We present the dataset and comprehensive manuscripts for the TROIKA (Triple Rotor Oscillator Interlinked with K-cycle Architecture) model. This framework establishes a bridge between discrete geometric entanglement and continuous rotational mechanics, exploring the hybrid coupling between directional degrees of freedom and continuous vibrational modes across three entangled sources. The provided documentation includes the full theoretical derivation of a cyclically entangled three-qutrit state that enforces a strict orthogonal trihedron upon projective measurement. By utilizing Weyl-Heisenberg shift operators and establishing a discrete phase variable through a classical Lagrangian, the model derives a novel directional-vibrational interaction Hamiltonian. Additionally, the full manuscripts detail the exact global symmetry operator (K) that confines the system's dynamics within an invariant subspace, drastically reducing the active Hilbert space dimensionality. Furthermore, this deposit includes the complete suite of MATLAB source codes used to solve the time-independent Schrödinger equation via exact matrix diagonalization and Kronecker products. The codebase implements advanced matrix contractions to compute partial traces, tracking non-Markovian Rabi revivals, Von Neumann entanglement entropy, and Wigner quasiprobability distributions—formally demonstrating the active generation of non-classical vibrational states with distinct negativity wells in phase space. Finally, an open-ended continuous extension is formulated, showcasing the discrete cyclic structure as an effective low-energy sector of an isotropic rigid rotor on the SO(3) configuration manifold. Files included:- Theoretical Manuscript in English (PDF)- Theoretical Manuscript in Spanish (PDF)- Core MATLAB source codes for dynamics, entropy, and Wigner functions (.m)
We present the dataset and comprehensive manuscripts for the TROIKA (Triple Rotor Oscillator Interlinked with K-cycle Architecture) model. This framework establishes a bridge between discrete geometric entanglement and continuous rotational mechanics, exploring the hybrid coupling between directional degrees of freedom and continuous vibrational modes across three entangled sources. The provided documentation includes the full theoretical derivation of a cyclically entangled three-qutrit state that enforces a strict orthogonal trihedron upon projective measurement. By utilizing Weyl-Heisenberg shift operators and establishing a discrete phase variable through a classical Lagrangian, the model derives a novel directional-vibrational interaction Hamiltonian. Additionally, the full manuscripts detail the exact global symmetry operator (K) that confines the system's dynamics within an invariant subspace, drastically reducing the active Hilbert space dimensionality. Furthermore, this deposit includes the complete suite of MATLAB source codes used to solve the time-independent Schrödinger equation via exact matrix diagonalization and Kronecker products. The codebase implements advanced matrix contractions to compute partial traces, tracking non-Markovian Rabi revivals, Von Neumann entanglement entropy, and Wigner quasiprobability distributions—formally demonstrating the active generation of non-classical vibrational states with distinct negativity wells in phase space. Finally, an open-ended continuous extension is formulated, showcasing the discrete cyclic structure as an effective low-energy sector of an isotropic rigid rotor on the SO(3) configuration manifold. Files included:- Theoretical Manuscript in English (PDF)- Theoretical Manuscript in Spanish (PDF)- Core MATLAB source codes for dynamics, entropy, and Wigner functions (.m)
This theoretical preprint develops a limited bridge from a specified two-state Hamiltonian with coherent coupling paths to a phase-dependent small-displacement stiffness. Reciprocal Ehrenfest forces and torque yield a conservative energy account; classical Langevin bath couplings and work-store or autonomous energy-depot controls extend the comparison to heat and supplied work. The autonomous free-depot branches have analytically determined rates that depend on the parameters. No independent selection of 0.218 Hz is obtained. A sustained mean rate and a shared phase are distinct, and the stochastic ensembles are independent repeats of one rotor, not a coupled rotor network. Illustrative scaled inputs are used throughout; physical Hz labels are conditional on a chosen reference time. The Born rule is an input. Material realization, global hybrid stability, full quantum Gibbs equilibrium and a complete Model-D embedding remain open. The package supplies the manuscript, source, figures, numerical records and integrity checks. No new empirical measurements or live Model-D integration are reported.
This repository contains the complete open-source algorithmic implementation, simulation testbenches, and cross-layer telemetry suites for the validation of the Ramanujan-Rao Quantum Soliton Protocol (RR-QSP). The software architecture is designed to reproduce the exact performance profiles, resource optimizations, and mathematical proofs presented in the manuscript titled: "The Ramanujan-Rao Quantum Soliton Protocol (RR-QSP): A Multi-Tier Asymmetric Architecture for Autonomous, Tail-Aware Quantum Wide-Area Networks" ### Core Functional Modules & Pipeline Architectures 1. Asymmetric [[384, 104]] Homological Q-LDPC Code Space Engine: Implements the asymmetric hypergraph Kronecker product configuration that systematically decouples the X-basis and Z-basis stabilizer parity-check matrices (enforcing M_Z = 2M_X). This module allocates double the check density to insulate volatile phase-flip (Z) errors over continental fiber waveguides, validating a 27.08% logical code rate efficiency. 2. 4th-Order Cornish-Fisher Syndrome Decoder: Replaces static linear thresholds with a moment-aware, physics-informed dynamic boundary loop. By mapping live syndrome skewness (γ) and excess kurtosis (κ) via an exact Edgeworth expansion, the decoder intercepts non-Gaussian error clustering tracks before they trigger catastrophic silent timing escapes. 3. Algebraic Chakravala Pellian Router: A deterministic, constant-time path deflection engine that solves the negative Pell equation (x² - D·y² = -1) over the ring of integers. The module achieves a loop-free path permutation layout within a O(√D log D) logarithmic tracking timeline, ensuring detour calculations complete inside a 0.21 ms window—well within the cryogenic T₂ coherence runway of stationary memory registers. 4. Multi-Tenant Coprime Graph Dispersion Layer (MT-CCRP): Leverages the regular spectral expansion bounds of a 3-regular Ramanujan graph to scatter concurrent user bursts across adjacent orthogonal sectors, maintaining a perfect 0% qubit drop rate under peak workload saturation. 5. High-Order Skewness Adversary Filter: Monitors third central moment shifts to instantly differentiate ambient thermal noise anomalies from active cryptographic projective measurement taps (Eve), executing real-time topological pruning and link isolation routines. ### Repository Contents & Out-of-the-Box Execution - main.py: The master functional execution script that runs the entire pipeline verification suite and outputs the core statistical benchmarks documented in the paper directly to the terminal workspace.- requirements.txt: Defines the software dependency framework required to run the simulation engine, including scientific computing libraries and graph neural learning packages.- README.md: Comprehensive user manual detailing installation steps, architectural parameterization guidelines, and module execution options.- LICENSE: Open-source release governed by the highly permissive MIT License rules. ### Metadata Tag KeywordsQuantum Internet; Asymmetric Q-LDPC Codes; Chakravala Pell Solvers; Cornish-Fisher Inversion; Multi-Tenant Congestion; Topological Security; Ramanujan Expander Graphs; Telemetry Predictors.
Beyond Binary: How Single-Transit Photonic Resonance and Dynamic Quantum Biophysics Solves the Decoherence Barrier ABSTRACT This work synthesizes a fundamental paradigm shift in computational physics and hardware architecture, uniting non-binary photonic wave interference with lossless, open-system quantum dynamics. Traditional computational paradigms suffer from two foundational bottlenecks: binary CMOS architecture bound by Landauer switching heat and von Neumann memory bus latency, and quantum simulations constrained by exponential Hilbert-space state blowup and environmental decoherence. Here, we establish the mathematical and physical foundations of Discrete Lattice Resonance Architecture (DLRA) operating alongside the Classical Quantum Biophysics (CQBP) simulation framework. 1. Photonic Substrate and Non-Binary Instruction Set We present a fixed-topology, single-transit optical core inscribed within an \\(\alpha\\)-quartz dielectric lattice containing core-shell Schottky junction nodes. Logic execution shifts from binary (1/0) gate switching to the continuous Octad Instruction Tuple \\(\Theta = (A, \phi_1, \phi_2, \phi_3, \phi_4, \sigma, \chi, \omega)\\), parameterizing pulse amplitude, quadrant phases, duration, polarization, and spatial carrier frequencies. Global matrix-vector transformations (\\(10^{12}\\) MACs/transit) execute within a single 21-picosecond optical transit (\\(\tau \approx 21\text{ ps}\\)), eliminating transistor gate-switching heat and memory bus delays entirely. 2. Diffeomorphic Reduction and State-Space Linearization We formalize the mathematical reduction of non-linear system flows and infinite-dimensional differential operators into exact, finite matrix updates. By establishing Lie algebra intertwining conditions over generator-closed invariant carriers, complex non-linear dynamics—including post-quantum cryptographic trapdoors (such as AES-128 key recovery)—are mapped via pullback transformations into exact linear-time (\\(\mathcal{O}(N)\\)) matrix-vector evolutions, resolving classical state-space search explosions. 3. Dynamically Coupled Open-System Quantum Biophysics To bridge classical molecular mechanics with open quantum systems, the CQBP engine introduces an object-oriented hierarchy inheriting from a unified `Simulator` base class. Classical particle dynamics (`MolecularDynamics`), driven by velocity Verlet integration of Lennard-Jones pair potentials with center-of-mass motion removal, operate in direct lockstep with open density matrix evolutions (`QuantumDynamics`): Dynamic Lindblad Coupling: Classical interatomic strain \\(r(t)\\) dynamically parameterizes the non-autonomous Lindblad superoperator \\(\mathcal{L}_t\\), modulating instantaneous dephasing rates \\(\gamma(t) = \max\left(\gamma_{\text{min}}, \gamma_0 \left[1 + \eta \left(\frac{r(t)-r_0}{r_0}\right)\right]\right)\\) and jump operators. Strict Conservation Laws: Fourth-order Runge-Kutta (RK4) and exact matrix exponential solvers explicitly enforce matrix Hermiticity \\(0.5(\rho + \rho^\dagger)\\) and unit trace normalization \\(\text{Tr}(\rho) = 1.0\\) at every integration step, preventing unphysical probability leakage or floating-point norm drift. Entropy Generation Kinetics: Real-time diagnostic observers measure quantum state populations \\(\rho_{ii}\\), off-diagonal coherences \\(\rho_{ij}\\), state purity \\(\text{Tr}(\rho^2)\\), and von Neumann entropy \\(S = -\text{Tr}(\rho \ln \rho)\\) with numerical zero-eigenvalue filtering (\\(\lambda > 10^{-10}\\)). Path Integral Discretization: Quantum statistical mechanics and zero-point thermal delocalization (mean radius of gyration \\(R_g = 0.0524\text{ nm}\\) at \\(T = 300\text{ K}\\)) are computed deterministically via ring-polymer bead chains (`PathIntegral`). 4. Quantitative Benchmarks & Physical Analysis Validation across a \\(3.0\text{ ps}\\) trajectory under a \\(1.5\text{ THz}\\) vibrational drive demonstrates real-time vibrational sideband imprinting onto off-diagonal quantum coherences \\(|\rho_{01}(t)|\\), thermal equilibration at \\(101.30\text{ K}\\), population relaxation to ground state \\(\rho_{00} = 0.5263\\) and excited state \\(\rho_{11} = 0.4737\\), and monotonic entropy stabilization at \\(S = 0.5789\\) with zero trace drift (\\(\text{Error} < 10^{-16}\\)). By uniting single-transit wave-interference hardware with zero-drift open quantum software, this framework replaces continuous differential singularities with exact discrete algebra, providing an exascale computational foundation for room-temperature quantum biology, bio-photonic engineering, and hardware security. Core Theoretical Specifications | Component / Subsystem | Primary Formulation / Mechanism | Numerical / Performance Boundary | | :--- | :--- | :--- | | Photonic Execution Core | Octad Tuple \\(\Theta = (A, \phi_{1..4}, \sigma, \chi, \omega)\\) | **\\(21\text{ ps}\\) single optical transit** | | Classical Engine (`MD`) | Velocity Verlet & Lennard-Jones Pair Forces | Maxwell-Boltzmann Velocity Sampling | | Quantum Engine (`QD`) | Lindblad Master Equation & RK4 Integration | Trace Preservation \\(\text{Tr}(\rho) \equiv 1.000000\\) [16–18] | | Decoherence Diagnostic | Von Neumann Entropy \\(S = -\text{Tr}(\rho \ln \rho)\\) | Eigenvalue Filtering \\(\lambda > 10^{-10}\\) | | Path Integral Engine (`PI`) | Ring-Polymer Bead Discretization (\\(P = 32\\)) | Quantum Delocalization \\(R_g = 0.0524\text{ nm}\\) |
We develop quantum information processing primitives for the planar rotor, the state space of a particle on a circle. The n-rotor Clifford group, $\text{U}{(1)}^{n(n+1)/2}\ensuremath{\rtimes}{\text{GL}}_{n}(\mathbb{Z})$, is represented by continuous U(1) gates generated by polynomials quadratic in angular momenta, as well as discrete GLn(Z) gates generated by momentum sign-flip and sum gates. Our understanding of this group allows us to establish connections between homological rotor error-correcting codes [Vuillot, Ciani, and Terhal, Commun. Math. Phys. 405, 53 (2024)] and oscillator quantum codes, including Gottesman-Kitaev-Preskill codes and rotation-symmetric bosonic codes. Inspired by homological rotor codes, we provide a systematic construction of multimode rotation-symmetric bosonic codes by making a parallel between oscillator Fock states and rotor states with fixed non-negative angular momentum. This family of homological number-phase codes protects against dephasing and changes in occupation number. Encoding and decoding circuits for these codes can be derived from the corresponding rotor Clifford operations. As a result of independent interest, we show how to nondestructively measure the oscillator phase using conditional occupation-number addition and postselection. We also outline several rotor and oscillator varieties of the Gottesman-Kitaev-Preskill-stabilizer codes [Phys. Rev. Lett. 125, 080503 (2020).].
Five-Part Validation-Grade Suite on Universal Spectral Determinism and Non-Commutativity Residues: An Unconditional Resolution and Precision-Certified Certification of the \text{BQP} \not\subseteq \text{PH} Separation via Topological, Variational, and Information-Theoretic Invariants --- This 5-package resolution suite delivers an unconditional, non-relativizing, and non-algebrizing proof establishing the strict mathematical separation of Bounded-Error Quantum Polynomial-Time (\bm{\text{BQP}}) from the classical Polynomial Hierarchy (\bm{\text{PH}}) and Polynomial Time (\bm{\text{P}}). Rather than approaching the problem through symbolic logic or localized circuit lower bounds—which are structurally constrained by historic complexity barriers—this framework shifts the question into the continuous, precision-certified domains of Differential Geometry, Algebraic Topology, Category Theory, Information Thermodynamics, and Matrix Perturbation Theory. By analyzing the intrinsic geometric and numerical properties of complexity reductions under finite-precision constraints, the suite isolates a hard computational singularity. This singularity serves as an absolute structural boundary, proving that classical alternating configurations lack the algebraic and dimensional capacity to track non-local multi-particle quantum updates. Individual and Interlinking Functional Dynamics The 5-package suite operates as a mathematically closed, self-contained replication pipeline, where each package executes a specific analytical or numerical enforcement that feeds directly into the next. Package A: Analytic Foundations & Definitions • Individual Mechanics: This package translates the discrete states of algorithmic complexity into the continuous geometry of Riemannian manifolds. It formalizes the structural constraints of the Unitary Manifold (\bm{\mathcal{M}_{\text{BQP}}}) against the Classical Hypercube Manifold (\bm{\mathcal{M}_{\text{PH}}}). By utilizing Ricci flow to evaluate the state space reduction, it identifies a non-invertible Phase-Collapse Singularity (a "Neck-Pinch" obstruction), mathematically proving that the projection fails to preserve the underlying manifold structure. • Interlinking Pipeline: It establishes the primary mathematical universe, smooth structures, and geometric boundaries used by all subsequent packages to measure computational state deformations. Package B: Geometric Mapping & The Final Proof • Individual Mechanics: Building on the continuous manifold foundations, this package details the exact differential-geometric embedding of the computational manifolds utilizing the Fubini-Study metric (\bm{g_{\text{FS}}}). It formally characterizes the absolute dimensional mismatch between exponential quantum phase spaces and polynomial classical coordinate systems. By calculating the metric pullback field via the mapping Jacobian matrix (\bm{J_{\Psi}}), it demonstrates that the scalar curvature (\bm{R}) blows up and diverges as \bm{\mathcal{O}(2^n)}, establishing a geometric impossibility for polynomial reduction. • Interlinking Pipeline: It bridges the smooth geometric structures of Package A with the global topological invariants needed to escape localized oracle dependencies. Package C: Categorical Asymmetry • Individual Mechanics: This package elevates the separation proof into the abstract realm of Category Theory, defining the Quantum Monoidal Category (\bm{\mathbf{Quant}_n}) and the Classical Alternating Category (\bm{\mathbf{Alt}_n}). It frames the candidate complexity reduction as a natural transformation between computational functors. By applying the Yoneda Lemma, the text proves that such a natural transformation cannot exist because the quantum and classical domains possess fundamentally incompatible limit and colimit preservation profiles, ensuring they cannot be mapped without violating the Mac Lane Pentagon Coherence Invariant. • Interlinking Pipeline: By mapping the exact functorial boundaries, Package C transforms geometric and topological asymmetry into strict algebraic constraints, setting up the exact limits required to analyze information-theoretic capacities. Package D: The Entropic Seal • Individual Mechanics: This package models computation as a continuous thermodynamic channel, mapping quantum density operators within the Von Neumann entropy space against classical joint probability distributions within the Shannon alternating space. By invoking the Data Processing Inequality alongside Landauer's Principle, it argues that compressing exponential quantum entanglement capacity into polynomial classical bounds triggers an Entropic Seal Singularity. This singularity is characterized by an unphysical, exponential thermodynamic dissipation requirement (\bm{\Delta Q \sim \Omega(2^n)}), which physically precludes the containment of \bm{\text{BQP}} within \bm{\text{PH}}. • Interlinking Pipeline: This package creates the thermodynamic boundary conditions—demonstrating that a polynomial classical reduction violates physical laws. It defines the continuous informational flow equations that serve as the direct mathematical input for the final numerical validation engine. Package E: Error Analysis & Replication Framework • Individual Mechanics: Package E serves as the definitive numerical and empirical verification asset. It evaluates the complete reduction within a finite-precision digital environment using Moore-Rump Interval Enclosures to rigorously track non-linear error accumulation. Through Singular Value Decomposition (SVD) of the Morphic Coherence Matrix, it documents that the minimum singular value collapses exponentially (\bm{\sigma_{\min} \sim \mathcal{O}(2^{-n})}), driving an explosion of the condition number (\bm{\kappa \sim \Omega(2^n)}). When propagated, the target classical enclosures expand past unity (\bm{\omega \ge 1}), causing a total loss of tracking capability. • Interlinking Pipeline: Package E closes the entire suite. By tracking the relaxation kinetics of the non-commutativity residue, it isolates the power-law halting of the stable execution time-step (\bm{\Delta t \to 0}) under a critical break exponent \bm{\alpha = 1.0000 \pm 0.0001}. This bridges the theoretical abstractions of Packages A–D with an exact, error-bounded, replicable digital certificate (The Numerical Interlock Break). How the Suite Resolves, Validates, Seals, and Enables Replication 1. Resolves: The suite completely bypasses the Relativization, Algebrization, and Natural Proofs barriers. It does not construct localized classifiers over boolean functions; instead, it establishes a global functional space exclusion inside a Bounded-Variation Numerical Space \bm{\mathcal{V}_{\text{mach}}}. It proves that a polynomial classical reduction requires an impossible infinite-energy or zero-precision mapping. 2. Validates: Validation is handled dynamically by monitoring invariant spectral properties, metric pullbacks, and the convergence kinetics of the naturality residue. The system does not rely on heuristic approximations; it enforces absolute mathematical containment via strict interval-enclosure walls. 3. Seals: The resolution is sealed by the simultaneous convergence of independent physical and mathematical invariants: the phase-collapse neck-pinch, the scalar curvature divergence, the Mac Lane coherence failure, the Landauer dissipation wall, and the finite-time numerical halting. The complexity separation is locked because breaking the proof would require violating the fundamental laws of matrix calculus, differential geometry, and thermodynamics. 4. Enables Replication: Package E contains fully pre-wired, compilable LaTeX source code environments, exact citation keys anchored directly to foundational peer-reviewed literature (e.g., Golub & Van Loan, Page, Rump, Aaronson), and explicit numerical stability criteria. Independent replication teams can immediately deploy automated verification scripts to track the SVD spectrum, the scalar curvature metrics, and the \bm{\alpha = 1} critical break exponent, ensuring a completely transparent peer-to-peer review process. --- Note: The accompanying Agnostic Replication Kit (ARK) and Standard Academic Core (SAC) 17-package operational suite will be uploaded in the forthcoming version release to enable down-stream cross-institutional simulation and formal peer review.