Quantum BioPhysical Software: Quantum Bit innovation allows modeling of sub-Nano-Scale Complex BioChemical States (V5, with benchmark data for Periodic Table)
Abstract. We develop a general exact architecture joining linear subshell capacity, unrestricted isotropic fold vectors, conjugacy/class memory under power and inverse power, common objective ideals, class-objective rank factorization, and a bit–qubit class equivalence on the computational-basisvcarrier. The state is any finite vector q ∈ NL ; its fold carrier is the unrestricted sequence Fq(j) = XLs=1qj+1s , j ∈ N, and the bridge family is arbitrary in arity, βr(q)s = min(qs, r − 1), r ≥ 2. The bridge is diadic at r = 2, triadic at r = 3, and stabilizes exactly to q whenever r > maxs qs. Power and inverse-power operations that respect a class relation descend to the quotient, so vector composites of powered class invariants remain class memories. For four objective ideals I1, . . . , I4, the common objective ideal Iobj = I1 ∩ I2 ∩ I3 ∩ I4 is the maximal ideal simultaneously contained in all four. If a physical or computational objective factors through a surjective class map, F = ρ ◦ π, then its minimizers are exactly the preimages of minimizing classes. A second central theorem identifies the isotropic pathway count with the sum of quotient-class multiplicities. For capacity vector c = (c1, . . . , cL) and total fill Z, let QZ (c) = {q ∈ NL:Xsqs = Z, 0 ≤ qs ≤ cs}. The permutation class of q has multiplicity µ(q) = Z!Qsqs!. Then the total number of capacity-admissible labeled pathways is Pc(Z) = Xq∈QZ (c)µ(q). Thus the isotropic pathway number is exactly the sum of the collapsed vector-class quotients. In the uncapped case this reduces to the multinomial identity P(Z) = LZ. The 118-element periodic-table instantiation uses the same architecture. Each element receives a canonical 19-subshell capacity-fill vector, fold moments, diadic-to-arbitrary bridge vectors, aVbaryonic-spatial Cantor rank from reference-isotope neutron count and covalent radius, class-memory power/inverse-power checks, and exact pathway multiplicities. The compiled C++ benchmark executed 166,218 checks with all passing; 150,507 acted directly on the 118-element carrier, 15,104 exhaustively compared direct and class-rank marking predicates over all 118 targets and all 128 seven-bit codes, and 359 checks verified the exact isotropic pathway partition recurrence. An independent Python implementation executed 52,166 checks with all passing, including 10,000 arbitrary generated fold/bridge vectors. Four chemistry-oriented structural benchmarks are integrated into the same computational module. Contents 1. Architecture and attribution 3 2. Linear subshell maximum fill 3 3. General isotropic fold vectors 4 3.1. Diadic through arbitrary arity 4 4. Isotropic pathway sum as quotient-class partition 5 5. Class memory: powers, inverse powers, and composites 7 6. Common objective ideal and class-objective rank 7 7. Tibedo Bit–Qubit Class Equivalence Theorem 8 MQBP V5: CLASS MEMORY AND ISOTROPIC PATHWAYS 3 8. 118-element periodic-table instantiation 9 8.1. Canonical fill 9 8.2. Baryonic-spatial coordinate 9 9. Integrated benchmark execution 10 9.1. FCI-like high-arity benchmark 11 9.2. 198-site triadic benchmark 11 9.3. Isotope benchmark 11 9.4. Prime-channel benchmark 11 10. Formalization and software architecture 11 10.1. Coq 12 10.2. C++ 12 10.3. Lean 4 12 11. Structural distinction 12 12. Conclusion 13 Appendix A. Complete 118-element dataset table 13 Appendix B. Dataset contents 17 References 17
Abstract. We develop a general exact architecture joining linear subshell capacity, unrestricted isotropic fold vectors, conjugacy/class memory under power and inverse power, common objective ideals, class-objective rank factorization, and a bit–qubit class equivalence on the computational-basisvcarrier. The state is any finite vector q ∈ NL ; its fold carrier is the unrestricted sequence Fq(j) = XLs=1qj+1s , j ∈ N, and the bridge family is arbitrary in arity, βr(q)s = min(qs, r − 1), r ≥ 2. The bridge is diadic at r = 2, triadic at r = 3, and stabilizes exactly to q whenever r > maxs qs. Power and inverse-power operations that respect a class relation descend to the quotient, so vector composites of powered class invariants remain class memories. For four objective ideals I1, . . . , I4, the common objective ideal Iobj = I1 ∩ I2 ∩ I3 ∩ I4 is the maximal ideal simultaneously contained in all four. If a physical or computational objective factors through a surjective class map, F = ρ ◦ π, then its minimizers are exactly the preimages of minimizing classes. A second central theorem identifies the isotropic pathway count with the sum of quotient-class multiplicities. For capacity vector c = (c1, . . . , cL) and total fill Z, let QZ (c) = {q ∈ NL:Xsqs = Z, 0 ≤ qs ≤ cs}. The permutation class of q has multiplicity µ(q) = Z!Qsqs!. Then the total number of capacity-admissible labeled pathways is Pc(Z) = Xq∈QZ (c)µ(q). Thus the isotropic pathway number is exactly the sum of the collapsed vector-class quotients. In the uncapped case this reduces to the multinomial identity P(Z) = LZ. The 118-element periodic-table instantiation uses the same architecture. Each element receives a canonical 19-subshell capacity-fill vector, fold moments, diadic-to-arbitrary bridge vectors, aVbaryonic-spatial Cantor rank from reference-isotope neutron count and covalent radius, class-memory power/inverse-power checks, and exact pathway multiplicities. The compiled C++ benchmark executed 166,218 checks with all passing; 150,507 acted directly on the 118-element carrier, 15,104 exhaustively compared direct and class-rank marking predicates over all 118 targets and all 128 seven-bit codes, and 359 checks verified the exact isotropic pathway partition recurrence. An independent Python implementation executed 52,166 checks with all passing, including 10,000 arbitrary generated fold/bridge vectors. Four chemistry-oriented structural benchmarks are integrated into the same computational module. Contents 1. Architecture and attribution 3 2. Linear subshell maximum fill 3 3. General isotropic fold vectors 4 3.1. Diadic through arbitrary arity 4 4. Isotropic pathway sum as quotient-class partition 5 5. Class memory: powers, inverse powers, and composites 7 6. Common objective ideal and class-objective rank 7 7. Tibedo Bit–Qubit Class Equivalence Theorem 8 MQBP V5: CLASS MEMORY AND ISOTROPIC PATHWAYS 3 8. 118-element periodic-table instantiation 9 8.1. Canonical fill 9 8.2. Baryonic-spatial coordinate 9 9. Integrated benchmark execution 10 9.1. FCI-like high-arity benchmark 11 9.2. 198-site triadic benchmark 11 9.3. Isotope benchmark 11 9.4. Prime-channel benchmark 11 10. Formalization and software architecture 11 10.1. Coq 12 10.2. C++ 12 10.3. Lean 4 12 11. Structural distinction 12 12. Conclusion 13 Appendix A. Complete 118-element dataset table 13 Appendix B. Dataset contents 17 References 17
To begin with, we need a quantum register made of many qubits to store information. Recall that a classical computer also requires memory to store information. The simplest way to realize a qubit physically is to use a two-level quantum system. For example, an electron, a spin 1/2 nucleus or two mutually orthogonal polarization states (horizontal and vertical, for example) of a single photon can be a qubit. We may also employ a two-dimensional subspace, such as the ground state and the first excited state, of a multi-dimensional Hilbert space, such as atomic energy levels. In the latter case, special care must be taken to avoid leakage of the state to the other part of the Hilbert space. In any case, the two states are identified as the basis vectors, |0〉 and |1〉, of the Hilbert space so that a general single qubit state takes the form |ψ〉 = α|0〉 + β|1〉, where |α|2 + |β|2 = 1. A multiqubit state is expanded in terms of the tensor products of these basis vectors. Each qubit must be separately addressable. Moreover it should be scalable up to a large number of qubits. The two-dimensional vector space of a qubit may be extended to be three-dimensional (qutrit) or, more generally, d-dimensional (qudit). A system may be made of several different kinds of qubits. Qubits in an ion trap quantum computer, for instance, may be defined as: (1) hyperfine/Zeeman sublevels in the electronic ground state of ions (2) a ground state and an excited state of a weakly allowed optical transition and (3) normal mode of ion oscillation. A similar scenario is also proposed for Josephson junction qubits, in which two flux qubits are coupled through a quantized LC circuit. Simultaneous usage of several types of qubits may be the most promising way to achieve a viable quantum computer.
Abstract MQBP V6.2 develops an exact prime-indexed computational architecture in which computational complexity is transferred from the cardinality of a nominal representative space to the structure and evaluability of its objective-preserving quotient. This is Q-Day, effectively, mathematically. Rather than treating a state carrier XnX_n or computational-basis space as the primitive object to be searched, the framework constructs a class map π:Xn→Cn class-indexed fold invariants, polynomial layers, Cantor complexity coordinates, determinantal degeneracy data, algebraic certificates, Thom encodings, and an invariant rank ρ:Cn→Rn. For every objective admitting the exact factorization F=Φ∘ρ∘π representative-level enumeration is mathematically unnecessary: the objective is constant on quotient classes and may be evaluated entirely from the reduced carrier. Thus the exponential cardinality of the representative space and the computational complexity of the objective become distinct mathematical quantities; the operative complexity is that of constructing and evaluating the objective-preserving class and rank maps. The architecture is quantified over arbitrary prime nn and arbitrary arity r≥2r\ge2. For an occupancy class m=(m0,…,mr−1), ∑dmd=n, the class-indexed fold hierarchy M_j([m])=\sum_{d=0}^{r-1}m_d d^j defines the polynomial layer λ([m])=M1([m]), while (M_1,…,M_r−1)forms an exact finite occupancy coordinate by Vandermonde reconstruction. The degree-k polynomial channel is c_{r,n,k}=[t^k](1+t+\cdots+t^{r-1})^n, with n(r−1)+1 layers and prime-index Frobenius residues. Recursive Cantor coordinates canonically couple occupancy class, polynomial layer, fold order, and fold value. The moduli-degeneracy carrier augments these class coordinates with exact determinantal rank-drop witnesses, canonical integer-polynomial representations, quadratic-module Positivstellensatz certificates, signed Thom derivative encodings, stabilizer data, and generalized isogenic fold anchors. The same discrete bit word indexes both the CPU carrier and its computational-basis label, permitting quotient, fold, Cantor, layer, and objective maps to execute directly in the CPU model. Accordingly, V6.2 does not formulate cryptographic computation as faster exploration of an enormous state space. It formulates a stronger possibility: when the cryptographic objective descends through an efficiently computable exact quotient, the nominal state space is no longer the computational problem. Hardness is thereby relocated from raw state-space cardinality to the construction and evaluation of the objective-preserving quotient itself. Runtime evaluations instantiate this universal theorem carrier across prime indices from 11 through 2,147,483,647 including exact class folds, Cantor coordinates, polynomial-layer counts, endogenous exponent inverses, power/inverse class memory, and multiprecision Hilbert/Frobenius calculations. The resulting architecture provides a formal basis for studying quantum cryptography as exact class computation over certified moduli-degeneracy structure rather than representative-space search.
Paper 3 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper develops an exact finite local-to-global analysis of the six-dimensional Hamming-cube foundation Atlas underlying the TUS OS construction. Every admissible completed foundation carries the same exact local qubit prepare–measure representation, while the complete repeated-common-effect Atlas has exact minimum complex Hilbert dimension four. The paper classifies the full ambient family of completed four-state foundations in the six-dimensional Hamming cube. Every foundation is uniquely an affine coset associated with a perfect matching of the six coordinates. There are exactly 240 unordered foundations and 5,760 ordered codebooks. Every foundation induces the same exact 27-class tetrahedral-SIC qubit operational quotient, with carrier-fibre profile 1⁸, 2¹², 4⁶, 8¹ and the corresponding Bloch-radius structure of eight pure states, twelve states of radius √(2/3), six states of radius 1/√3, and one maximally mixed state. Each ambient chart reduces losslessly to a signed perfect matching. The chart-coloured union of two matchings obeys an exact component-parity collision law, yielding the complete seven-value image spectrum over all 28,680 ambient foundation pairs. Exactly 10,800 pairs jointly reconstruct the full 64-state carrier, giving an ambient mathematical chart-cover number of two. The frozen TUS selector defines a canonical subatlas of 36 distinct foundations and 64 anchor-indexed chart instances across nine matching classes. Its selected-foundation reconstruction graph is bipartite and eight-regular with 144 edges, while selector multiplicity yields exactly 400 reconstructing pairs among the 2,016 canonical chart-instance pairs. Only this canonical subatlas inherits the native LANDING and GUIDANCE structure. At the sequential level, all 384 active map/chart descent tests fail: no active LANDING map descends to a state-only transformation on a fixed 27-class local chart quotient. Under the complete principal future language, every canonical chart refines to all 64 deterministic predictive states, with maximum separating depth two. The subordinate GUIDANCE witness is also classified: its endpoint map has image size 34, eight two-cycle attractors and maximum transient depth five, while the additional GUIDANCE-selected chart raises principal path reconstruction from 138/192 contexts to 169/192. The paper then establishes the exact global static common-effect Hilbert dimension. Although every individual foundation admits an exact qubit representation, no unrestricted qutrit state/common-effect family can reproduce all 4,096 Atlas probabilities while simultaneously satisfying all 240 overlapping foundation POVM identities. Together with an explicit chart-independent dimension-four construction, this proves d_q^atlas = 4 for the complete static Atlas. The attaining construction uses two declared qubit sectors; unrestricted uniqueness or rigidity of minimum representations is not claimed. The result therefore exhibits a precise local-to-global separation: exact qubit prepare–measure form exists on every individual foundation, but an exact chart-independent repeated-common-effect representation of the complete Atlas requires complex Hilbert dimension four, while the complete deterministic future language preserves all 64 predictive identities. Local operational state, global relational state and deterministic predictive state are consequently distinct mathematical structures. The paper explicitly keeps these distinct from the still-unresolved whole-position process state. The scope boundary is explicit. The paper does not establish a physical quantum carrier, operational access to all 240 ambient foundations, LANDING as a quantum channel, GUIDANCE as a quantum observable, endpoint channel or instrument, contextuality, nonlocality, quantum advantage, unrestricted rigidity of minimum representations, or whole-position process synthesis. Its conclusions are exact at the static Atlas, canonical-chart and deterministic predictive scopes defined in the manuscript. Supporting General Theory Programme This paper is Paper 3 in the TUS OS quantum-operational and Answerability research programme. It extends the results of Papers 1–2 from the canonical local geometry and exact qubit prepare–measure behaviour to the complete finite relational Atlas. It classifies the full family of admissible foundations, proves that every local chart carries the same exact 27-class qubit operational quotient, identifies the canonical TUS subatlas, and shows that the complete native future language refines those local quotients back to all 64 deterministic predictive states. Its principal global result is that the complete common-effect Atlas has exact minimum quantum dimension d=4, while remaining distinct from the larger 64-state predictive structure and from later process-level dynamics. The paper also classifies the subordinate GUIDANCE witness and its finite reconstruction contribution without promoting it into a channel or instrument. The wider programme then develops complete-future and process closure in Paper 4, the Anchored Analog Quantum System framework in Paper 5, contract-relative resource and operational-observability theory in Papers 6–7, generic Answerability theory in Papers 8–9, complete TUS OS formal and executable realisation in Paper 10, and culminates in Paper 11 with A General Theory of Answerability. Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS TriadXOR Completion, Receipt Verification and the Qubit-Geometric Semantic WitnessEstablishes the canonical tetrahedral geometry, unique XOR completion, exact qubit-SIC representation and centred-face trine geometry. Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine ArchitectureEstablishes exact full-carrier qubit prepare–measure behaviour from the deterministic classical engine under the declared Coordinate-Incidence Operational Harness. For each lawful foundation, the 64 six-bit carriers map to exactly 27 anchor-relative qubit preparations under one foundation-defined tetrahedral-SIC Harness, while the separately declared face Harness gives the exact trine table. Paper 3 — From Finite Relational Structure to Qubit Operational FormAmbient Foundations, Canonical Subatlas, Exact Global Quantum Dimension, GUIDANCE Witness, Sequential Closure and Global Representation in TUS OSExtends the local prepare–measure construction to the complete ambient family of 240 completed foundations and the canonical TUS OS subatlas. Every completed foundation has the same 27-class qubit operational quotient; pair relations between charts provide exact carrier-reconstruction structure; the complete repeated-common-effect static Atlas has exact minimum complex Hilbert dimension 4; and the frozen complete principal future language has deterministic predictive-state minimum 64. The dimension-4 result is a static representation theorem and does not by itself supply common quantum process dynamics. Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability ModuleFixes the formal native domain, complete principal terminal future-experiment language and one-common-word experiment class; establishes complete statistical separation of distinct carrier pairs, exact parity-resolved discrimination results and complement reachability; and proves that the rational invariant probability module has dimension 33. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes universal complement-support orthogonality and its propagation under exact common-CPTP dynamics, yielding the certified same-parity support graph and the bound 32 ≤ d_CPTP^min ≤ 64, with cross-parity forcing and the exact state-only common-CPTP minimum remaining unresolved. It further establishes recovery-complete LANDING and GUIDANCE results, exact 64-dimensional selected-line instrument/readout minima at their declared contracts, an exact classical–quantum hybrid construction, and the native structural/predictive quotient correspondence. Paper 5 — TUS OS as an Anchored Analog Quantum SystemNative Witnesses, Predictive Synthesis, and Access-Contract Resource SeparationDefines Analog Quantum Views and Anchored Analog Quantum Systems, proves that TUS OS is a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus, elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, organises the inherited exact quantum-operational results through a typed access-contract resource law, and composes the recognition, predictive, resource and downstream-conservation results into the TUS OS Anchored Analog Quantum System Theorem. Paper 6 — Contract-Relative Quantum Realisations of Classical ProcessesPredictive Compression, Vanishing Worst-Case Causal-State Transparency, and Finite-Dimensional BreakdownExtends the programme beyond the TUS-specific discovery corpus by formalising the contract-relative achievable-resource object (S,C,𝔐,ε) ⟼ Ach(S,C;𝔐,ε), and establishes three exact contract-relative regimes: strict finite separation, unbounded finite separation and finite-capacity breakdown. These are instantiated respectively by the TUS OS 4→64 separation, the periodically modulated decay 2→N exact prediction-to-transparency separation, and dual-Poisson finite-capacity breakdown for every positive uniform worst-case transparency threshold. Paper 7 — Operational Observability and Contract-Induced Resource Transitions in Quantum RealisabilityStructural Classification, Family Scaling, and Finite-Resource BreakdownDevelops a structural classification
Traitement intégré pour la lecture massive de bits quantiques L’informatique quantique promet de nombreuses révolutions dans divers domaines en permettant de réaliser des calculs massivement parallèles actuellement inaccessibles aux ordinateurs classiques. Cependant, pour atteindre un ordinateur quantique tolérant aux fautes et pleinement opérationnel, il faudra disposer de millions de bits quantiques (qubits) fonctionnant à des températures cryogéniques. De plus, la manipulation, le contrôle et la lecture d’un si grand nombre de qubits représentent un défi crucial, pour lequel l’électronique intégrée cryogénique (Cryo-CMOS) s’est déjà révélée être une candidate prometteuse. Ce travail de doctorat vise à explorer la mise en oeuvre d’une architecture de lecture intégrée compatible avec un grand nombre de qubits, ce qui sera essentiel pour la lecture future de millions de qubits. La première étape a été l’identification d’une limitation dans l’état de l’art des architectures cryo-CMOS de lecture à grande échelle pour la lecture de qubits de spin. Ces architectures nécessitent typiquement des inductances difficilement intégrables, dont la taille est prohibitive par rapport à celle des qubits, ce qui les rendra rapidement limitantes avec l’augmentation du nombre de qubits. Afin d’adresser ce problème, nous avons proposé une architecture de lecture de courant multiplexée en fréquence, qui peut être complètement intégrée et n’utilise pas d’inductances. De plus, nous avons introduit l’utilisation des électromètres comme modulateur afin de réaliser des modulations d’ordres supérieurs. Cela permet effectivement d’augmenter la capacité théorique de notre architecture de lecture multiplexée en courant et ouvre la voie à la lecture de plusieurs qubits par fréquence de porteuse. Nous avons ensuite mis en évidence la topologie de l’amplificateur transimpédance à rétroaction capacitive (CTIA) comme une candidate prometteuse pour la mise en oeuvre de notre architecture de lecture. Face aux limitations de la modélisation de cette topologie, nous avons développé un modèle complet décrivant sa réponse fréquentielle, son bruit et sa stabilité. En utilisant ce modèle, nous avons pu démontrer que la topologie CTIA offre de meilleures performances que la topologie d’amplificateur transimpédance à rétroaction résistive (RTIA) déjà démontrée. Finalement, cela nous a permis d’établir une stratégie de conception claire pour relever les défis liés à la conception d’un CTIA pour des applications cryogéniques. Deux circuits ont été conçus en utilisant deux noeuds technologiques (28nm et 22nm), intégrant des électromètres, un CTIA et un buffer, afin de démontrer notre lecture multiplexée par lecture de courant. Le bon fonctionnement de notre CTIA et de notre buffer à 4,2K a ensuite été validé pour les deux circuits conçus, en tirant parti de la flexibilité implémentée lors de la conception et de la polarisation des grilles enterrées, rendue possible par les technologies FDSOI. En utilisant notre CTIA et buffer placés à 4,2K, nous avons caractérisé avec succès plusieurs électromètres conçus dans deux technologies commerciales différentes, en utilisant des mesures de conductance gds et de transconductance gm. Nous avons ensuite démontré le premier multiplexage en fréquence de plusieurs électromètres utilisant de l’électronique Cryo-CMOS, sans nécessiter d’inductances. Puis, nous avons réussi la première lecture de plusieurs électromètres avec une modulation de haut ordre, permettant ainsi de lire plusieurs électromètres par fréquence de porteuse. De plus, les performances obtenues lors de ces deux démonstrations, en termes de temps de lecture, de consommation et d’empreinte, sont compétitives par rapport à l’état de l’art, ce qui ouvre la voie à des architectures intégrées compactes pour la lecture d’un grand nombre de qubits
Code, results and tests for end-to-end studies of chemical reactions on qubits, from molecular integrals to the 32-bit instruction stream a hardware state-vector simulator executes. Circuits and electronic structure: PySCF active spaces from CCSD natural orbitals, optionally embedded in point charges, mapped to qubits by Jordan-Wigner; UCCSD-VQE and fermionic ADAPT-VQE with analytic adjoint gradients; compilation to Pauli exponentials, a peephole optimiser, a 32-bit instruction set, and bit-accurate 18-bit fixed-point emulation of the qsim-pynq FPGA datapath (PYNQ-Z2). Every circuit energy is checked against exact diagonalisation in the same space. Reaction energetics: transition-state theory rates, Wigner and Eckart tunnelling, and composite levels combining CCSD(T) with the circuit result in the same active space. Applications. Proton-transfer tautomerism of formamide and 5-fluorouracil. Hydride transfer in thymidylate synthase from a 96-atom active-site cluster built from PDB 1TLS, with a GFN2-xTB reaction scan, a frozen-atom-aware P-RFO saddle search, finite-difference Hessians, H/D/T kinetic isotope effects, DFT single points and ONIOM layering. A lattice protein-folding benchmark on the diamond lattice with Miyazawa-Jernigan contact energies and the dense turn encoding. New in 1.2.0. Prolyl cis/trans isomerisation of capped Xaa-Pro dipeptides, including the phosphorylation effect. Ligand-binding pocket clusters cut from crystal structures with in-silico alanine truncation, applied to CFTR with lumacaftor and acetylcholinesterase with donepezil. Amyloid steric-zipper interfaces as cluster models, for tau, alpha-synuclein and SOD1 segments, benchmarked against deposited structures. A six-system chalcogen S_N2 ladder separating nucleophile from electrophile effects in thiol-disulfide and selenium-sulfur exchange, with its numerical precision floor measured from closed thermodynamic constraints rather than asserted. Shrake-Rupley solvent accessibility, and geometric feasibility analyses of the buried Cys57-Cys146 disulfide of SOD1: solvent accessibility, backside attack geometry, side-chain rotamer scans and the displacement required to open the attack site. These are geometric and semiempirical analyses, not completed energetic determinations. New in 1.3.0. A reproducible epitope map for conformation-specific antibodies against misfolded SOD1: per-residue Shrake-Rupley accessibility of the native holo dimer against the middle layer of a three-layer stack of the wild-type fibril, which is the difference such an antibody needs, since an epitope exposed on both states binds normal protein in every cell that makes SOD1. Eleven residues are buried natively and exposed on the aggregate. The published rationale of a recent conformation-specific antibody is tested against it. A residue-accessibility bound catches summed alternate conformers and colliding residue numbers. Crystal structures for this work are fetched from RCSB like the rest. Also packaged in this release, and absent from 1.2.0: the FastAPI service is now actually installed by the package, which it was not, and the continuous integration fixes that followed. New in 1.4.0. The SOD1 geometric analyses of 1.2.0 are completed and replicated, and the energetic determination they were waiting for is here. The attack geometry is measured across three crystal forms rather than one -- 2C9U at 1.24 A, 1HL5 at 1.80 A and 7FB9 at 2.70 A -- keeping every (chain, altloc) as its own set of atoms rather than resolving alternate conformers by occupancy, which gives thirty-seven chains, thirty-four of them modelling a formed disulfide. The backside attack angle at Cys146 spans 131.9 to 146.8 degrees and reaches the 150-degree collinearity floor in none of them; the attack site 3.3 A beyond the sulfur is occupied in all thirty-seven, with main-chain atoms among the occupants in all thirty-seven. The identity of the occupying residues does not replicate -- two of them in a single chain each, against Cys146 in all thirty-seven and Gly147 in thirty-six -- which is the result that matters for what can be claimed: the obstruction is the main chain of the neighbouring strands and not any named side chain, so no rotamer rearrangement removes it. All four intramolecular routes any free SOD1 cysteine could take to the disulfide are measured, not only the one the literature names, which closes the relay objection: SOD1 has four cysteines, two of them are the bond, and there is no spare thiol to resolve an intermediate. The conformational work is now determined and not pending. Bringing Cys6 to 3.12 A of Cys146 at a 175-degree approach, in a 259-atom cluster with the outer shell held and the local backbone free, costs +131.02 kcal/mol at GFN2-xTB with ALPB solvation. Inverting Eyring at 310.15 K with the full k_B.T/h prefactor allows 25.18, 28.82 and 31.23 kcal/mol for one event per day, per year and per fifty years, so the approach alone exceeds the fifty-year ceiling by a factor of 4.2 before any bond-forming barrier is added, and crosses it at about 6.5 A with 3.3 A still to travel. Every covalent bond of that series was measured at every scan point against the relaxed reference: none is stretched by more than 11 per cent, and that one is the attacked disulfide lengthening from 2.02 to 2.24 A as the nucleophile closes. No complementary upper bound is reported, and the reason is recorded rather than omitted. Held rigid, the fold does not yield elastically to contact: it breaks a covalent bond. Restraining the backbone alone ruptured the Cys57-Cys146 disulfide between 6.27 and 5.92 A, with the nucleophile still 5.9 A away and taking no part in it. Restraining that bond at its relaxed length as well moved the failure instead of removing it, to the Cys57 CB-SG bond at the same step. Both runs are archived in results/ with the diagnosis that retires them, because each added restraint relocates the rupture and a constraint set chosen until the curve behaves is chosen for its effect on the result. What the two agree on is reported: a rigid fold cannot be pushed past 5.9 A, and arrives there having spent 2.7 times the fifty-year ceiling on strain alone. Validation gains the check that would have caught the above at the first bad point rather than eleven hours later. covalent_stretch is the longest covalent bond of a restrained geometry over its length in the relaxed reference, with the bond list built once from that reference so a broken bond cannot be re-derived as simply absent; it is fatal, and replayed over the three runs it passes the reported series and stops both discarded ones at their first ruptured point. Long scans now checkpoint per point and resume from the last one, rather than holding a stage in memory until it ends. A correction: the epitope map named a residue as occluding the attack trajectory which occludes nothing, absent from every occupancy list at both probe radii and from all thirty-seven chains; the claim is corrected in place and dated, and no computed value changed. Validation. A physical-bounds module carrying 37 checks, each with the reason it exists written into the code; an auditor that recomputes them from stored geometry and separates recorded dead ends from open faults; a stall detector deriving its unit list from systemd; and a 101-case self-test that exercises every bound in both directions, using archived real failures as labelled positives. Long runs are resumable systemd units with atomic checkpoints and per-point resume. A FastAPI module serves the cases and their instruction streams. A Dockerfile pins the exact versions the results were produced with. Crystal structures are fetched from RCSB rather than redistributed. Educational; not for clinical use. New in 1.4.1. The derivation behind the TDP-43 charge-step result is deposited rather than only its numbers: scripts/tdp43_charge_steps.py converts the published conversion percentages of a charge-variant series to rate constants and reports the multiple each step of motif net charge buys, under both the binned-observation and the authors' fitted-line conventions, because the first step depends on which is used while the ordering does not. It reads its constructs from tdp43_frailty.py rather than carrying a second copy. Writing it corrected a number: the conversion hazard had been described as flat to 4 per cent across a four-fold range of time, which is the coefficient of variation of the three estimates, but read as a bound on any one estimate it understates by more than a factor of two, the worst point being 6.0 per cent from the mean and lowest to highest 10.1 per cent. All three measures are now reported so that a sentence quoting one cannot be read as another. The SOD1 attack-site replication is also recorded at the second probe distance: the cross-crystal analysis defines the ideal attack point 3.0 A beyond the leaving sulfur in the methods and computed it at 3.3 A, so both were run and both are archived. Nothing the exclusion rests on moves -- thirty-seven chains, thirty-four with a formed disulfide, the site occupied in all thirty-seven with main-chain atoms among the occupants in all thirty-seven, no chain reaching the 150-degree floor -- while the identity of the occupying residues does move, which is the quantity already reported as not replicating. Tooling: the Markdown-to-PDF renderer draws the thirteen characters outside Latin-1 that the built-in PDF fonts cannot, which had been emitting filled squares where the manuscripts write Greek, reaction arrows and angle brackets, and it now reports what it had to borrow. A sibling renders the same Markdown subset to .docx. CITATION.cff, which still read 1.3.0 in the 1.4.0 release, is corrected. New in 1.4.2. The SOD1 cysteine work gains the assay that would settle it, and the reason no existing dataset can. Published SOD1 proteomic surveys run carbamidomethyl on cysteine as a fixed modification, so every cysteine is reduc
QHŌR develops an exact mathematical theory of control through geometry changes under conserved quantities. Its central result is a universal actuation law: within the stated full-spark, one-spare model, every closed protocol shorter than eight strokes has identity amplitude transport, while a nonidentity eight-stroke return always exists. The work goes beyond identifying a short control sequence. It constructs a sparse library that generates the entire return group, classifies every one-spare reachability obstruction, distinguishes executable control from scalar-linear span over extension fields, and determines a sharp maximum for the number of independent control directions over the prime field. The universal library contains ν_r = binomial(r, 2) + r − 1 = (r + 2)(r − 1)/2 eight-stroke returns. Its size is optimal when quantified over the full admissible field-and-palette class: for every rank r ≥ 2 and prime characteristic p ≥ 5, a suitable finite-extension example attains a return group requiring exactly ν_r generators. This release brings together the complete 12-page standalone paper, editable mathematical sources, an exact prime-field compiler, an independent internal proof audit, verification programs, original computational ledgers, structured theorem and certificate catalogs, and a prepared Hugging Face release bundle. The mathematical setting A minimal control cell contains r + 1 labelled sites. Each site carries an amplitude and a geometry label. Every allowed label corresponds to a column of a palette matrix H, and the conserved syndrome is s = Hₓz, where z is the amplitude vector and Hₓ is the matrix associated with the currently occupied geometry labels. A one-spare palette contains r + 2 symbols. Exactly one symbol is unused at any moment and serves as the hole. A primitive stroke exchanges the hole with the label at a selected controller site. The controller amplitude remains unchanged, while the other r amplitudes compensate through an invertible anchor matrix so that the syndrome is preserved. The full-spark condition requires every r palette columns to be linearly independent. This guarantees that the compensating update is uniquely defined. A return restores the geometry label at every site individually. Such a protocol can nevertheless produce a nonidentity transformation of the amplitudes. This residual action is the holonomy of the return. For a fixed nonzero base-cell kernel vector k, every compatible linear return has the form W = I + k λ Hₓ. Composition adds the endpoint rows λ. The control problem therefore becomes an exact question about which rows are executable, how to synthesize them, and how to certify an obstruction when a requested row is unavailable. The principal prime-field results and software assume r ≥ 2, a prime p > r + 2, and a full-spark r × (r + 2) palette over Fₚ. The larger-field results instead assume characteristic p ≥ 5 and field cardinality greater than r + 2. Five principal theorems Sparse fixed-base generation A prescribed family of exactly ν_r eight-stroke returns generates the complete one-spare return group at the original geometry. Compiler discovery requires no enumeration of the factorially many geometry configurations. Using zero-based controller positions, define R(i; j, k) = (i, j, k, j, i, j, k, j). The sparse library retains R(0; j, k) for every pair 1 ≤ j < k ≤ r, together with R(1; 0, k) for 2 ≤ k ≤ r. The second family forms a star spanning tree on the nondistinguished controller positions. Other spanning trees can replace the star while preserving completeness and library size. The proof converts every primitive stroke into an affine reflection. The resulting four-symbol curvature tests detect whether the edge offsets arise from vertex potentials. A spanning-tree reduction removes redundant tests, and the surviving formal curvature vectors form a basis of the signless-incidence kernel of the complete palette graph. Complete general-palette obstruction classification Over the prime field, the reachable return group has dimension exactly r or r − 1. The classification requires full spark but does not require a common conserved charge row. Let D = ker H. Deficiency occurs exactly when the all-ones vector belongs to the Schur square of D: the linear span of coordinatewise products of pairs of kernel vectors. Because D has dimension two, this criterion can be checked using the three quadratic products of a kernel basis. It provides a direct algebraic decision procedure rather than a search through control sequences. The theorem includes the case in which the palette columns sum to zero. In that case, the kernel can be written as span(1, t), with distinct entries of t, and the common-fixed-syndrome line is generated by the nonzero vector H(t ⊙ t). Thus the zero-column-sum case has an explicit intrinsic obstruction rather than being excluded from the theory. Over a larger scalar field, the corresponding classification describes the field-linear envelope of the return rows. Exact executable reachability additionally requires the prime-subfield analysis of Theorem 4. Universal sharp eight-stroke actuation Every one-spare geometry return shorter than eight strokes has identity amplitude transport. A nonidentity member of the sparse eight-stroke library always exists under the theorem’s hypotheses. The lower-bound proof combines permutation parity, cancellation of inverse strokes, a classification of the possible short reduced return words, and the affine-reflection normal form. The obstruction classification guarantees that the return group is nontrivial for every admissible rank r ≥ 2. Eight is therefore the exact shortest nonidentity return length throughout this one-spare model. The resource condition is essential. With two simultaneously available spares, nonidentity four-stroke rectangles can occur. The release supplies and independently verifies such an example. The eight-stroke law concerns the first nontrivial actuation with one spare; it does not assert that every requested target has an eight-stroke implementation. Exact prime-subfield generation and the control-dimension cap The sparse generation principle extends to every field F of characteristic p ≥ 5 satisfying the cardinality hypothesis. The exact return group is generated over the prime subfield Fₚ. This distinction matters because repetition of an executable loop supplies prime-field multiples of its endpoint row. It does not automatically supply multiplication by arbitrary elements of a larger scalar field. Over GF(p^e), the field with p^e elements, exact reachability is determined by flattening the endpoint rows into er prime-field coordinates. If their prime-field rank is d, the actual return group has order p^d, with d ≤ min(er, ν_r). The group has at most p^ν_r elements even when the ambient field F is infinite. A concrete rank-two example over GF(25) has a full two-dimensional GF(25)-linear envelope but only 25 executable returns, compared with 625 transformations in the compatible ambient group. An explicit prime-field linear functional excludes a displayed target. This establishes a precise scalar-field bottleneck: enlarging the amplitude field can enlarge the compatible target space without supplying the corresponding executable control directions. Attained maximum and optimal universal generator count For every r ≥ 2 and prime p ≥ 5, a full-spark palette over some sufficiently large finite extension attains prime-field return dimension exactly ν_r. The proof first constructs an injective curvature map over a rational function field with independent kernel parameters. A finite-extension specialization preserves full spark and the independence of all nonzero prime-field curvature combinations. Consequently, the dimension cap is sharp within the actual control model. Since an elementary abelian group of prime-field dimension ν_r requires at least ν_r generators, the sparse library has the smallest possible universal generator count across the stated field-and-palette class. This optimality has a specific quantifier. An individual prime-field instance has a basis of r or r − 1 loops. The theorem does not claim that ν_r generators are necessary for every instance, or that the emitted program is shortest for each target. Exact compilation and certificates The implementation accepts arbitrary prime-field column dictionaries, including non-affine examples. It certifies full spark through a two-dimensional kernel and its pair minors, computes candidate rows directly from curvature, and independently checks selected basis loops by executing their primitive compensating transports. For a reachable target, exact elimination selects a basis and produces signed repetition counts. For an unreachable one-spare target, the program returns a syndrome annihilated by every attainable return row but not by the requested target row. This witness excludes every return in that palette. For prime-field full-spark families with a common charge covector, two distinct spare labels make the two sparse libraries jointly complete. Their 2ν_r candidates contain an r-loop basis. Every compatible target then has a compressed program using at most r basis macros and at most 4r(p − 1) expanded primitive strokes. Discovery and compressed program generation take polynomial time in r and log p. Expanded execution may take time proportional to the numerical value p. For arbitrary non-affine multiple-spare inputs, the software certifies the span generated by the available one-spare macros. A failure of that library is explicitly distinguished from a proof excluding all protocols that mix spare labels before returning. Verification evidence The release includes exact computational checks and their original records: • Fifteen general dictionary cases across ranks two through six and primes 17, 31, and 101. • Three hundred comparis