Synthesis

Synthesis

The architecture, assembled, reads as follows. Each line is a separate theorem, and each contributes something the others do not. $$\begin{aligned} &\text{physical reduction} && \Omega\longrightarrow S=\Omega/\!\simeq_{\mathrm{phys}} &&\text{(Def.\ \href{#def:source}{2.1})}\\ &\text{exact readout} && S\xrightarrow{\;p\;}T_1\cong S/\!\sim_p &&\text{(Lem.\ \href{#lem:induced}{2.3})}\\ &\text{descent test} && r=\bar r\circ p \iff r\ \text{fiber-constant} &&\text{(Lem.\ \href{#lem:descent}{3.2})}\\ &\text{completeness} && \text{(C1)}\Leftrightarrow\cdots\Leftrightarrow\text{(C5)} &&\text{(Thm.\ \href{#thm:completeness}{3.4})}\\ &\text{target obstruction} && p(s_0)=p(s_1),\ a_Q(s_0)\neq a_Q(s_1) &&\text{(Thm.\ \href{#thm:nonsource}{4.5})}\\ &\text{minimal completion} && E_Q=\mathop{\mathrm{Im}}(p,a_Q),\ \text{terminal} &&\text{(Thm.\ \href{#thm:universal}{4.6})}\\ &\text{geometric realization} && E_A^{\mathrm{geo}}=\mathop{\mathrm{Im}}(p_{\mathrm{geo}},J_A) &&\text{(Ex.\ \href{#ex:geo}{2.8})}\\ &\text{observable quotient} && \mathcal{H}_{\mathrm{src}}/\ker K\cong\mathop{\mathrm{Ran}}K &&\text{(Ex.\ \href{#ex:linear}{2.9})}\\ &\text{dynamical descent} && p\Phi_t=U_tp \iff \text{fibers preserved} &&\text{(Thm.\ \href{#thm:dichotomy}{5.2})}\\ &\text{linear form} && U(t)\ker K\subseteq\ker K,\quad PLQ^{\!\perp}=0 &&\text{(Prop.\ \href{#prop:invariance}{5.4})}\\ &\text{identifiability} && \textstyle\bigcap_{i}\sim_{P_i}\;=\;\Delta_S &&\text{(Thm.\ \href{#thm:identifiability}{6.2})}\\ &\text{internal ceiling} && P_i=\bar P_i\circ p \implies \sim_p\subseteq\sim_I &&\text{(Thm.\ \href{#thm:ceiling}{7.2})}\\ &\text{statistical ceiling} && \mu_s=\mathsf M_{p(s)} \implies \|\mu_{s_0}-\mu_{s_1}\|_{\mathrm{TV}}=0 &&\text{(Cor.\ \href{#cor:statceiling}{8.4})}\\ &\text{Markov projection} && \textstyle\sum_{j\in B_b}g_{ij}\ \text{independent of}\ i\in B_a &&\text{(Thm.\ \href{#thm:lumpability}{9.3})} \end{aligned}\tag{38}$$

Read as a whole, (38) makes four points that no single line makes alone.

The source/readout distinction is not inferred from the word “projection.” It is established, for a specified model, by a physically reduced nontrivial fiber carrying an essential distinction , and by nothing less.

Completion is not arbitrary extra structure. $E_Q$ is the terminal object among exact extensions sufficient for $Q$ (Theorem 4.6): it is forced, not chosen, and every adequate extension surjects onto it.

Geometric realization is a restriction, not an automatic step. In a geometrically realized source, the completion takes the form $E_A^{\mathrm{geo}}=\mathop{\mathrm{Im}}(p_{\mathrm{geo}},J_A)$ of (7), whose elements are exactly the completed signatures attained by compatible global geometric fields. Not every abstract signature is so realized: field equations, global compatibility, and topological obstructions restrict the image. Which signatures survive is a geometric question, answered by geometry and not by the quotient formalism.

The three failures are ordered, not independent. They stand in the exact relations of Proposition 8.6. Fiber-breaking dynamical nonclosure requires two distinct source states sharing a readout, so it implies kinematic incompleteness: noninjectivity of $p$ is necessary for it. The converse fails , Example 5.5 and Example 9.6(i) are kinematically incomplete yet dynamically closed , so noninjectivity is not sufficient, and, equivalently, autonomy of the projected law does not imply injectivity of $p$. Statistical nonidentifiability, by contrast, has no unconditional relation to $p$ in either direction; it acquires one only under the outcome-law factorization hypothesis (30), and under that hypothesis noninjectivity of $p$ implies statistical nonidentifiability within at least one fiber. The one moral that survives all three cases is negative: autonomy of the projected law is evidence of fiber preservation, never of fiber triviality.

The ceiling is exactly as wide as the input restriction. Theorem 7.2 bites only under Assumption 7.1, and a genuinely new probe , one not factoring through $p$ , lifts it. Source identification is possible exactly when the joint probe family separates physical source states, and not otherwise.