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.
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.
We study how calibrated support information on positive incidence records can certify a complete outcome-resolved operation. On a supplied four-dimensional complex carrier, the original null-supported preparations span a codimension-one operator hyperplane. Exact branch preservation is characterized by the adjoint incidence moment; retaining outcomes can expose a distinction cancelled by the averaged operation. The canonical failure of a faithful filter preserves the hyperplane exactly under a reciprocal-spectrum condition. We prove a quantitative six-plane coercivity estimate and use positive leakage to control the full successful map, including a feasible scale correction and success conditioning. A shared matrix effect constraint then classifies complete pairs of nonzero invertible faithful outcomes. Independently admissible fitted branches need not admit a small repair within their prescribed target family. For a prespecified nonsaturated canonical success/failure instrument, testing both outcome supports and controlling shared gains gives a reference-uniform flagged diamond bound. Complete comparisons compose through finite classical histories with probabilities retained. A fixed-sample binomial construction converts raw counts into a simultaneous confidence-qualified instrument estimate. Its present constants yield large sufficient sample budgets, and an exact contamination example demonstrates conservatism even at population probabilities. The results are conditional on the carrier, common process, preparation, readout, and statistical contract. They do not establish a physical apparatus, device-memory tolerance, efficient experimental certification, or gravitational dynamics. Selected supporting declarations have two recorded Lean 4.19.0 executions and declaration-level source, type and axiom review; this is not an end-to-end formalization of the argument.
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.
This preprint studies operational equivalence and recovery in a twenty-dimensional residual system with a fixed four-sector representation of dimensions 8, 8, 2 and 2. It distinguishes what a declared observation algebra can detect, what a channel preserves, and what a retained classical record makes recoverable. Record-resolved criteria separate useful reversible-error records from unrelated labels and measured-sector records, even when the discarded channel is the same. For a specified coarse-record model, the work identifies exactly recoverable observable algebras and proves an optimal entanglement-fidelity expression on a fixed four-state orbit code. Preparation-aware recovery is then classified over all one-use isometric subspaces in the stated family. An explicit ten-dimensional marker code attains exact recovery under a specified stochastic record where the whole-sector support restriction permits only eight dimensions. Its encoding, nonbaseline marker measurement, and conditional inverse operations remain supplied resources. Scoped Lean evidence verifies the explicit arbitrary-complex-matrix recovery construction and related finite identities; it does not formalize the general complete-positivity/reference framework or optimal-dimension theorem. The recorded-error model is not derived from the companion exchange-timing model, and no protection of its long pulse sequence is claimed.
Calibrated support observations can constrain a complete quantum instrument without identifying its action by unrestricted tomography. We study what those observations guarantee when the noise law is not supplied. On a fixed four-dimensional incidence-record experiment, we define the set of all complete two-outcome completely positive instruments compatible with the exact raw rejection and outcome probabilities, retaining the classical flags. Exact ideal data identify the target, but an explicit family of complete instruments reproduces the nonzero-data category laws of a low-error contamination device while admitting coherent error of square-root order. A target-ray mass certificate and a common complete-dilation comparison give a universal upper, while an exactly feasible rational instrument supplies a stronger lower witness. At the diagnostic parameter r = 1/300000000, the unhalved target-relative worst-case radius lies between 0.000258065115532250 and 0.000951566242566. The result separates avoidable inequality conservatism from information absent from the measured data. Fixed-functional and fixed-map optimizations have separately certified scopes; they do not solve the global radius. The conclusions are conditional on the supplied carrier, calibration and common-device model. They do not establish an apparatus, device-independent certification, a finite-sample power guarantee or complete formal verification. Selected supporting declarations have one recorded successful Lean 4.19.0 macOS execution and declaration-specific source, type and axiom review; this is not an end-to-end formalization.
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.
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.