Abstract
Quantum theory combines three structures that are often introduced separately: complex norm-preserving evolution, entanglement of composite systems, and quadratic weights for measurement outcomes. This paper asks whether a complementarity-first architecture can organize their relation without importing one structure under the name of another. The motivating image is the Taichi Diagram: two distinguishable sides and two internal “eyes,” with each side carrying a trace of its complement. The image formulates a question; it is neither a historical anticipation nor a physical proof.
On a supplied two-branch positive composite, a fixed-eye state family separates diagonal reciprocal inclusion from an off-diagonal coherent link. Positivity bounds that link, and within the declared two-qubit support the state is entangled exactly when the link is nonzero. On a supplied real two-plane, complement exchange as reversal admits elliptic, hyperbolic, nilpotent, and frozen generators. A non-fixed two-sided bounded orbit selects the elliptic branch, a positive invariant quadratic capacity, and a one-frequency Schrödinger normal form. One such plane is only one complex line and is projectively trivial, so relative phase and an entangling sector require an explicit aligned two-block lift. On that four-real-dimensional carrier, the complex-linear branch swap is distinguished from the conjugating reciprocal reversal . A supplied exchange/XY Hamiltonian is -even; its Schrödinger generator is -odd, its unitary flow is reciprocal under , and it produces an exact orbit from separability through maximal entanglement and back.
Separately, passive repeatable lossless separators become orthogonal projections, while a nonnegative modular readout axiom uniquely linearizes channel response and yields a finite pure-projective Born-form valuation. This result is conditional and does not establish the density-operator trace rule, general effects or instruments, one actual outcome, or objective chance. Environmental restriction suppresses accessible pair coherence by an overlap factor and explains how a larger entangled whole can cast a classical-looking local shadow. The contribution is an assumption-controlled synthesis and countermodel chain, including an explicit carrier bridge and a typed distinction between branch exchange and reciprocal reversal; the component two-qubit, planar-flow, exchange, projection, and additive-response formulas are not claimed as individually new.
The three-cut architecture
The proposal is not the flat claim that entanglement produces probability. The paper keeps three separate steps, and records what each one needs:
| step | what it takes | what it yields |
|---|---|---|
| reciprocal inclusion → complex motion | a non-fixed two-sided bounded orbit | the elliptic branch and a one-frequency Schrödinger normal form |
| one complex line → an entangling sector | an explicit aligned two-block lift | relative phase; a projectively non-trivial carrier |
| coherent whole → finite weights | a nonnegative modular readout axiom | a finite pure-projective Born-form valuation |
One complex line is projectively trivial. That is why the lift is stated as a supplied bridge rather than a derivation — the paper’s own boundary, not a gap left implicit.
Where it sits in the release
RIC-P1 is the foundational paper of Release III, Wave A. Its architecture is tested in a source-complete synthetic model by RIC-P2, and its finite formulas are bound to frozen fixtures by RIC-D1.
Download paper (Zenodo) — CC-BY-4.0.