This manuscript develops a mechanism-neutral structural theory of finite recurrent stability for closed descriptions of a universe. A closed system cannot rely on an external clock, observer, memory, origin label, or cycle counter to explain its own persistence. Beginning, ending, recurrence, and cycle number count as internal structure only when they are recoverable through relations admitted within the system itself. Persistent structured identity is defined as recoverable relational distinction rather than exact sameness, retained naming, or externally recognized continuity. The paper separates contraction, correction, carry-through, recurrence, collapse, and emergence as distinct recoverability conditions. Finite recurrent stability is classified by the structural condition S_min ≤ R_stab(Ξ, Γ, B_cond, P_stress, M_carry) ≤ S_max, together with the declared identity, correction, transformation, boundary, containment, comparison, recurrence, and carry-through conditions. The expression is classificatory, not a physical observable, evolution equation, or dynamical law. Collapse and emergence are treated as distinct exits from the same finite-stability architecture. Collapse requires lower-bound recoverability failure together with failure of the admitted correction and carry-through conditions. Emergence requires upper-bound containment failure together with loss of the prior containment relation and satisfaction of a declared boundary-readability condition. Neither threshold inequality alone establishes a physical collapse, physical emergence, or cosmological event. The paper develops a structural-horizon taxonomy covering collapse, emergence, external access, and cycle distinction. These horizons classify transitions in recoverability, containment, accessibility, readability, or recurrence-index distinction. They are not assumed to be physical surfaces, metric boundaries, event horizons, or cosmological objects. Recurrence need not imply exact repetition. Boundary-equivalent recurrence may preserve a declared boundary relation, π(Ξ(τ_max)) = π(Ξ(τ_0)), while the complete internal states remain unequal. This permits recurrent governing structure without requiring identical histories. Cycle-index indistinguishability arises when no admitted internal relation recovers an absolute cycle index. Beginning-index nonrecoverability arises when no admitted internal relation recovers firstness. Neither result proves that no beginning, ending, absolute ordering, or distinct history exists. The manuscript also defines weak oscillatory form as bounded recurrent correction, declared relational recurrence, and carry-through of recoverable structure without presupposing physical waves, frequency, propagation, energy transport, or temporal periodicity. It concludes with a realization-burden architecture separating structural classification from carrier selection, interface preservation, mechanism specification, mechanism operation, physical realization, theory-specific realization, and empirical validation. The paper therefore does not propose a physical stability operator or cosmological mechanism. It identifies the structural conditions and additional burdens that any later realization theory must satisfy. Version 2 revision note Version 2 is a substantial reconstruction rather than a minor editorial update. The derivational route was rebuilt and rechecked in full, with revised definitions, dependency ordering, governing conditions, failure conditions, and claim-scope boundaries. The manuscript was then rewritten throughout in new prose to reflect the reconstructed argument. Major changes include the formalization of finite recurrent stability through the revised RstabR_{\mathrm{stab}}Rstab framework; sharper separation of contraction, correction, carry-through, collapse, and emergence; a revised structural-horizon taxonomy; clearer treatment of boundary-equivalent recurrence, cycle-index indistinguishability, and beginning-index nonrecoverability; a more precise definition of weak oscillatory form; and an expanded realization-burden architecture separating structural classification from mechanisms, physical realization, theory-specific interpretation, and empirical validation. The revision also adds stronger nonpromotion controls, appendix-based claim and condition ledgers, and clearer limits on what the structural results do and do not establish. The central research program remains mechanism-neutral, but the argument, notation, scope discipline, and presentation have all been substantially revised.
This manuscript develops a mechanism-neutral structural theory of finite recurrent stability for closed descriptions of a universe. A closed system cannot rely on an external clock, observer, memory, origin label, or cycle counter to explain its own persistence. Beginning, ending, recurrence, and cycle number count as internal structure only when they are recoverable through relations admitted within the system itself. Persistent structured identity is defined as recoverable relational distinction rather than exact sameness, retained naming, or externally recognized continuity. The paper separates contraction, correction, carry-through, recurrence, collapse, and emergence as distinct recoverability conditions. Finite recurrent stability is classified by the structural condition S_min ≤ R_stab(Ξ, Γ, B_cond, P_stress, M_carry) ≤ S_max, together with the declared identity, correction, transformation, boundary, containment, comparison, recurrence, and carry-through conditions. The expression is classificatory, not a physical observable, evolution equation, or dynamical law. Collapse and emergence are treated as distinct exits from the same finite-stability architecture. Collapse requires lower-bound recoverability failure together with failure of the admitted correction and carry-through conditions. Emergence requires upper-bound containment failure together with loss of the prior containment relation and satisfaction of a declared boundary-readability condition. Neither threshold inequality alone establishes a physical collapse, physical emergence, or cosmological event. The paper develops a structural-horizon taxonomy covering collapse, emergence, external access, and cycle distinction. These horizons classify transitions in recoverability, containment, accessibility, readability, or recurrence-index distinction. They are not assumed to be physical surfaces, metric boundaries, event horizons, or cosmological objects. Recurrence need not imply exact repetition. Boundary-equivalent recurrence may preserve a declared boundary relation, π(Ξ(τ_max)) = π(Ξ(τ_0)), while the complete internal states remain unequal. This permits recurrent governing structure without requiring identical histories. Cycle-index indistinguishability arises when no admitted internal relation recovers an absolute cycle index. Beginning-index nonrecoverability arises when no admitted internal relation recovers firstness. Neither result proves that no beginning, ending, absolute ordering, or distinct history exists. The manuscript also defines weak oscillatory form as bounded recurrent correction, declared relational recurrence, and carry-through of recoverable structure without presupposing physical waves, frequency, propagation, energy transport, or temporal periodicity. It concludes with a realization-burden architecture separating structural classification from carrier selection, interface preservation, mechanism specification, mechanism operation, physical realization, theory-specific realization, and empirical validation. The paper therefore does not propose a physical stability operator or cosmological mechanism. It identifies the structural conditions and additional burdens that any later realization theory must satisfy. Version 2 revision note Version 2 is a substantial reconstruction rather than a minor editorial update. The derivational route was rebuilt and rechecked in full, with revised definitions, dependency ordering, governing conditions, failure conditions, and claim-scope boundaries. The manuscript was then rewritten throughout in new prose to reflect the reconstructed argument. Major changes include the formalization of finite recurrent stability through the revised RstabR_{\mathrm{stab}}Rstab framework; sharper separation of contraction, correction, carry-through, collapse, and emergence; a revised structural-horizon taxonomy; clearer treatment of boundary-equivalent recurrence, cycle-index indistinguishability, and beginning-index nonrecoverability; a more precise definition of weak oscillatory form; and an expanded realization-burden architecture separating structural classification from mechanisms, physical realization, theory-specific interpretation, and empirical validation. The revision also adds stronger nonpromotion controls, appendix-based claim and condition ledgers, and clearer limits on what the structural results do and do not establish. The central research program remains mechanism-neutral, but the argument, notation, scope discipline, and presentation have all been substantially revised.
This paper is Part II of The Pre-Emergence Origin of Probability trilogy. Part I develops the structural requirements for finite recurrent stability before spacetime. Part II asks when persistent boundary-readable structure can support numerical probability. Part III examines what probability-status remains after boundary loss, before numerical weighting, and under what narrower conditions Born-form weighting may follow. The paper analyzes how probability-measure status may arise within a closed deterministic description rather than being assumed as a primitive numerical rule. A closed description cannot appeal to an external observer, hidden weighting convention, retained label, post-outcome selection, or independently supplied probability measure to explain its own probability assignments. Numerical probability must instead be supported by structures and relations admitted within the described system. The analysis begins with the requirements for persistent structured identity. Persistent identity is treated as recoverable cross-state relational distinction rather than exact sameness, repeated naming, scalar aggregation, or externally preserved memory. Within the tested structural registry, persistent refinable identity requires ordered dependence, a sufficient stabilization architecture, distributed relational structure, recurrence, bounded recoverability, and noncollapse of the distinctions being preserved. Bounded recurrent relational correction is classified as weak oscillatory form in a deliberately structural sense. The classification does not establish a physical wave, sinusoidal motion, exact periodicity, frequency, propagation, dynamics, mechanism operation, or physical carrier. It identifies a recurrent recoverability architecture that a later realization would have to preserve. The paper then develops the transition from persistent structure to boundary observability. Representation-sensitive bulk descriptions may depend on phase origin, refinement history, decomposition, internal labels, subpart identity, or other distinctions that do not survive admissible boundary readout. Under admissible-observable indistinguishability, boundary-readable observables factor through an emergence-boundary quotient that preserves only the distinctions carried by the admitted observable family and surviving the declared transport relation. The boundary quotient does not itself provide probability. It supplies a local architecture of boundary-readable alternatives and permits the admission of candidate nonnegative contribution scalars. Neither the alternative carrier nor a candidate scalar establishes scalar uniqueness, scalar closure, additivity, measure realization, normalization, or probability interpretation. Numerical probability-measure status becomes available only under a narrower cumulative condition package. The package requires an exact finite carrier, a finite positive event measure, a normalized scalar component on the same carrier, exact scalar-measure compatibility, one-scalar factorization of every admitted weight-bearing map, exclusion of independent weight-bearing invariants, pre-outcome assignment, absence of an external weighting source, and a declared scalar-only predictive regime. Additivity, commensurability, regularity, positive scale, finite nonzero total, and normalization remain separately controlled conditions within that route. Under those conditions, normalized scalar weights may receive conditional probability-measure status inside the declared local regime. This is not an unconditional derivation of probability from determinism. The result does not establish unrestricted probability, universal scalar representation, a global probability law, objective chance, stochastic dynamics, collapse, measurement theory, physical realization, or a final physical theory. Within the trilogy, Part II therefore supplies the middle structural layer: persistent recurrent structuretoboundary-readable alternativestocandidate scalar contenttoconditional numerical probability-measure status. Part III begins from a different question. It examines what remains when a boundary-readable description loses the ability to guarantee which admissible alternative obtains, before any numerical scalarization is supplied. Part II provides the scalar, measure, normalization, compatibility, and realization-burden architecture inherited by that later analysis, but it does not pre-establish the boundary-loss results of Part III. ### Revision note This revision is a substantial reconstruction and trilogy integration rather than a minor editorial update. The derivational architecture, definitions, condition packages, failure boundaries, dependency order, and claim-scope controls have been revised throughout. The manuscript has also been rewritten in new prose to present the argument as Part II of The Pre-Emergence Origin of Probability trilogy. Major changes include a clearer separation of persistent identity, ordered dependence, stabilization, distributed relational recurrence, weak oscillatory form, boundary observability, candidate scalar status, scalar closure, measure realization, normalization, and conditional probability-measure status. The emergence-boundary quotient and transport-survival architecture have been clarified, and the numerical-probability result has been narrowed to an expressly declared scalar-only predictive regime satisfying the complete carrier, measure, compatibility, factorization, pre-outcome, and independent-invariant exclusion conditions. The revision also strengthens the distinction between candidate weights, normalized weights, conditional measures, and probability interpretation. It adds more precise failure localization, preserves the distinction between conditioned failure and universal impossibility, and expands the realization-burden architecture separating mechanism satisfaction, physical scalar selection, boundary enforcement, branch generation, refinement implementation, physical realization, theory-specific realization, empirical validation, and final physical-theory status. The paper is now explicitly positioned between Part I, which establishes the structural requirements for finite recurrent stability before spacetime, and Part III, which develops boundary-unresolved and pre-numerical probability-status after recoverability loss. The central research program remains mechanism-neutral. No physical mechanism, collapse process, specific realization theory, unrestricted probability law, or empirical validation is claimed.
The Origin of Probability: A Complete Derivation in Three Parts is the collected edition of a three-part theoretical investigation into the structural origin of probability within a deterministic closed-description framework. The work asks a deliberately narrow but foundational question: under what conditions can a system whose underlying description remains deterministic acquire a lawful probability-bearing description without inserting primitive randomness, probabilistic axioms, or the Born rule at the beginning of the construction? The trilogy develops that question cumulatively. Each part establishes a distinct layer of the argument, and later results depend on structures and limitations established earlier. The collected edition therefore preserves the full dependency order rather than treating the three papers as independent essays. It also includes a narrative overview and terminology guide intended to make the architecture of the complete derivation easier to follow. Part I, Finite Recurrent Stability Before Spacetime: Collapse, Emergence, Horizons, and Recurrence as Recoverability Boundaries, develops the realization-neutral structural foundation. It begins from persistent relational distinction and recoverability rather than assuming spacetime, metric distance, clock time, energy, probability, or quantum measurement. Within a declared finite-stability regime it distinguishes contraction, correction, carry-through, collapse, emergence, recurrence, cycle structure, and horizon-like boundaries by their effects on recoverable relational structure. Collapse and emergence appear as distinct lower- and upper-bound failures of one finite recurrent-stability architecture. Weak oscillatory form is subsequently obtained only at structural scope, without promoting recurrence into a physical wave, temporal oscillation, or cosmological mechanism. Part II, Stability, Boundary Observability, and Emergent Probability in Deterministic Systems, develops the transition from deterministic structural dynamics to probability-bearing description. Its central problem is not whether deterministic evolution can simply be renamed probabilistic, but whether loss of recoverability at an admissible boundary can support a lawful reduced description of unresolved alternatives. The paper therefore separates Boundary Loss from probability itself and distinguishes quotient formation, reduced observability, scalar closure, finite measure, normalization, and predictive interpretation as separate derivational burdens. A normalized set of branch weights is not treated as probability merely because it sums to one. Probability status is admitted only after the additional structural and interpretive conditions required for predictive use have been isolated. Part III, Boundary Loss and the Born Rule: The Origin of Probability, carries the construction into a bounded physical realization framework. Boundary Loss supplies unresolved alternatives but does not itself supply numerical probability. The derivation therefore proceeds through source-grounded scalar residues, finite positive total measure, normalization, local predictive admission, and independent source-state and event bridges into Hilbert-space representation. A minimum compatibility condition is then isolated, explicitly adopted at the appropriate stage, and applied separately. At the resulting local source-linked scope, the construction reaches the exact Born form p_i = ⟨J(Ψ), P_i J(Ψ)⟩ = ||P_i J(Ψ)||². The significance of the trilogy lies as much in the separation of these stages as in the terminal equation. Boundary Loss is not equated with probability. Scalarization is not assumed to be probability. Normalization is not treated as sufficient for probability. Hilbert-space representation is not allowed to retroactively supply the earlier structural derivation. Compatibility required for the final Born-form application is distinguished from structures already derived. The dependency chain is therefore designed so that later mathematical convenience cannot be silently imported backward as an earlier premise. Across the three parts, the argument develops the following bounded sequence: ordered dependence → recoverable distinction → finite recurrent stability → collapse/emergence boundary structure → boundary observability → Boundary Loss → unresolved alternatives → scalar closure → finite measure → normalization → predictive probability → Hilbert-space source and event bridges → minimum compatibility → exact local source-linked Born form. The collected edition is intended to function as the canonical integrated research record for this derivation. It contains the trilogy overview followed by Part I Version 4, Part II Version 3, and Part III Version 2. Internal bookmarks and navigation link the overview directly to the title page of each constituent work, while the individual papers remain separately citable publications with their own persistent identifiers. Constituent works: Part I: Finite Recurrent Stability Before Spacetime: Collapse, Emergence, Horizons, and Recurrence as Recoverability BoundariesDOI: 10.5281/zenodo.19966230 Part II: Stability, Boundary Observability, and Emergent Probability in Deterministic SystemsDOI: 10.5281/zenodo.19966289 Part III: Boundary Loss and the Born Rule: The Origin of ProbabilityDOI: 10.5281/zenodo.20102172 Taken as a whole, the trilogy proposes a disciplined route from recoverability and deterministic boundary structure to probability and, at its final declared scope, to the Born form. Its central methodological requirement is that every transition be earned at the layer where it is used, with unresolved burdens, adopted conditions, nonclaims, and realization boundaries kept explicit rather than compressed into the final result.
Description QSTH M.1B is the operational continuation of QSTH M.1A, Genealogy of Alpha-I-Dim and Equations E1–E10. M.1A established the algebraic genealogy of the reference Alpha-I-Dim construction and identified effective entropy S_eff and the structural parameter κ(Γ) as imported quantities whose physical meaning could not be supplied by algebra alone. M.1B addresses that open requirement by constructing a first explicit finite operational framework for structural configurations, equivalence, representation, statistical measure, dynamics, entropy and independent classification. The study defines typed relational configurations with a decidable admissibility predicate and equivalence under relabelling; constructs a faithful incidence representation; distinguishes representational equivalence, physical equivalence and indistinguishability under a limited readout; and states the conditions under which deterministic and probabilistic dynamics descend consistently to configuration classes. A finite operational statistical entropy is then constructed from a prespecified state space, measure and readout. Independently, a dynamical classifier based on closed recurrent sectors is introduced, together with an audit of exact charge protection versus finite escape time. The central integration result is deliberately restrictive. The independently defined sector count is not a universal entropic coefficient for Equation E10. The same entropy may coexist with different sector structures, while genuine internal degrees of freedom may increase entropy without creating an additional recurrent sector. The independent test is therefore allowed to reject the proposed identification rather than being normalized to reproduce it. Accordingly, the formal mathematical and statistical stage B1-B7 is complete within its declared domain, while physical horizon identification, M-independence and the continuation to E11+ remain open. This publication does not claim a new universal constant, a microscopic theory of horizon entropy, or empirical confirmation of QSTH. Its contribution is an auditable operational framework, explicit counterexamples, reproducible finite tests, and a precise boundary between mathematical construction and physical identification. A separate companion study, QSTH M.1B-SR1, examines whether a richer spectral response can provide a conditional bridge between an independently defined observable and the thermal entropy of specified interface carriers.
Abstract We present DLST Ver.11.0.0, a compact recurrence-classification framework for exact physical histories, relational time, and relativistic closure. Ver.11 explicitly corrects the earlier interpretation of DLST: finite signal speed, microscopic length, compactness, finite entropy, finite Hilbert dimension, or a discrete evolution label do not by themselves establish a fundamental discrete clock or exact recurrence. The central objects are instead the physical identity space, the stabilizer of an individual state, and the common global kernel of the evolution. This separates state-specific recurrence, universal recurrence, gauge stationarity, and observational indistinguishability. For autonomous discrete quantum evolution, we prove that a common exact projective return for all pure-state rays and density operators exists if and only if the one-step projective update [U] is a torsion element of PU(H), equivalently U^N = exp(i phi) I for some finite N. Membership in a finite projective unitary subgroup is therefore a sufficient special case. Finite-volume exact Clifford dynamics on finitely many prime-dimensional qudits provides an explicit conditional realization: the projective Clifford group is finite, so an exact Clifford quantum cellular automaton has a nonzero common recurrence period. This bridge does not require a finite quantum-state cardinality. Its scope is exact and finite-volume; infinite-lattice Clifford QCA or arbitrarily small irrational non-Clifford phases need not recur globally. Complementary positive criteria are given by finite same-law closure and spectral commensurability. For unitary evolution, exact recurrence of an occupied state requires its active eigenphase or energy differences to lie on a common arithmetic lattice. These results clarify why finite-dimensional Hilbert space alone is insufficient: exact recurrence is controlled by projective order and spectral arithmetic rather than vector-space dimension. The relativistic obstruction side is strengthened. In 1+d-dimensional Poincare-covariant quantum dynamics with d >= 2, if the scalar/projective translation kernel contains any nonzero translation—timelike, spacelike, or null—Lorentz covariance and additivity force the entire translation subgroup into the kernel. Hence a nontrivial translation representation cannot possess a nonzero universal spacetime period. A related connected simple-group obstruction excludes noncentral universal projective periods under the stated representation hypotheses. For relativistic scattering sectors, Haag-Ruelle wave-operator equivalence transfers the asymptotic energy spectral measure to the interacting sector. A nonzero non-atomic energy component is incompatible with the affine spectral lattice required for exact projective return, yielding an exact nonreturn criterion without assuming that every state belongs to the scattering sector. Independently, any strictly monotone extensional physical witness along the complete admitted history forbids exact recurrence. Ver.11 also proves a finite-window nonidentifiability result: for any finite observation horizon and tolerance, one can construct finite-dimensional Hamiltonians whose propagators remain arbitrarily close throughout the observed interval while belonging to different global recurrence classes. Finite measurement precision therefore cannot replace an exact structural bridge from observed dynamics to all-history recurrence. Related audits show that finite Hilbert dimension, Type II_1 finite-entropy language, gauge stationarity, and cosmological labels are not exact recurrence classifiers by themselves. Finite-dimensional completeness is also incompatible with a nontrivial exact continuous unitary representation of the standard connected de Sitter group under the stated assumptions. The resulting classification contains both positive and negative bridges. Positive exact routes include finite same-law closure, projective torsion, finite-volume exact Clifford QCA, and affine spectral lattices. Obstruction and nonreturn routes include Poincare symmetry, non-atomic scattering spectra, and strict extensional monotonicity. Gauge equivalence, compactness, finite observational windows, and coarse finiteness statements remain nonclassifying without additional physical input. The principal methodological change is therefore a shift from presumed temporal discreteness to fail-closed actuality promotion: a recurrence or nonrecurrence claim is promoted only after the physical identity relation, complete evolution law, and exact all-history bridge have been independently specified. DLST Ver.11.0.0 does not establish that the actual universe is periodic, aperiodic, fundamentally discrete in time, a finite Clifford automaton, or governed by a universal microscopic tick. C4 Ver.1.0.0 and OUROBOROS Ver.2.0.0 remain separate downstream research programs rather than premises of DLST. Deterministic reproducibility checks accompany the release for the finite-projective, Clifford-group, spectral, permutation, and finite-window constructions. These computational audits verify explicit finite examples and implementation details; the analytic theorems remain the scientific basis of the classification. DLST Ver.11.0.0 thus provides a compact framework connecting exact recurrence, dynamical stabilizers, projective unitary torsion, Clifford quantum cellular automata, Poincare symmetry, Haag-Ruelle scattering, operator-algebraic finiteness, de Sitter representation theory, and relational physical time under an explicit claim boundary.
This paper is the seventh in a sequence developing an interpretive framework that redescribes physical reality as a causally consistent history of information updates rather than as a collection of objects independently existing within spacetime. Two prior results set up its question. The third paper (Uchida, 2026c) characterized public finality—recoverability of a committed fact by many mutually independent accessors, non-disturbingly and repeatably—as a broadcast redundancy: an effectively classical pointer value proliferated across many disjoint environmental fragments, approximately in the form of spectrum broadcast structure. The fourth paper (Uchida, 2026d) read the evaporating black hole as the regime in which a committed interior record crosses, at the Page time, from hidden to recoverable encoding, reconstructible from the radiation through entanglement-wedge reconstruction; and the parent framework (Uchida, 2026a, §10) had flagged holographic quantum error correction as a place its language of protected, redundantly encoded, locally recoverable records might find content. The present paper makes that content precise as a structure theory of recoverable records, kept separate from any claim of new physics or any suggestion that the framework derives the underlying results. The central claim is conditional and structural. The framework does not claim that every recoverable piece of information is a code, that the universe is literally a quantum error-correcting code, or that AdS/CFT describes our universe. It claims that when information qualifies as an erasure-recoverable record because an encoding preserves a logical degree of freedom—or a logical operator algebra—against a specified family of physical erasures, then that recoverability is formalized as an exact or approximate quantum error-correcting structure, in which each correctable erasure leaves an authorized complementary region supporting a recovery map. That formalization is itself an established mathematical classification; the paper's interpretive contribution is to read the series' recoverable record through it, and to set it against a second, distinct architecture. Broadcast redundancy makes the same effectively classical value legible from many disjoint fragments; erasure-correcting redundancy protects noncommuting logical quantum information by keeping each correctable erased region ignorant of—approximately private with respect to—the protected logical content, while suitable remaining regions recover it. The formal engine of this distinction is the duality between correctability and privacy (the information–disturbance tradeoff), not the no-cloning theorem; no-cloning, with the access structure of quantum secret sharing, survives as the reason independently usable recovery regions for arbitrary unknown logical quantum information must overlap. Three commitments keep the reading honest. First, it separates three senses of "record"—protected logical quantum information, a committed classical record, and a publicly final record—and insists that recovery or decoding restores access to logical quantum information but does not, by itself, commit a classical value; commitment additionally requires measurement, pointer imprinting, decoherence, and stabilization (Uchida, 2026f), and public finality a still later broadcast. Second, holographic quantum error correction is offered as the sharpest worked model of the distinction, with exactly solvable tensor-network toy codes (exact within the model) kept distinct from realistic semiclassical holographic reconstruction beyond those limits (generally treated as approximate), and both from the further question of state-dependence. Third, code-recoverability does not by itself imply public finality: protected quantum information can be recoverable from authorized regions while remaining non-public. The framework relabels code structure as the architecture of recoverability and derives none of it, making no independent empirical prediction. Its disciplined thesis is that codes protect recovery; broadcasts create publicity—different architectures that a single physical history may nonetheless realize in sequence.
This paper presents Version 2 of “Meaning-Based Theory — Space 2.” The principal revision in this version is the reformulation of the generation of space from an explicitly quantum-generative perspective. In this formulation, quantum-generative structure does not arise inside an already completed space. Instead, a pre-spatial generative structure precedes positional and spatial establishment. The upstream sequence begins with Meaning, Being, Self-Reference, the distinction between Self and Other, and the repeated re-establishment of Self/Other role types. This recurrence is not periodic motion within space; it is a generative cycle that exists prior to spatial completion. When irreversible order of establishment is introduced, this recurrence becomes temporally differentiated. Distinct established states must remain distinguishable, leading to positional distinction. Space then first emerges as the structure that preserves such positional distinctions. A central feature of Version 2 is therefore the reversal of the conventional ordering: not Space → Quantum, but pre-spatial quantum-generative structure → position → emergence of space → quantum-relational interference → finite closure → completed space. The quantum-generative perspective is also applied to four-position generation. Multiple generatively admissible possibilities prior to closure are not prematurely fixed as a single completed classical configuration. A completed structure is established only after Relation consistency, four-point interference, and closure conditions are satisfied. This formulation does not assume probability amplitudes, a wave function, the Born rule, or measurement collapse. Continued positional establishment generates two-position and three-position structures, Spatial Direction, and finally a four-position structure consisting of A, B, C, and D. These four positions generate six two-position Relations and four three-position mutual Relations, forming the global four-position Relation Ω₄. The four-position structure becomes internally complete without requiring a fifth position. However, static relational completeness alone is not complete closure. Local actions preserving different shared Relations are generally noncommutative, producing four-point interference. For the circuit action Q = R₄R₃R₂R₁, the SU(2) representation yields the invariant I₄ = (1/2)ReTr(Q) = 1/2, together with the finite recurrence Q³ = −I and Q⁶ = I. Internal Relation completeness and finite closure of internal action are thereby simultaneously established. This is defined as I₄ Closure, and within MBT: I₄ Closure = completed local space. The defining advance of Space 2, Version 2 is therefore that completed space is no longer treated as the prior container in which quantum-generative processes occur. Rather, completed space itself is derived as the closure result of a generative process that begins prior to space, proceeds through recurrence and temporal differentiation, generates positional distinction and relational structure, and reaches completion through noncommutative four-point interference and finite I₄ closure.