Environmental Currying and Causal Internal Homs: Dimension-Profile and Representability Obstructions in Quantum, Classical, and Higher-Order Process Theories
Can a process with a fixed environmental input be represented naturally as one into anotherphysical system? Categorically, does tensoring with the environment have a right adjoint,or causal internal hom? The dimension-profile method introduced here records the numberof independent parameters in each family of processes. Any adjunction preserves convexmixtures and therefore these counts. The intercept principle turns systematic violations intogeometric or arithmetic no-go tests.Applied to quantum and classical channels, instruments, comb states, and superchan-nels, this yields complete or conditional classifications. Quantum channels fail an all-inputarithmetic test; even exceptional dimension matches have incompatible boundary geometry.Classical channel dimensions match, but products of simplices are not simplices. Nontrivialinstruments fail grade by grade. Adding a fixed channel type admits a right adjoint onlywhen the added type is trivial. Comb-state dimensions are derived separately from arbitraryhigher-order process spaces.Dimension is only the first layer. The analysis separates affine shape, positivity withancillas, and naturality; constructs the exact linear candidate for a causal internal hom; andproves that universal evaluators assemble into an adjunction. Finite tomography, symmetry,and robustness make the converse testable. Extensions treat generalized probabilistic the-ories, composites without local tomography, positive extension, and finite operator-systemrealization
Can a process with a fixed environmental input be represented naturally as one into anotherphysical system? Categorically, does tensoring with the environment have a right adjoint,or causal internal hom? The dimension-profile method introduced here records the numberof independent parameters in each family of processes. Any adjunction preserves convexmixtures and therefore these counts. The intercept principle turns systematic violations intogeometric or arithmetic no-go tests.Applied to quantum and classical channels, instruments, comb states, and superchan-nels, this yields complete or conditional classifications. Quantum channels fail an all-inputarithmetic test; even exceptional dimension matches have incompatible boundary geometry.Classical channel dimensions match, but products of simplices are not simplices. Nontrivialinstruments fail grade by grade. Adding a fixed channel type admits a right adjoint onlywhen the added type is trivial. Comb-state dimensions are derived separately from arbitraryhigher-order process spaces.Dimension is only the first layer. The analysis separates affine shape, positivity withancillas, and naturality; constructs the exact linear candidate for a causal internal hom; andproves that universal evaluators assemble into an adjunction. Finite tomography, symmetry,and robustness make the converse testable. Extensions treat generalized probabilistic the-ories, composites without local tomography, positive extension, and finite operator-systemrealization
We ask whether coupling a fixed finite-dimensional environment to a quantum process can be undone functorially: does tensoring with an environment E admit an adjoint in the category of quantum channels, and in related categories of instruments and stochastic maps? We prove that it does not, unless the environment is trivial. Our central new tool is an intercept principle: an enriched adjunction must preserve the affine dimension of hom-sets coefficient by coefficient, and probability normalization contributes a dimension-counting intercept that no environment-independent representing object can absorb. Specialized to the environment endofunctor, this gives a single coefficient-rigidity obstruction that governs channels, ordered instruments, finite stochastic maps, and higher-order superchannels uniformly, and yields an exact classification in each case: a left or right adjoint exists precisely when the environment is isomorphic to the tensor unit. We introduce an operational counterpart of this categorical obstruction. No finite-dimensional quantum memory with a fixed processor can store and exactly evaluate all channels on a nontrivial system, and any natural finite-memory encoding of processes would induce the adjoint just shown to be impossible. For approximate compression into insufficient memory we prove a new quantitative floor on the achievable diamond-norm error, sharp for the argument used and extended from linear to arbitrary continuous encoders by a Borsuk–Ulam argument. Finally, we place these obstructions against the standard theory of exact recovery: a channel is Petz-recoverable exactly when its complementary channel is constant, a morphism-level splitting that we show is strictly weaker than the functorial currying structure ruled out above. We delimit what these results do and do not establish, and accompany the mathematical claims with a machine-checkable verification suite.
Can an external environment be absorbed universally into a quantum process? This article identifies trace normalization as the obstruction and tracks it through higher orders. Exact ranks are derived for sequential-comb normalization at every order. Two concrete families of higher-order maps are defined, and their transformation-space dimensions are computed. In the family stable under arbitrary auxiliary combs, adding an environment to an input has neither a left nor a right adjoint from order three onward. Together with the channel and superchannel cases, this yields an all-order family of no-go theorems. The claim is restricted to specified process types: broader normalized type systems may still admit internal homs. Without normalization, a two-sided adjunction returns for normal completely positive maps. Exactly the finite-dimensional Hilbert spaces are dualizable, but trace scaling and rank-one evaluation prevent restriction to deterministic channels. The continuous scalar laws underlying composition- and tensor-additive dynamics are proved to be logarithmic boundary terms. Nonadditive quadratic actions are analyzed through exact stationarity equations and several exact qubit phase classifications. Finally, continuous channel encoding below the full channel-space dimension has worst-case diamond-norm error at least 1/(dAdB ), while normalized weighted cups have sharp defect 1 − 1/δ2. The results identify three ways around the obstruction: enlarge the type universe, remove normalization, or accept controlled approximation
We define the finite-dimensional one-slot deterministic superchannelcategory by its normalized Choi operators, and an environment-decorationfunctor on it by identity-wire insertion. A right adjoint to thisdecoration would internalize the environment universally, as a curryingoperation on processes. For unrestricted completely positive maps theanalogous construction exists through the compact-closed internal hom;we prove that probability normalization destroys it: the decorationfunctor has neither a right nor a left adjoint when the environment isnontrivial. The hom-sets of the category have explicit affine-dimensionpolynomials in the squared interface dimensions, and probabilitynormalization ties the constant term of each polynomial rigidly to onedistinguished coefficient. We call this the normalization-defect(intercept) principle. Environment decoration rescales precisely thedistinguished coefficient while leaving the constant term fixed, so norepresenting object satisfying its own normalization constraint canreproduce the decorated dimensions uniformly over all sources; hencethere is no right adjoint. A dual dimension count at the trivial objectforces a squared dimension strictly between consecutive squares,excluding every left adjoint. It follows that the canonical paralleltensor product is not closed at nontrivial environment objects. Anappendix constructs a category of deterministic combs of arbitraryfinite order and extends the same coefficient obstruction to show thatenvironment decoration has neither adjoint at any order. The obstructionis uniform over all sources and is consistent with the recoverability ofindividual processes.
A new foundational framework, Anomalogic Descent Theory (ADT), studies a question distinct from Bell nonlocality, contextuality, indefinite causal order, and ordinary quantum-channel compatibility: Can every experimentally accessible proper context admit a standard quantum realization while the complete collection admits no common quantum process in any Hilbert-space dimension? The work develops an implementation-sensitive notion of global quantum process descent based on positive block-Choi/operator-valued coherence kernels. Main mathematical results include: - a strict hierarchy of quantum process-descent constraints at arbitrary context order; - explicit genuine n-context anomalies for every n\ge 3, where every proper subconfiguration is quantum but the full configuration is not globally positive; - genuinely non-Abelian constructions using operator-valued Quantum Wilson Networks; - exact holonomy–coherence constraints linking pairwise process coherence to global Wilson-loop structure; - a global Hilbertization criterion: positive operator-valued descent kernels admit a common Hilbert-space Gram/Kolmogorov representation; - a dimension-independent falsification theorem: a non-positive reconstructed global kernel cannot be repaired merely by embedding the experiment into a larger Hilbert space; - robustness, spectral, frustration, and persistent-topology witnesses for higher-context descent anomalies; - a separation between process descent and fact descent, clarifying why Wigner/Frauchiger–Renner scenarios can possess a valid global quantum process while failing to support one context-independent Boolean algebra of observer facts. A central explicit family is quantum on every proper n-1 context while failing globally. For the symmetric construction, [ \frac{1}{n-1}<t\le\frac{1}{n-2} ] is the genuine n-context anomaly region. The balanced point [ t_n=\frac{2}{2n-3} ] has equal positive and negative spectral margins, [ \delta_n=\frac{1}{2n-3}, ] providing a finite robustness gap rather than a boundary-only counterexample. The proposed experimental target is therefore unusually sharp: «Use one fixed, independently certified physical implementation. Verify that every proper overlapping coherent-control context is compatible with ordinary CPTP quantum mechanics, while the full implementation-sensitive dataset excludes every common positive quantum-process realization.» Such an observation would not merely violate a Bell inequality or reveal contextuality. It would indicate that quantum mechanics remains valid locally but fails as a global composition principle. No experimental violation is claimed. The current contribution is a theorem-level mathematical framework, an operational post-quantum theory candidate, and a concrete falsification program. Claims of historical priority are also deliberately limited pending specialist review. Relevant fields and keywords: quantum foundations, generalized probabilistic theories, quantum channels, Choi matrices, complete positivity, positive-definite kernels, operator systems, quantum contextuality, Wigner’s friend, Frauchiger–Renner, quantum combs, process matrices, coherent control of channels, channel holonomy, non-Abelian Wilson loops, matrix completion, Hilbert-space dilation, locally PSD cones, higher-order compatibility, post-quantum theories. The release includes the full preprint, proofs, reproducibility code, claim-status ledger, prior-art audit, and experimental protocol. Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
The companion paper on effective composition characterises, for a finite substrate without a supplied subsystem factorisation, the possibilistic support layer completely and the preparation layer for two reference classes, and explicitly leaves open the states-and-effects layer of the resulting composition contract. We resolve that layer's finite, mono-system, and bipartite-composition questions, keeping the general preparation layer and the tensor product exactly as open as the companion paper left them. The primitive family arising from cylindrical conditioning of a strictly positive, rank-one reference measure is not itself convex: it is finite while its own convex hull is a continuum, unconditionally, for any local outcome set of size at least two. An explicit classical-randomisation postulate closes it onto the full classical probability simplex, independently of the reference measure's rank; rank governs composition of the primitive family, not the shape of its closure once randomisation is added. Simplicial geometry fixes the mathematical representation of affine effects as vectors in a cube, but not their physical availability as measurements; a weak coarse-graining axiom already yields the entire Boolean skeleton of sharp effects for free, while the full continuum of effects requires a further, dual randomisation postulate on the measurement side. Under two further, distinct, and logically independent axioms — one for a common classical control resource, one for the physical existence of a composed bipartite effect — the pointwise product of two local effects is proved unique, not postulated, and local tomography holds exactly when the local effect families linearly span their ambient functional spaces, equivalently when they separate their local states and contain the unit effect in their linear span. All results are finite, exact, and reproduced by committed scripts using rational arithmetic. Interpretive reading (structural, not a result): the finite hypotheses examined here force, successively and for free, a classical, simplicial operational skeleton — convex states once mixing is supplied, a Boolean effect algebra once coarse-graining is supplied, and exact local tomography once spanning local families are supplied — under precisely identified imports, and produce neither a quantum state geometry nor the Born rule. Locating exactly how much operational structure a finite admissibility contract yields for free, and where every further richness must instead be imported, is the intended service of this result to the non-injective foundations programme: it marks the boundary this classical skeleton cannot cross on its own, and hands that boundary, precisely typed, to whatever module or phase law is proposed next.
This paper extends the Ontological Totality Functional (OTF) programme into the domain of quantum causal structure. Its central claim is that physical admissibility cannot be exhausted by state-level consistency or dynamical stability alone. It must also constrain the admissibility of causal organization itself. The paper introduces a causal defect functional for higher-order quantum processes and argues that definite causal order can emerge as a limiting case of admissible structure. Working in a finite-dimensional process-matrix setting, the paper defines the causal defect as the trace-norm distance to the set of valid causally separable processes with fixed normalization. It proves that this defect is non-negative, vanishes exactly on the causally separable set, and is convex. A free class of admissible supermaps is then identified: linear maps on processes that preserve validity, preserve causal separability, and are contractive in trace norm. Under these maps, the causal defect is monotone. An exact dual operational characterization is established: the causal defect equals the maximal bounded-test advantage over the causally separable set for Hermitian tests of bounded operator norm. In this sense, bounded causal witnesses arise as certified lower-bound functionals for admissibility. To connect the formalism to concrete physical scenarios, the paper studies a dephased-switch family W_eta = W_diag + eta X, where W_diag is causally separable and X encodes interference between orders. For this family, the defect is bounded above linearly in |eta|, while a witness with positive overlap on X yields a matching linear lower bound. As eta tends to zero, the process approaches definite-order admissibility. Within the OTF series, this paper functions as a frontier extension. Paper I introduced physical admissibility mathematically, Paper II studied admissibility in a tensor-cosmology channel, and Paper III interpreted admissibility ontologically. Paper IV shows that the same programme can be pushed into higher-order causal structure, where admissibility constrains not only which states and channels are physically stable but also which causal architectures can count as physically realized. Companion records in the OTF series:OTF I — DOI: 10.5281/zenodo.19068375OTF II — DOI: 10.5281/zenodo.19075284OTF Master Edition — DOI: 10.5281/zenodo.19077303