Non-Source Projection and Internal Identifiability

The internal observation ceiling

Jeremy RodgersSection 8 of 14Original PDF ↗Publication DOI ↗

The internal observation ceiling

Assumption 7.1 (Internal-probe factorization).

Every admissible probe factors through the readout: for each $i\in I$ there is $\bar P_i\colon T_1\to Y_i$ with $P_i=\bar P_i\circ p .\tag{27}$

This is an assumption, supplied by the model, and it is the entire engine of what follows. It says that a probe may process, compare, transform, amplify, or combine readout information in any way whatever , $\bar P_i$ is an arbitrary map, subject to no continuity, computability, or complexity restriction , but that it receives no input from outside the readout state. A probe coupling directly to a fiber coordinate violates (27), and the theorem below then simply does not apply to it.

It is tempting to say that an observer “built out of Tier-1 material”, or “interacting only with Tier-1 degrees of freedom”, automatically satisfies (27). That is an unsupported inference, and we do not make it. An observer of that description is represented by Assumption 7.1 only if the physical model independently establishes that the complete input of every admissible probe factors through $p$ . Being called internal, or being composed of readout-level constituents, does not by itself prove the factorization: whether some residual coupling to a fiber coordinate exists is a substantive question about the model’s dynamics and interactions, and it must be settled there, not by the label. Assumption 7.1 is thus a hypothesis to be discharged, and §10 lists it among the things a physical instantiation must supply.

Theorem 7.2 (Internal identifiability limit).

Under Assumption 7.1, $\sim_p\;\subseteq\;\sim_I .\tag{28}$ Consequently:

If $p$ is noninjective, the internal probe family does not identify the source.
For every map $D\colon\mathop{\mathrm{Im}}P_I\to Z$ , statistic, post-processing rule, inference procedure, deterministic selector, or arbitrary function , and every $s,s'$ with $p(s)=p(s')$ , one has $D(P_I(s))=D(P_I(s'))$ .

Proof. Let $p(s)=p(s')$ . For every $i\in I$ , using (27), $P_i(s)=\bar P_i(p(s))=\bar P_i(p(s'))=P_i(s').$ Since this holds for all $i$ , $P_I(s)=P_I(s')$ , i.e. $s\sim_I s'$ . This is (28).

(L1): if $p$ is noninjective there are $s\neq s'$ with $p(s)=p(s')$ , hence $s\sim_I s'$ by (28), so $\sim_I\neq\Delta_S$ and Theorem 6.2(I4) fails.

(L2): $P_I(s)=P_I(s')$ was just shown; applying the function $D$ to equal arguments yields equal values. Note that no property of $D$ beyond being a function is used , which is exactly why no cleverness in post-processing can evade the conclusion. ◻

Corollary 7.3 (No internal selector).

If a readout fiber contains two physically inequivalent source states, then no inference procedure whose complete input factors through the readout can determine which member of that fiber was realized.

Proof. Immediate from (L2) with $Z=S$ and $D$ the putative selector: it returns the same value on $s$ and on $s'$ , so it is wrong on at least one of them. ◻

Remark 44 (Exactly what this ceiling is, and is not).

Five clarifications, each of which matters.

(a) The mechanism is equality of inputs. The obstruction in Theorem 7.2 arises because the probes’ arguments coincide: $P_I(s)$ and $P_I(s')$ are the same object. Nothing is being said about the difficulty, cost, or computability of any inference. A function of equal inputs returns equal outputs; that is all.

(b) It is not Gödel incompleteness. There is no self-reference, no arithmetization, no diagonalization, and no undecidable sentence anywhere in the proof. The result is an identifiability theorem in the sense of [9, 10], and conflating it with [6] is a category error.

(c) It does not apply to a genuinely additional probe. If some probe $P_j$ does not factor through $p$ , if it couples to the fiber coordinate , then Assumption 7.1 fails for the enlarged family, (28) need not hold, and by Proposition 6.4 the joint fibers may strictly shrink, possibly to the diagonal. The ceiling is a consequence of the assumed input restriction, and it is lifted exactly by lifting that restriction.

(d) It is conditional on completeness of the admissible family. Assumption 7.1 asserts that every probe available to the observer factors through $p$ . That assertion is a substantive physical claim about the model, and it can be false. Consequently Theorem 7.2 must not be inflated into a general claim that internal observers can never obtain evidence about a source. It concerns exactly those source distinctions that lie within the joint fibers of all allowed probes , and it says that those distinctions, and only those, are internally unrecoverable.

(e) It is a deterministic ceiling, and does not by itself yield the statistical one. Theorem 7.2 concerns deterministic probe values $P_i(s)$ and deterministic functions $D$ of them. What it rules out is accordingly a deterministic selector, and any post-processing, statistic, or inference rule applied to those probe values. It does not, on its own, license the far stronger assertions that “no test” or “no amount of data” can discriminate. Those are claims about outcome laws, and they require the separate hypothesis $\mu_s=\mathsf M_{p(s)}$ of (30) , namely, that the complete data law of the experiment factors through $p$ . One may of course stipulate that the complete experiment is itself one of the admissible probes, in which case (27) for that probe is (30); but that identification is an additional modelling step, and must be made explicitly. Absent it, a random experiment whose law depends on the fiber coordinate is perfectly compatible with Assumption 7.1 holding for every deterministic probe. The two ceilings are stated separately, and proved from separate hypotheses, in Theorem 7.2 and Corollary 8.4; see also Proposition 8.6(R3).