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.
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.
The papers released after Complementarity-First Foundational Release III developexplicit constructions in which a retained record, a comparison frame, an analytic domain, or apreparation map determines what can be composed and inferred. This review synthesizes fourteenresearch papers in five branches: autonomous source refresh; structured magnetic-core completionand local monopole dynamics; orientation-twisted composition and reciprocal quantum cells; non-linear superconducting response; and Feynman composition, configuration histories, and low-energyinterfaces. The source-refresh studies distinguish an exact spatial exchange from a reusable appara-tus, and an intended smooth model from its exact piecewise numerical source. The magnetic trilogyseparates extension of a structured bundle, observable-specific exterior recovery, energy coercivity,kinetic threshold behavior, and regular local evolution. The quantum-cell papers distinguish coher-ent changes of presentation from physically admitted processes, geometric cells from quantum statespaces, and boundary-word identities from unrestricted refinement invariance. The superconduct-ing comparison separates restricted density equivalence, a specified nonlinear field contrast, andexperimental identification. The four composition papers connect conditional amplitude rules tocontrolled one-dimensional histories, coherent coarse memory, induced metrics, common-windowpreparation, and explicit spectral-selection costs. Cross-branch comparisons identify genuine im-ports without assigning a common physical carrier to mathematically different models. Analyticarguments, finite certificates, historical replays, and partial formalization retain separate eviden-tiary roles. The contribution is a source-attributed release map and a synthesis of compositionand inference boundaries, not a new primitive derivation of quantum theory, matter, or gravity.Publication confirmation, access to the consulted edition, and deposited-file identity are recordedseparately.
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.
CT 4.0.2 is the foundational bundle for Complementarity Theory (CT): physical updating modeled as finite committed record crossing an interface aperture under a least-incoherence selection rule, with mathematical setting in synthetic differential geometry (SDG). The paper presents five contributions: (1) A conditional derivation of the local Minkowski interval at a stable aperture in the first-order single-aperture regime, from CT primitives together with named structural inputs (a smooth aperture model, an adapted-frame choice, a posited seam-budget axiom, and a saturating-history existence hypothesis). Every non-theorem input is stated where it enters. (2) A complex-amplitude structure for the soft-commit regime: composition and superposition as operational axioms; the Born probability map definitional, with a named, independently testable interference-order axiom (after Sorkin) as the stated route to its quadratic form; a finite-dimensional Hilbert structure following from the axioms. (3) A multi-subject geometric proposal: intersection-network gluing with subjects anchored at the primitive substrate. The transported Lorentz triangle product is classified as connection loop holonomy — identity marks a flat face — distinct from Čech patch consistency; the U(1) face holonomy is conditionally realized as the normalized third-order Bargmann invariant. (4) A research program for deriving the residue role H from τ₀-substrate structure in complexified SDG, with the Residue-to-Phase problem restated precisely (the in-paper result is narrowed to a valid minimal-compact-factor theorem) and a conjectural horizon-phase quantization ansatz. (5) A conditional phase-to-spin and two-spinor bridge: three orthogonal noncommuting phase flows on a distinguished C² amplitude sector generate SU(2) ≅ Spin(3), with the exact 3-2-3 correspondence and the standard Penrose two-spinor route to Lorentzian geometry — established mathematics whose CT premises are recorded as named open problems. Version note: 4.0.2 is a status-correction erratum revision of the 15 May 2026 deposit (that record carries the Zenodo version label v4.0.1; there is no intermediate content revision). Claims are re-labeled to match what is proved, in nearly every case restoring the labels the supporting notes themselves use. A new supporting note, Identity as History, proves the record/history recurrence dichotomy behind the paper's arrow-of-time remark; the P-bound diagnostic is added with a reflexive audit caveat. The full changelog is Appendix A of the paper. The bundle contains the paper, twelve supporting notes (LaTeX and PDF), two H-program notes, the standalone SDG note, the Malus note, and a navigation README. An AI-assistance disclosure is included in the paper's acknowledgments.
We distinguish the construction of an incidence geometry from its realization by operations on positive records. An admitted two-complex-dimensional carrier, an independent anti-dual summand and a specified cross-evaluation form give a signature-(2, 2) Hermitian space. Its transverse totally null two-planes are labelled by Hermitian 2 × 2 matrices, with their intersections governed by the Lorentzian determinant. A separately positive readout metric admits rank-two records from which the incidence label can be reconstructed. We prove that faithful support transport for six specified plane records by one common completely positive operation forces every Kraus amplitude to be proportional to the desired incidence isometry. Exact deterministic transport in this fixed encoding is therefore restricted to the positive-unitary incidence subgroup; noncompact transformations admit bounded success operations with retained failure outcomes. We then separate exact certification of a specified unitary from unrestricted tomography. As a code-subspace application of an established connected-spanning input criterion, complete restrictions on two nonorthogonal spanning codes identify the stipulated unitary channel. Yet all null-supported probes span only a fifteen-dimensional operator hyperplane and cannot distinguish an explicit pair of different channels. Finite-label recovery additionally requires output transversality to the chosen affine chart. The results are conditional on the carrier, forms, preparation, calibration and common-process model. They do not derive a physical record apparatus, primitive spacetime or gravitational dynamics, and do not prohibit passive Lorentz covariance or other encodings.
A supplied bounded reciprocal plane supports compatible elliptic, metric and symplectic geometry, but it does not itself select a physical quantum system or an absolute action scale. This paper organizes an assumption-controlled finite architecture of cell size, convex polarity, composition, paired transport and operational access. It distinguishes amplitude and phase-space interpretations, vector complementation and polarity of regions, local cell projections and global body composition, and geometric holonomy and measured response. The mathematical analysis includes a conditional polarity selector, a forward projection/section identity for supplied symplectic ellipsoids, an explicit counterexample to a fixed-splitting converse for general ellipsoids, inverse-transpose paired transport, a common-connection criterion and reference-dependent response comparisons. No primitive derivation of a quantum state cone, tensor product, minimum action, physical instrument or probability law is claimed. The accompanying source-and-evidence supplement contains the manuscript source, three preserved Python exact-check programs, and a partial Lean 4.19.0 formalization. Audited records of a local run support acceptance of 99 theorem/lemma declarations and 33 definitions/abbreviations. The actual set-level geometry of the concrete ellipsoid counterexample is included; the full convex-lattice, capacity, projective, all-rank geometric and smooth-connection results are not all formalized. This is not full-paper formal verification, external human peer review or a new compiler run. All selected command streams and type/axiom output remain verbatim. One local-run receipt is provided as a clearly labelled path-normalized derivative, with its original hash retained. The full private lineage is preserved separately. The current reading edition is version 1.0; the unchanged version 0.3 materials are retained only as verification baselines.