抄録 Unital completely positive maps preserve effects but need not preserve projections or commutative observable algebras. We systematize a sharp-context calculus around this obstruction. For composable maps we derive an exact kernel-corrected formula for the multiplicative domain of the composite. It yields the maximal partial transport among contexts on which the first map is multiplicative, and identifies how a later coarse-graining can erase an intermediate sharpness defect. In finite dimension every strict composition defect is detected by a binary context generated by one projection. Unitary detection extends to arbitrary commutative C*-contexts, while projection witnesses persist for von Neumann contexts. For normal maps, the von Neumann context transport is Scott-continuous on a directed-complete domain. We lift the partial transport to spans of covariant presheaf topoi carrying tautological internal commutative algebras and constructive Gelfand spectra. The lift is an identity-normal oplax functor into a bicategory of presented spans; its comparison cells record exactly the same sequential-domain inclusion as the operator-algebraic transport. These results give a precise separation between effects, which transport globally, and sharp classical contexts, which transport only on the multiplicative domain. They also make apparent recovery under nonfaithful coarse-graining visible at the level of contextual semantics. AI利用の開示 OpenAI Codex assisted with manuscript inventory, bibliographic extraction and preparation of this deposit. The uploaded PDF is an unchanged copy of the identified author manuscript. Authorship-stage AI assistance, where stated by the author, is described in the manuscript. No AI system is credited as an author. 検証範囲(Lean で形式検証した範囲と未検証部分) This deposit does not establish proof-assistant verification of the full manuscript. No new Lean or other proof-assistant verification, independent mathematical review, or empirical study was performed as part of this deposit preparation. Mathematical arguments, computational checks, educational design claims and any unresolved issues retain the scope stated in the manuscript; DOI assignment does not certify them.
抄録 Unital completely positive maps preserve effects but need not preserve projections or commutative observable algebras. We systematize a sharp-context calculus around this obstruction. For composable maps we derive an exact kernel-corrected formula for the multiplicative domain of the composite. It yields the maximal partial transport among contexts on which the first map is multiplicative, and identifies how a later coarse-graining can erase an intermediate sharpness defect. In finite dimension every strict composition defect is detected by a binary context generated by one projection. Unitary detection extends to arbitrary commutative C*-contexts, while projection witnesses persist for von Neumann contexts. For normal maps, the von Neumann context transport is Scott-continuous on a directed-complete domain. We lift the partial transport to spans of covariant presheaf topoi carrying tautological internal commutative algebras and constructive Gelfand spectra. The lift is an identity-normal oplax functor into a bicategory of presented spans; its comparison cells record exactly the same sequential-domain inclusion as the operator-algebraic transport. These results give a precise separation between effects, which transport globally, and sharp classical contexts, which transport only on the multiplicative domain. They also make apparent recovery under nonfaithful coarse-graining visible at the level of contextual semantics. AI利用の開示 OpenAI Codex assisted with manuscript inventory, bibliographic extraction and preparation of this deposit. The uploaded PDF is an unchanged copy of the identified author manuscript. Authorship-stage AI assistance, where stated by the author, is described in the manuscript. No AI system is credited as an author. 検証範囲(Lean で形式検証した範囲と未検証部分) This deposit does not establish proof-assistant verification of the full manuscript. No new Lean or other proof-assistant verification, independent mathematical review, or empirical study was performed as part of this deposit preparation. Mathematical arguments, computational checks, educational design claims and any unresolved issues retain the scope stated in the manuscript; DOI assignment does not certify them.
We study what happens when an exactly flat contextual operator-order transport is reducededgewise by a unital completely positive (UCP) map. The relevant obstruction is not nonunitarityby itself but failure of multiplicativity. For a UCP map Φ we use the standard multiplicativitydefect∆Φ(X, Y ) = Φ(XY ) − Φ(X)Φ(Y )and prove an exact decomposition of every reduced flat-loop defect into transported multiplicativity defects. The same defect obeys an exact composition law under nested reductions and hasa Stinespring leakage representation. For a uniformly continuous UCP semigroup Φt = etL, themultiplicativity defect satisfies an exact Duhamel formula, and for a bounded GKSL generatorthe corresponding Leibniz defect isΓL(X, Y ) = Xα[V∗α , X][Y, Vα].This yields an exact integral representation of reduced loop holonomy and explicit qubit braid-loopformulas for dephasing, depolarization, and amplitude damping.We then study the inverse problem. Closed-loop finite-channel data need not identify anarbitrary UCP channel, whereas the full infinitesimal Leibniz defect determines a boundedfinite-dimensional generator modulo Hamiltonian derivations. In Md(C) the dissipative quotienthas real dimension (d2 − 1)2, giving the face-count lower bound LCoxd ≥ ⌊d2/2⌋. Using blockextension, centered scalar-one tangent geometry, binder-compatible transversality, a transversenative-tilt argument, and a determinant-one odd-to-even transfer, we prove the matching upperbound. Thus, in the declared bounded finite-dimensional matrix-valued first-order Coxeter-facemeasurement model,LCoxd =d22, d ≥ 3.This sharp count is not a universal lower bound for arbitrary process tomography or arbitraryLindbladian-learning protocols. Conditioning, sample complexity, noise robustness, and finitetime channel reconstruction remain separate questions.
First preprint and reproducibility companion for bounded exact recovery of a nonzero trace-determinant tensor over an unknown binary extension field under corruption supported on a small common covector space. Includes the manuscript PDF and source, complete prose proofs, executable recovery and independent certificate checks, pinned partial Lean proofs, validation receipts, and three AI-model review reports with feedback dispositions. For s >= 2R+5, at most 8*2^(4R)*h trials return a verified compatible representation with probability at least 1-2^(-h). Accepted outputs are checked exactly; unique identification of a planted field is not claimed. The cubic-only bound does not establish the broader quantum application or a BPP/BQP result. Lean verification is partial. The complete recovery theorem, its probability and resource bounds, and the Python implementation are not fully formally verified. The reviews are AI-assisted internal reviews, not independent human peer review. Licenses: manuscript text and PDF are CC-BY-4.0; software and Lean sources are Apache-2.0. See LICENSE, LICENSES/CC-BY-4.0.txt and PROVENANCE.md. The package-level Other (Open) designation preserves these per-component licenses.
A partial-transpose image of the positive-semidefinite cone supplies an established alternative ordered presentation on a fixed composite carrier. Transporting both the cone and the processes preserves an abstract equivalence, whereas fixing the process representation can discriminate the two orders. We study the iteration of one specified right-argument transpose rule on supplied finite qubit carriers. With three standard leaves, its two bracketings have different cones, so no linear identification fixing every product operator pointwise can make them agree. Subset-orientation labels give an explicit repair. Relative to a separately supplied tensor-closed standard process category, we construct the raw tensor of arrows, associator, symmetry, asymmetric unit maps, and quasi-inverse tensorator, and prove symmetric monoidal equivalence by faithful transport. Complementary subsets define the same isolated cone, but their identification is not a congruence for the raw tensor product. At fixed rank, a precisely defined two-lift groupoid is a split central Z₂ extension; its odd lift is not an ordinary completely positive endomorphism at fixed representatives. A five-qubit example then separates matched reference transport, a fixed calibrated anchor, and an effect-based quotient. The latter loses a linear subspace rather than only a two-point transpose orbit and is preserved only by histories satisfying the standard kernel-invariance condition. A named real local unitary connects the anchor pair, while a real controlled-NOT (CNOT) gate exposes the limitation of an enlarged single-anchor history class. The contribution is an assumption-controlled analysis of one finite composition rule and its interfaces, not a new general coherence, orientation, or resource theorem, and not a derivation of quantum theory from primitive complementarity. Selected supporting statements and finite witnesses are accompanied by a partial Lean 4.19.0 formalization. A recorded local build and type/axiom audit accepted 117 theorem declarations and 57 definitions or abbreviations. The full symmetric monoidal category and equivalence in Theorems 4.5 and 4.6, and other explicitly listed manuscript links, remain incompletely formalized. The public source supplement also includes three preserved exact Python implementations, expected results, pinned dependencies and claim-to-code maps. Packaging and source/log audits are not another Lean compiler run or external peer review. The public evidence selection preserves the successful type/axiom output and the failure/success chronology. Machine-local account paths are omitted from derived run summaries, and one failed-build diagnostic has an explicitly normalized home-directory prefix. Accepted Lean proof sources remain unchanged; private review archives and original machine-local receipts are not public upload files.
In a finite-dimensional causal convex operational theory with closed physical effect sets and essentially unique purification, we derive pure products, general purity preservation and effect non-restriction. Exact binary steering follows from an informative noisy pointer; support order intervals, pure dilations and probabilistic returns then force unrestricted effects without local tomography, ideal compression, product domination or a bound on purifying partners. Finite conditioned control is also derived. Spectrality, pure support projections and symmetric transitions yield a Jordan representation. For an essentially small catalogue, we then classify compositions without local tomography. Exact pure-ancilla faces restrict simple factors to a common real or complex matrix field. Sector translations embed in a locally finite abelian group. Complete event representations and simultaneous phase normalizations identify the full sequential and symmetric monoidal structure. At fixed field, equivalence classes are determined by the effective charge group, the actual product-closed object catalogue and a parity character. The proof retains all rectangular event spaces and does not add charge objects or compressions to an incomplete catalogue. Local tomography selects the ordinary complex quantum case on the available multiplicative catalogue. Exact one-way teleportation supplies an operational comparison. In the specified protocol class, one state is shared on two copies of the input system, without an additional correlated record. Exact preservation of all reference correlations forces a pure balanced resource and reversible corrections on nonzero outcomes. A system with three rank-one sectors and Z3 fusion has exact one-copy outcome minima twelve over the reals and nine over the complexes, with corresponding message entropies, although both alphabets fit in four binary digits. Joint protocols have the same regularized rate. Alternative geometric proofs, restricted-model obstructions, optimal physical returns, a quantum-pointer application and an open-effect counter-theory delimit the assumptions and conclusions. Preparation note: This preprint builds on Yoann Dumont’s research programme, earlier manuscript, and research notes. ChatGPT assisted with proof development and checking, comparison with these prior materials, bibliographic searches, English drafting, and typesetting. AI-assisted checks do not constitute independent peer review. As author, Yoann Dumont takes responsibility for the claims, references, and any necessary corrections to the published text.
The Radon--Nikodym theorem for completely positive maps identifies the order interval below a map with positive contractions in the commutant of its minimal Stinespring representation. In this paper, we study the behavior of projection-valued Radon--Nikodym derivatives (sharp submaps) under composition. We prove that sharpness loss is determined by the multiplicative defect of the induced pullback map. In the finite-dimensional case, we show that this defect is determined by the orthogonal complement of the composite Kraus relation space, denoted EΛ,Ω. A sharp submap remains sharp if and only if EΛ,Ω reduces the lifted Radon--Nikodym projection. Furthermore, we establish a tensor-factor criterion for the global preservation of sharpness and explicitly quantify the sharpness-loss defect.
Reflection positivity is a positivity condition on a reflected half-space, and it is not inherited automatically by covariance pieces produced by coarse graining. We isolate a finite-dimensional obstruction for a common projection-type construction. Let H=H+⊕H− carry a unitary reflection Θ exchanging the two halves, let C=C∗ commute with Θ, and assume Osterwalder--Schrader (OS) positivity on H+. For a reflection-symmetric orthogonal projection Π that also preserves H+, we characterize exactly when the projection-sandwich residual C=C−ΠCΠ remains OS-positive: this holds if and only if the positive-half projection Π+ commutes with the OS Gram operator W=P+ΘCP+. Equivalently, the mixed positive/negative-time commutator block vanishes. This gives a direct algebraic test for projection-type coarse graining. We then record three consequences. First, in the scalar infinite-lattice setting a finite-range, fine-translation-invariant orthogonal projection is necessarily 0 or id, so nontrivial local projections must break fine translation invariance or leave this class. Second, a null-escape mechanism shows that a projection which sends an OS-null vector to a strictly positive OS-energy direction forces the residual to violate reflection positivity. For the massive free covariance with bond reflection we identify the OS null cone explicitly and obtain its dimension. Third, a complete 4×4 rational example gives a standalone exact failure certificate with positive original covariance and a negative residual OS direction. A larger audited 8×6 block-harmonic computation is retained only as supplementary provenance because the closure archive does not contain the complete projector table required for independent recomputation. The obstruction is deliberately narrow: it applies to projection-sandwich residuals and does not exclude non-projective Markov kernels, conditional expectations, completely positive maps, nonlinear coarse graining, or continuum constructions.