The NEXUS Quantum Compiler Mathematical-Space Execution and Constrained Molecular Folding
Dean Kulik
The NEXUS Quantum Compiler Mathematical-Space Execution and Constrained Molecular Folding Native reversible routes · quantum-sector reduction · legal molecular transformations · classical execution Driven by Dean A. Kulik · October 2026 ABSTRACT A quantum-compatible mathematical state space can be compiled and executed on ordinary classical hardware when its constraints, return paths, and observable symmetry sectors are explicitly represented. This manuscript integrates two complementary NEXUS studies without replacing either with a shorter summary. The first examines the exact native ring and E/G return machinery and audits an earlier quantum lattice-protein simulation whose eight-address mixer introduced four transitions through invalid padding states. It derives the short-time leakage coefficient 49/75, explains the distinction between finite penalties and exact subspace preservation, identifies the disconnected projected graph, and records the approximation and environmental-return requirements. The second advances to a self-avoiding honeycomb-chain compiler in which turn words construct geometry and admissible bit rewrites construct the transition graph. Global complement is an exact mirror symmetry; for a complement-stable Hamiltonian it separates the quantum space into even and odd sectors, retaining the necessary sheet sign and inverse readout. In the 12-residue HP example, 776 legal conformations form 388 mirror pairs. Classical propagation in the full Hilbert space and its independently assembled mirror-even sector agrees after reconstruction to machine precision under the reported sparse solver; the precise rounding error is implementation dependent. Local four-turn hexagon rewrites are reversible but fragment the legal graph; unrestricted pivots reconnect it while sacrificing bounded spatial support. Together these results establish a rigorous, finite compiler architecture and a precise reduction in the resources needed for symmetry-selected states. They do not establish experimentally validated molecular kinetics, a general quantum-simulation speedup, or measured low electrical power. The remaining link is an independently constrained molecular transition generator, its environmental coupling, and a resource benchmark carried out under the same input–output contract. Keywords: NEXUS; reversible routing; quantum-inspired classical simulation; parity decomposition; Hilbert-space reduction; self-avoiding walks; hydrophobic-polar models; constrained transitions; local moves; protein folding. Figure 1. The integrated compiler path: binary word → geometric legality → admissible transition graph → symmetry sectors → classical propagation. The route back to full state is explicit. PART I / THE NATIVE STATE SPACE AND WHAT “QUANTUM COMPILER” MEANS 1. Problem statement: computation before hardware The motivating hypothesis is that a sufficiently explicit mathematical relation space can execute useful quantum-style calculations on conventional, low-power hardware. In its strongest form, this would make special-purpose quantum machines unnecessary for the targeted tasks. A physically meaningful version of that hypothesis must distinguish three different propositions: an algebraic operator can be represented classically; a specified quantum evolution can be simulated faithfully; and the simulation has an advantageous cost in time, memory, or joules. The first two are demonstrated here in a finite structured family. The third remains subject to benchmarking. The Nexus program approaches problems from legal transformations and their return paths rather than from cataloged answers. Earlier One Inverter and twisted-partner-ring work constructed involutions, native addresses, temporal center reads, and odd-lane locks [1, 2]. The honeycomb extensions (§§144–146) supply a particularly instructive domain: a word is a conformation-generating route, while a molecular move is determined by geometry rather than by neighboring values in an address table [2]. Classical simulation of quantum subfamilies is established mathematical territory. Stabilizer circuits admit efficient classical simulation [6], and quantum-inspired linear algebra can dequantize some low-rank algorithms under explicit data-access assumptions [7, 8]. The point of the present work is narrower: show where the Nexus native-route and reflection machinery gives a concrete geometry-respecting Hilbert-space reduction, with complete return information and negative controls. The governing research hypothesis is stronger than an indexing engine. The compiler is proposed as a transformation architecture: a molecular sequence and surroundings determine lawful continuations, and a fold is the realized trajectory rather than a catalog match. The scientific question is whether a native, local, reversible transformation system reproduces a molecular process without placing the answer in a chosen transition matrix. Three different “one answer” statements must remain typed. A fully specified deterministic calculation has one result. A fully specified quantum experiment has a definite predicted probability distribution but may have multiple measurement outcomes. A protein may have an ensemble of conformations that changes with conditions. The route does not license substituting one for the others. For this paper, “compilation” has a testable meaning: construct a state domain, certify lawful transformations, select a representation that makes required relations visible, propagate that declared operator, and preserve an inverse or a residual route. Classical hardware is sufficient to carry out these finite algebraic steps. Whether it is efficient for an enlarging family depends on the structure of that family, not on a claim that all quantum amplitudes can be recovered from one address. 2. Native ring: partner, face, canonical address, and exact return For an n-bit word, J is bit reversal and C is bitwise complement. Define P(x)=J(x) unless x is palindromic, in which case P(x)=C(x). P²=I, P has no fixed word, and the native ring pairs addresses antipodally: PTP=T⁻¹, where T is the ring’s own successor. The inward E/G route ends at width two, retaining each branch and middle qualifier and a face bit. The complete code is n bits for an n-bit word; reconstruction is exact but not generic compression. P(x) = J(x) if J(x) ≠ x; P(x) = C(x) otherwise. The canonical address is an exact re-labeling of word space. T is a native address successor. A raw k-bit transport shift, a partner read, and T are separate operators. In particular, declaring an edge between successive catalog addresses does not establish that the corresponding conformations are molecular neighbors. Three interfaces must not be conflated. The native ring successor T is a step in canonical address order. The geometric word rewrite is a change of chain conformation. A measurement/readout R extracts a specified property of the state. In particular, T does not become a protein move simply because it is a reversible permutation. The twisted one-inverter clock and temporal reader The “One Inverter” paper derives the affine twisted rotation τ(x₀…xₙ₋₁)=x₁…xₙ₋₁¬x₀, so after k ticks the first k departing bits are returned complemented. At k=n it is global complement; at k=2n the state returns. The inward terminal reader is exactly the XOR of the two central word bits. Under τ, its temporal trace reads each adjacent difference around the cyclic word, with one known inverted crossing. τᵏ(x) = rotlₖ(x) ⊕ (2ᵏ − 1), 0 ≤ k ≤ n; τⁿ = C, τ²ⁿ = I. r(x) = xₙ/₂₋₁ ⊕ xₙ/₂ ; ⊕ₜ₌₀ⁿ⁻¹ r(τᵗ x) = 1. [even n] The n−1 independent adjacent differences plus one native face bit recover the word. The two-bit closed complement-feedback rule is τ², not a new primitive. None of this creates a physical molecule; it gives a provably executable, invertible substrate for how a local reader can obtain and return distinctions. Odd-lane locks and relational deviations For odd q and n=qs, the alternating-complement lift Λq(u)=u‖Cu‖u‖…‖u is invariant under the twisted clock and intertwines J, C, and P. A center trace of period s identifies this lock and supports exact reconstruction through the first s-bit lane. In the full q-lane state, de-staggered lane differences and one anchor preserve all n bits. τₙ Λq = Λq τₛ ; Pₙ Λq = Λq Pₛ ; dH(Λq u,Λq v) = q · dH(u,v). These differences were formerly called “defects.” They are lawful relational deviations from a selected lock—not failures of the transformation. Under common affine transport, differences rotate and preserve weight. An external input can alter them at a known boundary, but selecting an input law is not already derived by the native permutation. The native address T, the twisted clock τ, and the geometric pivot defined later remain separate typed operators. The exact E/G route has n bits for a general n-bit input. A locked multi-lane family instead requires only its independent local state and the stated lift rule; the reduction comes from a demonstrated invariant restriction, not from labeling an arbitrary large input “locked.” Which operations the embedding preserves The One Inverter temporal read is an operational return path, not just a yes/no label. For n=2h, define the cyclic adjacent differences Dᵢ=xᵢ⊕xᵢ₊₁ (indices modulo n) and the observed center trace yₜ=r(τᵗx). The twisted boundary contributes exactly one inverted crossing. Over n ticks the center presents n relations; their XOR equals one, and n−1 of them together with the retained canonical face bit reconstruct the entire original word. The E/G inward/outward route supplies a separate exact return by retained qualifiers. Neither return is compression of arbitrary n-bit words. yₜ = D₍ₕ₋₁₊ₜ₎ mod n ⊕ [t=h], ⊕ₜ₌₀ⁿ⁻¹ yₜ = 1; (y₀,…,yₙ₋₂, face) ↔ x. The two masks must not be conflated. The mirror shell lamps are gⱼ=xⱼ⊕xₙ₋₁₋ⱼ, the paired entries in x⊕J(x). The dist
•Oct 23, 2026•0 citations