Discussion and scope

Discussion and scope

What has been established

The results above are elementary in their machinery and, we would argue, correspondingly robust. Their content can be compressed into four statements.

Obstruction is target-relative. A noninjective readout is not, as such, defective. It fails a target exactly when the correct answer to that target varies within one of its fibers (Theorem 3.1, Corollary 3.2) , that is the general criterion, and it is stated without auxiliary hypotheses. When a mediating variable $r$ is nominated, so that $a_Q=\Phi_Q(p,r)$ as in (7), the same criterion localizes: the loss of $r$ obstructs $Q$ exactly when $r$ is active for $Q$ on some fiber (Theorem 4.3, Theorem 4.4). The same readout may pass one target and fail another (Example 9.2 versus Theorem 10.4). Consequently the inference from “information was lost” to “the description must be completed” is invalid without a nominated target, and we have been careful never to make it.

Impossibility is derived, not assumed. Every impossibility claim in this paper reduces to the same counting observation: a deterministic construction from the readout has the form $b\circ p$ and therefore assigns one answer per fiber, while the correct-answer map may assign two. No hypothesis of the form “no replacement exists” is anywhere invoked, and Proposition 3.4 closes the natural attempt to evade the conclusion by manufacturing derived readout variables.

Without completion, there is still a best answer, and it is a set. The compatible-answer set $\mathsf{A}_Q(t)=a_Q(F_t)$ is the pointwise-least uniformly sound set-valued rule (Theorem 5.4), and exactness is equivalent to its being everywhere a singleton (Corollary 5.5). This is a set-theoretic fact prior to any probabilistic modeling; a probability law over $\mathsf{A}_Q(t)$ ranks its elements but does not merge them and does not identify which source state obtained (Remark 5.6, Remark 12.11).

With completion, there is a least sufficient one. Every $Q$-sufficient extension must separate states with different answers (Theorem 7.1), and the joint target image $E_Q=\operatorname{Im}(p,a_Q)$ both is sufficient (Proposition 8.2) and receives a unique surjection from every sufficient extension (Theorem 8.3). It is thus the coarsest completion for the target, and it retains the readout value and the answer distinction and nothing else. In particular it does not reconstruct the source: Corollary 8.4 runs $E\to E_Q$, and the identity extension $\operatorname{id}_S$ is one of the $E$’s.

The example as a test of the theory

The flat $\mathrm{U}(1)$ example is not decoration. It exhibits, in a system where every object is computable in closed form, all four statements at once. The reduced source space $S_{\mathrm{flat}}\cong\mathrm{U}(1)$ is a genuine moduli space, obtained only after quotienting by large gauge transformations (30), so its points are physically inequivalent. The curvature readout (34) collapses it entirely. Against the flatness target the collapse is inert (Example 9.2); against the spectral target it is fatal (Theorem 10.4), and the fatality is a measurable one , the ground-state energy jumps from $0$ to $\pi^2/L^2$ between $\theta=0$ and $\theta=\pi$, with the eigenvalue multiplicities changing as recorded in Remark 10.2. Yet the completion demanded by the spectral target is strictly less than the source: Theorem 11.2 shows the spectrum determines $\theta$ only up to sign, so Theorem 11.4 identifies the coarsest sufficient completion as the inversion quotient, coordinatized by $\cos\theta$. And this coarsest completion is genuinely insufficient for a different, entirely physical target on the same space (Example 11.6): oriented transport requires the whole holonomy circle, because it can tell $e^{i\pi/2}$ from $e^{-i\pi/2}$ and the scalar spectrum cannot.

Scope, and what is not claimed

We close by delimiting the results, since each of the following overreaches is easy and none is supported.

We have not shown that every lossy description requires repair; Theorem 4.4 shows the opposite is routinely the case. We have not shown that the source is recoverable from any completion; Corollary 8.4 explicitly denies it. We have not shown that curvature is a poor physical observable, only that it is a poor readout for the spectral question, which is a claim about a pair (Remark 9.1). We have not shown that $\cos\theta$ is canonical among numerical coordinates or minimal for all targets (Remark 11.5, Example 11.6). We have not shown that stochastic reduced dynamics is illegitimate, only that it does not recover the realized trajectory and does not answer the target we posed (Remark 12.11). And we have not made, and do not make, any claim to have established that physical reality is a projection of a hidden source; the mathematics here constrains what a given projection can and cannot deliver, given a nominated target, and that is all it does.

Within the sequence of which this paper is a part, the role of the present results is to supply the discrimination that the descent theorem of [8] alone does not provide. Paper 1 identified when a source relation descends; this paper identifies when the failure to descend matters, and what a repair must contain. Two natural generalizations are deliberately deferred. Passing from a single correct-answer map to a family of realization relations, and constructing the canonical minimal completion for such a family, requires machinery not developed here and is the subject of [9]; the results above concern a single target throughout, and their minimality claims should be read accordingly. Passing from the finite-time statement of Theorem 12.3 to continuous-time or semigroup dynamics, and to the exact memory equations that govern projected evolution when (59) fails, likewise lies outside the present scope [10]; Section 12 proves exactly one thing about dynamics , the existence criterion for the finite-time propagator $U_\tau$ , and no more.