We embed the operational semantics of a strongly normalising quantum-programme calculus, the Enter calculus, into a discrete spatial substrate executed under partitioned block schedules of range r, and prove an exact domain-of-dependence theorem for the resulting dynamics. The causal cone of slope r arises from the geometry of the schedules, while the metatheoretic interface—strong normalisation, probabilistic confluence, internal no-signalling, records, and affinity, abstractly stated as (E1)–(E6)—ensures that programme semantics, including terminal laws, adaptive strategies, and readouts, factors through cone-local data. We establish a one-step domain-of-dependence lemma and an exact cone theorem for readout instruments, prove attainability through an explicit relay schedule, and show that the lattice signalling speed is exactly r. We also obtain a faithful-embedding pair for calculus-generated block instruments: soundness, together with simulation under a routing condition. We then prove necessity results for transmissive long-range primitives, realised as genuine repartitionings with a single exceptional block. These results include violation of the original-metric cone, sector transfer using explicit resources, and failure of metric-local autonomy despite identical non-selective quantum marginals. Nevertheless, fidelity monotonicity for completely positive maps shows that a linear long-range channel violates neither affinity, nor ensemble indistinguishability relative to the complete operational input, nor semantic no-cloning. Under two explicitly conditional hypotheses—device boost-closure and the two-clause causal-executability axiom (E7), which is provable on admissible substrates—bidirectional super-cone devices generate closed causal chains and are therefore excluded, yielding a retiming-stable causal bound; isotropy follows only under additional homogeneity assumptions. Finally, under explicit geometric, operational, algebraic, and frame-covariance hypotheses (C1)–(C7), the continuum limit satisfies a continuum domain-of-dependence theorem, operational no-signalling, algebraic microcausality, and, when the limiting cone is Euclidean and D ≥ 2, frame-invariance of the maximal speed via the Alexandrov–Zeeman theorem. The existence of a scaling family satisfying (C1)–(C7) remains open and is isolated in two sharply delimited conjectures.
We embed the operational semantics of a strongly normalising quantum-programme calculus, the Enter calculus, into a discrete spatial substrate executed under partitioned block schedules of range r, and prove an exact domain-of-dependence theorem for the resulting dynamics. The causal cone of slope r arises from the geometry of the schedules, while the metatheoretic interface—strong normalisation, probabilistic confluence, internal no-signalling, records, and affinity, abstractly stated as (E1)–(E6)—ensures that programme semantics, including terminal laws, adaptive strategies, and readouts, factors through cone-local data. We establish a one-step domain-of-dependence lemma and an exact cone theorem for readout instruments, prove attainability through an explicit relay schedule, and show that the lattice signalling speed is exactly r. We also obtain a faithful-embedding pair for calculus-generated block instruments: soundness, together with simulation under a routing condition. We then prove necessity results for transmissive long-range primitives, realised as genuine repartitionings with a single exceptional block. These results include violation of the original-metric cone, sector transfer using explicit resources, and failure of metric-local autonomy despite identical non-selective quantum marginals. Nevertheless, fidelity monotonicity for completely positive maps shows that a linear long-range channel violates neither affinity, nor ensemble indistinguishability relative to the complete operational input, nor semantic no-cloning. Under two explicitly conditional hypotheses—device boost-closure and the two-clause causal-executability axiom (E7), which is provable on admissible substrates—bidirectional super-cone devices generate closed causal chains and are therefore excluded, yielding a retiming-stable causal bound; isotropy follows only under additional homogeneity assumptions. Finally, under explicit geometric, operational, algebraic, and frame-covariance hypotheses (C1)–(C7), the continuum limit satisfies a continuum domain-of-dependence theorem, operational no-signalling, algebraic microcausality, and, when the limiting cone is Euclidean and D ≥ 2, frame-invariance of the maximal speed via the Alexandrov–Zeeman theorem. The existence of a scaling family satisfying (C1)–(C7) remains open and is isolated in two sharply delimited conjectures.
We prove an exact domain-of-dependence theorem for quantum program dynamics generated by a strongly normalizing calculus, called the Enter calculus, executed on a discrete substrate under partitioned block schedules of range r. The causal cone of slope r arises from the geometry of these schedules, while the metatheoretic interface—strong normalization, probabilistic confluence, internal no-signaling, records, and affinity, abstractly stated as conditions (E1)–(E6)—ensures that program semantics, including terminal laws, adaptive strategies, and readouts, factor through cone-local data. We establish a one-step domain-of-dependence lemma and an exact cone theorem formulated for readout instruments, prove attainability by an explicit relay schedule showing that the lattice signaling speed is exactly r, and construct a faithful-embedding pair consisting of soundness and simulation under a routing condition for calculus-generated block instruments. We then prove necessity results for transmissive long-range primitives realized as genuine repartitionings with a single exceptional block: such primitives violate the original-metric cone, transfer sectors using explicit resources, and destroy metric-local autonomy despite having identical non-selective quantum marginals. A complementary negative result follows from fidelity monotonicity for completely positive maps: a linear long-range channel violates neither affinity, ensemble indistinguishability relative to the complete operational input, nor semantic no-cloning. Under two explicitly conditional hypotheses—device boost-closure and the two-clause causal-executability axiom (E7), both provable on admissible substrates—bidirectional super-cone devices generate closed causal chains and are therefore excluded, yielding a retiming-stable causal bound; isotropy follows only under additional homogeneity assumptions. Finally, we replace the former informal continuum program with a conditional continuum limit: under explicit geometric, operational, algebraic, and frame-covariance hypotheses (C1)–(C7), the limit satisfies a continuum domain-of-dependence theorem, operational no-signaling, algebraic microcausality, and, when the limiting cone is Euclidean and D is at least 2, frame invariance of the maximal speed via the Alexandrov–Zeeman theorem. The existence of a scaling family satisfying (C1)–(C7) remains open and is isolated in two sharply delimited conjectures.
The Enter calculus is a linear quantum lambda calculus in which runtime qubit names refer to factors of a shared quantum state, and measurement is represented by an explicit operation, Enter, that consumes quantum resources and returns classical records. This paper gives a unified formal development of the calculus. The core language satisfies type preservation, progress, strong normalization, semantic no-cloning, adequacy for finite quantum circuits, confluence up to terminal distributions, and internal no-signaling. The Born probabilities are then removed from the operational rule and replaced by an abstract evaluation strategy. Five operational requirements are imposed: composability, no-signaling, nullity, continuity, and observational equivalence between direct evaluation and coherent record creation followed by delayed readout. The central result is that diagonal anchoring is not an additional axiom; it follows from the first four requirements through a decohered fine-graining construction in which a within-block index is exported to an additional register. The record principle then yields the Born rule for arbitrary coherent states, and comparison with all later experiments determines the projective state update. The development includes quantitative stability estimates, mixed-state and adaptive formulations, rotated projective measurements, POVM outcome laws, and an affine extension for quantum instruments. A single-level reflection operation compiles an agent’s protocol into coherent laboratory dynamics. A compatibility theorem for sharp measurement contexts then isolates the obstruction in the Frauchiger–Renner certainty chain. Finally, a static record-free functional equation is classified by Walsh-parity slack fields, and its remaining positivity problem is distinguished from the operational Born theorem.
The Enter calculus [EC] derives the Born rule and the projective update from five operational requirements: composability, no-signaling, nullity, continuity, and records. This paper equips the calculus with a strictly linear resource geometry and develops its finite-resource metatheory, stating each result at precisely the strength established. An update grammar of four primitive moves makes every cut-indexed causal potential a conservation law, as shown in Theorem E through a case analysis of the grammar, and derives scalar monotonicity rather than assuming it. A latent-active split with fully specified typing rules turns monogamy of causal change into a typing invariant, as established in Theorem F. Resource-relative admissibility, with its quantification made explicit, is downward closed along runs, ensuring that the forcing threshold is well defined. The minimal register cost of the decohered fine-graining construction is computed in closed form in Theorem H, and the resulting realizable rational net improves the capacity-four rigidity mesh from the vacuous value delta equals one to delta star equals one-sixth in Theorem C star. Two rigidity layers are proved unconditionally: boundary rigidity, consisting of monotonicity, antipodal symmetry, the Sandwich and Seal lemmas, and a descent proposition, which holds for every admissible strategy without an update hypothesis; and diagonal-faithful rigidity, established in Theorem I through a fully detailed single-qubit case with an explicit Pexider argument and a multi-qubit induction, forcing the Born rule on every sub-capacity configuration at every capacity N greater than or equal to three. The residual question—whether diagonal-unfaithful update slack can survive at full depth—remains unresolved. Instead, it is reduced in both directions to sharply delimited objects: an effective hierarchy of relaxations whose finite levels upper-bound the residual freedom and make rigidity refutable at a finite level, as shown in Theorem J, without claiming that the limit is attained; and a compensated transport family based on odds-ratio-preserving Plackett transports, for which the two-qubit layer, the record-family seal lift in closed form, and the full-support three-qubit extension on compact sets are proved. This family satisfies every axiom instance generated by record-chain protocols from two-qubit initial configurations at every depth, as established in Theorem M prime. Consequently, no capacity-three rigidity proof can proceed through record chains alone. Full capacity-three admissibility reduces to a single explicitly stated Extension Conjecture; under that conjecture, Open Problem G three has a positive answer and the minimum threshold N zero of one and the pair one, two equals four. Depth-two and depth-three record probes pass at machine precision, and an eta-scan Pareto front reaches the axis numerically. Part II specifies a reproducible artefact in which every major claim is either proved, reduced to a finite check, or implemented as a property test or golden regression over a rich class of instances, with each item’s status—Theorem, conditional result, conjecture, or probe—labelled explicitly.
The Enter calculus [EC] derives the Born rule and projective update from five operational requirements: composability, no-signaling, nullity, continuity, and records. This paper equips the calculus with a strictly linear resource geometry and develops its finite-resource meta-theory as a single coherent framework. An update grammar with four primitive moves makes every cut-indexed causal potential a conservation law, as stated in Theorem E, and derives scalar monotonicity rather than assuming it. A latent-active split turns monogamy of causal change into a typing invariant and reconciles token activation with monotone coarsening, as shown in Theorem F. Resourcerelative admissibility is downward closed along runs, as stated in Lemma 4.2; hence forcing is upward monotone in capacity and the forcing threshold is well defined. The minimal register cost of the decohered fine-graining construction is N_F^0(s, m⃗) = s + ceiling(log₂ M) + ceiling(log₂(max_k m_k)), as proved in Theorem H. The resulting realizable rational net strictly refines the dyadic mesh and improves the rigidity bound at capacity 4 from the vacuous value δ = 1 to δ⋆ = 1/6, as shown in Theorem C⋆. The central new result is a rigidity theorem: a multiplicative functional equation extracted from the sequential cocycle with one qubit of headroom forces every admissible strategy whose update diagonal is Lüders-faithful to agree exactly with the Born strategy on every sub-capacity configuration, for every capacity N ≥ 3, as established in Theorem I. The proof is domain-explicit: a witness state is constructed with specified amplitudes, normalisation, and Lüders weights; the cocycle is derived on its exact open domain of validity; an overlapping-interval argument reduces the relation to a function of the product; and the resulting continuous Pexider equation is solved from first principles. Consequently, all residual finite-capacity freedom is confined to full-capacity configurations and to diagonally unfaithful update slack, as stated in Corollary K. Open Problem 17.2 of [EC] is therefore equivalent to a diagonally unfaithful instance of Open Problem G at capacity 3, as shown in Corollary L, while an LP hierarchy converts the remaining question into a computable limit, as established in Theorem J. An exact Choi-rank ledger assigns costs to geometric instruments and identifies decoherence rate with capacity flux, as shown in Theorem D′. Part II specifies a reproducible computational artifact in the style of the Enter-calculus repository: every theorem is either executed as a derivation or enforced as a runtime invariant, and every open problem is represented by a numerical probe.
COS-CNS develops the causal-dynamics module of the Collapsing-Structure (COS) program. The manuscript studies a verifiable class of nonunitary, discrete-time dynamics on structures equipped with a causal partial order, focusing on operational no-signaling, finite-speed influence propagation, and scheduling robustness. The framework is formulated at the level of completely positive trace-preserving channels, instruments, local algebras, dependency neighborhoods, and locally implementable causal admissibility filters. It carefully separates unconditional branch-ignored dynamics from conditional trajectory or postselected descriptions, so that no-signaling is defined through operational marginal statistics rather than through selected branches. The main results apply to a restricted and auditable class of local, filtered channels satisfying explicit locality, support-propagation, causal-filtering, and scheduling/confluence assumptions. Within this class, the paper formulates no-signaling criteria, a strict discrete causal-cone statement, and robustness targets for approximate or leakage-affected dynamics. These results are not claimed to hold for arbitrary COS microdynamics or for nonlocal global projectors. COS-CNS also identifies failure modes in which the assumptions break down, including nonlocal Kraus support, nonlocal causal filtering, deterministic branch control, and nonconfluent scheduling. These counterexamples clarify why the locality and admissibility conditions are necessary for an operational causality minimum. The release includes a technical supplement with reproducibility protocols, numerical demonstrations, algorithmic details, supplementary proofs, negative controls, and audit conventions. The numerical material illustrates no-signaling tests, influence-cone diagnostics, and scheduling-robustness checks under controlled benchmark conditions. COS-CNS therefore serves as the causality and finite-speed-influence verification layer for COS-QD-like discrete open-system dynamics, while treating the connection to relativistic microcausality, Lorentz symmetry, and continuum causality as a restricted and future-facing interface rather than a completed derivation.
Pedagogical Guide, Foundations Audit, Uniqueness & Substitutability Analysis, Literature Comparison, and Reading Guidance : A Pedagogical Guide to Understanding the OT–GKSL Architecture This work develops a certified-domain theory of causality, locality, nonlocality, and relativistic frame covariance within the Einstein-locked optimal-transport/GKSL framework for open quantum systems. The native theory is formulated as an open-system dynamics on a finite effective spectral stratum, without assuming a primitive classical spacetime, Lorentzian metric, causal cone structure, or global relativity principle. Classical causal semantics is reconstructed only at readout level. A rank-four operational record map selects a horizontal sector of the native state manifold, while local sections and an independent soldering calibration define the physical coframe, Lorentzian readout metric, temporal orientation, and local causal cones. Record rank is therefore distinguished from physical coframe nondegeneracy, native entropic ordering from calibrated classical time, and bridge control from observable or dynamical causality. Certified locality is formulated on four-dimensional readout patches rather than on arbitrary neighborhoods of the higher-dimensional state space. Local causal preorders are then constructed from the section-induced Lorentzian geometry. The paper separates this kinematic result from stronger claims concerning microcausality, signal propagation, stable causality, or global hyperbolicity. Patchwise consistency is analyzed through proper orthochronous frame transformations, bridge-compatible transition laws, and triple-overlap cocycle conditions. Under exact compatibility assumptions, the local metric, cone field, and minimal causal preorder descend coherently across the certified atlas. Under defect-controlled compatibility, causal classification remains valid only for directions sufficiently separated from the null boundary. The paper also distinguishes native nonlocality, readout nonlocality, local certification failure, overlap or cocycle failure, global causal pathology, and propagation-level failure. These notions are shown to be logically and operationally inequivalent. Finally, under explicit coercivity, mixed-sector, spectral-regularity, and finite-budget assumptions, a conditional causal-local solvability theorem constrains calibrated temporal precision together with the local, bridge, spectral, orientation, overlap, and cocycle certification margins. The resulting framework identifies a finite corridor in which causal-local and relativistic readout semantics remains certifiable. Its failure suspends the classical causal interpretation without implying a breakdown of the native OT/GKSL dynamics or a state-dependent deformation of the Einstein–Hilbert kinetic sector. 1. Foundations of the Architecture: Foundations |GKSL/Lindblad ; Carlen–Maas ; Jacobson ; Sakharov ; Donoghue ; Lovelock) Establishes the core Einstein-locked OT/GKSL architecture for certified geometric readout and coherence-dependent gravitational sourcing. Optimal-Transport Gravity Trilemma | Identifies the certified operational boundary of geometric readout by proving the fundamental trade-off between temporal resolution, coframe stability, and bridge fidelity. 2. Emergence and Recovery of Classical Physics: Exact Reduced OT/GKSL Equations | Mori–Zwanzig/projection operators ; effective field theory ; Carlen–Maas ; Wilsonian reduction / Demonstrates the controlled recovery of classical Newtonian and gravitational sectors as exact non-linear reductions of the native OT/GKSL state dynamics. Certified Einstein Non-Linear Readout | Lovelock ; Bianchi identities ; Donoghue EFT ; Jacobson thermodynamic gravity// Develops the full non-linear Einstein-locked readout closure for the metric sector. Non-Linear Dynamics and Readout | Dynamical systems, center manifold/effective reduction ; quantum Markov semigroups ; non-linear open-system reductions // Explores the exact reduced non-linear evolution on collective state manifolds. The Seeley–DeWitt Bridge | Seeley–DeWitt heat-kernel ; Vassilevich // Formalizes the operational connection between native state dynamics and the effective classical readout. The SDW Bridge: Composite Brout–Englert–Higgs Dynamics, Spectral Separation, and the Emergent Graviton | Formalizes the emergence of the Brout-Englert-Higgs composite scalar and the spin-2 graviton via the Seeley-DeWitt expansion, strictly preserving the Einstein-Lock. Bridge between QCD and OT/GKSL Readout | Wilson lattice gauge theory ; Gross–Wilczek–Politzer asymptotic freedom ; Kogut–Susskind Hamiltonian lattice gauge theory // Connects the Optimal Transport / GKSL framework to Quantum Chromodynamics, exploring the constitutive bridge and effective low-energy dynamics. 3. The Certified Boundary and Structural Limits: Certified Spacetime Readout on Finite Support: A Unified Temporal and Geometric Boundary | Decoherence / Quantum Darwinism ; quantum reference frames ; finite information bounds ; Jacobson // Unifies the temporal and geometric branches of classical readout into a single certified spacetime problem. Introduces the unified spacetime readout burden and derives the central unified certified-budget inequality, proving that temporal precision, geometric coframe nondegeneracy, and bridge compatibility draw from the same finite entropic and informational resources and cannot be made simultaneously ideal. Certified Causality, Locality, Nonlocality, and Relativity in the Einstein-Locked OT/GKSL Framework | Algebraic QFT/locality ; operational quantum theory ; quantum reference frames ; relativistic causality tests // Determines the exact status of causality, locality, nonlocality, and the principle of relativity within the Einstein-locked OT/GKSL architecture. Shows that causal-local spacetime semantics is a certified readout property rather than a primitive native axiom; proves a patchwise gluing theorem for certified local causal structure; and derives a unified finite-budget inequality showing that temporal precision, geometric certification, bridge admissibility, and overlap compatibility all compete for a single residual causal-local headroom on finite effective support. Entropic Tick Cost and Certified Temporal Readout in the Einstein-Locked OT/GKSL Framework | Demonstrates that classical ticks are finite-resource readout objects extracted from native entropic ordering, rather than primitive background parameters. Decomposes the entropic tick cost into native, extraction, and certification branches, and derives a theorem-level certified temporal budget inequality connecting temporal resolution, finite effective support, and certification margins. Entropic Tick Cost & Spectral Budget | Page–Wootters time ; thermal time hypothesis ; quantum clocks ; Salecker–Wigner bounds // Establishes a theorem-strength certified boundary for classical spacetime by proving a fundamental trade-off between entropic tick resolution, coframe stability, and finite informational budget. Toy Certified Pipeline from Optimal Transport QCD | Provides a protocol-level implementation and scaling model for certified bridge margins. Certified Spectral Boundary from Heat-Kernel Budgets and Entropic Transport in the Einstein-Locked OT/GKSL Framework | Heat-kernel spectral budgets; entropic OT/GKSL transport; certified spectral boundary; Einstein-locked readout. Develops a spectral-geometric control layer for the OT/GKSL framework, where the native heat trace bounds finite spectral resources, the cutoff gap defines a certification margin, and entropic transport controls the drift of readout-support budgets without inducing a state-dependent Einstein–Hilbert kinetic term. Correlation Separation in the Einstein-Locked OT/GKSL Framework | Establishes a theorem-level distinction between native, readout, and causal-local correlations, and reframes the horizon information problem through certified-domain correlation layering 4. Cosmological Dynamics & Global Readout Constraints: Vacuum-like Residual Energy from Constitutive-Holonomic Balance in a Minimal Reduced OT-C3 Sector | Effective potentials ; Coleman-Weinberg ; Sakharov induced gravity ; vacuum energy problem // Demonstrates analytically that the macroscopic cosmological constant emerges as a non-zero vacuum-like residual energy resulting from the exact balance between scalar constitutive dissipation (source sector) and the non-commutative holonomic barrier of the Optimal Transport geometry. Homogeneous Closed Readout Dynamics under Finite Spacetime Budget | FLRW cosmology ; effective dark energy ; backreaction ; EFT of dark energy// Constructs a homogeneous and isotropic model (G-FLRW) demonstrating how the spacetime budget acts as a branch-selection mechanism, effectively identifying the vacuum-like sector (Λ) as the maintenance cost of certified spacetime solvability. 5. Experimental Protocols and Testability: Testing Source-Side State Dependence in Gravity with Lock-In Atom Interferometry | Kasevich–Chu ; Peters–Chung–Chu ; Rosi–Tino ; atom gravimetry // Proposes a concrete experimental protocol to falsify source-only emergent gravity at low energy. A Lock-in Atom-Interferometric Test (Clock) | Detailed operational implementation of the low-energy readout test for the Einstein-locked framework. Experimental Separation of Readout and Causal-Local Correlation Layers in the Einstein-Locked OT/GKSL Framework //Circuit QED / transmons ; readout fidelity ; mutual information ; quantum verification // Proposes a falsifiable experimental protocol (CLCP) to test the layered structure of correlation observables by separating certified readout and causal-local licensing thresholds on a controllable quantum platform . Repulsive Gravitationnel Force, Quantum Readout, and the Quantum Nature of Gravity: