Introduction

Introduction

The problem

A recurrent structure in physical modelling is the following. A theory posits a space of realizations, declares some of its differences to be pure redundancy (gauge, coordinate choice, relabelling), and then describes what is actually read off , measured, recorded, or lived , by a map into a smaller space of observable states. The observable description may be perfectly self-consistent: its states may evolve by autonomous laws, its probes may be mutually coherent, and its predictions may be exact. None of that settles the question of whether the observable space is the physical state space or merely a faithful quotient of a larger one.

This paper isolates the exact mathematical content of that question. We work with a surjection $$p\colon S\longrightarrow T_1,$$ where $S$ is a source space already reduced by the physical equivalence the model declares, and $T_1$ is the readout. Because the reduction has been performed first, a nontrivial fiber $p^{-1}(t)$ is not a residue of gauge freedom: its distinct points are physically inequivalent by the model’s own standard. The question is then whether such a fiber matters, and what an observer restricted to $T_1$ could ever learn about it.

Contribution

We prove a connected sequence of results, each stating a different location at which the source/readout distinction becomes visible or provably invisible.

1. Source completeness (§3). Under the hypothesis that a nominated family of invariant relations separates points of $S$, five conditions coincide: injectivity of $p$; singleton fibers; descent of every relation through $p$; bijectivity of the canonical completion projection $u_A\colon E_A\to T_1$; and existence of a two-sided inverse of $p$ (Theorem 3.4). The point-separating hypothesis is what makes the equivalence a theorem rather than a definition, and we mark precisely where it and surjectivity are used.

2. Non-source projection (§4). If some fiber carries a distinction that changes the correct answer to a target $Q$, then $p$ is noninjective, no deterministic function of the readout answers $Q$ correctly on all of $S$, the minimal target completion $E_Q$ properly refines $T_1$, and every exact extension sufficient for $Q$ maps uniquely and surjectively onto $E_Q$ (Theorem 4.5, Theorem 4.6). We prove the universal property directly, so that the manuscript is self-contained. Relation, bounded-linear, and equivariant-selector forms follow.

3. Projected-law dichotomy (§5). Deterministic source evolution either preserves readout fibers, in which case it descends uniquely to an autonomous readout law, or it does not, in which case no deterministic law on the instantaneous readout state reproduces every source trajectory (Theorem 5.2). We emphasize the negative corollary: coherent autonomous projected laws establish dynamical preservation of readout classes, not that those classes are singletons.

4. Identifiability, and two distinct ceilings (§§6–8). A probe family identifies the source iff its joint map separates points; otherwise the maximally identifiable object is $S/\!\sim_I\;\cong\;\mathop{\mathrm{Im}}P_I$, and adding probes refines monotonically. Two ceilings then follow from two separate hypotheses, and we are careful not to derive either from the other. The deterministic ceiling assumes that every admissible probe factors through $p$, i.e. $P_i=\bar P_i\circ p$; it rules out deterministic selectors and any post-processing of those probe values (Theorem 7.2). The statistical ceiling assumes instead that the complete outcome law factors through $p$, i.e. $\mu_s=\mathsf M_{p(s)}$; only this yields the stronger conclusion that no test, at any sample size, with any randomization, can discriminate (Corollary 8.4).

5. The exact relations among the obstructions (§8.1). Kinematic incompleteness, dynamical nonclosure, and statistical nonidentifiability are not logically independent. Fiber-breaking implies noninjectivity but not conversely; statistical nonidentifiability is unrelated to $p$ unless an outcome-law factorization is imposed, under which noninjectivity implies it within a fiber (Proposition 8.6).

6. Exact Markovian projection (§9). For a finite-state continuous-time chain, the projected process is Markov for every initial source distribution iff the block-sum (strong lumpability) condition holds; we prove necessity from instantaneous block rates and sufficiency at the process level, not merely for one-time marginals (Theorem 9.3).

7. The instantiation boundary (§10). We state exactly what a physical model must supply, and exactly what the theorems do and do not license.

Relation to established work

Every ingredient used below is classical, and we claim none of them. The factorization of a map through the quotient by its kernel pair, and the universal property of images, are elementary category- and set-theoretic facts [11]. Descent of functions through a quotient, and invariance of a subspace under a semigroup, are standard [5]. The Hilbert-space statement $\mathcal{H}_{\mathrm{src}}/\ker K\cong\mathop{\mathrm{Ran}}K$ for bounded $K$ with closed range is the closed-range theorem [14]. The projection formalism producing initial-state and memory terms for a non-invariant projector is due to Nakajima, Zwanzig and Mori [13, 15, 12]. The identification of observability with injectivity of the state-to-output map, which is the form our linear obstruction takes in Corollary 4.9, is Kalman’s [7]. Sufficiency, statistical identifiability, and the equivalence of equal laws with equal risk for all decision rules are Blackwell’s and Le Cam’s [2, 9, 10]. Strong lumpability of Markov chains is due to Kemeny and Snell [8], with the continuous-time form in Ball and Yeo [1] and Buchholz [3]. The obstruction to an equivariant section by empty fixed-point sets is standard equivariant topology [4].

What is new here is the integration: the assembly of these ingredients into a single source–readout architecture in which (i) the criterion for non-source projection is stated on a physically reduced domain, so that a nontrivial fiber cannot be dismissed as gauge; (ii) the target obstruction is integrated with a canonical minimal completion that is terminal among sufficient extensions; (iii) the exact logical relations among kinematic incompleteness, dynamical nonclosure, and statistical nonidentifiability are established , these are not independent, and Proposition 8.6 states precisely how they are ordered; (iv) the internal-probe factorization hypothesis is isolated, as an explicit hypothesis, as the exact source of the identifiability ceiling; and (v) the boundary between a model-internal theorem and a claim about the world is drawn explicitly and defended.

We claim no novelty for any component criterion. Quotient descent, invariant-subspace conditions for projected closure, statistical identifiability, and strong lumpability are established mathematics, due to the authors cited above; the descent lemma and the lumpability criterion in particular are classical and are reproved here only so that the manuscript is self-contained.

Companion papers of the same programme [16, 17, 18, 19, 20] are cited for context only. No proof below depends on any of them: where the source material appealed to a companion result, we supply the proof here.

Scope

We state the limitation once, in full, and repeat it where it bites.

Every theorem below is conditional on a supplied quadruple , a source domain, a physical-equivalence relation, a readout, and (as appropriate) a target, a dynamics, or a family of outcome laws. The mathematics does not generate these objects. In particular, Theorem 4.5 does not prove that any actual physical system possesses a non-source projection; it proves that a model exhibiting an essential non-gauge fiber distinction is one, relative to that model and that target. Noninjectivity of an arbitrarily impoverished descriptor is not a witness; neither is a map merely named a projection. The conversion of the theorem into a claim about nature requires physical instantiation of its hypotheses and empirical discrimination of its alternatives, and we do not perform either. Section 10 is devoted to this boundary.