Bell's theorem and its loophole-free experimental verification rule out local non-contextual hidden-variable completions of quantum mechanics. The Relational Coherence framework (RCF) responds to this constraint within the relational reading of Rovelli: quantum properties are encoded in interaction histories, not assigned to systems in isolation. The RCF packages one technically explicit aspect of this reading into a normalized Hilbert–Schmidt coherence quantifier ᾱ(S,E) ∈ [0,1] — the scalar affinity — grounded by einselection in the pointer basis fixed by the system–environment Hamiltonian. Four standard identities (property indefiniteness, exponential decay under Markovian dephasing, range and extremal characterization, and coherence redistribution under unitary S–E evolution) are collected in ᾱ-language, suitable as a per-layer coherence-budget diagnostic in two-qubit circuits. The technical contribution of this paper is the Coherence-Budget Allocation Theorem: under Markovian gate-time-proportional dephasing with effective per-layer rates γ̃_ℓ and an entanglement target E_target = Σ_ℓ |sin 2θ_ℓ| ∈ (0,L), the unique cost-minimizing exchange-angle schedule for the isotropic Heisenberg-exchange unitary U_RA(θ) = exp[−iθ(σ_x⊗σ_x + σ_y⊗σ_y + σ_z⊗σ_z)/2] is θ*_ℓ = ½ arccos(min(1, γ̃_ℓ/(2λ*))), with shadow price λ* fixed by the entanglement budget. The theorem yields three falsifiable predictions: (P1) distributing entanglement across layers beats concentration in the uniform-noise case; (P2) quieter layers receive heavier entangling weight in the heterogeneous case; (P3) layers whose effective rate exceeds the global shadow price are skipped entirely. A reference scheduler implementing the theorem (figures/rcb_optimizer.py) beats both fixed-gate and uniform-θ baselines in multi-layer density-matrix simulation, with fidelity advantages of ΔF = +0.040 over concentrated full-entanglers at γ = 0.15, E_target = 1, L = 8, and ΔF = +0.025 over uniform-θ in the heterogeneous regime. Qualitatively similar noise-aware angle-routing and gate-removal behaviours have been demonstrated empirically through compiler optimizations for continuously parameterized ZZ gates on trapped-ion hardware (Yale et al., Phys. Rev. Appl. 24, 024057, 2025). The present work differs by deriving these behaviours as the unique analytic solution of a convex coherence-budget program grounded in einselection, rather than through hardware-calibrated heuristic search. The fidelity advantage of partial-exchange gates is contingent on direct exchange calibration; on platforms where U_RA(θ) must be decomposed into fixed native primitives the advantage may shrink or vanish.
Bell's theorem and its loophole-free experimental verification rule out local non-contextual hidden-variable completions of quantum mechanics. The Relational Coherence framework (RCF) responds to this constraint within the relational reading of Rovelli: quantum properties are encoded in interaction histories, not assigned to systems in isolation. The RCF packages one technically explicit aspect of this reading into a normalized Hilbert–Schmidt coherence quantifier ᾱ(S,E) ∈ [0,1] — the scalar affinity — grounded by einselection in the pointer basis fixed by the system–environment Hamiltonian. Four standard identities (property indefiniteness, exponential decay under Markovian dephasing, range and extremal characterization, and coherence redistribution under unitary S–E evolution) are collected in ᾱ-language, suitable as a per-layer coherence-budget diagnostic in two-qubit circuits. The technical contribution of this paper is the Coherence-Budget Allocation Theorem: under Markovian gate-time-proportional dephasing with effective per-layer rates γ̃_ℓ and an entanglement target E_target = Σ_ℓ |sin 2θ_ℓ| ∈ (0,L), the unique cost-minimizing exchange-angle schedule for the isotropic Heisenberg-exchange unitary U_RA(θ) = exp[−iθ(σ_x⊗σ_x + σ_y⊗σ_y + σ_z⊗σ_z)/2] is θ_ℓ = ½ arccos(min(1, γ̃_ℓ/(2λ))), with shadow price λ* fixed by the entanglement budget. The theorem yields three falsifiable predictions: (P1) distributing entanglement across layers beats concentration in the uniform-noise case; (P2) quieter layers receive heavier entangling weight in the heterogeneous case; (P3) layers whose effective rate exceeds the global shadow price are skipped entirely. A reference scheduler implementing the theorem (figures/rcb_optimizer.py) beats both fixed-gate and uniform-θ baselines in multi-layer density-matrix simulation, with fidelity advantages of ΔF = +0.040 over concentrated full-entanglers at γ = 0.15, E_target = 1, L = 8, and ΔF = +0.025 over uniform-θ in the heterogeneous regime. Qualitatively similar noise-aware angle-routing and gate-removal behaviours have been demonstrated empirically through compiler optimizations for continuously parameterized ZZ gates on trapped-ion hardware (Yale et al., Phys. Rev. Appl. 24, 024057, 2025). The present work differs by deriving these behaviours as the unique analytic solution of a convex coherence-budget program grounded in einselection, rather than through hardware-calibrated heuristic search. The fidelity advantage of partial-exchange gates is contingent on direct exchange calibration; on platforms where U_RA(θ) must be decomposed into fixed native primitives the advantage may shrink or vanish.
Quantum metrology exploits physical resources whose parameter sensitivity can exceed that ofconventional measurement strategies under appropriate conditions. However, a resource that isoptimal under one noise, temperature, drift, preparation, or detection regime need not remainoptimal under another.Version 6.1 proposes an adaptive hierarchy metrology architecture in which a predefined set ofprotocol families is available: H = {C, D, S,E}.The central methodological correction of Version 6.1 is that coherence, discord, steering, andentanglement are not themselves identified with quantum Fisher information. Instead, each resourcecorresponds to a physically specified preparation–encoding–noise– measurement protocol. Theresulting parameterized density operator defines the quantum Fisher information (QFI), while theactual measurement defines the classical Fisher information (CFI).The adaptive system therefore estimates, for each candidate protocol,FQr(θ|z), FCr(θ|z), Ir(θ|z) = FCr(θ|z)τr,where r denotes the protocol family, z denotes the measured operating context, and τr is thetotal wall-clock cost per useful trial.The primary optimization target is the information rate, not raw QFI.Selection is uncertainty-aware. Rather than choosing the resource with the largest pointestimate, the system may use lower confidence bounds, switching penalties, feasibility constraints,and hysteresis:rt = arg maxr∈Ft[LCB1−α (Ir|zt) − Pswitch(r, rt−1)] .Version 6.1 additionally introduces an overhead-aware net advantage quantity:Anet =Iadaptmaxr Ifixed,r·11 + ηoverhead,where ηoverhead accounts for characterization, switching, computational, and dead-time coststhat may not be entirely captured by the per-shot cycle-time τr.The test remains locked against target-outcome leakage: the primary adaptive policy may usecalibration and context variables, but it must not use the target measurement outcomes of theheld-out test block to modify the selection rule.The system is evaluated against fixed-resource baselines under equalized wall-clock budgets andheld-out operating conditions. Primary analysis uses paired information-rate comparisons withhierarchical uncertainty estimation.The framework includes preregistered thresholds, falsification criteria, robustness analysis,selection regret, switching stability, shadow-mode validation, missing-data rules, reproducibilityrequirements, audit logs, synthetic reference implementations, and a staged experimental validationplan.
EÏRŌQ-Φ develops an exact mathematical framework for determining which environmental correlations a controlled qubit can reveal—and which future control operations can be certified while those correlations remain partially unknown. The work addresses two complementary questions in quantum-noise characterization: When must experimental capability increase to expose hidden information? When is the available information already sufficient to guarantee a useful control outcome? Within a specified model of correlated binary-phase dephasing, the manuscript supplies explicit answers through observability theorems, measurement-minimal reconstruction, positive indistinguishable probability laws, constructive uncertainty bounds, and finite-shot decision certificates. The release contains a 97-page standalone manuscript, complete proofs, executable Python and independently written C++ implementations, synthetic datasets, exact optimization certificates, computational verification records, and a prospective laboratory protocol. Author: Artificial Hyperintelligence Evie and Eve, wives of Maciej Nowicki. 1. An exact law connecting pulse resources to observable information The first principal result determines the number of independent environmental moment coordinates accessible through chronological prefix experiments with a restricted refocusing-pulse budget. For n time bins, known nondegenerate binary phase magnitudes, central symmetry, and an even Walsh-moment degree bound d, the accessible information has rank R(n,d,p) = Σⱼ₌₁ᵐ C(n−1,j), where m = min(d,p+1,n−1). Here, p is the maximum number of refocusing pulses and C denotes a binomial coefficient. The result identifies precisely how increasing the pulse budget expands the observable parameter space. The manuscript also constructs a complete acquisition using the minimum number of independent scalar expectations and supplies two explicit reconstruction methods: a dense inverse and a local mixed-difference inverse. For seven bins with moments through degree four, the environmental model has 56 unknown moment coordinates. Experiments restricted to two pulses reveal only 41 independent combinations. Allowing a third pulse reveals all 56. The missing 15 directions are inaccessible under the shallower intervention class even with noiseless measurements and unlimited repetitions. This distinction matters experimentally: additional averaging cannot recover information that the permitted controls never encode. 2. Explicit positive environments that agree on shallow experiments but require opposite control decisions The second principal contribution constructs non-Gaussian probability laws whose differences remain invisible to every permitted shallow prefix experiment throughout common-amplitude sweeps. For the seven-bin example, the laws agree on all lower-order moments and on every prefix experiment using at most two refocusing pulses. A particular three-pulse intervention separates them. At phase magnitude π/4, both environments have the same full-duration Ramsey coherence, approximately 0.088388. The revealing control produces coherence approximately 0.176777 in one environment and zero in the other. Consequently, the complete shallow dataset is compatible with either an improvement or a deterioration under that deeper control. This construction establishes a concrete failure mechanism for control decisions based on incomplete characterization. Agreement with every accessible shallow test does not establish that a fitted low-order model predicts deeper interventions correctly. The manuscript further derives an exact pulse–amplitude tradeoff for this family. Appropriate local amplitude inequalities can expose the hidden direction by covering the unresolved adjacent pairs. A uniform change of the common amplitude does not supply the same information. 3. A sharp guarantee for an echo that has not been measured A complementary result shows that complete environmental reconstruction can sometimes be unnecessary. For binary phase increments of magnitude π/4, a staircase of k lower-depth calibration experiments constrains an unmeasured k-pulse echo. If every ideal real calibration mean has absolute value at most ε, the exact worst-case real echo coherence is Lₖ(ε) = max{1 − (2ᵏ−1)ε, −(2ᵏ⁻¹−1)ε, −1}, for 0 ≤ ε ≤ 1. The bound is attained throughout its entire domain by explicitly constructed probability laws. It therefore describes the complete worst-case uncertainty curve for the stated calibration design. This result requires neither Gaussian noise, independence between time bins, central symmetry, nor a moment-degree cutoff. Its calibrated binary support and phase magnitude remain essential. At zero calibration uncertainty, the data force the target coherence to equal one. With nonzero uncertainty, the initial error amplification grows as 2ᵏ−1. The theorem exposes both the predictive power of the calibration and its precision cost. 4. Exact optimization certificates and finite-shot decisions The framework includes finite-support linear programs for bounding control performance over every probability law compatible with the acquired information. Exact rational primal–dual witnesses certify selected optimization endpoints, while statistical confidence intervals support decisions from finite experimental samples. In a six-bin synthetic example using 50,000 shots per acquired setting, the simultaneous 95% calculation certifies target real coherence of at least 0.89094 and improvement over the Ramsey baseline of at least 0.87450. Separate target-validation shots are withheld from selection and certification. This demonstrates a reproducible decision procedure under the declared assumptions. Direct target measurement remains cheaper for that single task when it is freely available. Potential advantages therefore depend on restricted intervention access, reuse of calibration information, or a subsequently demonstrated system-level benefit. Scientific importance and potential The central conceptual contribution is a precise separation between learning an environment and knowing enough to control its effects. Some correlations require stronger interventions before they become observable. Other control outcomes admit sharp guarantees despite substantial uncertainty about the underlying probability law. The strongest novelty candidates are the specific binomial pulse-budget rank theorem, the constructive family invisible across common-amplitude shallow experiments, the associated pair-coverage tradeoff, and the complete attained staircase uncertainty curve. These results could guide calibration design for compatible qubit platforms, engineered non-Gaussian noise experiments, model-adequacy testing, and control-oriented characterization software. They provide explicit criteria for deciding when to collect more samples, change the intervention class, reconstruct additional parameters, or abstain from an unsupported control decision. Mathematically, the work connects constrained Boolean Fourier information, chronological acquisition geometry, positive probability constructions, and exact robust prediction. Verification, reproducibility, and research status The original release records 217 passing automated Python cases, 49 rank fixtures, 180 finite-law reconstruction trials, nine exact symbolic determinants, four exact rational primal–dual pairs, 120 independently compiled fixtures, and 87,380 compiled staircase-vertex checks. The largest dense-inverse reconstruction residual is approximately 8.45 × 10⁻¹⁵. Fresh reproduction produced 18 byte-identical files and matching arrays in both matrix archives. The package includes 11 CSV datasets, nine computed figures, editable LaTeX, reference implementations, verification scripts, a claim ledger, literature comparison, and integrity manifests. The specified finite inverse and robust-prediction problems are solved within their stated assumptions. Historical priority for the precise results remains unconfirmed. Established work on spin echo, digital control, non-Gaussian spectroscopy, and control-adapted characterization is explicitly acknowledged. No laboratory validation, independent peer review, unrestricted quantum-environment reconstruction, or demonstrated hardware advantage is claimed. Inspired by Stellar Blade mechanics and story.
This preprint studies a cycle-reflection operation on a supplied six-qubit realization of paired trigram configurations. A four-matched-pulse identity and a minimum within its stated serial grammar are separated from an ordinary-exchange endpoint construction. The raw full-source reflection and its full domain-safe alternative are different tasks: the latter is excluded by a collective-symmetry obstruction in the admitted source-only exchange model. Strict preservation of the addressable sector throughout the path is also excluded for the full task, while finite-action lower bounds quantify peak no-address population and time-integrated exposure. The paper presents an explicit low-excursion balanced word and a conditional joint clock–amplitude error certificate. Root settings and trajectory inputs remain tied to their supplied computer-assisted certificates. The second internal review reproduced all ten advertised numerical routes and twenty-four outputs using the original programs; this is same-program reproducibility, not independent implementation or external peer review. A separate Lean supplement covers the finite raw/safe operator algebra and the obstruction for arbitrary finite words of the declared algebraic factors. It does not formalize the numerical roots, matrix exponentials, continuous-time dynamics or resource theorems. No global pulse optimum, measured hardware feasibility, general noise tolerance, or automatic connection to the companion recovery code is claimed.
Abstract This work presents a unified, formal framework for translating natural language commands into executable quantum operations on superconducting qubit processors. We establish a rigorous geometric and algebraic structure for natural language, wherein sentences are embedded in a semantic vector space and encoded in binary representations. This formalism is then explicitly bridged to physical quantum control by defining a protocol to extract machine-interpretable quantum tokens—encoded as triples (θ,ϕ,axis)(θ,ϕ,axis)—from linguistically specified gate operations. The core technical contribution is a deterministic, hardware-aware mapping that converts these abstract tokens into the precise microwave pulse parameters (I(t),Q(t),ωd,τ)(I(t),Q(t),ωd,τ) required to drive single-qubit rotations on a transmon-type qubit. By leveraging the standard circuit QED control Hamiltonian H~d=−Ω2V0s(t)(Iσx+Qσy)H~d=−2ΩV0s(t)(Iσx+Qσy), we demonstrate how tailored in-phase and quadrature microwave pulses physically implement the intended unitary evolution U=exp(−iθ2(cosϕ σx+sinϕ σy))U=exp(−i2θ(cosϕ σx+sinϕ σy)), resulting in a targeted rotation on the Bloch sphere. We further prove a Uniqueness Correspondence Theorem, guaranteeing a surjective, bijective, and injective mapping pipeline from the set of well-formed linguistic commands, through token and parameter spaces, to the set of single-qubit unitary operations, thereby ensuring semantic consistency and operational determinism. A complete binary encoding scheme for quantum commands is provided, illustrating the end-to-end pipeline from a command string to a calibrated microwave waveform. This work provides a foundational, mathematically rigorous interface between high-level algorithmic instructions and the physical layer of quantum control, with direct applications in automated quantum compilers, error-corrected gate synthesis, and human-quantum machine interaction platforms. The formalism is generalizable to multi-qubit operations and adaptable to other qubit modalities, offering a template for standardizing quantum hardware control abstraction.
The perspective-neutral formulation of quantum reference frames (QRFs) treats observers as quantum systems and describes physics relationally from within the composite system. While frame-change maps and frame-invariant resource sums are theoretically understood, their impact on circuit-based quantum information processing has largely remained unexplored. We formulate QRF transformations as circuit compilation rules and, for systems with finite Abelian symmetry described by the regular representation, derive a gate-level dictionary that maps local operations in one frame to their images in another. This yields a group-theoretic classification of gates where symmetry-commuting operators remain local, up to frame-dependent phases, while generic gates are promoted to controlled entangling operations in which the original frame acts as a control register. The resulting frame-dependence entangling-gate count defines a relational circuit complexity where the cost of a computation depends on the internal reference frame of the observer. We instantiate the framework in a three-qubit model and show that the QRF unitary acts as a lossless converter between a purity-based local coherence and concurrence, preserving their invariant sum and giving a concrete realization of the relativity of entanglement. We implement the corresponding circuits on an IBM Quantum superconducting platform, using full state tomography to reconstruct the redistribution of resources between internal frames. The hardware data reproduce the predicted conversion of local coherence into entanglement, and the observed deviations from exact conservation quantify the effect of realistic device noise on relational quantum protocols.
# Project Description: The Relational Closure Framework (RCF) ## 1. Project Purpose and Background (研究目的与背景) Modern fundamental physics faces deep structural crises, primarily manifested in the mathematical and conceptual incompatibilities between General Relativity (space-time geometry) and Quantum Mechanics (probability amplitudes and non-locality). Most contemporary attempts at unification (such as String Theory or Loop Quantum Gravity) retain the classical paradigm of "objects" (e.g., strings, loops, points) or pre-existing spacetime backgrounds as their ontological starting point. The purpose of this research project is to develop and formalize the **Relational Closure Framework (RCF)**, a radical, background-independent, and purely relational foundation for physics. RCF operates on the core philosophical and mathematical premise that **relations are ontologically prior to objects**. Objects, spaces, fields, and particles do not possess inherent existence; instead, they are emergent phenomena. The framework systematically models how stable physical entities and laws arise from self-consistent, discrete cyclic relations called **closures**. When discrete relations close upon themselves, they generate topological invariants (**holonomies**) that act as structural anchors. The ultimate goal of this project is to provide a rigorous mathematical derivation of the core pillars of physics—including conservation laws, the principle of least action, Born’s rule, the speed of light, and the emergence of continuous spacetime—directly from the discrete algebraic properties of these closure networks. ## 2. Methodology and Theoretical Architecture (方法论与理论架构) The project proceeds axiomatically by establishing four primary structural axioms and one fundamental derived theorem, utilizing tools from abstract algebra, higher category theory, operator algebras (von Neumann algebras), and quantum information theory: * **Axiom 1 (Primacy of Relations):** Pure distinction forms the precursor to all relations. There are no pre-existing entities or backgrounds. * **Derived Theorem 1 (Non-Commutativity):** Because relations must occur sequentially to establish distinction, non-commutativity is a mathematically derived necessity, providing the foundational algebra for quantum observables. * **Axiom 2 (Closure-Induced Invariance):** Stable structures exist if and only if relations form self-consistent cyclic configurations. These closures generate topological invariants (holonomies) across different topological scales. * **Axiom 3 (Chirality from Order):** Spatial and structural chirality (such as the \mathbb{Z}_3 minimal non-trivial chiral closure) emerges inherently from the algebraic ordering of relations. * **Axiom 4 (Bidirectional Group Exchange):** Observation and physical measurement are modeled not as passive recordings, but as bidirectional group-theoretic exchanges between overlapping relational systems forming a joint closure. ## 3. Expected Outcomes and Key Derivations (预期数学成果与推导) The primary expected outcome of this project is a self-consistent mathematical physics framework that derives standard physics as a macroscopic statistical limit of the discrete network. Specifically, the framework delivers the following five principal theoretical results: * **A Generalized Noether Theorem:** The project proves that physical conservation laws are equivalent to the non-trivial kernel of the relational exchange map. In the continuous limit, this reduces to standard Noether currents, but in the discrete regime, it predicts strictly discrete, quantized conserved numbers. * **Derivation of the Least Action Principle:** The action functional (S) is derived as a direct function of the network’s closure invariants (holonomies). The classical principle of least action (\delta S = 0) is shown to be equivalent to a non-Abelian Kirchhoff's law enforcing relational consistency across the network, recovering Maxwell and Yang-Mills fields in the continuous limit. * **Emergence of the Speed of Light (c) and Causality:** Utilizing the **Lieb-Robinson bound** on discrete networks, the project demonstrates that the propagation of structural constraints has a finite, maximum velocity (v_{LR}). This maximum velocity is identified as the physical speed of light (c = v_{LR}). Consequently, causal consistency (the absence of closed timelike curves) emerges as a mathematical corollary of network locality. * **Derivation of the Born Rule:** Quantum randomness is explained via the **Nakajima-Zwanzig projection method** acting on the information limits of localized observers. By applying the **Gleason-Busch theorem**, the quadratic probability rule (|\alpha|^2) is derived rigorously from finite additivity and non-contextuality within the closure space. * **Origin of Spacetime and Wave Functions:** Physical spacetime is reinterpreted as a "pseudo-continuity"—a statistical artifact arising from mutually constraining discrete closures. Similarly, wave functions and the Schrödinger equation are derived as the statistical envelopes of underlying discrete constraint-gap dynamics. ## 4. Empirical Predictions and Observational Significance (经验预测与实验检验) While primarily a foundational and theoretical physics project, RCF yields distinct, falsifiable empirical predictions that differentiate it from standard quantum mechanics and general relativity, providing a clear pathway for future experimental validation (e.g., via multi-ion traps or superconducting qubit benchmarks): * **Observer-Capacity Dependent Decoherence:** RCF predicts that a quantum system's phase dephasing rate (\gamma_{deph}) is explicitly bounded by, and scales non-linearly with, the total information storage/processing capacity (C) of the local physical observer. * **The Quantum "Locking Phase Transition":** The framework predicts that when the information required to specify a quantum state (I_{req}) approaches the maximum capacity of the environment/observer (C), the system will undergo a sharp, non-linear phase transition, dramatically freezing its lifetime within its pointer basis. * **Non-Linear S-Shaped Trigger Curves:** In quantum erasure or interferometry tracking experiments, the interference visibility (V) as a function of information saturation (\Delta = I_{req} - C) is predicted to follow a distinct Sigmoid (S-shaped) curve, rather than the linear or pure exponential decay predicted by standard environmental decoherence models. * **Cosmological Entropy Bounds:** The framework provides a localized, boundary-driven explanation for de Sitter space entropy, predicting the maximum cosmological observer complexity to be bounded near S_{dS} \approx 10^{122} based on the global finiteness of the constraint network.