Institute of Polymaths

QUANTUM MATHEMATICS

Title: Quantum Ramanujan Machine

Author: Hutan Ashrafian

Abstract: Automated conjecture generation is usually implemented as a loop that repeatedly proposes a candidate expression, evaluates it numerically, screens for near matches to a target quantity, refines the proposal distribution, and escalates survivors to high precision checks and proof attempts. The computational bottleneck is frequently the evaluator, especially when the target quantity is an expectation, integral, or simulation derived statistic whose classical estimation cost scales poorly with the required tolerance. This paper defines a “Quantum Ramanujan Machine” as a conjecture engine whose evaluator and screening stages are exposed as explicit operators inside an end-to-end pipeline, enabling quantitative predictions about when a quantum subroutine changes the pipeline bottleneck rather than merely accelerating a subroutine in isolation. The manuscript contributes three coupled elements. First, an operator level decomposition of conjecture generation that isolates the generator, evaluator, screening, verifier, and refinement update as composable maps acting on candidate distributions. Second, a local stability condition for refinement dynamics expressed as a spectral radius bound on the linearization of the update operator, which makes refinement noise and scoring curvature part of the algorithmic design constraints. Third, throughput and channel bounds that relate tolerance, precision, candidate volume, and stage capacities to end to end conjecture yield, and that predict the regime in which quantum mean estimation or amplitude amplified search shifts the limiting stage. The results are formulated to support empirical adjudication through measurable quantities, including yield scaling at fixed verification criteria and bottleneck shift predictions under controlled changes to evaluation accuracy and access costs.

DOI: 10.5281/zenodo.18432437

PDF: https://doi.org/10.5281/zenodo.18432437

License: CC BY-NC-ND 4.0 (© 2026 Hutan Ashrafian. All rights reserved.)







Title: The Quantum Huzita-Justin (Huzita-Hatori) Axioms, Algorithmic Reachability in Superposed Folding

Author: Hutan Ashrafian

Abstract: The Huzita-Justin axioms, often termed the Huzita-Hatori axioms, give an operational foundation for classical origami constructions by specifying which single-crease operations are permitted when points and lines are treated as exact geometric primitives. This paper proposes and formalizes a quantum analogue in which the choice of crease is not a classical branch but a coherent register, and a fold is implemented as a unitary reflection controlled on that register. The resulting model defines a “quantum folder” as a finite sequence of controlled fold unitaries and coherent axiom operators, together with a measurement rule that recovers classical folding as a commuting, decohered limit. Within this framework we introduce a notion of algorithmic reachability for geometric targets, defined by the probability of producing designated intersection events or alignment relations after interference over crease histories. The comparison protocol is stated on matched discretizations and resource measures, so that any change in reachability can be attributed to an explicit mechanism, whether generic amplitude amplification against a verifier or geometric interference specific to crease superposition.

DOI: 10.5281/zenodo.18432062

PDF: https://doi.org/10.5281/zenodo.18432062

License: CC BY-NC-ND 4.0 (© 2026 Hutan Ashrafian. All rights reserved.)








Title: Complexity-Bounded o-minimal Observers and Wasserstein-Indistinguishability of High-Frequency Superpositions

Author: Hutan Ashrafian

Abstract: Here we develop a measurement model in which "classicality" reflects a bound on the descriptive complexity available to an observer, rather than a separate dynamical regime. The observer is modeled by a compact class C(N,d) of probability measures on a bounded detector domain, whose densities have bounded representational format, e.g. piecewise polynomials of degree at most d on at most N cells (hence definable in an o-minimal structure). Given an incoming quantum probability law mu, the reported measurement outcome is defined as the Wasserstein-1 best approximation Phi(N,d)(mu) in argmin over nu in C(N,d) of W1(mu,nu). For the oscillatory family with density 1 + cos(2*pi*k*x) on [0,1], we prove a quantitative complexity-versus-frequency tradeoff: if k exceeds the observer's alternation budget O(Nd), then the optimal approximation cannot retain interference visibility and becomes Wasserstein-indistinguishable from the phase-averaged classical mixture at rate Theta(1/k). The proof uses a quantitative monotonicity budget for definable CDF deviations, showing that a tame CDF cannot track high-frequency oscillations beyond its format budget. We also justify using W1 (Kantorovich-Rubinstein) as the natural metric for complexity-bounded observation, since it corresponds to distinguishability by Lipschitz observables, whereas total variation corresponds to discontinuous probes requiring unbounded resolution.

DOI: 10.5281/zenodo.18432850

PDF: https://doi.org/10.5281/zenodo.18432850

License: CC BY-NC-ND 4.0 (© 2026 Hutan Ashrafian. All rights reserved.)








Title: Language as Societal Viruses - Cross-mapping Information Constructors in Engineering, Physics, Communication and Biology

Author: Hutan Ashrafian

Abstract: That languages behave as viruses is often treated as metaphor. This concept is examined and reframes it as a constrained scientific mapping between two classes of replicators that both persist by encoding compressible instructions, exploiting host resources, and propagating via transmissible packages. In biology, selection favors variants that replicate while evading host immunity via antigenic change and related mechanisms, and the present mapping asks whether an analogous balance between propagation and resistance operates for linguistic variants under cognitive, institutional, and platform level filtering. I formalize a cross-domain correspondence between biological viral life cycles and linguistic life cycles, define measurable invariants preserved under the mapping, and derive quantitative predictions using Shannon entropy rates, cross entropy, algorithmic complexity surrogates, constructive depth via assembly theory, and thermodynamic costs bounded by Landauer’s principle. The same mathematics that governs quasispecies, error thresholds, and reproduction numbers in virology can be adapted to linguistic diffusion and to autoregressive language modeling as a physical process that instantiates next token prediction. This yields testable claims about when linguistic variants behave like simple contagions versus complex contagions, how compressibility and surprisal shape transmissibility, and how energetic constraints couple to scaling in human and machine language production. I then place the formal program in dialogue with Wittgensteinian use, Chomskyan competence, Dawkinsian replicators, and contemporary arguments that large language models reveal properties of language as a learned system. Language as a transmissible informational agent and replication across minds, shifting cognitive and affective dynamics toward measurable behavioural and clinical correlates is also addressed. The resulting picture is neither that language is literally a virus nor that the analogy is purely poetic. Rather, the analogy is a partial isomorphism whose value is empirical. It generates falsifiable hypotheses, clarifies where the correspondence fails, and sharpens ethical questions about governance of high reproduction informational agents in human and machine ecologies.

DOI: 10.5281/zenodo.18433227

PDF: https://doi.org/10.5281/zenodo.18433227

License: CC BY-NC-ND 4.0 (© 2026 Hutan Ashrafian. All rights reserved.)

Title: Unitary Generative Design, a Quantum Formulation of Stiny's Shape Grammars

Author: Hutan Ashrafian

Abstract: Shape grammars provide a formal account of generative design in which rules act directly on spatial configurations rather than on symbolic strings. Quantum computation, in parallel, has developed mature rewriting traditions for circuits and string diagrams, where equivalence is witnessed by rule-based transformations. This paper gives a unitary formulation of spatial shape rewriting. Designs are modelled as quantum states over a configuration basis of embedded geometries, and shape rules are modelled as unitary operators that implement reversible replacement on matched subspaces. The framework separates matching from application by introducing a coherent match register, and it recovers classical derivations as a coarse grained limit. Two quantum specific phenomena then become available for generative design, interference controlled emergence in which an emergent subshape can be amplified or suppressed by phase, and entangled subdesign constraints in which distant regions exhibit correlated stylistic outcomes without explicit classical coordination. The interference signature is operationally visible when match or history information is not resolved by the readout, or is coherently erased before feature measurement. The result is a mathematically explicit bridge between visual calculation in design and operator-based evolution in quantum theory, together with a route to simulation and empirical evaluation via feature observables and distributional tests.

DOI: 10.5281/zenodo.18433819

PDF: https://doi.org/10.5281/zenodo.18433819

License: CC BY-NC-ND 4.0 (© 2026 Hutan Ashrafian. All rights reserved.)

Title: Quantum Region Connection Calculus for operator defined region relations

Author: Hutan Ashrafian


Abstract: Quantum theory typically presupposes a fixed factorisation of the total Hilbert space into subsystems, or an equivalent fixed choice of commuting subalgebras. This convention hard-codes a classical notion of boundary between "System A" and "System B". Recent work has formalised subsystem ambiguity by defining partitions directly at the level of C*-subalgebras, including non-tensor-factor subsystems, and by studying representability and the role of centres. In parallel, work under the label "quantum mereology" studies how subsystems can emerge top-down from spectral structure or from changes of tensor product structure. The aim here is different. We define RCC-style region relations directly from unitary invariants of pairs of sharp region projectors, so that boundary indefiniteness is a relation-level observable rather than a chosen decomposition. A quantum mereotopology is developed in which "regions" are represented by projection operators and their topological relations are defined intrinsically, without background space and without a privileged tensor product structure. The central object is the commutator of region projectors, whose spectrum and operator norms quantify the extent to which a putative boundary is definite or indefinite. A quantum extension of region connection reasoning is proposed in which the commuting regime yields a classical Boolean core of region relations, while a noncommuting regime captures superposed and dynamically entangled partitions through an explicit interface sector. This yields a coordinate-free separability criterion expressed purely in terms of region relations, and it provides a framework in which boundary fuzziness becomes an observable notion tied to measurement compatibility.

DOI: 10.5281/zenodo.18433878

PDF: https://doi.org/10.5281/zenodo.18433878

License: CC BY-NC-ND 4.0 (© 2026 Hutan Ashrafian. All rights reserved.)








Title: Constraint Rigidity, A Combinatorial Framework for Controlled Hamiltonian Stability

Author: Hutan Ashrafian

Abstract: A recurring engineering problem in quantum many body systems is to anticipate when an intended interaction pattern admits a controlled low energy sector that remains stable within a specified, constraint defined family of local deformations. Existing gap stability results are powerful but model dependent and typically do not yield a simple graph level diagnostic that can be applied before any diagonalisation. This work proposes a combinatorial notion of constraint rigidity for Hamiltonians specified by an interaction graph. The central construction is a rigidity style linearisation of how local basis degrees of freedom and edge level interaction constraints trade off, producing a Maxwell type count and a sparsity condition that plays the role of a quantum analogue of minimal rigidity for a specified constraint model. The framework yields graph level necessary conditions for eliminating constraint preserving local mechanisms under that model, proves a local spectral invariance statement for constraint preserving drifts in a controlled family, and motivates a conjectural link between gap preservation along calibrated paths and generic full rank of an associated “Hamiltonian rigidity matrix”. The result is a concrete interface between classical rigidity theory and Hamiltonian engineering that isolates when stability can plausibly be attributable to interaction structure under an explicit constraint model, rather than to fine tuned couplings.

DOI: 10.5281/zenodo.18433924

PDF: https://doi.org/10.5281/zenodo.18433924

License: CC BY-NC-ND 4.0 (© 2026 Hutan Ashrafian. All rights reserved.)

Title: Homotopic Concurrency in Hilbert Space, Quantum Higher-Dimensional Automata for Entangled Processes

Author: Hutan Ashrafian

Abstract: Standard quantum circuit descriptions represent processes by linear wiring and sequential composition, while concurrency is typically reduced to commutation checks and tensor factorisation. This manuscript introduces Quantum Higher-Dimensional Automata, a geometric model in which executions are directed paths on a cubical complex and quantum coherence is carried by structured projective unitary data on cells. The construction defines a quantum cubical set whose vertices represent configurations and whose higher cells represent coherent multi-parameter evolutions that witness compatibility of concurrent actions. Within this framework, compatibility is promoted from an algebraic condition on operators to a directed homotopy invariant condition on executions, and geometric phase is interpreted as curvature on execution surfaces when smooth lifts exist. The model yields a homotopy based compatibility criterion and supplies obstruction based notions of deadlock grounded in phase holonomy on cubical loops. The aim is an operational semantics in which concrete families of quantum controls induce computable invariants on an explicit concurrency geometry.

DOI: 10.5281/zenodo.18433964

PDF: https://doi.org/10.5281/zenodo.18433964

License: CC BY-NC-ND 4.0 (© 2026 Hutan Ashrafian. All rights reserved.)

 

Title: Beyond Heredity, Axiomatic Foundations of Quantum Greedoids for Time Ordered Circuits

Author: Hutan Ashrafian

Abstract: Greedoid theory generalizes matroids to feasible systems where the natural operation is extension along a trajectory rather than closure under all deletions. Quantum circuit families, especially variational ansatze defined by time ordered gate words, can violate the heredity property in a structural way. Feasibility can depend on interference and cancellation, so that a feasible circuit word need not have any feasible single deletion. This paper introduces an axiomatic framework for “quantum greedoids” on the free monoid of gate words, explicitly incorporating order and multiplicity. We define word accessibility and a suffix exchange axiom adapted to repeated gates, give minimal separations between heredity, accessibility, and exchange on one qubit, and identify stabilizer regimes where hereditary behaviour reappears. We then connect accessibility to pruning based training and to local information geometry diagnostics through the quantum Fisher information, framing greedy constructability as a structural correlate of navigability rather than a guarantee of optimization success.

DOI: 10.5281/zenodo.18434022

PDF: https://doi.org/10.5281/zenodo.18434022

License: CC BY-NC-ND 4.0 (© 2026 Hutan Ashrafian. All rights reserved.)

 


Title: Superposed Topology - a formulation of Quantum Discrete Morse Theory

Author: Hutan Ashrafian

Abstract: Discrete Morse theory compresses a finite cell complex by cancelling matched cells and produces a smaller complex that is homotopy equivalent to the original. Classical reductions are selection-based. A matching is chosen and cancellation follows, so any attempt to place matchings into superposition collapses coherence at readout. This paper introduces a coherent analogue by defining an admissible superposed gradient as a graded operator acting on a Hilbert chain complex, together with a contraction identity and a finiteness condition that makes all reductions algebraic. A critical projector defines quantum Morse numbers as critical subspace dimensions in each degree and yields a reduced differential whose homology agrees with the original. The resulting quantum Morse inequalities bound Betti numbers by quantum Morse numbers and reduce to the classical Morse inequalities when the gradient encodes a deterministic matching. A key distinction from probabilistic mixing is that coherent superpositions produce interference terms inside the critical projector, so phase can change critical statistics while leaving homology invariant.

DOI: 10.5281/zenodo.18445473

PDF: https://doi.org/10.5281/zenodo.18445473

License: CC BY-NC-ND 4.0 (© 2026 Hutan Ashrafian. All rights reserved.)

Title: A von Neumann Bottleneck Theorem for Fault-tolerant Quantum Machines

Author: Hutan Ashrafian

Abstract: Automated conjecture generation is usually implemented as a loop that repeatedly proposes a candidate expression, evaluates it numerically, screens for near matches to a target quantity, refines the proposal distribution, and escalates survivors to high precision checks and proof attempts. The computational bottleneck is frequently the evaluator, especially when the target quantity is an expectation, integral, or simulation derived statistic whose classical estimation cost scales poorly with the required tolerance. This paper defines a “Quantum Ramanujan Machine” as a conjecture engine whose evaluator and screening stages are exposed as explicit operators inside an end-to-end pipeline, enabling quantitative predictions about when a quantum subroutine changes the pipeline bottleneck rather than merely accelerating a subroutine in isolation. The manuscript contributes three coupled elements. First, an operator level decomposition of conjecture generation that isolates the generator, evaluator, screening, verifier, and refinement update as composable maps acting on candidate distributions. Second, a local stability condition for refinement dynamics expressed as a spectral radius bound on the linearization of the update operator, which makes refinement noise and scoring curvature part of the algorithmic design constraints. Third, throughput and channel bounds that relate tolerance, precision, candidate volume, and stage capacities to end to end conjecture yield, and that predict the regime in which quantum mean estimation or amplitude amplified search shifts the limiting stage. The results are formulated to support empirical adjudication through measurable quantities, including yield scaling at fixed verification criteria and bottleneck shift predictions under controlled changes to evaluation accuracy and access costs.

DOI: 10.5281/zenodo.18741261

PDF: https://doi.org/10.5281/zenodo.18741261

License: CC BY-NC-ND 4.0 (© 2026 Hutan Ashrafian. All rights reserved.)