Complementarity Before Quantum Theory: Assumption Control, Countermodels, and Conditional Complex Closure

Abstract

Complementarity can organize a relational starting point for quantum reconstruction, but a completion relation does not yet supply preparations, effects, probabilities, continuous transformations, composites, or a scalar field. We formulate the missing passage as an assumption-control problem. Beginning from the typed primitive $\mathcal{C}=[a\dashv b]$, we separate incidence, operationalization, continuity, scalar coherence, transition probability, effects, instruments, composition, purification, and process identification. Every negative reconstruction bridge is tied to an exact premise package and a typed target; geometric or pre-operational witnesses are not promoted into countermodels of a larger theory they do not instantiate. At exact complex-closure resolution, let $A_{\mathrm{OPT}}$ denote the finite-dimensional causal operational framework of an external reconstruction theorem, $L$ local equivalence, and $E$ equivalent-system purification. Finite classical theory witnesses $A_{\mathrm{OPT}}+L$ without $E$, while finite-dimensional real quantum theory witnesses $A_{\mathrm{OPT}}+E$ without $L$; the external theorem closes only $A_{\mathrm{OPT}}+L+E$. Neither selector is derived from primitive complementarity here. The real-quantum stress test is sharpened by the two-rebit decomposition $\mathrm{Sym}(4)=\mathrm{Sym}(2)\otimes \mathrm{Sym}(2)\oplus\mathbb{R}(J\otimes J)$: product effects see a nine-dimensional shadow of a ten-dimensional carrier. The identity setting together with one additional algebraically transverse joint-interaction setting gives complete terminal-product state exposure. This later observability remains distinct from original local tomography, universal process tomography, and complex-field selection. The contribution is a source-generated dependency architecture, exact finite witnesses, and a precise selector-origin problem: complex closure is formally conditional, while the physical origin of local equivalence and equivalent-system purification remains open.

The selector structure

The compressed logical shape of the complex-closure result:

Neither $L$ nor $E$ is a descendant of primitive complementarity. That is the precise sense in which the route to complex quantum theory is conditional: the theorem is valid, and the origin of its premises is the open problem.

Where it sits in the release

CFQF-Q4 controls the premise ledger for the whole quantum branch. It receives the primitive from CF-F1 and supplies reconstruction context to the three focused studies: two-rebit readout, certificate-relative objectivity, and relational time.

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