This manuscript presents a Lean-formalised, discrete null-lattice light-cone construction that derives an auxiliary field and rapidity ladder directly from combinatorial structure. The resulting framework supplies an explicit, lossless interpretation of quantum superposition on an octonionic ℝ⁸ carrier while exactly reproducing the inner-product invariants of standard quantum mechanics. It simultaneously enables an exact sparse, horizon-causal update rule that supports efficient classical simulation in structured regimes. The model establishes a horizon-limited bridge between quantum mechanics and quantum field theory. A finite measurement layer implements Born-rule normalisation together with closed energy accounting through auxiliary, birefringence, and redshift channels. The discrete carrier is ultraviolet-finite by construction; infrared behaviour is controlled by a causal horizon. Lattice combinatorics force the shell-wise curvature-imprint ratio α = 3/5 and the complementary unit-horizon fraction γ = 2/5 exactly. The same auxiliary bookkeeping naturally accommodates polarisation-rotation observables, with direct implications for cosmic-microwave-background birefringence at large scales. Applications to quantum chemistry and protein folding are demonstrated through nonnegative traceable site-energy contracts and refinement algebras. The auxiliary carrier wave supplies full local bookkeeping of energy flows and probability currents within the horizon-limited causal structure. This deterministic local accounting explicitly rules out globally branching many-worlds interpretations and instead provides an algebraic foundation that aligns the model with a pilot-wave ontology: measurement selection emerges from an explicit, calibrated rule that enforces Born-rule statistics without invoking dynamical collapse. The construction remains a representation-level reformulation focused on operational completeness and conservation.
This manuscript presents a Lean-formalised, discrete null-lattice light-cone construction that derives an auxiliary field and rapidity ladder directly from combinatorial structure. The resulting framework supplies an explicit, lossless interpretation of quantum superposition on an octonionic ℝ⁸ carrier while exactly reproducing the inner-product invariants of standard quantum mechanics. It simultaneously enables an exact sparse, horizon-causal update rule that supports efficient classical simulation in structured regimes. The model establishes a horizon-limited bridge between quantum mechanics and quantum field theory. A finite measurement layer implements Born-rule normalisation together with closed energy accounting through auxiliary, birefringence, and redshift channels. The discrete carrier is ultraviolet-finite by construction; infrared behaviour is controlled by a causal horizon. Lattice combinatorics force the shell-wise curvature-imprint ratio α = 3/5 and the complementary unit-horizon fraction γ = 2/5 exactly. The same auxiliary bookkeeping naturally accommodates polarisation-rotation observables, with direct implications for cosmic-microwave-background birefringence at large scales. Applications to quantum chemistry and protein folding are demonstrated through nonnegative traceable site-energy contracts and refinement algebras. The auxiliary carrier wave supplies full local bookkeeping of energy flows and probability currents within the horizon-limited causal structure. This deterministic local accounting explicitly rules out globally branching many-worlds interpretations and instead provides an algebraic foundation that aligns the model with a pilot-wave ontology: measurement selection emerges from an explicit, calibrated rule that enforces Born-rule statistics without invoking dynamical collapse. The construction remains a representation-level reformulation focused on operational completeness and conservation.
This report develops the mathematical spine of a discrete information-theoretic space-cell framework. A finite region of the substrate is represented by a tensor product of local Hilbert spaces over a graph or cell complex. Closed dynamics are modeled by finite-depth circuits of bounded-range unitary gates; open-system, measurement-inclusive, or coarse-grained dynamics are represented by local completely positive trace-preserving maps. Under these assumptions, norm or trace preservation is exact, local observables acquire support only inside finite update cones, and unconditional local statistics are invariant under distant trace-preserving operations. A one-dimensional symmetric split-step quantum walk is used as an explicit baseline. Selecting a quasienergy branch of its finite-step update operator defines an effective Hamiltonian. In the long-wavelength, small-step regime, the leading generator has Dirac form, with the cell spacing and update interval fixing an effective propagation speed and the local component-mixing angle fixing a mass scale. The Schrödinger equation is then obtained as a secondary positive-energy, low-momentum envelope approximation after factoring out the rest-energy phase. This hierarchy prevents the discrete update, Dirac continuum regime, and nonrelativistic Schrödinger regime from being treated as equivalent claims. The report also proves reduced-state no-signaling for local trace-preserving operations, states the corresponding probability-level result for local POVMs, and shows how Bell/CHSH correlations can coexist with no controllable superluminal communication. Entanglement is interpreted as distributed nonfactorization generated by local interaction history, not as measurement-time signaling. The report closes with explicit proof obligations: Born-rule and measurement derivation, interacting relativistic fields, spin-statistics, gauge structure, curved geometry, exact or experimentally bounded Lorentz recovery, local conservation beyond global norm, and the causal safety of χ or deformation variables.
This paper develops a discrete, information-theoretic framework in which physical reality is modeled as a lattice or relational graph of interacting “space cells,” each evolving by local update rules. The framework treats space as an active information-processing substrate rather than as a passive background. Time is modeled as an ordering of state transitions rather than as a fundamental substance. Energy is provisionally associated with event density, update rate, action flow, or structured information-processing intensity, with Planck’s constant interpreted as a conversion scale between discrete transitions, phase, and action. The paper combines formal mathematical development with a summary of toy-model probes used to test whether the framework can support the kinds of structures required of a physical theory. Mathematically, it distinguishes theorem-level consequences of local/unitary dynamics from continuum-limit derivations and from simulation-supported but still speculative model-selection branches. Finite-radius local updates imply causal cones; unitary updates preserve norm; local trace-preserving operations preserve no-signaling at the reduced-state level; split-step quantum-walk updates support Dirac-like low-momentum continuum behavior; and Schrödinger-like dynamics arise as a nonrelativistic positive-energy envelope approximation rather than as the most immediate continuum limit. The space-cell framework is organized around two candidate geometric realizations. A distortable-cell model assigns deformation, strain, connection, or topology-bearing degrees of freedom to cells and links. An alternative extrinsic χ-compression model assigns an additional internal scalar coordinate , interpreted as compression, event-density elevation, hidden displacement, or update-rate modulation. The χ option is not initially treated as a literal fourth spatial dimension; it is a parsimonious internal state variable to be compared against deformation-based models under causal, weak-field, Lorentz-like, source-specific, and false-source controls. The toy-model program summarized here tests whether local update rules can generate stable, bounded, identity-preserving informational structures that survive perturbation, scattering, false-carrier pressure, and geometry-like coupling. These simulations do not prove that nature uses space cells, but they have produced mechanism-level evidence for relational identity preservation, local reconstruction, false-carrier rejection, conservation-ledger requirements, photon-like interaction bookkeeping, and χ/deformation model-selection criteria. This paper does not claim to have derived the Standard Model, full quantum field theory, or the Einstein field equations. It presents a staged mathematical and computational research framework. The strongest current claim is that local information-processing rules can be formalized in ways that recover several necessary structural features—locality, conservation-like invariants, causal propagation, no-signaling compatibility, Dirac/Schrödinger continuum regimes, and candidate weak-field geometry equations—while toy-model branches provide constrained tests of whether those structures can support physically relevant informational excitations.
Finite Reversible Closure: A Discrete Substrate Framework for Quantum Theory, Relativistic Kinematics, Gauge Curvature, Renormalisation and Emergent Geometry (Paper 8) Abstract This work presents an eight paper programme establishing a finite reversible closure (FRC) framework in which quantum theory, relativistic kinematics, gauge structure, renormalisation and geometric language arise as scale-dependent compressions of a finite discrete substrate. The primitive layer consists of discrete sequencing (tₚ), discrete adjacency (ℓₚ), finite-dimensional local Hilbert spaces, strictly local finite-depth unitary update, exact gauge constraints on links and a finite propagation bound. No continuum ontology is assumed primitive. From these assumptions, quantum representation and Born-rule compatibility emerge as consistent linear structure; bounded locality induces a causal cone and relativistic dispersion regimes; gauge covariance and plaquette holonomy define discrete curvature observables; renormalisation appears as channel-valued coarse-graining; and under explicit infrared (IR) regime conditions, smooth affine geometry can be constructed from Wilson loop statistics. Continuum quantum field theory and classical geometric language are treated as IR correspondence layers rather than fundamental ontology. The framework is internally finite, structurally disciplined and explicitly falsifiable. Introduction Modern theoretical physics relies heavily on continuum models whose mathematical structure often exceeds the domain of directly observable invariants. Divergences, renormalisation prescriptions and geometric reinterpretations frequently arise as artefacts of these representations rather than as primitive physical necessities. This programme begins from a different premise: assume only a finite discrete substrate with reversible local dynamics and bounded propagation. No continuum space, no background manifold and no infinite local degrees of freedom are taken as fundamental. Papers 1–8 develop this premise into a closed structural architecture;- Paper 1 establishes finite reversible closure and discrete gravitational bookkeeping; Paper 2 demonstrates compatibility with quantum linear representation and Born-rule probability assignment; Papers 3–4 show how bounded locality yields causal cones and emergent relativistic dispersion in infrared regimes; Papers 5–6 introduce exact gauge constraints and plaquette holonomy as discrete curvature observables; Paper 7 reframes renormalisation as channel-valued structural compression and this Paper 8 constructs an explicit holonomy-to-geometry bridge and provides falsification criteria. The central claim is not that continuum physics is wrong, but that it is not primitive. Quantum mechanics, relativistic kinematics, gauge curvature, renormalisation flow and geometric language all emerge as scale-dependent compressions of a finite reversible substrate.
We develop the quantum layer of the finite ring cosmology framework: observation as the interaction of a Subject and an Object, phase-coherent cyclic subsystems of a common finite Carrier shell. Four identifications carry the construction, each a derivation rather than an interpretation. The wave function is phase: pure states are windings of the Object's cycle, selected by stationarity under the global drive, with E=hf an identity. The probability amplitude is Object-to-Subject projection onto a forced orthogonal basis, so that completeness, the selection rules, and the Lüders update are finite character theory rather than postulates. Probability is a coincidence count in the Subject's ledger - the group ring Z[i] of the shared quarter-turn core - which derives the Born rule below the coherence horizon and forces complex amplitudes. Composition is conservation: the single drive conserves the relative offset of a pair, synchronised orbits are the Bell states, and the framework saturates the Tsirelson bound exactly, S=2(ζ8+ζ8−1)=22, while reproducing the network behaviour of the real-versus-complex discrimination. The same conservation makes gravity an entangling channel whose distinctive parameter-free signature is equal force but unequal entanglement - a source gravitates by its full mass yet entangles only through its coherent fraction --- with entanglement forming at the Newtonian rate under the complementarity V2+C2=1. Every claim is verified by exact arithmetic in three worked configurations (eleven validation suites, eighty-nine checks), and the experimental confrontation closes in a single falsifiability ledger on one committed carrier scale.
This preprint develops a discrete, fully relational framework in which spacetime structure, classical dynamics, and quantum-like behavior emerge from three minimal principles: relational existence without background manifold, local deterministic update, and finite per-element update capacity. The central organizing primitive is finite update capacity. At each update step, elements can realise only a bounded number of relational changes. Competition among admissible updates induces “blocked realisations,” which are defined micro-ontologically as energy. Proper time is identified with realised record growth. Spatial distance and effective dimension arise from intrinsic graph structure and shell scaling. Under broad locality conditions, marginal frontier stability constrains viable scaling regimes. Requiring compatibility between flux balance and the observed inverse–square law selects an effective three-dimensional scaling regime among integers. Classical force structure emerges from anisotropies in blocked realisation. A statistical layer appears through coarse-graining of deterministic microhistories. Composition and norm stability require unit-modulus complex weights; refinement invariance and classical coherence fix the phase rule θ=ℏ−1∫Edτ, with the same blocking-defined energy that generates intrinsic force. In decohered regimes, quadratic (Born-type) probability scaling follows from additive consistency under random-phase aggregation. The framework is programmatic: it does not yet derive general relativity or quantum field theory from explicit microscopic rule families. However, it establishes structural coherence linking geometry, inertia, inverse–square interaction, phase accumulation, and probabilistic outcomes under a single finite-capacity relational principle. The work also identifies structural empirical constraints, including compatibility between shell scaling and classical force behavior, stability of inertial response, and the possible appearance of capacity-induced high-energy dispersion within the intrinsic causal-cone bound. Explicit construction and simulation of proposal/selection rule families constitute the next stage of investigation.
We present the final audited-closure formulation of CRQR, a principle-based reconstruction of finite-dimensional complex quantum theory from a pre-operational relational substrate. The reconstruction keeps structural possibility, physical test membership, quantitative statistical identity, classical control, topological closedness, finite-dimensionality, causal normalisation, and specifically quantum selection principles logically distinct. A possibility-only continuation quotient is shown to define only a structural proto-state; full operational states and transformations appear only after a Stable Statistical Experiment Sector and a regime-indexed all-context statistical quotient have been introduced. Stable Classical Control yields dyadic mixtures. A weak operational topology is then defined on actual deterministic preparation transformations, and Physical Operational Closedness requires operationally convergent physical preparations to have a physical preparation realising their limit. This yields full convexity and affine dynamics without adjoining nonphysical state points. Finite statistical separability gives a finite-dimensional affine representation and a canonical ordered real linearisation without assuming no-restriction or local faithfulness of process representations. The final bridge to the finite-dimensional causal operational theory used by Nakahira is rebuilt at the level of physical finite indexed tests. A single regime-indexed quotient is used throughout. Transformations are precisely quotient classes represented by outcomes of physical tests; deterministic and normalised states are distinguished and then proved equivalent; state-branch normalisability is formulated operationally; finite classical outcome processing is explicitly required to be statistically additive in every downstream context; and a Finite Sandwich Expansion theorem proves equivalence between CRQR complete-context equality and ancillary-sandwich equality. A canonical real-linear isomorphism between the CRQR state-space linearisation and Nakahira’s real state span supplies the missing finite-dimensionality/closedness match. These results yield an Exact Physical-OPT Extraction Theorem that verifies the required operational background before Local Equivalence or purification is invoked. Local Equivalence and Equivalent-System Purification, with one purifying system fixed for each system type, are introduced only afterwards. Conditional on Nakahira’s 2026 reconstruction theorem, the resulting sector is standard finite-dimensional complex quantum mechanics, with measurement no-restriction derived rather than assumed. A dependency audit makes explicit that physical operations precede their linear representations and that no Hilbert-space state, POVM, completely positive map, or complex tensor-product structure is inserted into the bridge.