in $\mathfrak{P}$ records the declared object/preparation-label identity that the task permits one to hold fixed; it does not by itself assert equality in an external physical state space. Elements of $\mathfrak{A}$ label record-forming operations or conditions. This role assignment, and any physical anchoring later attached to it, is supplied by the operational specification or an antecedent theory and is not inferred from an unlabelled collection of outcomes.

2.

$\mathcal{Y}$ is a common set of actual record values, $\perp \notin \mathcal{Y}$, and

$${\mathcal{Y}}_{\perp }:=\mathcal{Y}\bigsqcup \{\perp \}.$$

(2.7)

3.

The total map

$${\mu }_0:\mathfrak{P}\times \mathfrak{A}⟶{\mathcal{Y}}_{\perp }$$

(2.8)

assigns every pair in the declared comparison universe $\mathfrak{P}\times \mathfrak{A}$ either its certified record value or certified nonimplementability. Formal cross-pairs that cannot be realized receive ⊥; none are left unspecified. If the status of any pair is unknown, the data are open and Definition 2.2 does not yet apply. The symbol ⊥ never means unknown, unmeasured, or missing data.

4.

The presentation is empirically adequate when it reproduces every certified outcome and infeasibility statement in the declared task while preserving the object/preparation and record-forming roles used to certify them.

The word closed-world is therefore relative to the declared task: its comparison universe is the typed product $\mathfrak{P}\times \mathfrak{A}$, and it asserts completeness on that product, not completeness of all physically conceivable observations.

A deterministic primitive observation record is consequently written as

$${\mu }_0:\mathfrak{P}\times \mathfrak{A}⟶{\mathcal{Y}}_{\perp },(p,a)⟼{\mu }_0(p,a).$$

(2.9)

Expression (2.9) retains only the two typed labels and the certified table entry. It requires no linear, topological, smooth, group, or probabilistic structure on $\mathfrak{P}$, $\mathfrak{A}$, or $\mathcal{Y}$, and it does not presume that the primitive labels are already irreducible theoretical states.

Definition 2.3

(Certified fixed-label co-implementability and record-forming non-triviality). Distinct $a_0,a_1\in \mathfrak{A}$ arecertified co-implementable at the fixed first label $p\in \mathfrak{P}$when the protocol independently certifies both pairs $(p,a_0)$ and $(p,a_1)$ as implementable under the same declared object/preparation identity. Equality of p is task metadata supplied by that protocol; it is not inferred from equality or inequality of table values. When an external physical-state interface is supplied, the certification may additionally assert equality of the corresponding physical state. The presentation isrecord-forming non-trivial at pwhen, more generally,

$${\mu }_0(p,a_0)\ne {\mu }_0(p,a_1)$$

for some $a_0,a_1\in \mathfrak{A}$; the witness isstrongwhen both entries lie in $\mathcal{Y}$ andweakwhen at least one entry is ⊥. A strong witness records co-implementability and an indexed actual-record difference at the declared fixed label; it does not by itself establish an intervention or a causal effect. A weak witness establishes a distinction in the complete law, including feasibility, but not a difference in the restricted actual-output law.

This fixed-label certification is external protocol metadata about the typing and feasibility of the comparison. It is neither statistical independence nor a causal conclusion inferred from output values, and it does not claim that every element of $\mathfrak{P}$ can be combined with every element of $\mathfrak{A}$. In particular, the admissible joint domain derived below may be a proper subset of $\mathcal{X}\times \mathcal{S}$. Set theory does not manufacture the typing or its physical anchor; it determines the exact representational consequences conditional on an empirically adequate typing. An unlabelled outcome collection cannot determine that typing: the same two values may be represented either by a unary map on two object labels or by two formation operations at one fixed object label.

Definition 2.4

(Closed completion of partial task data). Let $D\subseteq \mathfrak{P}\times \mathfrak{A}$ and let $m:D\to {\mathcal{Y}}_{\perp }$ contain the entries already certified for fixed typed sets $\mathfrak{P}$, $\mathfrak{A}$, and ${\mathcal{Y}}_{\perp }$. Aclosed completionof m is a total map

$$\overline{m}:\mathfrak{P}\times \mathfrak{A}⟶{\mathcal{Y}}_{\perp }$$

such that $\overline{m}{|}_D=m$. Thus a completion preserves every certified entry and fills only the previously unresolved pairs, without changing the label types, the output resolution, or the meaning of the fresh symbol ⊥.

Counterexample 2.5

(An open table does not determine the task-global quotient). Let $\mathfrak{P}=\{p_0,p_1\}$, $\mathfrak{A}=\{a_0,a_1\}$, and $\mathcal{Y}=\{0,1\}$. Suppose a partial record $m:D\to {\mathcal{Y}}_{\perp }$ has three certified entries

$$m(p_0,a_0)=m(p_0,a_1)=m(p_1,a_0)=0,$$

while the status of $(p_1,a_1)$ is unknown. One closed completion assigns the fourth entry 0, making the two columns behaviorally equivalent. Another assigns it 1, making them inequivalent. The partial data therefore do not determine a unique task-global formation-side quotient. Replacing the unknown entry by ⊥ would add a certification of nonimplementability; it is a legitimate completion only if that added certification is empirically warranted.

Definition 2.6

(Implementability and Two Types of Formation Difference). For $(p,a)\in \mathfrak{P}\times \mathfrak{A}$, the object–operation pair is calledimplementableif ${\mu }_0(p,a)\ne \perp$. With p fixed:

1.

if ${\mu }_0(p,a_0)\ne {\mu }_0(p,a_1)$, the operations $a_0,a_1$ are distinguishable in complete-table behavior; the difference may arise from recorded value or from implementability;

2.

if both entries lie in $\mathcal{Y}$ and ${\mu }_0(p,a_0)\ne {\mu }_0(p,a_1)$, the operations have animplementable-output difference.

The symbol ⊥ is only an implementability marker; it is not treated as an ordinary observed value. Complete-table behavior is used to construct behavioral equivalence and minimal state. After the totalized table is restricted to the admissible domain $F:\mathcal{B}\to \mathcal{Y}$, however, whether the output law itself can be represented by the first state alone must be decided by the second notion: a difference between two implementable outputs in the same first-state fiber.

Classical measurement theory emphasizes that numerical representations must faithfully preserve antecedently specified empirical relations and then asks about the existence and uniqueness of such representations. Krantz, Luce, Suppes, and Tversky gave a systematic account of this program [13]. We follow the same order: first determine which observational facts must be preserved, then ask how states and maps can represent them without redundancy.

In the stochastic case, each implementable table entry may be replaced by a conditional distribution

$${\mathsf{W}}_0(\mathrm{d}y\mid p,a).$$

(2.10)

If implementability itself depends on $(p,a)$, one may either retain ⊥ as a deterministic atom in an extended outcome space or record the admissible domain separately. In such an application, the values compared by behavioral equivalence are the specified conditional laws or complete task-level statistical behaviors; inequality of two uncontrolled realized samples is not by itself a behavioral witness. This foundational section uses only the deterministic totalized table (2.8) to establish the set-theoretic skeleton. Expression (2.10) shows that the double indexing is not a consequence of determinism; the full statistical lift belongs to a later layer.

2.5. Impossibility of Single-Axis Representation

Theorem 2.7

(Impossibility theorem for a single-axis representation preserving the declared first-state label). Let ${\mu }_0:\mathfrak{P}\times \mathfrak{A}\to {\mathcal{Y}}_{\perp }$. If there exist $p_0\in \mathfrak{P}$ and $a_0,a_1\in \mathfrak{A}$ such that

$${\mu }_0(p_0,a_0)\ne {\mu }_0(p_0,a_1),$$

(2.11)

then no single function $h:\mathfrak{P}\to {\mathcal{Y}}_{\perp }$ can represent the complete table while retaining p as its declared first-state argument, that is, while satisfying

$$h\left(p\right)={\mu }_0(p,a),\forall (p,a)\in \mathfrak{P}\times \mathfrak{A}.$$

(2.12)

Proof. 

If such an h existed, then at the same $p_0$ one would have simultaneously

$$h\left(p_0\right)={\mu }_0(p_0,a_0),h\left(p_0\right)={\mu }_0(p_0,a_1),$$

and hence ${\mu }_0(p_0,a_0)={\mu }_0(p_0,a_1)$, contradicting (2.11). □

Example 2.8

(Minimal two-column record). Suppose there is only one object label $p_0$ and the record table is

Any function depending only on $p_0$ can return only one value and therefore cannot preserve both columns. The numerical values 1 and $-1$ carry no additional physical meaning here; the table displays only the minimal logical form in which a single-axis representation fails.

Counterexample 2.9

(Boundary at which the formation operation can be eliminated). Assume $\mathfrak{A}\ne ⌀$. If, for every $p\in \mathfrak{P}$ and all $a,a^{\prime }\in \mathfrak{A}$,

$${\mu }_0(p,a)={\mu }_0(p,a^{\prime }),$$

then all operation columns are identical. Choose any $a_0$ and define $h\left(p\right)={\mu }_0(p,a_0)$; the entire record is then represented without loss. A second variable position is not unconditionally necessary. Its necessity arises precisely when the formation-operation side exhibits nontrivial behavioral differences.

Corollary 2.10

(Implementable-output difference rules out a first-state-only output map). If there exist $p_0,a_0,a_1$ such that ${\mu }_0(p_0,a_i)\in \mathcal{Y}$ for $i=0,1$ and

$${\mu }_0(p_0,a_0)\ne {\mu }_0(p_0,a_1),$$

then even after restricting attention to implementable records, no single-valued output function independent of the formation operation can represent both entries while preserving the declared first-state identity of $p_0$.

The theorem does not require every observation to have two varying coordinates. It says that once a formation-side difference exists at the same empirically anchored first-state label, every exact representation preserving that typed role must retain the witnessed distinction. In the external-interface branch that label may additionally carry certified physical-state identity. At this stage the second position is still occupied by the primitive operation label a. Whether that label contains redundancy, how it should be reduced, and what theoretical name the reduced state should receive have not yet been decided.

2.6. Coordinate Concatenation Is Not Structural Reduction

One may of course encode the pair of labels into a single enlarged variable

$$z=(p,a)$$

and rewrite (2.8) as a unary function $G\left(z\right)$. This alters only the notation, not the problem. As long as the model must still express both “hold p fixed while varying a” and “hold a fixed while varying p,” the enlarged state space must retain projections that recover the two roles.

Two operations must therefore be distinguished:

1.

syntactic unarization: concatenating several coordinates into one variable;

2.

structural reduction: proving that the observation table factors through a representation that omits one typed state role without loss.

The former is always possible; the latter requires a factorization or behavioral-equivalence theorem. If z is simply renamed the “physical state” while the projection to the original first-state or object-label identity is discarded, then fixed-first-state comparisons across formation conditions have also been discarded. It is a fixed-physical-state question only when the external physical-state interface supplies that semantics. Deleting the projection does not prove the second state redundant; it changes the problem.

2.7. Domain of Validity and Axiomatic Boundary

This part of the analysis uses only sets, maps, and proof by contradiction. The following boundaries must therefore be observed.

First, in (2.8), a is only a primitive formation-operation label; it cannot yet be called an observer state. Distinct operation labels may induce exactly the same complete behavior, and this label redundancy must first be removed.

Second, the argument proves the necessity of retaining a witnessed formation-side distinction while the first-state role is preserved; it does not yet construct or name a task-global observer-state space. A single-axis model remains exact when the formation condition is fixed, uniquely determined by the object state, or behaviorally invisible at the output.

Third, no topology, manifold, metric, group action, connection, or probabilistic experiment has been introduced. The “double indexing” used here signifies only two classes of primitive labels; it is not yet the “observational dual axis” obtained later through behavioral reduction and identification of state roles.

Fourth, the present argument does not postulate Physobser Axiom. Axiomatization must follow behavioral reduction, minimal realization, and identification of the second state’s role. Otherwise the observer state that ought to be derived would be inserted into the model in advance.

2.8. Interim Conclusion

When the formation condition is fixed, the ordinary map $y=h\left(x\right)$ is the exact deterministic slice proved above; if a later stochastic model supplies measurable spaces and a Markov kernel, its analogous fixed-condition object is $P_{Y\mid X}^{\left(a_0\right)}$. When two observation-forming operations are certified at the same first-state label and their table entries differ, an exact presentation that keeps both primitive roles explicit is

$${\mu }_0:\mathfrak{P}\times \mathfrak{A}⟶{\mathcal{Y}}_{\perp }.$$

(2.13)

If two operations have unequal entries under the same p, no single-axis representation preserving the declared first-state label can be complete. Concatenating $(p,a)$ into one enlarged variable merely hides the coordinate form of the two roles; it does not prove that either structure has disappeared.

The next stage begins with complete row and column behaviors. It removes redundancies separately from the object labels and the formation-operation labels, constructs the two effective state spaces and their admissible joint domain, and then tests the canonical status of the second state through the family of observation maps and behavioral minimal realization. Only after these steps have been completed can the theoretical identity of the second state be fixed.

3. Two-Sided Behavioral Reduction and Behavior-Minimal State Realization

3.1. Complete Row and Column Behaviors and Label Redundancy

The preceding analysis established the following fact: unequal entries under the same declared object/preparation label cannot be represented by a unary map that preserves that label as its argument. The primitive operation labels $a\in \mathfrak{A}$ may nevertheless still contain redundancy. Two labels being distinct in name does not imply that they generate different observational behavior under every object condition; the same issue arises on the object-label side.

For each $a\in \mathfrak{A}$, define its column behavior by

$$h_a:\mathfrak{P}⟶{\mathcal{Y}}_{\perp },h_a\left(p\right):={\mu }_0(p,a),$$

(3.1)

and denote the family of all column behaviors by

$${\mathcal{H}}_0:=\{h_a:a\in \mathfrak{A}\}\subseteq {\mathcal{Y}}_{\perp }^{\mathfrak{P}}.$$

(3.2)

If $h_a=h_{a^{\prime }}$, then a and $a^{\prime }$ have no distinguishable formation behavior on the present set of admissible objects. Conversely, whenever $h_a\ne h_{a^{\prime }}$, at least one object label distinguishes them.

On the object side, define the row behavior analogously by

$$k_p:\mathfrak{A}⟶{\mathcal{Y}}_{\perp },k_p\left(a\right):={\mu }_0(p,a).$$

(3.3)

If $k_p=k_{p^{\prime }}$, then no record available under the present set of admissible formation operations can distinguish p from $p^{\prime }$. Primitive object and operation labels therefore cannot be assumed to be irredundant theoretical states merely because their names differ. At the present observational interface, effective state identity is determined by complete behavior rather than by label names.

The principle of identifying an internal state by its external behavior has a long history in automata theory and realization theory. Moore’s experiments on sequential machines, Myhill–Nerode-type behavioral equivalence, and Kalman’s irreducible realizations all express, in different settings, the same underlying point: labels indistinguishable by complete external behavior are identified when the declared state notion is behavioral [14,15,16,17,18]. Only the set-theoretic core of that principle is used here; no temporal evolution, linear structure, finite dimensionality, or controllability is assumed.

3.2. Two-Sided Behavioral Quotients

Define on $\mathfrak{P}$ the relation

$$p{\sim }_X{p}^{\prime }⟺{\mu }_0(p,a)={\mu }_0(p^{\prime },a),\forall a\in \mathfrak{A},$$

(3.4)

and on $\mathfrak{A}$ the relation

$$a{\sim }_S{a}^{\prime }⟺{\mu }_0(p,a)={\mu }_0(p,a^{\prime }),\forall p\in \mathfrak{P}.$$

(3.5)

Both are equivalence relations. Denote the quotient maps and quotient sets by

$$q_X:\mathfrak{P}\twoheadrightarrow \mathcal{X}:=\mathfrak{P}/{\sim }_X,q_S:\mathfrak{A}\twoheadrightarrow \mathcal{S}:=\mathfrak{A}/{\sim }_S,$$

(3.6)

and write

$$x=q_X\left(p\right)={\left[p\right]}_X,s=q_S\left(a\right)={\left[a\right]}_S.$$

(3.7)

Here $\mathcal{X}$ is the effective state set of complete row behaviors on the object side, whereas $\mathcal{S}$ is the effective state set of complete column behaviors on the formation-operation side. Both are constructed relative to the given observation table and the admissible label sets; neither is an ontological classification prescribed independently of the observation relation. In particular, s is still called only the second effective state in this section. It has not yet acquired the theoretical name “observer state.”

Proposition 3.1

(Well-Definedness of the Two-Sided Behavioral Quotient). The relations (3.4) and (3.5) induce a unique map from the primitive observation table,

$$\tilde{F}:\mathcal{X}\times \mathcal{S}⟶{\mathcal{Y}}_{\perp },\tilde{F}({\left[p\right]}_X,{\left[a\right]}_S):={\mu }_0(p,a).$$

(3.8)

This map is independent of the chosen representatives and is characterized by

$${\mu }_0=\tilde{F}\circ (q_X\times q_S).$$

(3.9)

Proof. 

If $p{\sim }_X{p}^{\prime }$ and $a{\sim }_S{a}^{\prime }$, then

$${\mu }_0(p,a)={\mu }_0(p^{\prime },a)={\mu }_0(p^{\prime },a^{\prime }).$$

The first equality follows from behavioral equivalence on the object side, and the second from behavioral equivalence on the operation side. Thus (3.8) is well defined. The product $q_X\times q_S$ is surjective, so (3.9) fixes every value of the descended map; hence the induced map is unique. □

The quotient removes static label redundancy. It already guarantees that distinct elements of $\mathcal{S}$ correspond to distinct complete column behaviors, but it does not yet compare arbitrary state parameterizations or prove that all nonredundant realizations differ only by a behavior-preserving relabeling. Those questions require a general theory of state realization.

3.3. Joint Universal Property of the Two-Sided Behavioral Quotient

Definition 3.2

(Surjective Typed Exact Factorization). A surjective typed exact factorization of the primitive table ${\mu }_0:\mathfrak{P}\times \mathfrak{A}\to {\mathcal{Y}}_{\perp }$ consists of sets $U,V$, surjections

$$u:\mathfrak{P}\twoheadrightarrow U,v:\mathfrak{A}\twoheadrightarrow V,$$

(3.10)

and a map $G:U\times V\to {\mathcal{Y}}_{\perp }$ such that

$${\mu }_0=G\circ (u\times v).$$

(3.11)

Here typed means that the two primitive roles are represented by separate maps u and v, rather than by an arbitrary map on the joint-event set. Exactness is required on the complete totalized table, and surjectivity excludes states of U or V unused by the primitive presentation.

A morphism from $(U,V,u,v,G)$ to another such factorization $(U^{\prime },V^{\prime },u^{\prime },v^{\prime },G^{\prime })$ is a pair of maps $\alpha :U\to U^{\prime }$ and $\beta :V\to V^{\prime }$ satisfying

$$u^{\prime }=\alpha \circ u,v^{\prime }=\beta \circ v,G=G^{\prime }\circ (\alpha \times \beta ).$$

Thus a morphism preserves both primitive anchors and the complete law.

The factorization is biextensional when G has neither duplicate rows nor duplicate columns:

$$\begin{array}{cc}\hfill \left[G(\xi ,\eta )=G({\xi }^{\prime },\eta )\mathit{for}\mathit{every}\eta \in V\right]& ⟹\xi ={\xi }^{\prime },\hfill \end{array}$$

(3.12)

$$\begin{array}{cc}\hfill \left[G(\xi ,\eta )=G(\xi ,{\eta }^{\prime })\mathit{for}\mathit{every}\xi \in U\right]& ⟹\eta ={\eta }^{\prime }.\hfill \end{array}$$

(3.13)

Theorem 3.3

(Joint Universal Property of the Two-Sided Quotient). Let $(U,V,u,v,G)$ be a surjective typed exact factorization of ${\mu }_0$. Then there exist unique maps

$$\alpha :U⟶\mathcal{X},\beta :V⟶\mathcal{S}$$

(3.14)

such that

$$q_X=\alpha \circ u,q_S=\beta \circ v.$$

(3.15)

Both maps are surjective, and the intermediate law necessarily satisfies

$$G=\tilde{F}\circ (\alpha \times \beta ).$$

(3.16)

Moreover, their fibers are exactly the row and column redundancies of G:

$$\begin{array}{cc}\hfill \alpha \left(\xi \right)=\alpha \left({\xi }^{\prime }\right)& ⟺G(\xi ,\eta )=G({\xi }^{\prime },\eta )\mathit{for}\mathit{every}\eta \in V,\hfill \end{array}$$

(3.17)

$$\begin{array}{cc}\hfill \beta \left(\eta \right)=\beta \left({\eta }^{\prime }\right)& ⟺G(\xi ,\eta )=G(\xi ,{\eta }^{\prime })\mathit{for}\mathit{every}\xi \in U.\hfill \end{array}$$

(3.18)

Consequently, α is bijective precisely when G has distinct rows, β is bijective precisely when G has distinct columns, and both are bijective precisely when G is biextensional. Equivalently, the canonical two-sided quotient is terminal in the category of surjective typed exact factorizations with law-preserving typed morphisms.

Proof. 

If $u\left(p\right)=u\left(p^{\prime }\right)$, then for every $a\in \mathfrak{A}$ exactness gives

$${\mu }_0(p,a)=G\left(u\left(p\right),v\left(a\right)\right)=G\left(u\left(p^{\prime }\right),v\left(a\right)\right)={\mu }_0(p^{\prime },a).$$

Thus $p{\sim }_X{p}^{\prime }$ and $q_X\left(p\right)=q_X\left(p^{\prime }\right)$. The formula $\alpha \left(u\left(p\right)\right):=q_X\left(p\right)$ is therefore well defined; surjectivity of u makes it unique. Interchanging the two roles gives the unique $\beta$ satisfying $q_S=\beta \circ v$. Surjectivity of $q_X$ and $q_S$ makes $\alpha$ and $\beta$ surjective.

For every $(p,a)\in \mathfrak{P}\times \mathfrak{A}$,

$$G\left(u\left(p\right),v\left(a\right)\right)={\mu }_0(p,a)=\tilde{F}\left(q_X\left(p\right),q_S\left(a\right)\right)=\tilde{F}\left(\alpha \left(u\right(p\left)\right),\beta \left(v\right(a\left)\right)\right).$$

The product $u\times v$ is surjective, so cancellation proves (3.16).

For the first fiber identity, choose $p,p^{\prime }\in \mathfrak{P}$ with $u\left(p\right)=\xi$ and $u\left(p^{\prime }\right)={\xi }^{\prime }$. If $\alpha \left(\xi \right)=\alpha \left({\xi }^{\prime }\right)$, then $q_X\left(p\right)=q_X\left(p^{\prime }\right)$, so the primitive rows agree. Given $\eta \in V$, choose $a\in \mathfrak{A}$ with $v\left(a\right)=\eta$; exactness gives $G(\xi ,\eta )={\mu }_0(p,a)={\mu }_0(p^{\prime },a)=G({\xi }^{\prime },\eta )$. Conversely, equality of the two G-rows implies equality of the primitive rows after evaluation at every $v\left(a\right)$, hence $\alpha \left(\xi \right)=\alpha \left({\xi }^{\prime }\right)$. The second fiber identity is symmetric. The final assertions follow because $\alpha$ and $\beta$ are already surjective. □

Corollary 3.4

(Unique Isomorphism of Biextensional Exact Factorizations). For $i\in \{1,2\}$, let

$$u_i:\mathfrak{P}\twoheadrightarrow U_i,v_i:\mathfrak{A}\twoheadrightarrow V_i,G_i:U_i\times V_i\to {\mathcal{Y}}_{\perp }$$

be biextensional surjective typed exact factorizations of the same primitive table. Then there exist unique bijections $\phi :U_1\to U_2$ and $\psi :V_1\to V_2$ such that

$$u_2=\phi \circ u_1,v_2=\psi \circ v_1,G_2\circ (\phi \times \psi )=G_1.$$

(3.19)

Thus every biextensional surjective typed exact factorization is uniquely isomorphic, with its primitive anchors preserved, to the canonical two-sided behavioral quotient.

Proof. 

For each i, Theorem 3.3 supplies bijections ${\alpha }_i:U_i\to \mathcal{X}$ and ${\beta }_i:V_i\to \mathcal{S}$. Define

$$\phi :={\alpha }_2^{-1}\circ {\alpha }_1,\psi :={\beta }_2^{-1}\circ {\beta }_1.$$

The quotient triangles give the two anchor identities, and (3.16) gives the law identity. Surjectivity of $u_1$ and $v_1$ makes any maps satisfying the anchor identities unique. □

Counterexample 3.5

(Surjectivity Is Essential). Let $\mathfrak{P}=\mathfrak{A}=\{*\}$, $\mathcal{Y}=\{0,1\}$, and ${\mu }_0(*,*)=0$. The canonical quotient sets are singletons and the descended law has value 0. Take

$$U=\{u_0,u_1\},V=\left\{v_0\right\},u(*)=u_0,v(*)=v_0,$$

and set $G(u_0,v_0)=0$ and $G(u_1,v_0)=1$. Then ${\mu }_0=G\circ (u\times v)$ on the primitive table, but u is not surjective. Any map from U to the canonical singleton collapses $u_0$ and $u_1$, so its composite with the canonical descended law cannot equal G. The unused state $u_1$ is unconstrained by the primitive data. Hence exact agreement only on the image of $u\times v$ does not imply the universal factorization.

3.4. Invariance Under Typed Changes of Presentation

Proposition 3.6

(Invariance under isomorphic empirical presentations). Let

$$\mathfrak{E}=(\mathfrak{P},\mathfrak{A},\mathcal{Y},\perp ,{\mu }_0),{\mathfrak{E}}^{\prime }=({\mathfrak{P}}^{\prime },{\mathfrak{A}}^{\prime },{\mathcal{Y}}^{\prime },{\perp }^{\prime },{\mu }_0^{\prime })$$

be empirically typed closed-world presentations. Suppose there are bijections

$$\varphi :\mathfrak{P}\to {\mathfrak{P}}^{\prime },\psi :\mathfrak{A}\to {\mathfrak{A}}^{\prime },\chi :{\mathcal{Y}}_{\perp }\to {\mathcal{Y}}_{{\perp }^{\prime }}^{\prime }$$

(3.20)

such that $\chi (\perp )={\perp }^{\prime }$, $\chi \left(\mathcal{Y}\right)={\mathcal{Y}}^{\prime }$, and

$${\mu }_0^{\prime }\left(\varphi \left(p\right),\psi \left(a\right)\right)=\chi \left({\mu }_0(p,a)\right)\mathit{for}\mathit{all}(p,a)\in \mathfrak{P}\times \mathfrak{A}.$$

(3.21)

Let

$$(q_X,q_S,\tilde{F},\mathcal{B},F)\mathrm{and}(q_X^{\prime },q_S^{\prime },{\tilde{F}}^{\prime },{\mathcal{B}}^{\prime },F^{\prime })$$

be the canonical reduced interfaces of $\mathfrak{E}$ and ${\mathfrak{E}}^{\prime }$, respectively, and write ${\chi }_{\mathcal{Y}}:\mathcal{Y}\to {\mathcal{Y}}^{\prime }$ for the corestriction of ${\chi |}_{\mathcal{Y}}$; it is a bijection because $\chi \left(\mathcal{Y}\right)={\mathcal{Y}}^{\prime }$. Then there are unique bijections

$$\overline{\varphi }:\mathcal{X}\to {\mathcal{X}}^{\prime },\overline{\psi }:\mathcal{S}\to {\mathcal{S}}^{\prime }$$

(3.22)

satisfying

$$\overline{\varphi }\circ q_X=q_X^{\prime }\circ \varphi ,\overline{\psi }\circ q_S=q_S^{\prime }\circ \psi .$$

(3.23)

They intertwine the descended laws:

$${\tilde{F}}^{\prime }\circ (\overline{\varphi }\times \overline{\psi })=\chi \circ \tilde{F}.$$

(3.24)

Moreover, $\overline{\varphi }\times \overline{\psi }$ restricts to a bijection $\mathcal{B}\to {\mathcal{B}}^{\prime }$ and

$$F^{\prime }\circ \left((\overline{\varphi }\times \overline{\psi }){|}_{\mathcal{B}}\right)={\chi }_{\mathcal{Y}}\circ F.$$

(3.25)

Thus the reduced dual-axis interface depends on the empirical table, not on the names chosen for its labels or records.

Proof. 

Because $\psi$ and $\chi$ are bijective, complete-row equality for $p,p^{\prime }\in \mathfrak{P}$ holds if and only if complete-row equality holds for $\varphi \left(p\right),\varphi \left(p^{\prime }\right)\in {\mathfrak{P}}^{\prime }$. Hence

$$\overline{\varphi }\left({\left[p\right]}_X\right):={\left[\varphi \left(p\right)\right]}_{X^{\prime }}$$

is well defined and bijective. Interchanging rows and columns gives the well-defined bijection

$$\overline{\psi }\left({\left[a\right]}_S\right):={\left[\psi \left(a\right)\right]}_{S^{\prime }}.$$

These formulas imply (3.23); surjectivity of $q_X$ and $q_S$ makes the induced maps unique. Evaluating both sides of (3.24) on quotient representatives and using (3.21) proves the equality. Finally, $\chi \left(\mathcal{Y}\right)={\mathcal{Y}}^{\prime }$ implies that a reduced pair is implementable in one presentation exactly when its image is implementable in the other, so the product bijection restricts to $\mathcal{B}\to {\mathcal{B}}^{\prime }$ and intertwines the restricted laws. □

Corollary 3.7

(Invariance under Redundant Exact Presentations). Let

$$\mathfrak{E}=(\mathfrak{P},\mathfrak{A},\mathcal{Y},\perp ,{\mu }_0),{\mathfrak{E}}^{\prime }=({\mathfrak{P}}^{\prime },{\mathfrak{A}}^{\prime },{\mathcal{Y}}^{\prime },{\perp }^{\prime },{\mu }_0^{\prime })$$

be empirically typed closed-world presentations, with canonical reduced interfaces

$$(q_X,q_S,\tilde{F},\mathcal{B},F)\mathit{and}(q_X^{\prime },q_S^{\prime },{\tilde{F}}^{\prime },{\mathcal{B}}^{\prime },F^{\prime }).$$

Suppose there are surjections

$$\varphi :\mathfrak{P}\twoheadrightarrow {\mathfrak{P}}^{\prime },\psi :\mathfrak{A}\twoheadrightarrow {\mathfrak{A}}^{\prime },$$

and a bijection $\chi :{\mathcal{Y}}_{\perp }\to {\mathcal{Y}}_{{\perp }^{\prime }}^{\prime }$ such that $\chi (\perp )={\perp }^{\prime }$, $\chi \left(\mathcal{Y}\right)={\mathcal{Y}}^{\prime }$, and

$${\mu }_0^{\prime }\left(\varphi \left(p\right),\psi \left(a\right)\right)=\chi \left({\mu }_0(p,a)\right)\mathit{for}\mathit{every}(p,a)\in \mathfrak{P}\times \mathfrak{A}.$$

(3.26)

Write ${\chi }_{\mathcal{Y}}:\mathcal{Y}\to {\mathcal{Y}}^{\prime }$ for the induced bijective corestriction of ${\chi |}_{\mathcal{Y}}$. Then there are unique bijections $\overline{\varphi }:\mathcal{X}\to {\mathcal{X}}^{\prime }$ and $\overline{\psi }:\mathcal{S}\to {\mathcal{S}}^{\prime }$ satisfying

$$\overline{\varphi }\circ q_X=q_X^{\prime }\circ \varphi ,\overline{\psi }\circ q_S=q_S^{\prime }\circ \psi .$$

(3.27)

They obey

$${\tilde{F}}^{\prime }\circ (\overline{\varphi }\times \overline{\psi })=\chi \circ \tilde{F}.$$

(3.28)

Moreover, $\overline{\varphi }\times \overline{\psi }$ restricts to a bijection $\mathcal{B}\to {\mathcal{B}}^{\prime }$ and

$$F^{\prime }\circ \left((\overline{\varphi }\times \overline{\psi }){|}_{\mathcal{B}}\right)={\chi }_{\mathcal{Y}}\circ F.$$

(3.29)

Thus inserting or deleting redundant typed labels through a surjective exact cover does not change the canonical reduced interface.

Proof. 

Define

$$u:=q_X^{\prime }\circ \varphi :\mathfrak{P}\twoheadrightarrow {\mathcal{X}}^{\prime },v:=q_S^{\prime }\circ \psi :\mathfrak{A}\twoheadrightarrow {\mathcal{S}}^{\prime },G:={\chi }^{-1}\circ {\tilde{F}}^{\prime }:{\mathcal{X}}^{\prime }\times {\mathcal{S}}^{\prime }\to {\mathcal{Y}}_{\perp }.$$

Equation (3.26) and the quotient descent for ${\mu }_0^{\prime }$ give $G(u\left(p\right),v\left(a\right))={\mu }_0(p,a)$. Hence $({\mathcal{X}}^{\prime },{\mathcal{S}}^{\prime },u,v,G)$ is a surjective typed exact factorization of ${\mu }_0$. The canonical law ${\tilde{F}}^{\prime }$ has distinct rows and columns by the defining behavioral quotient, and composition with ${\chi }^{-1}$ preserves this property. Thus G is biextensional.

Theorem 3.3 supplies bijections $\alpha :{\mathcal{X}}^{\prime }\to \mathcal{X}$ and $\beta :{\mathcal{S}}^{\prime }\to \mathcal{S}$ such that

$$q_X=\alpha \circ q_X^{\prime }\circ \varphi ,q_S=\beta \circ q_S^{\prime }\circ \psi ,{\chi }^{-1}\circ {\tilde{F}}^{\prime }=\tilde{F}\circ (\alpha \times \beta ).$$

Set $\overline{\varphi }:={\alpha }^{-1}$ and $\overline{\psi }:={\beta }^{-1}$. These identities give (3.27) and (3.28); uniqueness follows from surjectivity of $q_X$ and $q_S$.

Finally, $\mathcal{B}={\tilde{F}}^{-1}\left(\mathcal{Y}\right)$, ${\mathcal{B}}^{\prime }={\left({\tilde{F}}^{\prime }\right)}^{-1}\left({\mathcal{Y}}^{\prime }\right)$, and $\chi \left(\mathcal{Y}\right)={\mathcal{Y}}^{\prime }$. Hence (3.28) makes $\overline{\varphi }\times \overline{\psi }$ a bijection from $\mathcal{B}$ to ${\mathcal{B}}^{\prime }$. Restriction of the same equality proves (3.29). □

Proposition 3.6 covers reversible changes of empirical presentation, while Corollary 3.7 also covers insertion or deletion of purely redundant typed labels when an explicit surjective exact cover is supplied. Neither result identifies arbitrary non-isomorphic tables. Changed task domains, output resolutions, operational typings, or certified feasibility claims require an explicit typed comparison and may produce different reduced state objects. When an external physical-state interface is used, a presentation comparison must additionally commute with the supplied physical-state maps; otherwise preservation of physical identity has not been established.

Counterexample 3.8

(Untyped re-factorization can change the axes). Let $\mathfrak{P}=\mathfrak{A}=\mathcal{Y}=\{0,1\}$ and ${\mu }_0(p,a)=p\oplus a$, where ⊕ is addition modulo 2. Both behavioral quotients have two elements. Define another complete table on ${\mathfrak{P}}^{\prime }={\mathfrak{A}}^{\prime }=\{0,1\}$ by ${\mu }_0^{\prime }(p^{\prime },a^{\prime })=a^{\prime }$. The bijection of untyped joint-event sets

$$\theta :\mathfrak{P}\times \mathfrak{A}\to {\mathfrak{P}}^{\prime }\times {\mathfrak{A}}^{\prime },\theta (p,a)=(p,p\oplus a)$$

satisfies ${\mu }_0^{\prime }\circ \theta ={\mu }_0$. Nevertheless, all rows of ${\mu }_0^{\prime }$ agree, so $|{\mathcal{X}}^{\prime }|=1$, while $\left|\mathcal{X}\right|=2$; both column quotients have two elements. The map θ does not factor as a product of a bijection on object labels and a bijection on operation labels. Thus untyped joint records do not determine the two roles, and arbitrary re-factorization is not presentation invariance.

Proposition 3.9

(External Physical-State Interface). Suppose an independent physical theory supplies a state space ${\mathcal{X}}_{\mathrm{phys}}$ together with a surjection $r:\mathfrak{P}\twoheadrightarrow {\mathcal{X}}_{\mathrm{phys}}$, and suppose that the physical identity represented by r is required to be preserved. If

$$r\left(p\right)=r\left(p^{\prime }\right)⟹{\mu }_0(p,a)={\mu }_0(p^{\prime },a),\forall a\in \mathfrak{A},$$

(3.30)

then there exists a unique totalized observation table

$${\mu }_{\mathrm{phys}}:{\mathcal{X}}_{\mathrm{phys}}\times \mathfrak{A}\to {\mathcal{Y}}_{\perp }$$

(3.31)

such that ${\mu }_0={\mu }_{\mathrm{phys}}\circ (r\times {id}_{\mathfrak{A}})$. In this branch one may take $\mathcal{X}={\mathcal{X}}_{\mathrm{phys}}$ directly and minimize only the formation-operation side by complete column behavior. The subsequent results on observation-map families, behavior-minimal realization, and state non-hiding remain valid relative to this fixed first-state semantics.

Proof. 

For $x=r\left(p\right)$, define ${\mu }_{\mathrm{phys}}(x,a):={\mu }_0(p,a)$. Condition (3.30) makes the definition independent of the representative p, and surjectivity of r gives uniqueness. Moreover, for $a,a^{\prime }\in \mathfrak{A}$,

$$a{\sim }_S{a}^{\prime }⟺{\mu }_0(p,a)={\mu }_0(p,a^{\prime })\mathrm{for}\mathrm{all}p⟺{\mu }_{\mathrm{phys}}(x,a)={\mu }_{\mathrm{phys}}(x,a^{\prime })\mathrm{for}\mathrm{all}x\in {\mathcal{X}}_{\mathrm{phys}},$$

where the final equivalence again uses surjectivity of r. Thus the formation-side behavioral quotient is the same quotient whether it is computed before or after passage to ${\mathcal{X}}_{\mathrm{phys}}$, and the later realization arguments apply with ${\mathcal{X}}_{\mathrm{phys}}$ as the fixed first-state set. □

Remark 3.10

(Behavioral equivalence is not ontological physical identity). Without an independent physical theory, $\mathcal{X}=\mathfrak{P}/{\sim }_X$ should be interpreted only as the object-side behavioral state resolved by the present observation family. If an external physical state space must be preserved, the interface above should be used; observational indistinguishability alone does not license the conclusion that two primitive objects are physically identical.

Proposition 3.9

(Behavioral shadow of an anchored physical state). Assume the hypotheses of Proposition 3.9. Let

$$q_X^{\mathrm{beh}}:\mathfrak{P}\twoheadrightarrow {\mathcal{X}}_{\mathrm{beh}}:=\mathfrak{P}/{\sim }_X$$

be the row-behavior quotient of the same primitive table. Then there exists a unique surjection

$$\rho :{\mathcal{X}}_{\mathrm{phys}}\twoheadrightarrow {\mathcal{X}}_{\mathrm{beh}}$$

(3.32)

such that

$$q_X^{\mathrm{beh}}=\rho \circ r.$$

(3.33)

The map ρ is bijective if and only if the current observation family separates the physical states represented by r, namely if and only if

$$\left[{\mu }_0(p,a)={\mu }_0(p^{\prime },a)\mathrm{for}\mathrm{every}a\in \mathfrak{A}\right]⟹r\left(p\right)=r\left(p^{\prime }\right).$$

(3.34)

Thus the behavioral first-state quotient is canonically a quotient of the anchored physical state, not a substitute for it.

Proof. 

If $r\left(p\right)=r\left(p^{\prime }\right)$, compatibility of the physical interface implies equality of the complete rows, hence $q_X^{\mathrm{beh}}\left(p\right)=q_X^{\mathrm{beh}}\left(p^{\prime }\right)$. Therefore

$$\rho \left(r\left(p\right)\right):=q_X^{\mathrm{beh}}\left(p\right)$$

is well defined. Surjectivity of r makes $\rho$ unique, while surjectivity of $q_X^{\mathrm{beh}}$ makes it surjective. The map $\rho$ is injective exactly when equality of row classes forces equality of the corresponding physical states, which is condition (3.34). □

For uniform notation in the remainder of the paper, both branches are now written with a totalized map

$$\tilde{F}:\mathcal{X}\times \mathcal{S}\to {\mathcal{Y}}_{\perp }.$$

In the row-quotient branch this is the map already defined in (3.8). In the external-physical-state branch, set

$$\tilde{F}(x,{\left[a\right]}_S):={\mu }_{\mathrm{phys}}(x,a).$$

This is well defined because equality of formation-state classes is exactly equality of the induced columns on ${\mathcal{X}}_{\mathrm{phys}}$, as proved above. Thus all subsequent formulas may be read literally in either branch.

3.5. The Admissible Joint Domain and Canonical Generation of the Dual-Axis State Structure

Define the admissible joint domain by

$$\mathcal{B}:=\{(x,s)\in \mathcal{X}\times \mathcal{S}:\tilde{F}(x,s)\in \mathcal{Y}\}.$$

(3.35)

Restricting $\tilde{F}$ to $\mathcal{B}$ gives the observation law

$$F:\mathcal{B}⟶\mathcal{Y},F(x,s):=\tilde{F}(x,s).$$

(3.36)

and retains the natural projections

$${\pi }_X:\mathcal{B}\to \mathcal{X},{\pi }_S:\mathcal{B}\to \mathcal{S}.$$

Equivalently, the jointly injective coordinate map is the canonical inclusion

$$j_B=({\pi }_X,{\pi }_S):\mathcal{B}\hookrightarrow \mathcal{X}\times \mathcal{S}.$$

(3.37)

Define the active first-state set by

$${\mathcal{X}}_B:={\pi }_X\left(\mathcal{B}\right).$$

Define the active formation-state set symmetrically by

$${\mathcal{S}}_B:={\pi }_S\left(\mathcal{B}\right).$$

Whenever the codomain matters, use the canonical corestriction

$${\pi }_X^{\mathrm{act}}:\mathcal{B}\twoheadrightarrow {\mathcal{X}}_B,{\pi }_X^{\mathrm{act}}\left(b\right):={\pi }_X\left(b\right).$$

and, symmetrically,

$${\pi }_S^{\mathrm{act}}:\mathcal{B}\twoheadrightarrow {\mathcal{S}}_B,{\pi }_S^{\mathrm{act}}\left(b\right):={\pi }_S\left(b\right).$$

These corestrictions have the same fibers as the original projections and make later factorization statements type-correct.

Definition 3.12

(Typed Dual-Axis Precursor and Physobser Structure). The state structure determined by the two typed projections on the admissible joint domain $\mathcal{B}$,

$${\pi }_X:\mathcal{B}\to \mathcal{X},{\pi }_S:\mathcal{B}\to \mathcal{S},$$

is called a typed dual-axis structure. The projection ${\pi }_X$ carries the first-state role. When $\mathcal{X}$ is supplied by the external physical-state interface, it is the physical-state axis that the model is required to preserve; when $\mathcal{X}$ is obtained only from row behavior, it is the object-side behavioral axis resolved by the present observation family. The projection ${\pi }_S$ carries the observation-formation-state role, while $F:\mathcal{B}\to \mathcal{Y}$ records the observation outcome. In the physically anchored branch, the ordered pair of roles $({\mathcal{X}}_{\mathrm{phys}},\mathcal{S})$ is called the physical–observation, or Physobser, dual-axis structure.

Here “axis” denotes two typed state roles whose reducibility and non-degeneracy must be tested separately. It does not mean one-dimensional, linear, orthogonal, statistically independent, or symmetric coordinates, and it does not presuppose

$$\mathcal{B}=\mathcal{X}\times \mathcal{S}.$$

The admissible joint domain may contain constraints, gaps, or discrete strata. The term dual-axis is used whenever the two projections are retained as distinct typed state roles, even if one role is degenerate on a particular model class. The space $\mathcal{Y}$ is the observation-result domain, not a third state axis.

Theorem 3.13

(Typed Dual-Axis Generation Theorem). Given an empirically typed closed-world presentation ${\mu }_0:\mathfrak{P}\times \mathfrak{A}\to {\mathcal{Y}}_{\perp }$, quotienting separately by complete row behavior and complete column behavior canonically induces

$$q_X:\mathfrak{P}\twoheadrightarrow \mathcal{X}=\mathfrak{P}/{\sim }_X,q_S:\mathfrak{A}\twoheadrightarrow \mathcal{S}=\mathfrak{A}/{\sim }_S,$$

(3.38)

together with

$$\mathcal{B}\subseteq \mathcal{X}\times \mathcal{S},\tilde{F}:\mathcal{X}\times \mathcal{S}\to {\mathcal{Y}}_{\perp },F:\mathcal{B}\to \mathcal{Y}.$$

(3.39)

Furthermore:

1.

if there exist $(x,s_0),(x,s_1)\in \mathcal{B}$ with $F(x,s_0)\ne F(x,s_1)$, then there is no map $h:{\mathcal{X}}_B\to \mathcal{Y}$ satisfying $F=h\circ {\pi }_X^{\mathrm{act}}$; in this implementable-output sense, the second axis cannot be deleted while the first-state projection is preserved;

2.

distinct formation states may remain distinguishable in the totalized map solely because their implementability domains differ. If, after restriction to $\mathcal{B}$, F is constant on every nonempty first-state fiber, then the restricted output law factors through ${\pi }_X^{\mathrm{act}}$. Complete-table irreducibility and implementable-output irreducibility are therefore distinct claims.

Proof. 

Existence and well-definedness of the quotient sets and induced totalized map follow from the two-sided behavioral quotient. For the first assertion, if $F=h\circ {\pi }_X^{\mathrm{act}}$, then every two points in the same ${\pi }_X$-fiber must have the same F-value, contradicting $F(x,s_0)\ne F(x,s_1)$. For the second assertion, an implementability difference is retained by the value ⊥ in the totalized map and therefore yields distinct complete column behaviors. After ⊥ is removed by restriction to $\mathcal{B}$, however, the remaining F-values in each nonempty first-state fiber may all coincide. The fiberwise-constant factorization theorem below then permits F to depend only on the first state. □

For any fixed $s_0\in \mathcal{S}$, define

$${\mathcal{X}}_{s_0}:=\{x\in \mathcal{X}:(x,s_0)\in \mathcal{B}\}$$

(3.40)

and

$$F_{s_0}:{\mathcal{X}}_{s_0}⟶\mathcal{Y},F_{s_0}\left(x\right):=F(x,s_0).$$

(3.41)

Thus, when the formation state is fixed, the ordinary relation $y=h\left(x\right)$ is recovered exactly. The typed dual-axis structure enlarges the class of questions that can be expressed; it does not abolish the classical single-axis model.

3.6. From the Dual-Axis Relation to a Family of Observation Maps

Because $\mathcal{B}$ need not be a full Cartesian product, the admissible first-state set may vary with s. To compare the observation behavior associated with each state in a common function space, define for every $s\in \mathcal{S}$ the totalized map

$${\widehat{F}}_s:\mathcal{X}⟶{\mathcal{Y}}_{\perp },{\widehat{F}}_s\left(x\right):=\tilde{F}(x,s)$$

(3.42)

This gives the family of observation maps

$${\mathcal{H}}_F:=\{{\widehat{F}}_s:s\in \mathcal{S}\}\subseteq {\mathcal{Y}}_{\perp }^{\mathcal{X}}.$$

(3.43)

Definition 3.14

(Family of Observation Maps). Given sets $\mathcal{X}$ and ${\mathcal{Y}}_{\perp }$, any subset

$$\mathcal{H}\subseteq {\mathcal{Y}}_{\perp }^{\mathcal{X}}$$

is called a family of observation maps. An element $h\in \mathcal{H}$ represents a complete observation behavior, including information about states at which implementation is impossible.

The role of a state s is now explicit: its value indexes the complete observation map represented in the model. The fact that two states happen to produce the same result at one x does not make them behaviorally equivalent. They represent the same complete behavior only when their totalized maps agree throughout $\mathcal{X}$. No topology, distance, manifold structure, or group structure is required on $\mathcal{H}$ at this stage.

3.7. State Realizations and Behavioral Quotients

The same map family can be parameterized by different state sets. An engineering system may represent its operating state by mode numbers, control words, register contents, or high-dimensional internal variables, any of which may retain duplicate labels that have no effect on observation behavior.

Definition 3.15

(State Realization). Let $\mathcal{H}\subseteq {\mathcal{Y}}_{\perp }^{\mathcal{X}}$. A state realization is a pair $(\Sigma ,\Gamma )$, where Σ is a state set and

$$\Gamma :\Sigma ⟶\mathcal{H}$$

(3.44)

maps each state σ to the complete observation map that it realizes. The realization is called complete if Γ is surjective.

If

$$\Gamma \left(\sigma \right)=\Gamma \left({\sigma }^{\prime }\right),\sigma \ne {\sigma }^{\prime },$$

(3.45)

then the two state labels are distinct but cannot be distinguished by any complete observation behavior.

Definition 3.16

(Behavioral Equivalence). For a state realization $(\Sigma ,\Gamma )$, define

$$\sigma {\sim }_{\Gamma }{\sigma }^{\prime }⟺\Gamma \left(\sigma \right)=\Gamma \left({\sigma }^{\prime }\right).$$

(3.46)

Denote the behavioral quotient by

$${\Sigma }_{\mathrm{eff}}:=\Sigma /{\sim }_{\Gamma },$$

(3.47)

and the natural quotient map by

$$q_{\Gamma }:\Sigma \twoheadrightarrow {\Sigma }_{\mathrm{eff}},q_{\Gamma }\left(\sigma \right)={\left[\sigma \right]}_{\Gamma }.$$

(3.48)

Proposition 3.17

(Unique Factorization through the Quotient). There exists a unique map

$$\overline{\Gamma }:{\Sigma }_{\mathrm{eff}}⟶\mathcal{H}$$

(3.49)

such that

$$\Gamma =\overline{\Gamma }\circ q_{\Gamma }.$$

(3.50)

The map $\overline{\Gamma }$ is necessarily injective; if the original realization is complete, then $\overline{\Gamma }$ is bijective.

Proof. 

Define $\overline{\Gamma }\left({\left[\sigma \right]}_{\Gamma }\right):=\Gamma \left(\sigma \right)$. Behavioral equivalence makes this definition independent of representatives, and (3.50) follows immediately. Since $q_{\Gamma }$ is surjective, that factorization determines $\overline{\Gamma }$ uniquely. If two equivalence classes have the same image under $\overline{\Gamma }$, then their representatives are behaviorally equivalent, so the two classes coincide; hence $\overline{\Gamma }$ is injective. If the original realization is complete, $\Gamma$ is surjective, and therefore so is $\overline{\Gamma }$. □

If $(\Sigma ,\Gamma )$ is complete, then

$${\Sigma }_{\mathrm{eff}}\cong \mathcal{H}$$

(3.51)

as a bijection of sets. This bijection is induced by $\Gamma$ itself and does not depend on an arbitrary selection of representatives; it does not assert preservation of any additional topological, smooth, or geometric structure.

3.8. Canonical Realization and Behavioral Minimality

Definition 3.18

(Morphism of Realizations). Let $({\Sigma }_1,{\Gamma }_1)$ and $({\Sigma }_2,{\Gamma }_2)$ be two state realizations of the same map family $\mathcal{H}$. A map

$$\phi :{\Sigma }_1\to {\Sigma }_2$$

satisfying

$${\Gamma }_2\circ \phi ={\Gamma }_1,$$

(3.52)

makes φ an observation-behavior-preserving morphism of realizations. If φ is bijective, it is called an isomorphism of realizations.

For a fixed map family $\mathcal{H}$, write

$$Real\left(\mathcal{H}\right):=\mathbf{Set}/\mathcal{H}.$$

Its objects are realization maps $\Gamma :\Sigma \to \mathcal{H}$, and its morphisms are exactly the maps $\phi :{\Sigma }_1\to {\Sigma }_2$ satisfying ${\Gamma }_2\circ \phi ={\Gamma }_1$. If such a morphism is bijective as a set map, then ${\Gamma }_1\circ {\phi }^{-1}={\Gamma }_2$, so its inverse is also a realization morphism. Thus the categorical isomorphisms are precisely the bijective behavior-preserving morphisms.

The map family itself supplies the canonical realization

$${\Sigma }_{\mathrm{can}}:=\mathcal{H},{\Gamma }_{\mathrm{can}}:={id}_{\mathcal{H}}.$$

(3.53)

It is terminal in $Real\left(\mathcal{H}\right)$: for every $\Gamma :\Sigma \to \mathcal{H}$, the map $\Gamma$ itself is the unique morphism from $(\Sigma ,\Gamma )$ to $(\mathcal{H},{id}_{\mathcal{H}})$ [22].

Definition 3.19

(Behavior-Minimal Realization). A state realization $(\Sigma ,\Gamma )$ is called behavior-minimal if

$$\Gamma \left(\sigma \right)=\Gamma \left({\sigma }^{\prime }\right)⟹\sigma ={\sigma }^{\prime },$$

(3.54)

that is, if Γ is injective.

Behavioral minimality does not compare the dimensions of state coordinates. It requires only that the realization retain no duplicate states with identical complete behavior.

Theorem 3.20

(Canonical Minimal-State Realization Theorem). Let $\mathcal{H}\subseteq {\mathcal{Y}}_{\perp }^{\mathcal{X}}$ be a family of observation maps. Then:

1.

the canonical realization $(\mathcal{H},{id}_{\mathcal{H}})$ is complete and behavior-minimal;

2.

for any complete realization $(\Sigma ,\Gamma )$, the quotient realization $(\Sigma /{\sim }_{\Gamma },\overline{\Gamma })$ is complete and behavior-minimal;

3.

for every complete and behavior-minimal realization, $\Gamma :\Sigma \to \mathcal{H}$ is bijective;

4.

between any two complete and behavior-minimal realizations there exists a unique state isomorphism that preserves observation behavior.

Proof. 

The identity map is both injective and surjective, proving the first assertion. The second follows because, under completeness, the induced realization map $\overline{\Gamma }$ is bijective. The third follows directly from the surjectivity supplied by completeness and the injectivity supplied by behavioral minimality.

Let $({\Sigma }_1,{\Gamma }_1)$ and $({\Sigma }_2,{\Gamma }_2)$ both be complete and behavior-minimal. Their realization maps are bijective. Define

$$\phi :={\Gamma }_2^{-1}\circ {\Gamma }_1.$$

(3.55)

Then ${\Gamma }_2\circ \phi ={\Gamma }_1$, so $\phi$ is an isomorphism of realizations. If $\psi$ is another behavior-preserving isomorphism, then ${\Gamma }_2\circ \psi ={\Gamma }_1$ gives $\psi ={\Gamma }_2^{-1}\circ {\Gamma }_1=\phi$, proving uniqueness. □

The theorem uses only classical properties of equivalence relations, quotient sets, and bijections. Its purpose here is not to claim a new result in set theory, but to fix a standard of state identity: once the complete family of observation maps is given, every complete and nonredundant state realization differs from every other only by a unique behavior-preserving relabeling.

Definition 3.21

(Anchored Realization of Primitive Formation Behavior). Let

$$c_0:\mathfrak{A}\twoheadrightarrow {\mathcal{H}}_0,c_0\left(a\right):=h_a.$$

An anchored realization of $c_0$ is a triple $(\Sigma ,\beta ,\Gamma )$ with

$$\beta :\mathfrak{A}\twoheadrightarrow \Sigma ,\Gamma :\Sigma \to {\mathcal{H}}_0,\Gamma \circ \beta =c_0.$$

A morphism $\phi :(\Sigma ,\beta ,\Gamma )\to ({\Sigma }^{\prime },{\beta }^{\prime },{\Gamma }^{\prime })$ is a map $\phi :\Sigma \to {\Sigma }^{\prime }$ satisfying

$$\phi \circ \beta ={\beta }^{\prime },{\Gamma }^{\prime }\circ \phi =\Gamma .$$

Because $c_0$ is surjective, every anchored realization is complete; it is behavior-minimal when Γ is injective.

Proposition 3.22

(Anchored Universal Property of the Formation Quotient). Define

$${\overline{c}}_0:\mathcal{S}⟶{\mathcal{H}}_0,{\overline{c}}_0\left({\left[a\right]}_S\right):=h_a.$$

Then ${\overline{c}}_0$ is bijective, and $(\mathcal{S},q_S,{\overline{c}}_0)$ is terminal among anchored realizations of $c_0$. More explicitly, for every anchored realization $(\Sigma ,\beta ,\Gamma )$ there exists a unique surjection $\alpha :\Sigma \twoheadrightarrow \mathcal{S}$ such that

$$q_S=\alpha \circ \beta ,\Gamma ={\overline{c}}_0\circ \alpha .$$

(3.56)

If Γ is injective, then α is bijective. Hence every behavior-minimal anchored realization is uniquely isomorphic to the formation quotient while preserving both primitive labels and complete behavior.

Proof. 

The map ${\overline{c}}_0$ is well defined because ${\left[a\right]}_S={\left[a^{\prime }\right]}_S$ is equivalent to $h_a=h_{a^{\prime }}$. The same equivalence proves injectivity, and the definition ${\mathcal{H}}_0=\{h_a:a\in \mathfrak{A}\}$ proves surjectivity.

Define $\alpha \left(\beta \left(a\right)\right):={\left[a\right]}_S$. If $\beta \left(a\right)=\beta \left(a^{\prime }\right)$, then

$$h_a=c_0\left(a\right)=\Gamma \left(\beta \left(a\right)\right)=\Gamma \left(\beta \left(a^{\prime }\right)\right)=c_0\left(a^{\prime }\right)=h_{a^{\prime }},$$

so $a{\sim }_S{a}^{\prime }$ and $\alpha$ is well defined. The first identity in (3.56) follows immediately, and surjectivity of $q_S$ implies surjectivity of $\alpha$. Precomposition with the surjection $\beta$ gives the second identity and also proves uniqueness. If $\Gamma$ is injective and $\alpha \left(\sigma \right)=\alpha \left({\sigma }^{\prime }\right)$, then

$$\Gamma \left(\sigma \right)={\overline{c}}_0\left(\alpha \left(\sigma \right)\right)={\overline{c}}_0\left(\alpha \left({\sigma }^{\prime }\right)\right)=\Gamma \left({\sigma }^{\prime }\right),$$

so $\sigma ={\sigma }^{\prime }$. Thus $\alpha$ is bijective in the behavior-minimal case. □

The primitive family ${\mathcal{H}}_0\subseteq {\mathcal{Y}}_{\perp }^{\mathfrak{P}}$ and the reduced family ${\mathcal{H}}_F\subseteq {\mathcal{Y}}_{\perp }^{\mathcal{X}}$ generally have different function domains and are not asserted to be literally equal. Each is canonically parameterized by the same formation quotient $\mathcal{S}$ in its corresponding branch.

3.9. Consistency Between the Two-Sided Behavioral Quotient and the Canonical Realization

From the second-state quotient $\mathcal{S}=\mathfrak{A}/{\sim }_S$, define

$${\Gamma }_F:\mathcal{S}⟶{\mathcal{H}}_F,{\Gamma }_F\left(s\right):={\widehat{F}}_s.$$

(3.57)

Proposition 3.23

(Behavioral Minimality of the Second Effective State Space). The map ${\Gamma }_F$ is bijective. Consequently, the space $\mathcal{S}$ obtained by the formation-side behavioral quotient is a complete, behavior-minimal realization of ${\mathcal{H}}_F$ and is uniquely isomorphic to the canonical realization ${\mathcal{H}}_F$, whether the first state is the row-behavior quotient or an externally supplied physical state space satisfying Proposition 3.9.

Proof. 

Surjectivity follows directly from the definition of ${\mathcal{H}}_F$. Suppose ${\Gamma }_F\left(s\right)={\Gamma }_F\left(s^{\prime }\right)$ and choose representatives $s={\left[a\right]}_S$, $s^{\prime }={\left[a^{\prime }\right]}_S$. In the row-quotient branch, for arbitrary $p\in \mathfrak{P}$ let $x={\left[p\right]}_X$; equality of the totalized maps gives

$${\mu }_0(p,a)={\widehat{F}}_s\left(x\right)={\widehat{F}}_{s^{\prime }}\left(x\right)={\mu }_0(p,a^{\prime }),$$

so $a{\sim }_S{a}^{\prime }$. In the external-physical-state branch, Proposition 3.9 shows that the same equivalence relation on $\mathfrak{A}$ is obtained by comparing the induced columns $x\mapsto {\mu }_{\mathrm{phys}}(x,a)$ on ${\mathcal{X}}_{\mathrm{phys}}$. Equality ${\widehat{F}}_s={\widehat{F}}_{s^{\prime }}$ therefore again implies $a{\sim }_S{a}^{\prime }$. Hence $s=s^{\prime }$ in either branch, and ${\Gamma }_F$ is injective. □

Theorem 3.24

(Equivalent Criteria for Non-Triviality of the Formation-Side Quotient). Let $\mathfrak{E}$ be an empirically typed closed-world presentation and let $\mathcal{X}$, $\mathcal{S}$, $\tilde{F}$, $\mathcal{B}$, F, ${\mathcal{H}}_F$, and ${\Gamma }_F$ be its complete formation-reduced realization, using either the row-behavior quotient or a compatible external physical-state interface for the first axis. Write ${pr}_X:\mathcal{X}\times \mathcal{S}\to \mathcal{X}$ for the first product projection. Then the following are equivalent:

1.

the table is record-forming non-trivial, so

$${\mu }_0(p_0,a_0)\ne {\mu }_0(p_0,a_1)$$

for some fixed $p_0\in \mathfrak{P}$ and $a_0,a_1\in \mathfrak{A}$;

2.

$\left|\mathcal{S}\right|\ge 2$;

3.

$|{\mathcal{H}}_F|\ge 2$;

4.

there is no $h:\mathcal{X}\to {\mathcal{Y}}_{\perp }$ such that

$$\tilde{F}=h\circ {pr}_X.$$

(3.58)

The following stronger statements are also equivalent:

(a) 

the table has a strong record-forming witness at a fixed primitive object/preparation label;

(b) 

there exist $(x,s_0),(x,s_1)\in \mathcal{B}$ such that $F(x,s_0)\ne F(x,s_1)$;

(c) 

there is no $h:{\mathcal{X}}_B\to \mathcal{Y}$ such that

$$F=h\circ {\pi }_X^{\mathrm{act}}.$$

(3.59)

More generally, any partial certified table that already contains such a fixed unequal pair has at least two formation classes in every closed completion in the sense defined above. Closed-world completeness is additionally required to determine the entire task-global quotient, its full map family, and its canonical complete realization.

Proof. 

A primitive witness makes $a_0{\cancel{\sim }}_S{a}_1$, so their quotient classes are distinct and $\left|\mathcal{S}\right|\ge 2$. Conversely, two distinct quotient classes have different complete columns by definition, so some fixed primitive object label witnesses record-forming non-triviality. Bijectivity of ${\Gamma }_F$ gives the equivalence with $|{\mathcal{H}}_F|\ge 2$. If a factorization as in (3.58) existed, every totalized slice ${\widehat{F}}_s$ would equal h, so the map family and $\mathcal{S}$ would both be singletons. Conversely, because $\mathfrak{A}$ is nonempty, $\left|\mathcal{S}\right|=1$ makes all totalized slices identical and defines the required h. This proves the first cycle of equivalences.

A strong primitive witness descends to two points of $\mathcal{B}$ with the same first coordinate and unequal F-values. Theorem 3.25 gives the equivalence of (b) and (c). Conversely, an effective witness in (b) lifts to primitive representatives. In the row-quotient branch, equality of the first quotient class lets one use either row representative for both operations. In the external physical-state branch, choose $p\in \mathfrak{P}$ with $r\left(p\right)=x$ and $a_i\in \mathfrak{A}$ with $q_S\left(a_i\right)=s_i$; then

$${\mu }_0(p,a_i)={\mu }_{\mathrm{phys}}(x,a_i)=\tilde{F}(x,s_i)=F(x,s_i).$$

The lifted entries lie in $\mathcal{Y}$ and are unequal, hence form a strong primitive witness. □

Every equivalence in Theorem 3.24 is relative to the fixed declared task, common output type, and closed table. It does not assert that every x admits both states, that $\mathcal{B}=\mathcal{X}\times \mathcal{S}$, or that the quotient survives a change of task or output resolution. Counterexample 2.5 shows that an open table need not determine the entire quotient, although an already certified unequal pair remains unequal in every closed completion. A weak witness involving ⊥ makes the totalized formation-side quotient non-trivial but does not by itself obstruct factorization of the actual-output law F.

The primitive formation-operation type $\mathfrak{A}$ is supplied by the observational task. What is canonically determined rather than appended is its nonredundant state identity: $\mathcal{S}$ is the complete behavior-minimal realization of ${\mathcal{H}}_F$, and every other complete behavior-minimal realization of that same family is uniquely behaviorally isomorphic to it.

3.10. When an Exact Return to a First-State Function Is Possible

We begin with a classical factorization fact.

Theorem 3.25

(Fiberwise-Constant Factorization). Let $\varpi :E\twoheadrightarrow X_0$ be surjective and let $f:E\to Y_0$ be any map. There exists a unique map $\overline{f}:X_0\to Y_0$ satisfying

$$f=\overline{f}\circ \varpi$$

(3.60)

if and only if f is constant on every fiber of ϖ, that is,

$$\varpi \left(e\right)=\varpi \left(e^{\prime }\right)⟹f\left(e\right)=f\left(e^{\prime }\right).$$

(3.61)

Proof. 

If $f=\overline{f}\circ \varpi$, then two points in the same fiber plainly have the same f value. Conversely, suppose f is constant on every fiber. For $x\in X_0$, choose any $e\in {\varpi }^{-1}\left(x\right)$ and define $\overline{f}\left(x\right):=f\left(e\right)$. Fiberwise constancy makes the definition independent of the chosen representative, while surjectivity of $\varpi$ guarantees uniqueness of $\overline{f}$. □

Recall ${\mathcal{X}}_B={\pi }_X\left(\mathcal{B}\right)$. The theorem immediately gives the following result.

Corollary 3.26

(Necessary and Sufficient Condition for Representation by the First State Alone). There exists a unique map $h:{\mathcal{X}}_B\to \mathcal{Y}$ such that

$$F=h\circ {\pi }_X^{\mathrm{act}}$$

(3.62)

if and only if every state pair satisfying $(x,s),(x,s^{\prime })\in \mathcal{B}$ also satisfies

$$F(x,s)=F(x,s^{\prime }).$$

(3.63)

Whether the model can be reduced to $y=h\left(x\right)$ is therefore a precise factorization question. The second state may exist while having no visible effect on the current output, or the admissible domain may contain only one point in each first-state fiber. In either case the single-axis model is exact. Outside these cases, whenever one first-state fiber contains two states that yield different results, the second axis cannot be removed while preserving the semantics assigned to the first state.

Proposition 3.27

(Separate non-degeneracy criteria for the two axes). In the two-sided row-and-column quotient branch,

$$\begin{array}{cc}\hfill \left|\mathcal{X}\right|\ge 2& \iff \exists p_0,p_1\in \mathfrak{P},a\in \mathfrak{A}:{\mu }_0(p_0,a)\ne {\mu }_0(p_1,a),\hfill \end{array}$$

(3.64)

$$\begin{array}{cc}\hfill \left|\mathcal{S}\right|\ge 2& \iff \exists p\in \mathfrak{P},a_0,a_1\in \mathfrak{A}:{\mu }_0(p,a_0)\ne {\mu }_0(p,a_1).\hfill \end{array}$$

(3.65)

For the restricted outcome law,

$$\begin{array}{cc}\hfill F\mathit{fails}\mathit{to}\mathit{factor}\mathit{through}{\pi }_X^{\mathrm{act}}& \iff \exists (x,s_0),(x,s_1)\in \mathcal{B}:F(x,s_0)\ne F(x,s_1),\hfill \end{array}$$

(3.66)

$$\begin{array}{cc}\hfill F\mathit{fails}\mathit{to}\mathit{factor}\mathit{through}{\pi }_S^{\mathrm{act}}& \iff \exists (x_0,s),(x_1,s)\in \mathcal{B}:F(x_0,s)\ne F(x_1,s).\hfill \end{array}$$

(3.67)

Consequently, a fixed-first-state witness establishes formation-side non-degeneracy only. Non-degeneracy of both quotient axes requires witnesses in both directions.

Proof. 

The first two equivalences are the definitions of inequality of complete row and column functions. The last two are the fiberwise-constant factorization theorem applied to the surjections ${\pi }_X^{\mathrm{act}}$ and ${\pi }_S^{\mathrm{act}}$, respectively. □

Neither side implies the other. The table with one object label and two operation labels carrying distinct entries has a singleton $\mathcal{X}$ and a non-degenerate $\mathcal{S}$; interchanging the two typed roles gives the converse. In the external physical-state branch, the identity and possible non-degeneracy of the first axis are supplied by the external theory rather than proved by the row quotient. Proposition 3.27 therefore prevents a formation-side witness from being promoted silently into a claim that both typed axes are non-degenerate.

3.11. Extended Encodings and State Non-Hiding

It is always formally possible to encode $(x,s)$ as one enlarged variable. Yet, so long as the original first-state semantics retain theoretical meaning, the enlarged state must preserve the first-state fiber to which it belongs.

Definition 3.28

(Extended Encoding that Preserves First-State Semantics). Given $F:\mathcal{B}\to \mathcal{Y}$, an extended encoding that preserves the semantics of the first state consists of a set E, maps

$${\varpi }_X:E\to {\mathcal{X}}_B,F_E:E\to \mathcal{Y},$$

and an encoding $\iota :\mathcal{B}\to E$ satisfying

$${\varpi }_X\circ \iota ={\pi }_X^{\mathrm{act}},F_E\circ \iota =F.$$

(3.68)

Theorem 3.29

(State Non-Hiding Theorem). Every extended encoding $(E,{\varpi }_X,F_E,\iota )$ that preserves first-state semantics satisfies the kernel constraint

$$Eq\left(\iota \right)\subseteq Eq\left({\pi }_X^{\mathrm{act}}\right)\cap Eq\left(F\right).$$

(3.69)

Consequently, if $(x,s_0),(x,s_1)\in \mathcal{B}$ and

$$F(x,s_0)\ne F(x,s_1),$$

(3.70)

then

$$\iota (x,s_0)\ne \iota (x,s_1),$$

(3.71)

while both encoded points lie in the same first-state fiber ${\varpi }_X^{-1}\left(x\right)$.

Proof. 

If $\iota \left(b\right)=\iota \left(b^{\prime }\right)$, applying ${\varpi }_X$ and $F_E$ to the common encoded value gives

$${\pi }_X^{\mathrm{act}}\left(b\right)={\pi }_X^{\mathrm{act}}\left(b^{\prime }\right),F\left(b\right)=F\left(b^{\prime }\right).$$

This proves (3.69). The stated consequence follows immediately: unequal outputs cannot arise from one encoded point, whereas ${\varpi }_X\circ \iota ={\pi }_X^{\mathrm{act}}$ places both images over x. □

The theorem separates “hiding” from “eliminating.” Concatenating coordinates can hide the notation of the second typed state role, but whenever two formation-state values at fixed first state index unequal outputs, every first-state-preserving exact encoding must contain distinct points in that fiber.

Definition 3.30

(State-Faithful Extension). If the extended encoding also admits a map $d:E\to \mathcal{S}$ satisfying

$$d\circ \iota ={\pi }_S,$$

(3.72)

then the extension is called faithful to both classes of state.

Theorem 3.31

(Observation-State Decoder Criterion). Let $\iota :\mathcal{B}\to E$ be any encoding. A map $d:E\to \mathcal{S}$ satisfying $d\circ \iota ={\pi }_S$ exists if and only if

$$Eq\left(\iota \right)\subseteq Eq\left({\pi }_S\right).$$

(3.73)

Whenever it exists, the decoder is uniquely determined on $\iota \left(\mathcal{B}\right)$; it is unique on all of E when ι is surjective.

Proof. 

Necessity follows by applying d to two equal encoded values. Conversely, assume (3.73). Define

$$d_0\left(\iota \left(b\right)\right):={\pi }_S\left(b\right)(b\in \mathcal{B}).$$

The kernel condition makes $d_0$ well defined on $\iota \left(\mathcal{B}\right)$. Because $\mathcal{S}$ is nonempty, choose one $s_{*}\in \mathcal{S}$ and extend $d_0$ to $E\setminus \iota \left(\mathcal{B}\right)$ by the constant value $s_{*}$. Every decoder must agree with $d_0$ on the image, proving uniqueness there; if the image is all of E, global uniqueness follows. □

Theorem 3.32

(State-Faithful Embedding Theorem). Let

$$j_B^{\mathrm{act}}:\mathcal{B}\hookrightarrow {\mathcal{X}}_B\times \mathcal{S},j_B^{\mathrm{act}}(x,s):=(x,s)$$

be the canonical inclusion. If an extended encoding satisfies both ${\varpi }_X\circ \iota ={\pi }_X^{\mathrm{act}}$ and $d\circ \iota ={\pi }_S$, then ι is injective and

$$({\varpi }_X,d)\circ \iota =j_B^{\mathrm{act}}.$$

(3.74)

Thus the original joint state domain embeds into the extended space in a manner that preserves both task-level state coordinates.

Proof. 

For every $(x,s)\in \mathcal{B}$,

$$({\varpi }_X,d)\left(\iota (x,s)\right)=(x,s)=j_B^{\mathrm{act}}(x,s),$$

so (3.74) holds as an equality of maps into ${\mathcal{X}}_B\times \mathcal{S}$. If two joint states have the same image under $\iota$, applying $({\varpi }_X,d)$ and using injectivity of $j_B^{\mathrm{act}}$ shows that the two joint states coincide; hence $\iota$ is injective. □

3.12. The Behavioral Identity of the Second Effective State

The preceding results can now be combined. The second effective state s is obtained from complete column behavior; it is in bijection with the family of totalized observation maps ${\mathcal{H}}_F$; every complete, behavior-minimal realization of that family is uniquely isomorphic to it; and, whenever two implementable states in one first-state fiber produce different outputs, their distinction cannot be eliminated by an encoding that preserves the first-state projection. These properties determine the mathematical responsibility of the second state without assigning it an ontological interpretation.

Theorem 3.33

(Behavioral Identity and Uniqueness of the Second Effective State). Suppose the first-state set $\mathcal{X}$ has either been obtained by the object-side behavioral quotient or supplied through the external physical-state interface of Proposition 3.9, and suppose the complete observation facts induce

$$\mathcal{B}\subseteq \mathcal{X}\times \mathcal{S},F:\mathcal{B}\to \mathcal{Y}.$$

Relative to the specified $\mathcal{X}$ and the complete admissible map family ${\mathcal{H}}_F$, the second effective state space $\mathcal{S}$ is a complete behavior-minimal realization of ${\mathcal{H}}_F$ and is unique, among all complete behavior-minimal realizations, up to the unique behavior-preserving isomorphism of Theorem 3.20. Its mathematical responsibility is to select the complete observation-formation map ${\widehat{F}}_s$ currently realized.

Proof. 

In the row-quotient branch, the two-sided behavioral quotient makes ${\Gamma }_F:\mathcal{S}\to {\mathcal{H}}_F$ bijective. In the external-physical-state branch, Proposition 3.9 shows that the formation-side equivalence relation is unchanged after passage from primitive labels to ${\mathcal{X}}_{\mathrm{phys}}$, and the preceding proposition again gives bijectivity of ${\Gamma }_F$. The canonical minimal-state realization theorem then supplies the unique behavior-preserving isomorphism between $\mathcal{S}$ and any other complete behavior-minimal realization. If two implementable second states in one first-state fiber have unequal F-values, the state non-hiding theorem additionally shows that no encoding preserving the first-state projection can merge those two states. □

The uniqueness in this theorem is relative to a fixed task, first-state semantics, common output type and resolution, complete test family, and exact behavioral equality. It is uniqueness of a complete nonredundant realization within that realization problem, not an absolute microscopic identity.

Proposition 3.34

(Task relativity and canonical coarse-graining maps). Let $\mathcal{S}$ be the formation-side quotient for ${\mu }_0:\mathfrak{P}\times \mathfrak{A}\to {\mathcal{Y}}_{\perp }$.

1.

If $⌀\ne {\mathfrak{P}}_0\subseteq \mathfrak{P}$ and ${\mathcal{S}}_0$ is computed from the restricted table ${\mu }_0{|}_{{\mathfrak{P}}_0\times \mathfrak{A}}$, then there is a canonical surjection

$$c_{{\mathfrak{P}}_0}:\mathcal{S}\twoheadrightarrow {\mathcal{S}}_0,c_{{\mathfrak{P}}_0}\left({\left[a\right]}_S\right)={\left[a\right]}_{S,0}.$$

(3.75)

2.

Let ${\perp }^{\prime }\notin \mathcal{Z}$ and ${\mathcal{Z}}_{{\perp }^{\prime }}:=\mathcal{Z}\bigsqcup \left\{{\perp }^{\prime }\right\}$. If $\kappa :{\mathcal{Y}}_{\perp }\to {\mathcal{Z}}_{{\perp }^{\prime }}$ is an output coarse graining satisfying $\kappa (\perp )={\perp }^{\prime }$ and $\kappa \left(\mathcal{Y}\right)\subseteq \mathcal{Z}$, and ${\mathcal{S}}_{\kappa }$ is computed from $\kappa \circ {\mu }_0$, then there is a canonical surjection

$$c_{\kappa }:\mathcal{S}\twoheadrightarrow {\mathcal{S}}_{\kappa },c_{\kappa }\left({\left[a\right]}_S\right)={\left[a\right]}_{S,\kappa }.$$

(3.76)

Thus restricting the object tests or coarsening the output resolution can merge observer states; adding tests or refining output resolution can split them. These operations change the task and are not presentation isomorphisms.

Proof. 

Equality of two complete columns on all of $\mathfrak{P}$ implies equality on ${\mathfrak{P}}_0$, and equality before applying $\kappa$ implies equality afterward. Hence each displayed formula is well defined. Surjectivity follows because every restricted or coarse-grained class contains an operation label from $\mathfrak{A}$. □

For sequential observation, “complete behavior” must quantify over every continuation admitted by the declared task. Two configurations that agree on the present one-step table may be separated by a future continuation; a quotient based only on current outputs is therefore a one-step, task-relative realization.

Definition 3.35

(Observer State). Within this paper, the second effective state identified by Theorem 3.33, whose responsibility is to index the complete observation-formation map represented in the model, is called anobserver state, written $s\in \mathcal{S}$; $\mathcal{S}$ is theobserver-state space. It is an empirically anchored, task-relative interface state determined by complete record-forming behavior. The definition does not identify behaviorally equivalent implementations microscopically or assert that the present task resolves every physical distinction.

“Observer” is therefore the name of a state responsibility, not a device class. It does not require a human observer and does not mean a sensor external to the information system. Array weights, pose, reference phase, receiver mode, sensor configuration, or protocol state can instantiate the observer-state role only when their values index the observation-formation maps represented in the model under study. Such states may be physical, algorithmic, or hybrid; the terminology itself carries no further ontological conclusion.

3.13. The Set-Theoretic Physobser Representation Theorem

The general row–column quotient supplies a typed behavioral precursor. The following theorem states the physically anchored form. It is the point at which the first axis receives physical semantics from an antecedent theory, while the second axis receives its state identity from complete observation-forming behavior.

Theorem 3.36

(Set-Theoretic Physobser Representation Theorem). Let $\mathfrak{E}=(\mathfrak{P},\mathfrak{A},\mathcal{Y},\perp ,{\mu }_0)$ be an empirically typed closed-world presentation, and let

$$r:\mathfrak{P}\twoheadrightarrow {\mathcal{X}}_{\mathrm{phys}}$$

be a compatible physical-state interface satisfying (3.30). Let

$$q_S:\mathfrak{A}\twoheadrightarrow \mathcal{S}:=\mathfrak{A}/{\sim }_S$$

be the formation-side behavioral quotient. Then the following statements hold.

1.

There exists a unique map

$${\tilde{F}}_{\mathrm{phys}}:{\mathcal{X}}_{\mathrm{phys}}\times \mathcal{S}⟶{\mathcal{Y}}_{\perp }$$

(3.77)

such that

$${\mu }_0={\tilde{F}}_{\mathrm{phys}}\circ (r\times q_S).$$

(3.78)

With

$${\mathcal{B}}_{\mathrm{phys}}:={\tilde{F}}_{\mathrm{phys}}^{-1}\left(\mathcal{Y}\right),F_{\mathrm{phys}}:={\tilde{F}}_{\mathrm{phys}}{|}_{{\mathcal{B}}_{\mathrm{phys}}},$$

(3.79)

the joint domain is exactly the implementable part of the totalized physical–observation law.

2.

The behavior map

$${\Gamma }_{\mathrm{phys}}:\mathcal{S}⟶{\mathcal{Y}}_{\perp }^{{\mathcal{X}}_{\mathrm{phys}}},{\Gamma }_{\mathrm{phys}}\left(s\right):={\tilde{F}}_{\mathrm{phys}}(·,s)$$

(3.80)

is injective, hence bijective onto its image. Thus $\mathcal{S}$ is the complete behavior-minimal realization of the observation-map family carried by the anchored physical state.

3.

Let $\beta :\mathfrak{A}\twoheadrightarrow \Sigma$ be any surjection and let

$$G:{\mathcal{X}}_{\mathrm{phys}}\times \Sigma \to {\mathcal{Y}}_{\perp }$$

satisfy ${\mu }_0=G\circ (r\times \beta )$. Then there exists a unique surjection

$$\alpha :\Sigma \twoheadrightarrow \mathcal{S}$$

(3.81)

such that

$$q_S=\alpha \circ \beta ,G={\tilde{F}}_{\mathrm{phys}}\circ ({id}_{{\mathcal{X}}_{\mathrm{phys}}}\times \alpha ).$$

(3.82)

Its fibers are exactly the duplicate G-columns:

$$\alpha \left(\sigma \right)=\alpha \left({\sigma }^{\prime }\right)⟺G(x,\sigma )=G(x,{\sigma }^{\prime })\mathrm{for}\mathrm{every}x\in {\mathcal{X}}_{\mathrm{phys}}.$$

(3.83)

Hence α is bijective precisely when the realization G has distinct columns.

4.

The following conditions are equivalent:

$$\begin{array}{cc}\hfill & \exists p\in \mathfrak{P},a_0,a_1\in \mathfrak{A}:{\mu }_0(p,a_0)\ne {\mu }_0(p,a_1)\hfill \\ \hfill & ⟺\left|\mathcal{S}\right|\ge 2\hfill \\ \hfill & ⟺{\tilde{F}}_{\mathrm{phys}}\mathit{does}\mathit{not}\mathit{factor}\mathit{through}{pr}_{{\mathcal{X}}_{\mathrm{phys}}}:{\mathcal{X}}_{\mathrm{phys}}\times \mathcal{S}\to {\mathcal{X}}_{\mathrm{phys}}.\hfill \end{array}$$

(3.84)

These equivalent statements express non-triviality of the complete totalized law. Separately, let

$${\mathcal{X}}_{\mathrm{phys},B}:={\pi }_X\left({\mathcal{B}}_{\mathrm{phys}}\right),{\pi }_X^{\mathrm{act}}:{\mathcal{B}}_{\mathrm{phys}}\twoheadrightarrow {\mathcal{X}}_{\mathrm{phys},B}$$

be the active physical-state projection. Then the following stronger conditions are equivalent:

(a) 

there exist $p\in \mathfrak{P}$ and $a_0,a_1\in \mathfrak{A}$ such that ${\mu }_0(p,a_i)\in \mathcal{Y}$ for $i=0,1$ and ${\mu }_0(p,a_0)\ne {\mu }_0(p,a_1)$;

(b) 

there exist $(x,s_0),(x,s_1)\in {\mathcal{B}}_{\mathrm{phys}}$ with $F_{\mathrm{phys}}(x,s_0)\ne F_{\mathrm{phys}}(x,s_1)$;

(c) 

no map $h:{\mathcal{X}}_{\mathrm{phys},B}\to \mathcal{Y}$ satisfies $F_{\mathrm{phys}}=h\circ {\pi }_X^{\mathrm{act}}$.

Proof. 

For $x=r\left(p\right)$ and $s=q_S\left(a\right)$, define

$${\tilde{F}}_{\mathrm{phys}}(x,s):={\mu }_0(p,a).$$

Compatibility with r removes dependence on the representative p, while the definition of ${\sim }_S$ removes dependence on a. Surjectivity of $r\times q_S$ gives uniqueness and proves the first assertion.

If two states $s=q_S\left(a\right)$ and $s^{\prime }=q_S\left(a^{\prime }\right)$ determine the same map on ${\mathcal{X}}_{\mathrm{phys}}$, then for every $p\in \mathfrak{P}$,

$${\mu }_0(p,a)={\tilde{F}}_{\mathrm{phys}}(r\left(p\right),s)={\tilde{F}}_{\mathrm{phys}}(r\left(p\right),s^{\prime })={\mu }_0(p,a^{\prime }).$$

Thus $a{\sim }_S{a}^{\prime }$ and $s=s^{\prime }$, proving injectivity of ${\Gamma }_{\mathrm{phys}}$.

For the universal property, if $\beta \left(a\right)=\beta \left(a^{\prime }\right)$, exactness gives equality of the corresponding primitive columns, so $a{\sim }_S{a}^{\prime }$. Therefore

$$\alpha \left(\beta \left(a\right)\right):=q_S\left(a\right)$$

is well defined, unique, and surjective. For arbitrary $x\in {\mathcal{X}}_{\mathrm{phys}}$ and $\sigma \in \Sigma$, choose p and a with $r\left(p\right)=x$ and $\beta \left(a\right)=\sigma$. Then

$$G(x,\sigma )={\mu }_0(p,a)={\tilde{F}}_{\mathrm{phys}}(x,q_S\left(a\right))={\tilde{F}}_{\mathrm{phys}}(x,\alpha \left(\sigma \right)),$$

which proves (3.82). If $\alpha \left(\sigma \right)=\alpha \left({\sigma }^{\prime }\right)$, choose $a,a^{\prime }$ with $\beta \left(a\right)=\sigma$ and $\beta \left(a^{\prime }\right)={\sigma }^{\prime }$. Equality of their quotient classes gives equality of the primitive columns; surjectivity of r then gives $G(x,\sigma )=G(x,{\sigma }^{\prime })$ for every x. Conversely, equality of the two G-columns implies equality of the primitive columns and hence $\alpha \left(\sigma \right)=\alpha \left({\sigma }^{\prime }\right)$. This proves (3.83).

A primitive unequal pair gives two distinct ${\sim }_S$-classes, hence $\left|\mathcal{S}\right|\ge 2$. Conversely, two distinct classes have unequal complete columns and are separated by some $p\in \mathfrak{P}$. If ${\tilde{F}}_{\mathrm{phys}}=h\circ {pr}_{{\mathcal{X}}_{\mathrm{phys}}}$, every totalized slice is the same function h, so injectivity of ${\Gamma }_{\mathrm{phys}}$ forces $\left|\mathcal{S}\right|=1$. Conversely, because $\mathcal{S}$ is nonempty, a singleton $\mathcal{S}$ makes all slices identical and defines such an h. This proves (3.84).

A strong primitive witness descends to two admissible points over the same physical state with unequal $F_{\mathrm{phys}}$-values. Conversely, any such effective pair lifts through the surjections r and $q_S$ to a strong primitive witness. The equivalence with failure of factorization through ${\pi }_X^{\mathrm{act}}$ is exactly the fiberwise-constant factorization theorem. □

Corollary 3.37

(Unique Physobser realization at fixed physical semantics). Let $({\Sigma }_i,{\beta }_i,G_i)$, $i=1,2$, be two surjective physical-state-preserving exact realizations of the same primitive table on ${\mathcal{X}}_{\mathrm{phys}}$, and suppose both $G_i$ have distinct columns. Then there is a unique bijection

$$\phi :{\Sigma }_1\stackrel{\sim }{⟶}{\Sigma }_2$$

satisfying

$${\beta }_2=\phi \circ {\beta }_1,G_2\circ ({id}_{{\mathcal{X}}_{\mathrm{phys}}}\times \phi )=G_1.$$

Thus the observer-state axis is unique up to a unique behavior-preserving relabeling once the physical-state semantics and complete task are fixed.

Proof. 

Theorem 3.36 gives bijections from each ${\Sigma }_i$ to the same canonical quotient $\mathcal{S}$. Composing one with the inverse of the other yields $\phi$, and surjectivity of ${\beta }_1$ gives uniqueness. □

The theorem separates the origins of the two axes. Physical identity is not obtained by quotienting observation records; it is supplied by r. Observation-state identity is not appended as an arbitrary hidden variable; it is forced by the behavior-minimal quotient of the complete observation-forming family. Their joint appearance is therefore asymmetric in origin but exact in representation.

3.14. Behavioral Minimality Is Not Single-Observation Identifiability

Behavioral minimality says only that distinct observer states index distinct complete observation maps. It does not imply that one output identifies the joint state.

Counterexample 3.38

(A Behavior-Minimal Realization Whose Joint State Is Not Distinguished by One Output). Let $\mathcal{X}=\mathcal{Y}=\{0,1\}$ and take

$$h_0\left(0\right)=0,h_0\left(1\right)=0,$$

$$h_1\left(0\right)=0,h_1\left(1\right)=1.$$

Because $h_0\ne h_1$, the pair $(\{h_0,h_1\},{id}_{\{h_0,h_1\}})$ is a behavior-minimal realization of the map family $\{h_0,h_1\}$. Yet $h_0\left(0\right)=h_1\left(0\right)=0$, so the single observation $y=0$ cannot distinguish the joint states $(0,h_0)$ and $(0,h_1)$.

Thus

$$\begin{array}{c}\mathrm{the}\mathrm{state}\mathrm{realization}\mathrm{is}\mathrm{free}\mathrm{of}\mathrm{redundancy}\\ \cancel{⟹}\mathrm{joint}-\mathrm{state}\mathrm{identifiability}\mathrm{from}\mathrm{a}\sin \mathrm{gle}\mathrm{output}.\end{array}$$

(3.85)

Behavioral minimality asks whether entire functions are duplicated; identifiability asks which states can still produce a specified output. The quantifiers differ and must be treated separately.

3.15. Domain of Validity and the Interface with Classical Realization Theory

Four boundaries of the present construction must be retained throughout the theory.

3.15.1. Observer State Uniquely Determined by the First State

If ${\mathcal{X}}_0\subseteq \mathcal{X}$ and there exists $\sigma :{\mathcal{X}}_0\to \mathcal{S}$ such that

$$\mathcal{B}=\{(x,\sigma \left(x\right)):x\in {\mathcal{X}}_0\},$$

(3.86)

then ${\mathcal{X}}_B={\mathcal{X}}_0$, and the active corestriction

$${\pi }_X^{\mathrm{act}}:\mathcal{B}\twoheadrightarrow {\mathcal{X}}_B={\mathcal{X}}_0$$

is bijective. Defining $h:{\mathcal{X}}_0\to \mathcal{Y}$ by $h\left(x\right):=F(x,\sigma (x\left)\right)$ gives

$$F=h\circ {\pi }_X^{\mathrm{act}}.$$

Thus the present restricted problem can be written exactly as $y=h\left(x\right)$. This is not because observer state has been eliminated in general, but because the admissible domain contains exactly one observer-state value over each active first state.

3.15.2. Observer State with No Output Distinction on the Current Task

If F is constant on every fiber of ${\pi }_X$—equivalently, on every fiber of the surjection ${\pi }_X^{\mathrm{act}}:\mathcal{B}\twoheadrightarrow {\mathcal{X}}_B$—then Theorem 3.25 supplies a unique map $h:{\mathcal{X}}_B\to \mathcal{Y}$ such that

$$F=h\circ {\pi }_X^{\mathrm{act}}.$$

The restricted value map therefore depends only on the active first-state variable. If ${\mathcal{X}}_B\ne \mathcal{X}$, an extension of h to all of $\mathcal{X}$ is additional data not determined by F; it may be nonunique, and when $\mathcal{Y}=⌀$ and $\mathcal{X}\setminus {\mathcal{X}}_B\ne ⌀$, no such extension exists. Observer state may still occur in an underlying system. The factorization shows only that it may be omitted from the codomain-valued map $h:{\mathcal{X}}_B\to \mathcal{Y}$; it need not be removable from the admissibility structure $\mathcal{B}$ or from a different declared task.

3.15.3. Deleting the Original First-State Projection Changes the Type of Problem

If $z=(x,s)$ is treated as one enlarged state and the projection to the original x is deleted, then the observation law can of course be written as $y=G\left(z\right)$. Such a representation can no longer express comparisons with the declared first state held fixed. It is a fixed-physical-state question only when an external physical-state interface supplies that semantics. Deleting the projection is therefore a change in the object of study, not a proof that the second typed role is redundant.

3.15.4. An Incomplete Realization Is Not a Complete Minimal Realization

An injective realization may contain no redundancy, but if it covers only a proper subset of $\mathcal{H}$, it cannot represent all admissible observation behaviors. The uniqueness statement in the canonical minimal-state realization theorem requires both completeness and behavioral minimality.

In the row-and-column quotient branch, the static two-sided table construction is the separated–extensional, or biextensional, reduction of the associated Chu space, and the formation-side realization is the image of the curried behavior map [19,20,21]. The external-physical-state branch instead preserves the independently supplied first-state interface and performs only the compatible formation-side reduction; it need not be separated on the first side. No novelty is claimed for the quotient or image-factorization operations themselves. The contribution is their empirically typed use in observation formation, the separation of feasibility from actual-output difference, and the resulting typed representational responsibility chain. Automata theory adds inputs, transitions, and future continuations; linear realization theory adds linear dynamics, controllability, observability, and state dimension. The present theorem concerns only the behavior-minimal realization of the fixed complete task declared here.

3.16. Interim Conclusion

In the row-and-column quotient branch, the primitive observation table undergoes the two-sided behavioral quotient

$$\mathcal{X}=\mathfrak{P}/{\sim }_X,\mathcal{S}=\mathfrak{A}/{\sim }_S.$$

In the external-physical-state branch satisfying Proposition 3.9, the supplied space $\mathcal{X}={\mathcal{X}}_{\mathrm{phys}}$ is retained through $r:\mathfrak{P}\twoheadrightarrow {\mathcal{X}}_{\mathrm{phys}}$, and only $\mathcal{S}=\mathfrak{A}/{\sim }_S$ is behaviorally minimized. In either branch one obtains

$$\mathcal{B}\subseteq \mathcal{X}\times \mathcal{S},F:\mathcal{B}\to \mathcal{Y}.$$

The common result is a dual-axis interface whose formation side is behavior-minimal; biextensional reduction on the first side is asserted only in the row-and-column quotient branch. The observer-state space $\mathcal{S}$ is in bijection with the complete family of totalized observation maps ${\mathcal{H}}_F$ and is unique, among all complete behavior-minimal realizations of that family, up to a behavior-preserving isomorphism. If one first-state fiber contains two implementable observer states with unequal outputs, every extended encoding that preserves the first-state semantics must retain them as distinct points in that fiber. If the only difference is implementability, that distinction remains part of the totalized observation behavior, but it does not by itself imply that the restricted output law F is irreducibly two-axis.

At this point the reduced observer-state space has been identified by an empirically typed closed-world table, behavioral reduction, and minimal realization. The primitive formation-operation type was supplied by the task; its behavior-minimal state identity was not. The next section adopts this proved structure as Physobser Axiom for downstream theories. Within that interface it then studies observational equivalence, conditional and joint identifiability, fixed-physical-state comparison, and the comparison domain available to later transport models.

4. Physobser Axiom: Identifiability and Physical-State Comparison

4.1. From the Representation theorem to Physobser Axiom

The preceding derivation did not assume a physical–observation dual axis. It began with primitive typed records, separated the two behavioral roles, retained an independently supplied physical-state identity, and then proved that the observation-forming side has a canonical behavior-minimal realization. Theorem 3.36 therefore supplies a theorem-derived interface. In Physobser Theory this interface is adopted as the foundational structural axiom below; the dual-axis structure it defines is a basic structure on which later geometric, dynamical, statistical, and application-specific developments may be built without repeating the reduction.

Axiom 4.1

(Physobser Axiom). A complete Physobser model is a physical–observation dual-axis interface consisting of nonempty typed state sets $\mathcal{X}$ and $\mathcal{S}$, an actual-output set $\mathcal{Y}$, a fresh symbol $\perp \notin \mathcal{Y}$ with

$${\mathcal{Y}}_{\perp }:=\mathcal{Y}\bigsqcup \{\perp \},$$

(4.1)

and maps

$$j_B=({\pi }_X,{\pi }_S):\mathcal{B}\hookrightarrow \mathcal{X}\times \mathcal{S},F:\mathcal{B}\to \mathcal{Y},\tilde{F}:\mathcal{X}\times \mathcal{S}\to {\mathcal{Y}}_{\perp },$$

(4.2)

satisfying the following conditions.

(P1) 

$\mathcal{X}$ carries physical-state identity supplied by an antecedent physical theory, while $\mathcal{S}$ carries the state of observation formation. They are distinct sorts even if their underlying carrier sets happen to be equal.

(P2) 

The coordinate map $j_B$ is injective. Thus the two projections jointly determine every admissible joint state, but no equality $\mathcal{B}=\mathcal{X}\times \mathcal{S}$ is assumed.

(P3) 

With $i_{\mathcal{Y}}:\mathcal{Y}\hookrightarrow {\mathcal{Y}}_{\perp }$ the canonical inclusion,

$$\tilde{F}\circ j_B=i_{\mathcal{Y}}\circ F,j_B\left(\mathcal{B}\right)={\tilde{F}}^{-1}\left(\mathcal{Y}\right).$$

(4.3)

Hence $\mathcal{B}$ is exactly the implementable part of the totalized law.

(P4) 

The behavior map

$${\Gamma }_F:\mathcal{S}⟶{\mathcal{Y}}_{\perp }^{\mathcal{X}},{\Gamma }_F\left(s\right):=\tilde{F}(·,s)$$

(4.4)

is injective. Equivalently, after defining ${\mathcal{H}}_F:={\Gamma }_F\left(\mathcal{S}\right)$, the corestriction ${\Gamma }_F:\mathcal{S}\to {\mathcal{H}}_F$ is bijective. Thus the observer-state axis is complete for the family it realizes and contains no duplicate complete behaviors.

The term Physobser refers precisely to the ordered typed pair $(\mathcal{X},\mathcal{S})$ in this axiom: physical state on the first axis, observation-formation state on the second. The outcome set $\mathcal{Y}$ is not a third state axis. Nor does the axiom assert symmetry, orthogonality, statistical independence, controllability, or a globally rectangular domain.

Remark 4.2

(Logical status and behavioral precursor). Within this foundation paper, Physobser Axiom is the compressed consequence of Theorem 3.36; within a downstream theory it functions as an axiom. If no independently validated physical-state interface is available, the row–column quotient still yields the canonical typed behavioral precursor proved earlier, but its first axis should not be called physical merely because it is observationally nonredundant. Proposition 3.11 gives the exact relation between an anchored physical state and that behavioral shadow.

From this point onward, $\mathcal{B}$ is identified with its image $j_B\left(\mathcal{B}\right)\subseteq \mathcal{X}\times \mathcal{S}$, so its elements are written as pairs $(x,s)$. Under Physobser Axiom, x is a physical state and s is an observation-forming state. A singleton $\mathcal{S}$ is the exact degenerate boundary of the second axis; Theorem 3.36 characterizes when the non-degenerate axis is activated. A weak witness involving ⊥ activates the totalized structure, while irreducibility of the actual-output law requires two implementable states at one physical state with unequal outputs.

The axiom is a structural interface, not a universal empirical theorem. Its applicability to a physical domain depends on the adequacy of the operational typing, the closed-world certification, and the antecedent physical-state semantics. Once these are supplied, the remaining identifiability results are exact consequences of the set maps and projections.

4.2. Set-Theoretic Domain and Three Basic Equivalence Relations

The analysis remains entirely at the set-theoretic level. It uses only sets, maps, projections, equivalence relations, quotient sets, and fiber products; it invokes no probability, topology, differentiation, metric, group action, or connection. These criteria apply to the underlying carrier sets of discrete, continuous, hybrid, or singular models; they make no claim about continuity, smoothness, singular structure, or any other additional regularity. This ordering prevents differential-geometric language from being introduced before the observer state has acquired any smooth structure.

Let the two state sets that actually occur in the admissible joint domain be denoted by

$${\mathcal{X}}_B:={\pi }_X\left(\mathcal{B}\right),{\mathcal{S}}_B:={\pi }_S\left(\mathcal{B}\right).$$

(4.5)

and let the natural projections be

$${\pi }_X:\mathcal{B}\to \mathcal{X},{\pi }_S:\mathcal{B}\to \mathcal{S}.$$

(4.6)

Because $\mathcal{B}$ may be a proper subset of the full product, every subsequent discussion at fixed x or fixed s takes place within the corresponding admissible fiber. No arbitrary cross-combination is presumed to be realizable.

No nonemptiness assumption is imposed on $\mathcal{B}$. If $\mathcal{B}=⌀$, then

$${\mathcal{X}}_B={\mathcal{S}}_B=⌀,F:⌀\to \mathcal{Y}$$

is the empty map. Consequently, identifiability predicates expressed through injectivity of F, quantification over active slices, or quantification over $y\in F\left(\mathcal{B}\right)$ hold vacuously and carry no empirical distinguishing content. A nonvacuous identification claim therefore requires at least one admissible joint state; a pointwise or slice-specific claim additionally requires the relevant active fiber to be nonempty. Under condition (P4), $\mathcal{B}=⌀$ also forces $\left|\mathcal{S}\right|=1$, because condition (P3) then gives $\tilde{F}\equiv \perp$ and all totalized formation-side behaviors coincide.

For the remainder of this section, the $\mathcal{X}$-coordinate is the physical state specified by Physobser Axiom. The same set-theoretic identities may be applied to the earlier behavioral precursor, but only the physically anchored branch supports the physical interpretation used here.

For any set map $f:U\to V$, define the equality relation

$$Eq\left(f\right):=\{(u,u^{\prime })\in U\times U:f\left(u\right)=f\left(u^{\prime }\right)\}.$$

(4.7)

which is an equivalence relation on U. Applied to the dual-axis joint domain, this gives

$$\begin{array}{cc}\hfill Eq\left({\pi }_X\right)& =\{(b,b^{\prime }):{\pi }_X\left(b\right)={\pi }_X\left(b^{\prime }\right)\},\hfill \end{array}$$

(4.8)

$$\begin{array}{cc}\hfill Eq\left({\pi }_S\right)& =\{(b,b^{\prime }):{\pi }_S\left(b\right)={\pi }_S\left(b^{\prime }\right)\},\hfill \end{array}$$

(4.9)

$$\begin{array}{cc}\hfill Eq\left(F\right)& =\{(b,b^{\prime }):F\left(b\right)=F\left(b^{\prime }\right)\}.\hfill \end{array}$$

(4.10)

The first relation means equality of the physical-state projection, the second equality of observer state, and the third equality of observation. Under the external physical-state interface, equality of the first projection is exactly equality of physical state. Their origins are different, and none can replace another.

Let the diagonal relation on the joint domain be

$${\Delta }_{\mathcal{B}}:=\{(b,b):b\in \mathcal{B}\}.$$

(4.11)

Since the two projections jointly determine a joint state,

$$Eq\left({\pi }_X\right)\cap Eq\left({\pi }_S\right)={\Delta }_{\mathcal{B}}.$$

(4.12)

Definition 4.3

(Observational equivalence). For $b,b^{\prime }\in \mathcal{B}$, if $F\left(b\right)=F\left(b^{\prime }\right)$, the two states are said to be observationally equivalent under the present observation law, written $b{\sim }_F{b}^{\prime }$. The corresponding equivalence class is

$${\left[b\right]}_F=F^{-1}\left(\left\{F\left(b\right)\right\}\right).$$

(4.13)

The quotient set

$$\mathcal{B}/{\sim }_F$$

(4.14)

retains only those classes of joint states that can be distinguished by the current output value. By the classical quotient factorization, F factors uniquely through this quotient as an injection into $\mathcal{Y}$. Thus the quotient records the finest joint-state distinction supported by the observation.

4.3. Physical-State Slices and Conditional Identifiability

For each $s\in {\mathcal{S}}_B$, define the admissible physical-state slice

$${\mathcal{X}}_s:=\{x\in \mathcal{X}:(x,s)\in \mathcal{B}\},$$

(4.15)

and

$$F_s:{\mathcal{X}}_s\to \mathcal{Y},F_s\left(x\right):=F(x,s).$$

(4.16)

For each $x\in {\mathcal{X}}_B$, define the admissible observer-state slice

$${\mathcal{S}}_x:=\{s\in \mathcal{S}:(x,s)\in \mathcal{B}\},$$

(4.17)

and

$$F_x:{\mathcal{S}}_x\to \mathcal{Y},F_x\left(s\right):=F(x,s).$$

(4.18)

The two kinds of slice answer two complementary questions: when the observer state is known, can the physical state be distinguished from the output; and when the physical state is known, can the observer state be distinguished from the output?

Definition 4.4

(Conditional Physical-State Identifiability). Given $s\in {\mathcal{S}}_B$, if $F_s$ is injective, the physical state is said to be globally identifiable on the observer-state slice s. For a particular $(x,s)\in \mathcal{B}$, if

$$F_s^{-1}\left(\left\{F(x,s)\right\}\right)=\left\{x\right\},$$

(4.19)

then the physical state x is said to be identifiable from that observation point conditional on the known value s.

Definition 4.5

(Conditional observer-state identifiability). Given $x\in {\mathcal{X}}_B$, if $F_x$ is injective, the observer state is said to be globally identifiable on the physical-state slice x. For a particular $(x,s)\in \mathcal{B}$, if

$$F_x^{-1}\left(\left\{F(x,s)\right\}\right)=\left\{s\right\},$$

(4.20)

then s is said to be identifiable from that observation point conditional on the known value x.

Only set-theoretic global and pointwise identifiability are used here. Local identifiability requires a neighborhood structure; differential identifiability requires smoothness and rank conditions. Rothenberg’s local parameter identifiability and Hermann–Krener local weak observability both depend on mathematical structure stronger than that assumed at the present set-theoretic level [24,25].

Proposition 4.6

(Relational criterion for conditional identifiability). The observer state is identifiable on every fixed-physical-state slice if and only if

$$Eq\left(F\right)\cap Eq\left({\pi }_X\right)={\Delta }_{\mathcal{B}}.$$

(4.21)

The physical state is identifiable on every fixed-observer-state slice if and only if

$$Eq\left(F\right)\cap Eq\left({\pi }_S\right)={\Delta }_{\mathcal{B}}.$$

(4.22)

Proof. 

We prove the first identity. Suppose every $F_x$ is injective, and take

$$(b,b^{\prime })\in Eq\left(F\right)\cap Eq\left({\pi }_X\right).$$

Write the two points as $b=(x,s)$ and $b^{\prime }=(x,s^{\prime })$. From $F\left(b\right)=F\left(b^{\prime }\right)$ we obtain $F_x\left(s\right)=F_x\left(s^{\prime }\right)$, and injectivity gives $s=s^{\prime }$; hence $b=b^{\prime }$. Conversely, suppose the intersection equals the diagonal and some $F_x$ is not injective. Then there exist $s\ne s^{\prime }$ satisfying $F(x,s)=F(x,s^{\prime })$, which gives a point of the intersection off the diagonal, a contradiction. The second identity is proved in the same way. □

Conditional identifiability does not require F to be injective on the entire joint domain. It requires only that observational equivalence contain no nontrivial pair inside the relevant fixed-state fibers.

4.4. Identifiability When the Other State Is Unknown

Conditional identifiability assumes that the other state axis is given. If only y is observed and no observer-state label accompanies it, recovery of the physical state is a stronger requirement.

For any $y\in F\left(\mathcal{B}\right)$, define

$$\begin{array}{cc}\hfill {Amb}_{\mathcal{B}}\left(y\right)& :=F^{-1}\left(\left\{y\right\}\right),\hfill \end{array}$$

(4.23)

$$\begin{array}{cc}\hfill {Amb}_{\mathcal{X}}\left(y\right)& :={\pi }_X\left(F^{-1}\left(\left\{y\right\}\right)\right),\hfill \end{array}$$

(4.24)

$$\begin{array}{cc}\hfill {Amb}_{\mathcal{S}}\left(y\right)& :={\pi }_S\left(F^{-1}\left(\left\{y\right\}\right)\right).\hfill \end{array}$$

(4.25)

These sets give, respectively, the joint states, physical states, and observer states that can produce y.

Definition 4.7

(Physical-State Identifiability without an Observer-State Label). If every $y\in F\left(\mathcal{B}\right)$ satisfies

$$|{Amb}_{\mathcal{X}}\left(y\right)|=1,$$

(4.26)

then the physical state is said to be identifiable from the observation independently of the observer-state label.

Definition 4.8

(Observer-State Identifiability without a Physical-State Label). If every $y\in F\left(\mathcal{B}\right)$ satisfies

$$|{Amb}_{\mathcal{S}}\left(y\right)|=1,$$

(4.27)

then the observer state is said to be identifiable from the observation independently of the physical-state label.

Definition 4.9

(Joint identifiability). If

$$|{Amb}_{\mathcal{B}}\left(y\right)|=1,\forall y\in F\left(\mathcal{B}\right),$$

(4.28)

equivalently, if F is injective, the joint state is said to be globally identifiable.

Theorem 4.10

(Relational criteria for the three forms of label-free identifiability). The following statements hold:

1.

The physical state is identifiable from y independently of the observer-state label if and only if

$$Eq\left(F\right)\subseteq Eq\left({\pi }_X\right);$$

(4.29)

2.

The observer state is identifiable from y independently of the physical-state label if and only if

$$Eq\left(F\right)\subseteq Eq\left({\pi }_S\right);$$

(4.30)

3.

The joint state is identifiable if and only if

$$Eq\left(F\right)={\Delta }_{\mathcal{B}}.$$

(4.31)

If the first two conditions both hold, then the third holds.

Proof. 

Suppose the physical state is identifiable independently of the observer-state label, and take arbitrary $(b,b^{\prime })\in Eq\left(F\right)$. The two states produce the same y, while ${Amb}_{\mathcal{X}}\left(y\right)$ is a singleton; hence ${\pi }_X\left(b\right)={\pi }_X\left(b^{\prime }\right)$. The converse follows immediately from the definition. The second assertion is identical with the roles reversed.

The third assertion is the standard criterion for injectivity of a set map. If the first two assertions both hold, then

$$Eq\left(F\right)\subseteq Eq\left({\pi }_X\right)\cap Eq\left({\pi }_S\right)={\Delta }_{\mathcal{B}}.$$

The diagonal is always contained in $Eq\left(F\right)$, so equality follows. □

Identifiability of the physical state does not require identifiability of the observer state, nor conversely. The joint state is identifiable only when the output uniquely determines both state axes.

4.5. Response Sets and Cross-State Confounding

For each $x\in {\mathcal{X}}_B$, define the set of admissible observations

$${\mathcal{Y}}_x:=\{F(x,s):s\in {\mathcal{S}}_x\}\subseteq \mathcal{Y},$$

(4.32)

For each $s\in {\mathcal{S}}_B$, define

$${\mathcal{Y}}_s:=\{F(x,s):x\in {\mathcal{X}}_s\}\subseteq \mathcal{Y}.$$

(4.33)

Proposition 4.11

(Separation criterion for response sets). The physical state is identifiable independently of the observer-state label if and only if

$$\forall x,x^{\prime }\in {\mathcal{X}}_B,x\ne x^{\prime }⟹{\mathcal{Y}}_x\cap {\mathcal{Y}}_{x^{\prime }}=⌀.$$

(4.34)

The observer state is identifiable independently of the physical-state label if and only if

$$\forall s,s^{\prime }\in {\mathcal{S}}_B,s\ne s^{\prime }⟹{\mathcal{Y}}_s\cap {\mathcal{Y}}_{s^{\prime }}=⌀.$$

(4.35)

Proof. 

If there exist $x\ne x^{\prime }$ with ${\mathcal{Y}}_x\cap {\mathcal{Y}}_{x^{\prime }}\ne ⌀$, then a single output can be produced by two distinct physical states, so physical-state identifiability fails. Conversely, if an output corresponds to two distinct physical states, it belongs to the intersection of their response sets. The observer-state statement is proved dually. □

For $x,x^{\prime }\in {\mathcal{X}}_B$, define the physical-state confounding relation on ${\mathcal{X}}_B$ by

$$x{\bowtie }_{\mathcal{X}}{x}^{\prime }⟺{\mathcal{Y}}_x\cap {\mathcal{Y}}_{x^{\prime }}\ne ⌀.$$

(4.36)

A dual relation may be defined on the observer-state side. A confounding relation is reflexive and symmetric, but need not be transitive; it therefore cannot be used to form a quotient space without a separate proof.

Counterexample 4.12

(A Confounding Relation Need Not Be Transitive). Let

$$\mathcal{X}=\{x_1,x_2,x_3\},\mathcal{S}=\{s_0,s_1\},\mathcal{Y}=\{0,1\},$$

and define the totalized table

with $\mathcal{B}:={\tilde{F}}^{-1}\left(\mathcal{Y}\right)$ and $F:=\tilde{F}{|}_{\mathcal{B}}$. Then

$${\mathcal{Y}}_{x_1}=\left\{0\right\},{\mathcal{Y}}_{x_2}=\{0,1\},{\mathcal{Y}}_{x_3}=\left\{1\right\}.$$

Hence $x_1{\bowtie }_{\mathcal{X}}{x}_2$ and $x_2{\bowtie }_{\mathcal{X}}{x}_3$, but $x_1{\cancel{\bowtie }}_{\mathcal{X}}{x}_3$. Thus the relation “there exists some combination of observer states for which the outputs agree” cannot in general be quotiented out directly.

Observational equivalence ${\sim }_F$ on the joint state is a genuine equivalence relation. Projecting it onto either state axis usually yields only a confounding relation and does not automatically preserve transitivity.

4.6. Conditional Identifiability Does Not Imply Joint Identifiability

Even when observer state is identifiable on every fixed physical-state slice and physical state is identifiable on every fixed observer-state slice, the joint state may still fail to be identifiable.

Counterexample 4.13

(Both Conditional Identifiabilities Hold but Joint Identifiability Fails). Take

$$\mathcal{X}=\mathcal{S}=\{0,1\},\mathcal{B}=\mathcal{X}\times \mathcal{S},\mathcal{Y}=\{0,1\},$$

and define

$$F(x,s)=x\oplus s,$$

(4.37)

where ⊕ denotes addition modulo 2. For every fixed s, $x\mapsto x\oplus s$ is a bijection, and for every fixed x, $s\mapsto x\oplus s$ is likewise a bijection. Yet

$$F(0,0)=F(1,1)=0,F(0,1)=F(1,0)=1,$$

so F is not injective on the joint domain.

This example shows that “identifiable when the other state is known” and “still identifiable when the other state is unknown” are different propositions. It also shows that two joint states differing in both coordinates may nevertheless have the same observation value.

4.7. The Same-Source Relation and Fixed-Physical-State Comparison

Definition 4.14

(Same-Source Relation). For $b_a,b_b\in \mathcal{B}$, if

$${\pi }_X\left(b_a\right)={\pi }_X\left(b_b\right),$$

(4.38)

then the two states are said to form a same-source pair.

Throughout the present framework, “same-source” is a purely structural name meaning equality of the physical-state projection. When the first axis is supplied by an external physical state space, this is exactly equality of physical state. The term does not require the two records to be simultaneous, does not assume common noise, and carries no further identity in time, space, or causality. Its purpose is to isolate pairs with equal physical-state projection from pairs differing in that projection; no intervention or temporal variation is implied.

All ordered same-source pairs form the set-theoretic fiber product

$$\mathcal{B}{\times }_{\mathcal{X}}\mathcal{B}:=\{(b_a,b_b)\in {\mathcal{B}}^2:{\pi }_X\left(b_a\right)={\pi }_X\left(b_b\right)\}.$$

(4.39)

If $b_a=(x,s_a)$ and $b_b=(x,s_b)$, the corresponding pair of observation results is

$$\left(F(x,s_a),F(x,s_b)\right).$$

(4.40)

At the present set-theoretic level, one may assert only that the two results lie on the same physical-state slice. The space $\mathcal{Y}$ still provides no difference, quotient, phase difference, group displacement, or transport operator.

Proposition 4.15

(Same-Source Nonidentifiability Set). The set of all nontrivial pairs for which observer state is unidentifiable at fixed physical state is exactly

$$\left(Eq\left(F\right)\cap Eq\left({\pi }_X\right)\right)\setminus {\Delta }_{\mathcal{B}}.$$

(4.41)

Proof. 

Every pair in this set has the same physical-state projection and the same observation result but consists of distinct joint states; the two points can therefore differ only in observer state. The converse follows directly from the definitions. □

Being same-source is not the same as being observationally equivalent, nor does it guarantee identifiability of observer state. It asserts only equality of the physical-state projection; under the physical-state interface this is equality of physical state. Whether any output difference remains is determined by $F_x$.

4.8. Comparison at Fixed Observer State and General Joint Variation

Dually to the same-source relation, state pairs satisfying

$${\pi }_S\left(b_a\right)={\pi }_S\left(b_b\right)$$

(4.42)

lie in

$$\mathcal{B}{\times }_{\mathcal{S}}\mathcal{B}:=\{(b_a,b_b)\in {\mathcal{B}}^2:{\pi }_S\left(b_a\right)={\pi }_S\left(b_b\right)\}.$$

(4.43)

They have equal observer-state projections and may have different physical-state projections.

Any two distinct admissible joint states stand in one of three basic relations: only their physical-state coordinates differ, only their observer-state coordinates differ, or both coordinates differ. The third case cannot in general be decomposed into a sequence of the first two. If

$$b_a=(x_a,s_a),b_b=(x_b,s_b),$$

then the formal intermediate point $(x_b,s_a)$ or $(x_a,s_b)$ may not belong to $\mathcal{B}$ at all.

Counterexample 4.16

(A Constrained Joint Domain Need Not Be Rectangular). Let

$$\mathcal{X}=\{x_0,x_1\},\mathcal{S}=\{s_0,s_1\},$$

but admit only

$$\mathcal{B}=\{(x_0,s_0),(x_1,s_1)\}.$$

(4.44)

The comparison between the two admissible points cannot be decomposed within $\mathcal{B}$ through either cross-pair $(x_1,s_0)$ or $(x_0,s_1)$, because neither cross-pair is admissible.

Even when both cross-states are admissible, $\mathcal{Y}$ is still only a set at the present set-theoretic level. The expression $F(x_b,s_b)-F(x_a,s_a)$ cannot in general even be formed, much less decomposed into a “purely physical-state term”—physical only under the external interface—and a “purely observer-state term.” Any such operation must be supported by additional algebraic or geometric structure on the response space.

4.9. Same-Source Comparison as the Point of Departure for Observational Geometry

After fixing $x\in {\mathcal{X}}_B$, one has

$$F_x:{\mathcal{S}}_x⟶\mathcal{Y}.$$

(4.45)

For $s_a,s_b\in {\mathcal{S}}_x$, the set-theoretic structure supplies only the two endpoints

$$y_a=F_x\left(s_a\right),y_b=F_x\left(s_b\right).$$

(4.46)

If $y_a=y_b$, the two observer states are indistinguishable on that particular physical-state slice, although their complete maps may differ elsewhere; if $y_a\ne y_b$, the two observer-state values index unequal outputs on that slice. Neither case by itself asserts a causal effect or temporal change, and set structure alone cannot define a canonical relative quantity.

Physobser Axiom supplies no ready-made transport equation; it supplies only the canonical set-theoretic fixed-physical-state comparison domain on which a later transport model may be posed once the required additional structure is given:

$$\mathcal{B}{\times }_{\mathcal{X}}\mathcal{B}.$$

Differences, ratios, phases, group displacements, connections, and transports between the endpoints become meaningful only when the observational response has additional comparable algebraic or geometric structure. The structural identifiability of Bellman–Åström, Rothenberg’s statistical identifiability, Hermann–Krener observability, and system identification theory answer, in their respective settings, whether states can be distinguished from input–output behavior [23,24,25,26,27]. They do not automatically provide a consistent conversion rule between same-source representations.

The preceding results constitute the set-theoretic foundations of observational geometry: the dual axes specify two typed state roles, the admissible joint domain records which task-declared pairs are certified implementable, and the same-source fiber product specifies the comparison domain at fixed physical state under the external physical-state interface. Phase, group-action, bundle, and connection structures do not follow from these set-level objects; their additional inputs are stated below.

4.10. Domain of Validity, Counterexample Boundaries, and Terminological Constraints

The conclusions of this section rest only on the set map $F:\mathcal{B}\to \mathcal{Y}$ and its two natural projections. The following boundaries therefore apply.

First, behavioral minimality does not imply identifiability from a single observation. Joint identifiability may fail even when conditional identifiability holds in both directions.

Second, conditional identifiability differs from label-free identifiability. Injectivity of $F_s$ shows only that physical states can be distinguished when observer state is known. If response sets under different observer states overlap, erasing the state label still creates physical-state confounding.

Third, the confounding relation obtained after projection is generally not an equivalence relation. Only ${\sim }_F$ on the joint state is guaranteed to be transitive by equality of function values. A quotient on either state axis requires a separate proof of a genuine equivalence relation.

Fourth, same-source means only equality of the physical-state projection. It is not observational equivalence, identifiability, or the prior existence of a transport. Under the physical-state interface, equality of the physical-state projection is equality of physical state; no causal source of variation is thereby identified.

Fifth, Physobser Axiom does not require observer state to be controllable, known, or variable. A fixed observer state is an exact slice; a hidden or unrecorded observer state may become a nuisance or erasure variable once a statistical model is supplied; an actively selected observer state may enter protocol design once admissible control operations are specified. These cases share the formal dual-axis interface but belong to different later models.

Sixth, the present section does not invoke local identifiability, rank deficiency, Fisher information, or stabilizers. Local and differential conclusions require topology or smooth structure; Fisher information requires a statistical model; stabilizers require a group action. Mathematical structure should be introduced only when the objects under study actually possess the corresponding properties.

4.11. Summary and the Interface with Geometric Transport

Physobser Axiom is summarized by the typed span

$$\mathcal{X}\stackrel{{\pi }_X}{\leftarrow }\mathcal{B}\stackrel{{\pi }_S}{\to }\mathcal{S},F:\mathcal{B}\to \mathcal{Y},$$

where $\mathcal{X}$ carries physical-state identity and $\mathcal{S}$ carries the behavior-minimal state of observation formation. Observational equivalence is $Eq\left(F\right)$; equality of physical state and equality of observer state are $Eq\left({\pi }_X\right)$ and $Eq\left({\pi }_S\right)$. Their intersections and inclusions give exact criteria for conditional, label-free, and joint identifiability.

All ordered comparisons at fixed physical state form the fiber product

$$\mathcal{B}{\times }_{\mathcal{X}}\mathcal{B}.$$

This object specifies where a later same-source transport law may be posed; it does not itself supply a difference, ratio, phase, group displacement, or connection. A geometric theory must add the structures that make such relative quantities meaningful. In particular, a complex response may support phase only after nonvanishing and normalization are specified, while parallel transport requires a base, a bundle, and a connection. None of these follows from the set-theoretic axiom alone.

5. Discussion

The proofs use elementary objects—functions, equivalence relations, quotients, factorization through fibers, and fiber products—but their order is decisive. The operational task first supplies two roles. A fixed-physical-state witness then tests whether the observation-forming role can be deleted. Behavioral quotienting removes label redundancy without erasing the type distinction. The physical interface preserves antecedent physical identity, while the formation quotient supplies the unique nonredundant observer-state realization. Only after these steps is the physical–observation interface adopted as an axiom.

This order resolves three recurrent ambiguities. First, concatenating $(x,s)$ into one variable is syntactic unarization, not proof of structural reducibility. Any exact encoding that preserves physical-state semantics must retain output-distinguishable points within the same physical fiber. Second, observational equivalence is not physical identity. The row-behavior quotient is a behavioral shadow of physical state and may be strictly coarser; Proposition 3.11 measures that distinction exactly. Third, behavior-minimality is not single-observation identifiability. Distinct observer states can represent different complete maps while agreeing at one physical state, and both conditional identifiabilities can hold while joint identifiability fails.

The nonimplementability value ⊥ is mathematically useful only because its empirical meaning is fixed. It marks certified nonimplementability, not ignorance. A difference involving ⊥ is sufficient to distinguish totalized observation behaviors and therefore to prevent collapse of the complete formation quotient. It is not sufficient to make the restricted output law irreducibly two-axis; that stronger conclusion requires unequal implementable outputs in one physical-state fiber.

The observer-state terminology names a responsibility, not a species of device. A physical, algorithmic, or hybrid state occupies the role when it selects the complete observation-formation map represented at the interface. The state may be fixed, recorded, hidden, estimated, or controlled in later models. Those regimes require different statistical or dynamical assumptions, but they share the same set-theoretic responsibility.

Physobser Axiom is therefore neither a claim of symmetry nor a declaration that every conceivable observation is two-dimensionally Cartesian. It is the foundational structural statement of Physobser Theory under the hypotheses proved here. The axes are typed, generally asymmetric in origin, and may live on a constrained joint domain. The physical axis is supplied by antecedent physics; the observation axis is derived from behavior. Their conjunction defines a basic dual-axis structure of the theory, and its empirical adequacy remains testable in each application.

6. Conclusion

Starting from a complete empirically typed table, we proved the representational boundary of a physical-state-only observation law and constructed the canonical reduction that lies beyond it. The row–column quotient is terminal among surjective typed exact factorizations; biextensional realizations are uniquely isomorphic; and typed isomorphisms or redundant exact covers preserve the reduced interface. When physical identity is supplied independently, it is retained through a compatible surjection rather than replaced by observational equivalence. The resulting observer-state quotient is terminal among physical-state-preserving exact realizations and is unique up to a unique behavior-preserving relabeling.

The Set-Theoretic Physobser Representation Theorem isolates the central equivalence: observation-forming non-triviality, non-degeneracy of the state set $\mathcal{S}$, and failure of physical-projection factorization of the totalized law are the same statement at the set-theoretic level. A strong witness gives the corresponding obstruction for the actual-output law. The admissible domain is the pullback of actual outputs, and the kernel conditions for exact encodings state precisely what may be hidden and what cannot be merged.

Within Physobser Theory, Physobser Axiom compresses these results into the ordered physical–observation interface

$$\mathcal{X}\stackrel{{\pi }_X}{\leftarrow }\mathcal{B}\stackrel{{\pi }_S}{\to }\mathcal{S},F:\mathcal{B}\to \mathcal{Y}.$$

Within this interface, conditional, label-free, and joint identifiability are exact relations among the kernel equivalences of F, ${\pi }_X$, and ${\pi }_S$. The fiber product $\mathcal{B}{\times }_{\mathcal{X}}\mathcal{B}$ is the canonical domain of comparison at fixed physical state. Geometry begins only after additional algebraic, topological, or smooth structure is supplied. The physical–observation dual-axis structure is thus established as a basic set-theoretic structure of Physobser Theory. The present foundation ends where those further geometric, probabilistic, or dynamical choices begin.