Complementarity can organize a relational starting point for quantum reconstruction, but a completion relation does not yet supply preparations, effects, probabilities, continuous trans- formations, composites, or a scalar field. We formulate the missing passage as an assumption-control problem. Beginning from the typed primitive C= [a⊣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 countermod- els of a larger theory they do not instantiate. At exact complex-closure resolution, let AOPT denote the finite-dimensional causal operational framework of an external reconstruction theo- rem, L local equivalence, and E equivalent-system purification. Finite classical theory witnesses AOPT + L without E, while finite-dimensional real quantum theory witnesses AOPT + E without L; the external theorem closes only AOPT + L+ E. Neither selector is derived from primitive complementarity here. The real-quantum stress test is sharpened by the two-rebit decomposition Sym(4) = Sym(2) ⊗Sym(2) ⊕R(J ⊗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.
Complementarity can organize a relational starting point for quantum reconstruction, but a completion relation does not yet supply preparations, effects, probabilities, continuous trans- formations, composites, or a scalar field. We formulate the missing passage as an assumption-control problem. Beginning from the typed primitive C= [a⊣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 countermod- els of a larger theory they do not instantiate. At exact complex-closure resolution, let AOPT denote the finite-dimensional causal operational framework of an external reconstruction theo- rem, L local equivalence, and E equivalent-system purification. Finite classical theory witnesses AOPT + L without E, while finite-dimensional real quantum theory witnesses AOPT + E without L; the external theorem closes only AOPT + L+ E. Neither selector is derived from primitive complementarity here. The real-quantum stress test is sharpened by the two-rebit decomposition Sym(4) = Sym(2) ⊗Sym(2) ⊕R(J ⊗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.
We present the final audited-closure formulation of CRQR, a principle-based reconstruction of finite-dimensional complex quantum theory from a pre-operational relational substrate. The reconstruction keeps structural possibility, physical test membership, quantitative statistical identity, classical control, topological closedness, finite-dimensionality, causal normalisation, and specifically quantum selection principles logically distinct. A possibility-only continuation quotient is shown to define only a structural proto-state; full operational states and transformations appear only after a Stable Statistical Experiment Sector and a regime-indexed all-context statistical quotient have been introduced. Stable Classical Control yields dyadic mixtures. A weak operational topology is then defined on actual deterministic preparation transformations, and Physical Operational Closedness requires operationally convergent physical preparations to have a physical preparation realising their limit. This yields full convexity and affine dynamics without adjoining nonphysical state points. Finite statistical separability gives a finite-dimensional affine representation and a canonical ordered real linearisation without assuming no-restriction or local faithfulness of process representations. The final bridge to the finite-dimensional causal operational theory used by Nakahira is rebuilt at the level of physical finite indexed tests. A single regime-indexed quotient is used throughout. Transformations are precisely quotient classes represented by outcomes of physical tests; deterministic and normalised states are distinguished and then proved equivalent; state-branch normalisability is formulated operationally; finite classical outcome processing is explicitly required to be statistically additive in every downstream context; and a Finite Sandwich Expansion theorem proves equivalence between CRQR complete-context equality and ancillary-sandwich equality. A canonical real-linear isomorphism between the CRQR state-space linearisation and Nakahira’s real state span supplies the missing finite-dimensionality/closedness match. These results yield an Exact Physical-OPT Extraction Theorem that verifies the required operational background before Local Equivalence or purification is invoked. Local Equivalence and Equivalent-System Purification, with one purifying system fixed for each system type, are introduced only afterwards. Conditional on Nakahira’s 2026 reconstruction theorem, the resulting sector is standard finite-dimensional complex quantum mechanics, with measurement no-restriction derived rather than assumed. A dependency audit makes explicit that physical operations precede their linear representations and that no Hilbert-space state, POVM, completely positive map, or complex tensor-product structure is inserted into the bridge.
Complementarity-First Foundational Release I is a ten-record research release consisting of nine technical papers and this overview. The technical corpus comprises 246 date-neutral public-manuscript pages and spans a conceptual-formal primitive, four quantum-foundations studies, a paired-incidence and transport architecture for finite/local gravity, two distinct local Regge descendants, and a source-controlled Unified Dynamics synthesis. This article supplies the release- wide paper map, dependency graph, claim/evidence classification, artifact register, and open-problem ledger. Its organizing principle is anti-flattening: conceptual proposals, conditional theorems, exact countermodels, finite computations, prospective tests, negative results, and source-controlled syntheses are not treated as interchangeable evidence. The foundational paper defines a completed relation with internal role distinction and shows by elementary independence models that the primitive does not itself determine probability, continuity, geometry, dynamics, or complex quantum theory. The quantum branch separates a conditional reconstruction audit from three focused problems: readout of the two-rebit tomographic defect, certificate-relative operational objectivity underanoninjectivepublicinterface, andtheresourcerequirementsforrecordsandfiniteautonomous maintenance without a free thermodynamic arrow. The gravity branch begins only after a rank-two paired-chiral carrier, Lorentzian real structure, orientation, affine displacement sector, and transport law are selected; common-coframe data then arise only conditionally through the admitted simplicity, gluing, and nondegeneracy chain, after which the branch reaches conditional Palatini–Regge structure and exact finite periodic endpoints but not arbitrary-mesh or continuum closure. Two later gravity papers address different questions: descriptor-local H2/H2R transfer and a conditional curved-Regge identity, versus an intrinsic six-sector Reynolds cancellation mechanism for local zero germs. Unified Dynamics organizes controlled quantum and gravity descendants around a finite dual-pair carrier and a common BF-type kernel while retaining a variation-space obstruction, open complement-operation coherence, no absolute scale, and no Level-4 prediction. The strongest defensible conclusion is therefore a disciplined finite/local reconstruction architecture with exact technical advances, explicit failures, and auditable selectors. Release I is not a completed theory of physics, a continuum unification, a derivation of the Born rule or complex scalars from primitive complementarity, a quantized-gravity result, or an externally peer-reviewed and independently replicated corpus.
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 S is distinguished from the conjugating reciprocal reversal Theta = S C. A supplied exchange/XY Hamiltonian is Theta-even; its Schrödinger generator is Theta-odd, its unitary flow is reciprocal under Theta, 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.
Complex multiplication, coherent addition, and quadratic readout are related ingredients of quantum calculation, but they answer different reconstruction questions. We separate them on a supplied reciprocal carrier. The centralizer of a complex structure on a real two-plane is the complex algebra; its invertible transports are nevertheless not closed under the addition needed for destructive interference at a selected output. A one-outcome dephaser and a reversible phase kick show that certainty of a public outcome does not imply nondisturbance. Within an explicitly assumed positive quadratic response class, calibrated zero-leakage comparisons on a connected tree select the all-ones Gram kernel. We prove quantitative stability estimates and retain the distinction between unconditional record-local operations, selective conditioning, and a controlled inverse interaction. A local normalized-weight ansatz, arbitrary anchored orthogonal refinement, and additive coarse-graining of recorded outcomes force weights proportional to quadratic size. Equal binary refinement alone does not: a positive continuous increasing log-periodic function supplies a counterexample, while binary and ternary refinement at every size, with continuity, recover linearity in size. Real rotations implement the requisite refinements and controlled tagging without full complex control. Fixed-complement preservation and reciprocal reversal admit different generator sectors; independently switchable real mixers on a connected graph plus one addressable local phase generate the unitary Lie algebra. A common scalar phase does not. Supplementary finite action–phase and history examples expose the remaining calibration assumptions. The contribution is an assumption-controlled synthesis of conditional selectors and obstructions, not the introduction of the standard component constructions. The carrier, comparison response, control resources, tensor interpretation, and empirical probability interface are not derived from primitive complementarity.
The Admissibility of Natural Physics, Part I begins where an admissibility-first reconstruction of single-system quantum mechanics ends. Its central question is not whether the familiar tensor-product formalism can be assumed, but whether any physical composition law can be derived without importing composite systems, global state spaces, entanglement, locality, or signalling as primitives. The book develops a disciplined reconstruction of binary quantum composition from operational and structural requirements. It first records the exact content and limitations of the inherited single-system closure, establishes dependency and non-circularity statutes, and introduces the possible object of natural physics only as a provisional target ledger rather than as an assumed ontology. It then examines joint preparation, product states and effects, marginal consistency, operational no-signalling, minimal and maximal probabilistic composites, local tomography, and the classification of admissible tensor-product representations. Particular attention is given to the distinction between a mathematical representation theorem and the uniqueness of a physical composition rule. Real, complex, quaternionic, Jordan-algebraic, minimal, maximal, restricted, and non-tomographic alternatives are compared under explicitly declared admissibility conditions. The complex Hilbert-space tensor product is recovered only on branches satisfying the required factorization, coherence, scalar, and operational closure conditions. Where several inequivalent composites survive, the result is a classified family rather than a falsely asserted uniqueness theorem. Entanglement is introduced only after composition has been commissioned. It is defined as the failure of an admitted joint state to belong to the convex closure of admitted product preparations. The analysis proves, within the resulting predictive grammar, that entanglement is a property of composite-state geometry, not an additional force, substance, interaction mechanism, or signalling channel. Throughout, the argument enforces quotient primacy, refinement stability, finite diagnostic closure, regeneration stability, explicit dependency receipts, local termination, and the possibility of Undefined as a legitimate scientific outcome. Part I therefore supplies neither a speculative ontology nor a presupposed many-world formalism. It provides a conditional classification theorem for binary physical composition and identifies precisely what remains to be established before interaction, repeated composition, particle identity, locality, fields, symmetries, and particle content may be addressed.
Complementarity-First Foundational Release III develops reciprocal phase architecture, conditional gauge and incidence geometry, finite Lorentzian connection selection, and relational quantum-information tasks. This paper synthesizes the eight research preprints and one reproducibility dataset in Waves A–C without transferring theorem ownership or treating their separately admitted carriers as a single derived physical system. The release is organized by four distinctions: structural admission versus conditional selection; geometric covariance versus operational implementation; restricted agreement versus complete identification; and mathematical existence versus a resource-accounted realization. The reciprocal quantum construction separates local inclusion, coherent completion, entanglement, finite projective valuation, and individual outcomes. Its synthetic biphoton companion distinguishes image intensity from phase-sensitive entanglement evidence. The geometric papers distinguish a derived compact phase-frame type from electromagnetic gauge interpretation and constitutive dynamics, anti-linear opposite-weight pairing from same-bundle reversal, an indefinite incidence form from positive records, and geometric-section agreement from independent connection variation. The Wave C papers supply finite addressing classifications, task- and record-relative recovery, encoded-preparation advantages, and exact exchange constructions with trajectory and timing bounds. Cross-wave comparisons locate genuine imported interfaces and identify bridges that remain absent, including the passage from finite Born-form valuation to general instruments and from a recorded group-error model to a physical timing-error model. Analytic proofs, finite calculations, numerical certificates, recorded replays, and scoped Lean verification are assigned distinct evidentiary roles. The contribution is a source-attributed scientific synthesis and an explicit dependency and residual-premise map, not a new primitive derivation of quantum theory, electromagnetism, or gravity.