Cosmological beginning problems often begin with an early classical state of the universe and then ask what that state can tell us about earlier structure, initial conditions, or origin. This paper studies a prior finite-dimensional question: under what explicit conditions can a stable low-entropy classical description be initiated from more primitive data? The paper proves finite sufficient conditions under which history selection, boundary extraction, projective reduction, and action-unit calibration produce a stable classical boundary law from an upstream finite carrier. The resulting boundary law is a first accessible classical chart generated by the finite datum. It is not assumed as a primitive classical state and is not identified with the origin of the carrier from which it descends. The construction begins with a finite total carrier, a finite boundary Hilbert space, a finite history-selection family, a boundary channel, and a finite projective boundary context. Under a positive rigidity gap, the selected boundary state converges exponentially to an equilibrium boundary state. The projective boundary context then determines a commutative outcome algebra, and the corresponding outcome weights form a classical probability law. The theorem also gives an explicit entropy concentration result. When the equilibrium boundary support has finite rank, the boundary entropy satisfies a concentration bound with exponentially small off-sector weight. The resulting boundary law is therefore stable and effectively classical. The boundary-accessibility quotient determines exactly which distinctions are represented by the initiated chart and which distinctions remain inaccessible through the declared boundary channel. In the noninjective case, the initiated chart is necessarily compressed: it represents precisely the distinctions separated by the boundary law and no more. The finite theorem proceeds through four stages: history selection, boundary extraction, projective reduction, and action-unit calibration. Boundary accessibility determines the content of the initiated chart, while Planck action-unit closure supplies the final calibration step needed to make the chart physically reportable by fixing the energy-time reporting scale associated with the boundary law. The theorem supplies finite sufficient conditions for initiating a classical cosmological chart from upstream finite data. It identifies a first-principles distinction between carrier origin and chart initiation: the initiated boundary law is certified from explicit selection, accessibility, projection, quotient, entropy, and calibration data. For Big-Bang-type applications, the theorem functions as an audit condition for the initial classical chart itself. Any downstream use of that chart in spacetime realization, inflationary dynamics, singularity resolution, or microscopic quantum-gravity modeling must supply the additional hypotheses required for that application. With causal-chart certification data added, the initiated boundary law becomes the finite boundary-channel component of a classical cosmological chart. License note: Distributed under CC BY-NC-ND 4.0.
Cosmological beginning problems often begin with an early classical state of the universe and then ask what that state can tell us about earlier structure, initial conditions, or origin. This paper studies a prior finite-dimensional question: under what explicit conditions can a stable low-entropy classical description be initiated from more primitive data? The paper proves finite sufficient conditions under which history selection, boundary extraction, projective reduction, and action-unit calibration produce a stable classical boundary law from an upstream finite carrier. The resulting boundary law is a first accessible classical chart generated by the finite datum. It is not assumed as a primitive classical state and is not identified with the origin of the carrier from which it descends. The construction begins with a finite total carrier, a finite boundary Hilbert space, a finite history-selection family, a boundary channel, and a finite projective boundary context. Under a positive rigidity gap, the selected boundary state converges exponentially to an equilibrium boundary state. The projective boundary context then determines a commutative outcome algebra, and the corresponding outcome weights form a classical probability law. The theorem also gives an explicit entropy concentration result. When the equilibrium boundary support has finite rank, the boundary entropy satisfies a concentration bound with exponentially small off-sector weight. The resulting boundary law is therefore stable and effectively classical. The boundary-accessibility quotient determines exactly which distinctions are represented by the initiated chart and which distinctions remain inaccessible through the declared boundary channel. In the noninjective case, the initiated chart is necessarily compressed: it represents precisely the distinctions separated by the boundary law and no more. The finite theorem proceeds through four stages: history selection, boundary extraction, projective reduction, and action-unit calibration. Boundary accessibility determines the content of the initiated chart, while Planck action-unit closure supplies the final calibration step needed to make the chart physically reportable by fixing the energy-time reporting scale associated with the boundary law. The theorem supplies finite sufficient conditions for initiating a classical cosmological chart from upstream finite data. It identifies a first-principles distinction between carrier origin and chart initiation: the initiated boundary law is certified from explicit selection, accessibility, projection, quotient, entropy, and calibration data. For Big-Bang-type applications, the theorem functions as an audit condition for the initial classical chart itself. Any downstream use of that chart in spacetime realization, inflationary dynamics, singularity resolution, or microscopic quantum-gravity modeling must supply the additional hypotheses required for that application. With causal-chart certification data added, the initiated boundary law becomes the finite boundary-channel component of a classical cosmological chart. License note: Distributed under CC BY-NC-ND 4.0.
This paper proves a finite-dimensional structural theorem for boundary accessibility compression. A boundary channel determines which distinctions of a total carrier remain distinguishable at a boundary and which become invisible under boundary access. The resulting compression invariant measures the difference between the total distinction structure and the distinction structure accessible at the boundary. The main theorem provides exact dimension formulas linking accessible distinctions, boundary-invisible distinctions, and the corresponding accessibility quotient. Compression vanishes exactly when the boundary law separates all states of the total carrier, equivalently when it is informationally complete for the total state space. The paper distinguishes three notions that are often conflated: full-state separation, restricted-sector accessibility, and physical recovery. A restricted-state theorem shows how separation on a chosen class differs from compression on the full carrier, thereby keeping accessibility, reconstruction, and recovery conceptually distinct. The theorem is applied to partial trace, channel composition, projective measurement, and finite history selection. In each case, the compression invariant can be computed exactly. The results show that compression is monotone under channel composition, quantify the distinguishable structure retained under nonselective measurement, and characterize the loss of independent distinctions induced by finite history quotients before any additional boundary compression is applied. As a holographic application, the paper formulates holographic reduction as exact boundary representation on an accessibility quotient. A boundary description carries precisely the distinctions separated by its accessible observable structure, while the remaining distinctions lie in boundary-invisible fibers. The theorem therefore provides a finite-dimensional criterion for determining when a boundary law is full-state separating, when it is compressed, and what additional structure is required if physical reconstruction or completely positive trace-preserving recovery is claimed. The companion theorem, Recursive Holographic Descent and Source Recognition: Composed Accessibility Quotients and Survivor Invariants, extends the one-step accessibility-compression theorem developed here to composed channel chains. It establishes the corresponding factorization criterion under which terminal survivor data recognize source-level structure through composed accessibility quotients, survivor spaces, and source-completeness. Together, the two papers provide a carrier-level audit framework for boundary, measurement, selection, holographic, and source-recognition claims. License note: Distributed under CC BY-NC-ND 4.0.
This paper gives a finite-dimensional, first-principles derivation of boundary invariance from completion-locked projective phase and channel data. No-signaling and conserved boundary structure are derived from fixed carrier and channel structure, rather than imposed as external constraints. The central invariant is the Heisenberg fixed-point condition Φ*(X) = X for every boundary observable X in the locked boundary algebra. Equivalently, boundary expectations and boundary POVM laws are preserved uniformly over input states exactly when the admissible complement channel fixes the boundary algebra in the Heisenberg picture. Inside a completion-locked datum, a label encoded only through an admissible complement channel has zero statistical channel into fixed boundary outcomes. Here completion-locked means that the projective phase carrier, boundary algebra, admissible boundary-observable charts, state and normalization data, admissible channel class, and channel-extraction maps are fixed before Bell tables, tensor-factor partitions, random-unitary selections, or quotient channels are compared. The primitive carrier is the projective phase bundle U(1) → S(H) → P(H). Tensor factorizations, Bell partitions, product tests, and local boundary observable algebras enter downstream as locked charts. The theorem sequence is: projective phase carrier → locked boundary algebra → admissible complement channel → Heisenberg fixed point → invariant boundary law. The theorem is realized in three explicit boundary-channel settings: projective quantum no-signaling; Gibbs-weighted random-unitary symmetry-selection channels, whose invariant algebra is the active commutant; and finite equilibrium-combinatorial quotient realizations, whose barycentric extraction is a boundary channel with diamond-norm control of the selected equilibrium limit. Taken together, these results place projective no-signaling, random-unitary active-commutant conservation, and equilibrium-quotient channel convergence under one exact finite-dimensional fixed-point law: admissible complement dynamics preserve boundary data exactly when their Heisenberg dual fixes the locked boundary algebra. The result is a finite-dimensional boundary framework for exact no-signaling, steering without unconditioned boundary signaling, conserved commutants, zero label-to-boundary information, and quotient-selected channel convergence. License note: Distributed under CC BY-NC-ND 4.0.
This paper proves a finite-dimensional boundary-accessibility theorem for apparent information loss. A boundary channel does not merely produce a reduced state; it induces an accessibility quotient on the total state carrier. Distinct total states become operationally indistinguishable whenever they define the same boundary law, and recovery is possible only for distinctions separated by the resulting quotient. The main result gives an exact Heisenberg-range criterion for complete recovery. A boundary channel separates all total density states if and only if its Heisenberg adjoint has full operator range. Equivalently, pulled-back boundary effects span the full real space of total observables. A restricted version characterizes recovery on arbitrary state classes, code sectors, or history sectors. A dimension obstruction follows: no channel into a smaller boundary Hilbert space can be informationally complete for all total states. The theorem is applied to total unitary evolution followed by boundary restriction. Unitary dynamics preserve total-state distinctions, but do not make a noninjective boundary channel injective. In the canonical tensor-factor model, exterior partial trace is noninjective whenever the complementary factor is nontrivial. Consequently, mixed boundary states, increased boundary entropy, and reconstruction failure are compatible with total purity and total unitarity. As an application, the black-hole information problem is reformulated as a theorem of boundary accessibility. Apparent information loss is not identified with failure of total unitarity, but with failure of the boundary-accessible quotient to separate total states. A model claiming exterior recovery must either make the boundary channel injective on the relevant carrier or state class, or explicitly supply additional recovery data such as a code subspace, complementary access, records, histories, islands, a recovery map, a selected quotient section, or another carrier extension. More broadly, the theorem identifies apparent information loss as an accessibility-level quotient phenomenon. Total unitarity acts on the total carrier, while exterior recovery is governed by the distinctions separated by the boundary-accessible carrier. The boundary law recovers exactly those distinctions, and no others, unless additional recovery structure is supplied. License note: Distributed under CC BY-NC-ND 4.0.
This paper gives a first-principles structural derivation of Planck’s constant within the completion-locked closure framework. The derivation begins from a completion-locked phase–time–energy structure in which admissible phase-frequency comparisons are fixed before energy observables are compared. It does not assume the numerical value of Planck’s constant, the Planck–Einstein energy–frequency law, the standard Hamiltonian phase convention, or canonical commutator normalization. Instead, the action scale is derived from a projective phase carrier, a strongly continuous time-translation representation with Stone angular-frequency generator, an admissible class of energy observables for the same deformation, a sector-rich and spectrally exhaustive comparison structure, and, for dimensional reporting only, a unit convention. The main representation theorem proves that any admissible energy observable satisfying spectral affiliation, direct-sum naturality, tensor-product additivity, zero normalization, continuity, positivity orientation, spectral exhaustivity, and common-source descent must be linear in the Stone generator. This yields a unique reduced action scale, from which the full-cycle quantum of action follows by circle normalization. The Planck–Einstein energy–frequency law is then derived as a consequence of the same structure rather than assumed as an input. The paper also establishes conditional action-unit closure pathways. A certified electromagnetic response branch supplies one action-unit closure path, while a certified curvature–vacuum branch supplies a second through an independently derived source vacuum invariant. A single certified dimensional branch determines the action unit within its unit convention; two same-source certified branches overdetermine the same action unit and therefore must agree. Since Planck’s constant is dimensional, its decimal value becomes meaningful only after an action-unit convention or certified dimensional branch has been specified. Under the post-2019 SI convention, the numerical value of Planck’s constant is exact by definition. The structural contribution of the paper is to identify Planck’s constant not as an empirical insertion into quantum dynamics, but as the derived action-conversion invariant relating phase-frequency generation to physical energy observables, with independent electromagnetic and curvature–vacuum closure paths for the same action unit. License note: Distributed under CC BY-NC-ND 4.0.
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.