The Two-Sided Memory Law I: Schur-Complement Viability Collapse and the Acuity Firewall Minimum Viability at the Floor, Maximum Fisher Budget at the Ceiling
Abstract The Canonical Semantic Framework (CSF) assigns finite acuity to the physical carrier itself rather than to an external measurement apparatus. A retained sinc/Paley–Wiener network is then admissible only inside a two-sided viability domain on its information frame : a lower floor (the minimum-memory law) and an upper ceiling (the maximum-memory law), with the covariance frame. Finite acuity is not measurement error; it is an admissibility structure, and the two constraints are the two faces of one variational law. The load-bearing core is a Schur-complement viability-collapse theorem. Profiling out an unresolved coupled sector replaces the bare frame by the Schur complement , and the lower frame bound can only fall, by an amount bounded above by (Weyl plus submultiplicativity). Consequently a globally coupled network can lose causal viability while every local sector stays strictly above the floor at finite local energy: collapse is a memory-solvency failure, not an energy singularity, and forward (Schur margin) and inverse (Cramér–Rao variance) pictures are one fact read two ways. We separate the operator-valued viability gradient flow , the geometric flow of the Fifth Interaction, from the scalar KKT/Pontryagin boundary multiplier that binds only on the active viability boundary, with complementary slackness . Conflating the two was a genuine vulnerability of the first draft; the operator current flows everywhere, the scalar toll is exacted only on the boundary. The identification of the economic surface with the geometric floor is neither an identity nor a scalar calibration but a boundary-functional matching condition on the band-edge leakage functional ; we state it as a theorem and isolate exactly when it carries physical content. Closing the law from above gives a water-filling principle, the depth–optionality split , and the Acuity Firewall: an answer-independent invertible amplifier conserves the Fisher information about a hidden bit (data-processing equality), while finite acuity caps that information at . Reading a -separated oracle bit requires Fisher information of order , so the semiclassical Schrödinger–Newton NP oracle of the Abrams–Lloyd lineage cannot be read at polynomial bandwidth, not by damping the amplifier, but because invertible amplification cannot manufacture the Fisher budget the oracle requires. Under stated continuous-admissibility hypotheses, this is a physical mechanism for an escape class already conceded in that literature, not a refutation. The same ceiling yields an operational null band: a finite-acuity observer cannot resolve a null readout more sharply than . CSF therefore does not first modify spacetime; it modifies the resolvability of causal order. We develop, as explicitly conjectural control theory, an acuity-softened cone in which precedence is graded, the cone band is allocable at fixed budget, and a closed control loop may accumulate a signed cone-twist holonomy with a loop-reversal-odd fingerprint. Throughout, theorem, proposition, corollary, conjecture, and open protocol are distinguished; the Schur theorem and the Fisher ceiling carry the load and depend on none of the frontier extensions. This manuscript makes no claim on the Riemann Hypothesis.
Abstract The Canonical Semantic Framework (CSF) assigns finite acuity to the physical carrier itself rather than to an external measurement apparatus. A retained sinc/Paley–Wiener network is then admissible only inside a two-sided viability domain on its information frame : a lower floor (the minimum-memory law) and an upper ceiling (the maximum-memory law), with the covariance frame. Finite acuity is not measurement error; it is an admissibility structure, and the two constraints are the two faces of one variational law. The load-bearing core is a Schur-complement viability-collapse theorem. Profiling out an unresolved coupled sector replaces the bare frame by the Schur complement , and the lower frame bound can only fall, by an amount bounded above by (Weyl plus submultiplicativity). Consequently a globally coupled network can lose causal viability while every local sector stays strictly above the floor at finite local energy: collapse is a memory-solvency failure, not an energy singularity, and forward (Schur margin) and inverse (Cramér–Rao variance) pictures are one fact read two ways. We separate the operator-valued viability gradient flow , the geometric flow of the Fifth Interaction, from the scalar KKT/Pontryagin boundary multiplier that binds only on the active viability boundary, with complementary slackness . Conflating the two was a genuine vulnerability of the first draft; the operator current flows everywhere, the scalar toll is exacted only on the boundary. The identification of the economic surface with the geometric floor is neither an identity nor a scalar calibration but a boundary-functional matching condition on the band-edge leakage functional ; we state it as a theorem and isolate exactly when it carries physical content. Closing the law from above gives a water-filling principle, the depth–optionality split , and the Acuity Firewall: an answer-independent invertible amplifier conserves the Fisher information about a hidden bit (data-processing equality), while finite acuity caps that information at . Reading a -separated oracle bit requires Fisher information of order , so the semiclassical Schrödinger–Newton NP oracle of the Abrams–Lloyd lineage cannot be read at polynomial bandwidth, not by damping the amplifier, but because invertible amplification cannot manufacture the Fisher budget the oracle requires. Under stated continuous-admissibility hypotheses, this is a physical mechanism for an escape class already conceded in that literature, not a refutation. The same ceiling yields an operational null band: a finite-acuity observer cannot resolve a null readout more sharply than . CSF therefore does not first modify spacetime; it modifies the resolvability of causal order. We develop, as explicitly conjectural control theory, an acuity-softened cone in which precedence is graded, the cone band is allocable at fixed budget, and a closed control loop may accumulate a signed cone-twist holonomy with a loop-reversal-odd fingerprint. Throughout, theorem, proposition, corollary, conjecture, and open protocol are distinguished; the Schur theorem and the Fisher ceiling carry the load and depend on none of the frontier extensions. This manuscript makes no claim on the Riemann Hypothesis.
Abstract From a single physical postulate, that every observer has finite resolution, together with standard regularity conditions on the observation kernel, we derive a chain of results whose logical status ranges from unconditionally closed theorems through conditional closures to explicitly labeled conjectures and engineering programs. The chain culminates, in the present version, in three structural identifications that elevate the framework from a collection of spectral facts into a single operator-level grammar on the active reality fiber. (i) Two-layer observation architecture. Observation is split cleanly into an amplitude-level reconstruction operator (Definition 2.2a) and an induced Born-rule probability density (Definition 2.2b). Under axioms (A1)–(A3), is literally an orthogonal projection onto the retained band, and is its standard projection-valued Born density. The 2001 sinc activation function of Alcocer et al. is not the ancestor of CSF’s probability-layer axioms; it is a literal instance of the amplitude-layer axioms. (ii) Fisher–Rao holonomic convolution. Proposition 4.Z lifts the ordinary convolution theorem from the substrate to the deck-group algebra of the local reality fiber. Every -equivariant observation operator is a circulant matrix on the four-sheet carrier and diagonalizes in the discrete Fourier basis . When the transport operator is also -equivariant, observation, holonomy, and admissible transport resolve simultaneously in the same four channels. Classical signal convolution is the trivial-fiber shadow of this fiber-level convolution theorem. (iii) The dihedral completion of transport. The NJV field of Theorem 6.Z, previously described only as a “sparse orientation-reversing correction”, is identified structurally (Remark 6.Z.5) as the unique off-diagonal coupling between the two nontrivial holonomic channels and . Under the -equivariance of the active sector, the transport current decomposes as , where the -term is -diagonal (Corollary 6.Z.1) and is the dihedral mode-exchange operator . The ad-hoc NJV term becomes a forced structural consequence of -equivariance, and Theorem 6.Z rises to its proper status: a -equivariant constitutive law on the active sector. Theorem 4(E) is simultaneously tightened. The old absolute-uniqueness claim for as a subgroup of is replaced by minimal spectral closure of the active quarter-turn sector: within the active sector, the four-sheet orbit generated by is the unique minimal faithful carrier closed under monodromy and anti-pairing, and it is the carrier on which holonomic convolution, modal transport, and the spectral constitutive law are simultaneously diagonal in the discrete Fourier basis. A forward pointer closes the preprint: on spinors, the calculation of Section 4.2 establishes the covering group (order 16) with . The spinor-level holonomic convolution therefore lives on , giving eight spectral modes rather than four. The extra four modes encode the spinor Berry-phase structure and are the natural home for Conjecture 3.1 (curvature as fifth interaction). This spinor-level extension is flagged as the spine of the next paper and is not installed in the present chain. No Hilbert space is assumed. Stochasticity is not primitive; it is the geometric form of limited visibility on a deterministic substrate. Every claim is stratified by its exact logical status; the status table in Section 6 is the authoritative record. The structural prediction is that ququart-native circuits should outperform qubit-native decompositions for -symmetric algorithms, with the advantage now attributable not merely to modal alignment (Proposition 5.X) but to pipeline-native diagonalization of the entire filter-transport-control chain (Proposition 4.Z together with Corollary 6.Z.1). Engineering consequence programs remain explicitly stratified: the reactor is an engineering proposal contingent on SAFG coupling to nuclear matter; the propulsion architecture is labeled speculative.
Abstract Paper 3 fixed the conserved carrier layer of the Canonical Semantic Framework and forced the retention map M(mu) = sqrt(1 - mu). The question that remained was how admissible transports are actually selected once the retention law is fixed. The present paper constructs the minimal operator-level mechanism for that selection. We formulate the Traffic Law as a constrained primal-dual problem. The primal form minimizes a reconstruction cost subject to amplitude-level retention on the carrier and admissibility budgets on three defect functionals: rank defect, boundary defect, and invariant defect. The dual form introduces regulator fields associated with those constraints. Slack variables quantify the operational margin between admissible transport and boundary saturation. Complementary slackness identifies the regulator as dormant in the interior and active exactly where admissibility saturates. Two structural results follow. First (Theorem 6.1): no selection rule depending only on the contextual leakage scalar mu can pick admissible sectors on the Z4 fiber, because mu has nontrivial level sets and carrier admissibility is phase-sensitive. Leakage is necessary but not sufficient. Second (Proposition 8.1): the elementary lift at maximal misalignment is characterized as a boundary-activated admissible point in the primal-dual system, at which the delta_partial slack saturates, the corresponding regulator activates, and the retention map evaluates to 1/sqrt(2). This reading replaces the earlier contextual-extremum interpretation of the factor 1/sqrt(2) with its correct operator-theoretic status: the first point at which complementary slackness and forced retention coincide. The paper is intentionally narrow. It does not yet specify the microscopic forms of the defect functionals; it does not derive phenomenological observables; it does not fully identify the regulator with any physical field. Its role is structural: it completes the minimal operator form of the Traffic Law, identifies the point at which sector selection becomes well-defined, and isolates the dual regulator as the operator-theoretic object underlying the framework's admissibility-enforcing sector.
Abstract Paper 5 of the Canonical Semantic Framework (CSF) established the minimal coherent grammar: a retained spectral band, a -graded microscopic fiber, an admissible operator class, five residual channels of admissibility failure, and an elementary lift realizing the minimal coherent defect profile. Those results were microscopic. They fixed the operator grammar of the theory but did not yet specify what a macroscopic observer, one who cannot resolve the internal quarter-turn cocycle of the fiber, actually measures. The present paper provides that step. It develops the macroscopic reduction of the grammar and shows that several central structures of observed physics arise as controlled shadows of the microscopic coherent transport. The reduction map. The canonical phase-averaging operator is the Hilbert–Schmidt orthogonal projection and completely positive, trace-preserving conditional expectation onto the grade-invariant observable algebra. It is not one choice among many decoherence prescriptions. It is the unique symmetry-compatible projection encoding macroscopic blindness to the grading, and it is forced by the microscopic grammar. The density operator. Paired with the positive observable state , the reduction produces which is a density operator. The density-matrix formalism of quantum mechanics is not assumed; it is recovered as the reduced description of a more structured object. The effective mass functional. The retained-band projector of Paper 5 is read as a bandlimit with cutoff on an action channel. The effective mass is then the out-of-band Hilbert–Schmidt load Mass is not primitive. It is the structural cost of maintaining coherent transport outside the retained admissible band. Kinematics. Under spectral dilation the exact formula for yields monotone growth of the out-of-band load with . Exact relativistic -scaling follows from a constitutive edge-density ansatz, stated as a modeled special case rather than as a universal theorem. Inertial stabilization. The Negative Jacobian Variational (NJV) compensator of Paper 5 is given its reduced dynamical role. Under an out-of-band dissipativity hypothesis, NJV contributes a linear damping term to the evolution of . The reduced observer experiences this damping as inertia; the stability bound is the precise mathematical content. Non-propagating correlation structure. A sector of microscopic structure supported at and commuting with the grading generator is fixed by and not generated by transport on the action base. This provides the geometric locus for entanglement-type correlations in the reduced theory. The propagation-speed parameter associated with such correlations is absent rather than superluminal; the construction is compatible with no-signaling by design. Two-fold loss. The Hilbert–Schmidt content invisible to the macroscopic observer decomposes orthogonally into two independent pieces: spectral loss (the out-of-band content measured by ) and algebraic loss (the inter-grade content, which we name residual coherence ). These are independent; one may vanish while the other does not. The reduced observer is blind to their sum. Worked example. Appendix D evaluates the reduction operator, the mass functional, and the residual coherence on the elementary lift of Paper 5 and on the leakage-only and pure-geometry counterexample pairs. The quantitative profile of each defect channel through the reduction is exhibited in closed form and matches the Paper 5 Appendix A numerics. What this paper does not claim. No full solution of the measurement problem. No universal relativistic scaling law. No derivation of the Standard Model spectrum, the up-quark anomaly and its expected vortex-core resolution are deferred to Paper 7, which develops the -scaling mass law. No Einstein field equations, the gravitational slot is stated as a structural statement, not derived. No engineering proposals. What this paper does claim. A precise reduction framework, canonical, positivity-preserving, symmetry-forced, that carries the microscopic CSF grammar into macroscopic effective phenomenology. From that reduction there emerge: density operators, a Born-compatible probability calculus, positive effective mass as out-of-band load, inertial stabilization as NJV dissipativity, and non-propagating correlation structure as invariant-sector persistence. The universe the macroscopic observer measures, on this reading, is what a coherent finite-acuity grammar looks like once its internal grading is projected out and its action channel is bandlimited. That is not a small claim. It is narrower than a theory of everything. It is also more defensible.
Abstract Paper 2 established the contextual leakage scalar mu_{epsilon,C}(lambda) as the first positive observable generated by projective winding under finite acuity, together with the two-angle factorization mu(theta, phi) = sin^2(phi) * sin^2(theta) and the rank-one identity mu = 1 - |alpha|^2, where alpha = <psi|U|psi>. The interpretive question left open was which physical layer of the carrier is conserved across the finite-acuity boundary, and therefore which function of mu plays the role of the resolved-sector retention map M(mu). The present paper resolves that question in two steps. First, we observe (Proposition 3.1) that rank-one unitarity already supplies the complete algebraic dichotomy: amplitude-level retention equals sqrt(1 - mu), while intensity-level retention equals 1 - mu. This dichotomy is an observation, not a choice. Second, we prove (Theorem 5.2) that within the Canonical Semantic Framework's carrier architecture -- linear holonomic transport, phase-sensitive admissibility, and contextual squaring as the sole origin of operational intensity -- the primary conserved layer prior to contextual projection is necessarily the amplitude layer. The intensity law is its derived contextual shadow. The retention map is therefore forced: M(mu) = sqrt(1 - mu). At lowest winding and maximal misalignment, this yields the factor 1/sqrt(2) as a structural prediction of the framework rather than a contextual extremum. The result identifies the precise operator-level point at which the Canonical Semantic Framework first becomes structurally selective.
A finite joint-query interface admits a single shared classical record if and only if its tested continuation-word algebra embeds faithfully into a commutative algebra of records — if and only if it admits a finite Boolean-record defender — if and only if a finite system of unit, product, recovery, positivity, and state–effect equations is solvable. The failure of every commuting completion is the precise definition of irreducible joint constraint. This is the Sep/IJC Representation Theorem, and it is the centerpiece of The Enforceability of Distinction. What this paper is A standalone formal-foundation paper for finite-interface physics. Self-contained: no prior exposure to the Admissibility Physics Framework is required. The mathematical core is a single biconditional theorem about finite joint-query interfaces, stated and proved in §11 using only finite linear-algebraic content over ℝ. The paper is built on a deliberately small primitive spine of three commitments: Physical identity is finite admissible continuation identity. A physical state is what it can still do. A physical distinction is a finite enforceable separator of continuation profiles. Distinctions are operational, not abstract. Physical distinctions carry positive enforcement cost. The positive cost floor is the only foundational scalar input. Everything else — the physical quotient, descent of distinctions and filters, the enforcement ledger, convex mixtures from finite classical control, and the continuation-word quotient algebra — is a forced construction. The finite-interface ingredients (finite query family Q, finite tested word set 𝒯, operational resolution, positive resolved floor) are not postulates: they are consequences of a finite act of interrogation plus the positive-cost spine. The Sep/IJC Representation Theorem Let (Γ, Q, 𝒯) be a finite joint-query interface with finite query family Q, finite tested word set 𝒯, finite operational resolution, and positive resolved floor. Let 𝒜Q,𝒯 be the tested continuation-word quotient algebra. Then the following are equivalent: (Γ, Q, 𝒯) admits a commuting-extension defender; 𝒜Q,𝒯 embeds faithfully into ℝΛ as a commutative record representation, for some finite labelling Λ; The associated finite Boolean-feasibility system over unit, product, recovery, positivity, and state–effect equations is solvable. The Sep branch is exactly the case where the queried interface fits inside one shared classical record. The IJC branch is the precise failure of every faithful commuting continuation completion. IJC is not a synonym for “quantum.” It is an interface-relative algebraic verdict on whether a single Boolean-record completion exists for the tested word behavior. Quantum interfaces are an empirical instance of the IJC branch — but the IJC classification is independent of any specific physical realization, and there is no postulated identification of IJC with quantum mechanics anywhere in the paper. Surrounding results Refinement stability. Sep is monotone under coarsening of 𝒯; IJC can be revealed by refinement but cannot be erased by it. Branch verdicts are stated relative to both 𝒯 and the chosen operational resolution. Directed finite-test limits. A coherent projective family of finite-stage defenders is required for a global Sep limit; Sep at every finite stage alone is not sufficient. Compactness and bounded-atom hypotheses recover a projective-limit defender. ε-faithful defenders and finite-margin robustness. An approximate-defender theory with explicit operational-norm tolerances; the IJC margin m is preserved under finite-table perturbations of size below m/2. Continuation-word quotient algebra. Typed partial-product structure, incompatible-concatenation rule, contextual congruence, and the two-sided ideal property proved before the quotient is invoked. No-added-resource criterion. Operational sufficient tests and an explicit checklist precede the cost-monotonicity theorem; passive post-processing is the canonical witness. Worked models. A fully worked Sep defender (two classical bits); a Sep with positive interface cost (shared bus); a sequential IJC example (the explicit two-query table); a coupled-but-classical example (shared classical bath). A numerical ledger/floor case study and an explicit κΓ-calibration sensitivity table accompany the foundational sections. Process-category axiom block. A finite APF process-category axiom block; the word algebra realised as a typed path algebra equivalent to a free word algebra modulo contextual-equivalence relations; the defender restated as a partial Boolean-record functor. Trigger-condition discipline Every load-bearing structural theorem is paired with the explicit conditions under which it holds, named at point of use: Cost monotonicity — no-added-resource coarse-graining; Ledger additivity — factorized support; Independent counting — ledger-independent resolution; Robust-floor extension beyond explicitly finite protocols — compact robust-interface conditions R1–R4 (continuity, lower-semicontinuity, bounded atoms, projective coherence); ε-faithful defenders — explicit operational-norm tolerances and finite-margin half-perturbation regime. Each theorem-local condition is stated alongside an explicit failure-mode description and conservative-counting examples (shared coarse record, independent records, coupled discriminator). Reviewers see what the conclusion depends on without searching the paper. Bridges to standard frameworks Section 14 supplies a full finite-static bridge from APF to the published landscape of foundations programmes: a generalized-probabilistic-theory translation (effect algebras, ordered ∗-algebras, accessible GPT fragments, cone equivalence) and finite-static equivalence propositions for Spekkens-style noncontextuality, Fine-style joint distributions, joint-measurability and commuting-dilation criteria, sheaf-theoretic contextuality, sequential instruments and order algebras, the resource-theoretic hierarchy, epBA minimal Boolean embeddings, and the NPA hierarchy with finite-level commuting certificates. In the static finite case: Sep ⇔ commuting dilation ⇔ noncontextual model ⇔ faithful Fine joint distribution ⇔ global Bochner section. The Sep/IJC verdict is a single algebraic invariant that all of these standard frameworks witness from different sides. The novelty boundary is explicit: this paper’s contribution is the four-fold combination — continuation identity + positive cost + ledger bookkeeping + word quotienting — not the standard equivalence between finite Boolean models and commutative algebras. What this paper does not claim This paper does not derive Hilbert space, complex amplitudes, the Born rule, tensor products, density matrices, or the Standard Model. Promoting the commutative record representation to a complex-Hilbert-space construction requires record-positivity and composite-tomography assumptions that are isolated and named in §15. Those assumptions are deferred to a companion paper on quantum structure. This paper does not assert that quantum systems are the only IJC instance, or that classical systems are the only Sep instance. The classification is interface-relative. This paper does not commit to a substance-level reading of admissibility. The Admissible Possibility Space is set-theoretic; the framework’s commitment is structural, not metaphysical. Code and reproducibility The companion repository contains the v8.8 manuscript, the bundled apf/ codebase (every theorem traces to a named check function, including the Phase 21 inseparable-IJC bridge check check_T_inseparable_IJC), an interactive D3.js dependency DAG for the full Admissibility Physics Framework, and a one-click Colab walkthrough notebook. The repository and Colab notebook are being refreshed alongside this v8.8 deposit and will carry the updated v8.8 vocabulary plus the Phase 22 codebase rev (Sep/IJC representation theorem and continuation-word algebra checks) as those land. The links below are stable across the refresh; their content updates in place. GitHub repository — full codebase, manuscript .tex/.pdf, AI-onboarding bundle, audit-native context pack, theorem-by-theorem traceability. One-click Colab walkthrough — reproduces every Paper 1 result from first principles, no setup required, runs end-to-end in your browser. Interactive dependency DAG — explore the bank-registered theorem graph; click any node to surface its check function and dependency chain. Why this paper matters for the series Paper 1 is the bedrock of the Admissibility Physics Framework. v8.8 fixes that bedrock as a finite-interface algebraic theorem with an explicit primitive boundary — positive enforcement cost — and explicit theorem-local triggers, in language an external reviewer can audit without prior APF exposure. The Sep/IJC dichotomy, stated entirely within finite linear algebra over ℝ, is the regime classifier that downstream APF papers (Paper 2 on Standard Model gauge structure, Paper 5 on quantum reconstruction, Paper 6 on dynamics and gravity, Paper 8 on the Admissibility-Capacity Ledger) inherit and refine. The companion quantum-structure paper promotes the commutative record representation to a complex-Hilbert-space construction by adding record-positivity and composite-tomography hypotheses. The argument-first companion volume to this paper — the Paper 1 main body, currently v4.8 at 10.5281/zenodo.18604678 — is being brought up to v4.9 alignment with the v8.8 spec on a parallel track. About the APF series. The Admissibility Physics Framework is a ten-paper derivation chain plus core infrastructure, building physics from finite enforcement of distinction taken seriously as a constitutive commitment. Paper 1 (this paper) is the formal bedrock; downstream papers extend the chain through the Standard Model gauge group, fermion content, quantum formalism, Loren
This paper tells the quantum-mechanical account of Admissibility Physics in one place, from the first declared fact to the last committed record. From a single finite-capacity axiom (A1) plus one declared occupancy fact, the story runs through the Δ calculus that prices jointness (Δ = ε(J − R), derived, with two fences: Δ is not an information functional, and Δ > 0 is not a quantum criterion); the separable / irreducibly-joint fork, decided by an exact Boole-polytope feasibility test; the Boolean-defender bridge that forces noncommutativity where the global Boolean section fails; the gate pipeline that earns complex Hilbert space and the finite Born trace rule as endpoint theorems rather than postulates; the correlation ladder from product states to Tsirelson, with a native family ceiling 2√2·102/103 carried at honest reading grade with a live falsifier; resolution as a class transition, with the free/spend/permanent triad proved as a composition of banked theorems and measurement, decoherence, the Page curve, and the Diósi–Penrose scale as four worked instances of one mechanism; and the arrow of time as the one-way ratchet of the admissibility floor — confinement, the second law, and the quantum as one axis read at graded strengths. Every load-bearing step is anchored to a named, machine-verified check function in the APF theorem bank; the grade of each step is stated where the step is made; and everything the story does not claim is stated in one place. The Technical Supplement (deposited alongside and linked from this record) is self-contained for this paper: every definition, proof, and grade the story rests on, in one document, with a worked end-to-end instance and a reproducibility appendix pinning the theorem bank to a public repository and an archived release. About the series. Admissibility Physics is a structural realist reconstruction program built on a single commitment: any bounded region of the world can hold only finitely many distinctions. A principle of least enforcement cost then selects, among all the ways those distinctions might be arranged, the configurations a finite world can sustain. From this the content of physics is reconstructed rather than postulated: the Standard Model's gauge group and matter content, the Hilbert-space and Born-rule architecture of quantum mechanics, the geometry and field equations of general relativity, the four laws of thermodynamics, and the large-scale budget of cosmology emerge as different readings of one finite-capacity ledger. The unification is quantitative as well as conceptual: the framework's dimensionless constants appear not as free parameters but as counts read off that ledger — among them the dark-energy fraction ΩΛ = 42/61 and the weak mixing angle sin2θW = 3/13 — fixed, in the strongest cases, to within current measurement. The core derivation runs Papers 0–11; the extension papers carry the individual sectors. Each paper's main text and Technical Supplement is deposited separately; every quantitative claim traces to a named, executable check in the public theorem bank (over 3,800 bank-registered theorems, 48 quantitative predictions); and each result is carried at an explicit, graded level of confidence — from analytically forced to structurally grounded — with the measurements that would refute it named. Admissibility Physics is offered as a candidate, not a finished theory. If the program holds, the laws of physics could not have been otherwise: they are what a finite world permits. Website: admissibilityphysics.com. Engine and theorem bank: https://github.com/Ethan-Brooke/APF-Engine. Full series and latest deposits: Zenodo community admissibility_physics. The Engine — Admissibility Physics: An Executable Constraint Framework and Constants Map — DOI 10.5281/zenodo.18529115. Paper 0 — What Physics Permits: A Constraint-First Framework for Physics — DOI 10.5281/zenodo.18439523 Paper 1 — The Enforceability of Distinction — DOI 10.5281/zenodo.18439200 Paper 2 — Finite Admissibility and the Failure of Global Description — DOI 10.5281/zenodo.18439274 Paper 3 — Entropy, Time, and Accumulated Cost — DOI 10.5281/zenodo.18439363 Paper 4 — Admissibility Constraints and Structural Saturation — DOI 10.5281/zenodo.18439397 Paper 5 — Quantum Structure from Finite Enforceability — DOI 10.5281/zenodo.18439433 Paper 6 — Dynamics and Geometry as Optimal Admissible Reallocation — DOI 10.5281/zenodo.18439445 Paper 7 — A Minimal Quantum of Action from Finite Admissibility — DOI 10.5281/zenodo.18439513 Paper 8 — The Admissibility-Capacity Ledger — DOI 10.5281/zenodo.19721384 Paper 9 — The Geometric Substrate as Cost Structure of Comparison Continuations — DOI 10.5281/zenodo.20041675 Paper 10 — The Calculus of Finite Continuability — DOI 10.5281/zenodo.20041680 Paper 11 — Forced Universality from Capacity-Bounded Admissibility — DOI 10.5281/zenodo.20684198 Paper 12 — The Gauge Sector of Admissibility Physics — DOI 10.5281/zenodo.21212521 Paper 13 — The Minimal Admissibility Core — DOI 10.5281/zenodo.18361446 Paper 14 — The Quantum Story — DOI 10.5281/zenodo.21212188 Paper 16 — Markov Breakdown and the Hard Problems — DOI 10.5281/zenodo.20684207 Paper 18 — The Electroweak Sector as a Capacity Equilibrium — DOI 10.5281/zenodo.20684209 Paper 19 — Large-Angle CMB Suppression as a Finite-Continuability Constraint — DOI 10.5281/zenodo.21199218 Paper 20 — The Enforcement Crystal — DOI 10.5281/zenodo.18531732 Paper 23 — Magnetically Dark Winding-Class Quantum Processors — DOI 10.5281/zenodo.21199220 Paper 24 — The Recruitment-Radius Extension — Foundations — DOI 10.5281/zenodo.20684211 Paper 25 — The Recruitment-Radius Extension: Applications — DOI 10.5281/zenodo.21199222 Paper 28 — Absolute Mass Scales from Electroweak Capacity Saturation — DOI 10.5281/zenodo.20684215 Paper 29 — Plaquette Representation Dominance and Confinement — DOI 10.5281/zenodo.20684218 Paper 30 — A Tube Mechanism for the Lattice Mass Gap — DOI 10.5281/zenodo.20684220 Paper 31 — Osterwalder–Schrader Structure of Lattice Yang–Mills — DOI 10.5281/zenodo.20684222 Paper 32 — Quark Anchors as APF Trace Masses — DOI 10.5281/zenodo.21199226 Paper 33 — Trace-to-Scheme Export Architecture — DOI 10.5281/zenodo.20684224 Paper 35 — The Dark Sector as a Two-Role Capacity Decomposition — DOI 10.5281/zenodo.20684228 Paper 36 — The Missing Floor — DOI 10.5281/zenodo.21199228 Paper 37 — Collapse as Realignment — DOI 10.5281/zenodo.21199231 Paper 38 — Unification as Ledger — DOI 10.5281/zenodo.21199233 Paper 39 — Coherent Phase Regimes in the APF Interface Engine — DOI 10.5281/zenodo.21199235 Paper 40 — The Thermodynamics of Finite Distinction (Between Symmetry and the Void) — DOI 10.5281/zenodo.20684235 Paper 41 — The Horizon as a Continuation Ledger — DOI 10.5281/zenodo.20684241 Paper 42 — The Weak Mixing Angle Is Not Free — DOI 10.5281/zenodo.20684245 Paper 43 — The Neutrino Sector — DOI 10.5281/zenodo.21194191 Paper 44 — What a Finite Theory Owes the Continuum — DOI 10.5281/zenodo.21195874 Paper 45 — What the Vacuum Can Keep — DOI 10.5281/zenodo.21194195 Paper 46 — The Strong CP Angle Is Not Free — DOI 10.5281/zenodo.21194197 Paper 47 — The Confinement Antecedent Is Not Forced — DOI 10.5281/zenodo.21194199 Paper 48 — The Light-Bending Weight Is Not Free — DOI 10.5281/zenodo.21199627 Technical Supplements cross-link to their main paper via isSupplementTo and to the engine via isDocumentedBy. 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