added; unclosed consequences remain explicitly OPEN. This paper develops the interaction, phase-transport and identification components of the All-Domain Framework (ADF), which seeks a common account of microscopic and macroscopic formation and maintenance. The framework takes multi-frequency immaterial angular-momentum dynamic structures as the sole fundamental constituents of existence and states: dynamics is existence; without dynamics, there is no existence. Within a declared lowest-order local two-axis second-order tensor class, seven independent operator forms are obtained for non-collinear axes. Their unital real associative algebra is the full real three-by-three matrix algebra; this is not a completeness theorem for quantum states. Integrable real linear evolution has positive fundamental-matrix determinant at finite times, a constraint distinct from algebraic generation and from time irreversibility. The ideal phase–vector interface gives an inverse geometry, a chord-factorization obstruction and a complete conditional circle-map family. On a fixed-axis radial branch, an explicit formation-phase correction transports supplied initial phase data while retaining its winding. Oriented finite cross geometry identifies sampled relative phases; finite samples do not identify an unrestricted phase function or its winding. Tensor-side descriptors are complete in the declared class, but the frequency-side audit does not supply the matching layer actions. The physical layer–form mapping therefore remains OPEN / STOP. Coefficient selection, actual initial preparation, global probability weights, experimental correlations and single-record generation are not derived. The front comparison states precisely which answers are mathematical, which are conditional and which remain research objectives. Keywords: All-Domain Framework; quantum-generation interface; multi-frequency immaterial angular-momentum dynamic structures; helical self-organization; operator-form classification; phase transport; identifiability Preprint release: v1.0, 11 September 2026. This release is derived from research candidate v14. It aligns the series name with All-Domain Framework (ADF), removes the framework name from the title, and states the series-wide no-phenomenon-specific-patching rule. It does not upgrade any mathematical, physical or experimental result; all declared OPEN interfaces remain OPEN
added; unclosed consequences remain explicitly OPEN. This paper develops the interaction, phase-transport and identification components of the All-Domain Framework (ADF), which seeks a common account of microscopic and macroscopic formation and maintenance. The framework takes multi-frequency immaterial angular-momentum dynamic structures as the sole fundamental constituents of existence and states: dynamics is existence; without dynamics, there is no existence. Within a declared lowest-order local two-axis second-order tensor class, seven independent operator forms are obtained for non-collinear axes. Their unital real associative algebra is the full real three-by-three matrix algebra; this is not a completeness theorem for quantum states. Integrable real linear evolution has positive fundamental-matrix determinant at finite times, a constraint distinct from algebraic generation and from time irreversibility. The ideal phase–vector interface gives an inverse geometry, a chord-factorization obstruction and a complete conditional circle-map family. On a fixed-axis radial branch, an explicit formation-phase correction transports supplied initial phase data while retaining its winding. Oriented finite cross geometry identifies sampled relative phases; finite samples do not identify an unrestricted phase function or its winding. Tensor-side descriptors are complete in the declared class, but the frequency-side audit does not supply the matching layer actions. The physical layer–form mapping therefore remains OPEN / STOP. Coefficient selection, actual initial preparation, global probability weights, experimental correlations and single-record generation are not derived. The front comparison states precisely which answers are mathematical, which are conditional and which remain research objectives. Keywords: All-Domain Framework; quantum-generation interface; multi-frequency immaterial angular-momentum dynamic structures; helical self-organization; operator-form classification; phase transport; identifiability Preprint release: v1.0, 11 September 2026. This release is derived from research candidate v14. It aligns the series name with All-Domain Framework (ADF), removes the framework name from the title, and states the series-wide no-phenomenon-specific-patching rule. It does not upgrade any mathematical, physical or experimental result; all declared OPEN interfaces remain OPEN
Macroscopic and microscopic organization are studied here as parts of one proposed relational formation– maintenance framework, the All-Domain Framework (ADF). Its foundational claim is that multi-frequency immaterial angular-momentum dynamic structures are the sole fundamental constituents of existence: dynamics is existence; without dynamics, there is no existence. Across the series, no phenomenon-specific physical postulate, outcome-specific operator or post hoc parameter is added; unclosed consequences remain explicitly OPEN. This paper establishes mathematical consequences of specified premises, not empirical confirmation of that ontology. Theorem C gives a bijection between finite-common-return frequency classes modulo common positive scaling and primitive positive integer representatives; the quotient is at most countable while within-class scales remain continuous. On specified closed orbits, invariant measures and Fourier projection statistics are derived. Constant-rate phase–vector compatibility is separated from mere phase recurrence. Conditional formation transport in the companion paper relates the remaining circle-map function to supplied initial data. Synchronized maintained-circle families obey an exact positive-semidefinite, unit-diagonal, rank-at-most-two Gram characterization. These results connect recurrence, maintained motion and geometric statistics without identifying a tensor count with quantum states or an orbit measure with Born probabilities. A front-loaded comparison distinguishes established quantum calculations from the generative questions addressed here. Global preparation weights, two-body experimental correlations, single-record generation and the physical frequency-layer-to-form mapping remain open.
Starting from the single-axis relational update established in the preceding papers, we analyze a fixed-axis, radial-coefficient branch in which formation, sustained maintenance, and phase transport are generated by the same law. We derive memberwise and family recovery bounds, sufficient times to prescribed tolerances, phase-matched maintenance trajectories, and initial-condition sensitivity of the maintained phase. The fixed-axis spectrum yields exact reconstruction identities for the transverse radial rate, circulation magnitude, and axial excess rate, together with a maintenance-condition factorization. For coupled source-detector blocks, an invariant linear graph d=Kx requires the Riccati-type compatibility condition C+EK-KA-KBK=0; this condition is not a unique constitutive selector. A residual projection criterion characterizes preservation of the inherited single-axis three-channel operator class, while the nonlinear graph-error equation gives sufficient local contraction and recovery bounds. A target-adaptive kernel condition separates full-state reconstruction from recovery of a declared observable. The appendices delimit the observation interface: rotation-representation compatibility does not identify physical spin, and symmetric second-moment shape does not uniquely determine polarization. EC-IV-01 is retained separately as a fixed-setting held-out experimental protocol. No laboratory measurement, unique constitutive law, or completed quantum measurement derivation is claimed.
When the conditions under which a record is formed remain fixed, a physical theory may represent that record completely by a unary law of the state observed. This paper asks what structure is forced when the same independently certified physical identity gives different complete records under different formation conditions. The initial datum distinguishes two typed empirical roles, while the state identity of the formation side is derived from its complete behavior. Failure of factorization through the first projection proves that the formation-side role cannot be deleted. The complete row and column behaviors of the primitive law then determine two kernel pairs and their coequalizers. Unique descent retains every observable distinction while removing every redundant label, and the resulting realization is terminal among all exact role-preserving realizations. The product pairs the reduced roles; a pullback extracts the combinations on which records can actually be formed. Only after these steps does exponential transposition identify each surviving formation class with one complete response map. The resulting bijection supplies the state criterion and establishes the observation-state object. Compatibility with the antecedent theory preserves the independently given physical state space and yields the physical–observation law in which a fact is generated jointly by the state observed and the state of observation. The same construction gives exact criteria for conditional, label-free, and joint identifiability and selects the same-source pullback on which transport may subsequently be defined. A nonzero coherent response on that pullback yields the phase-transport fundamental equation, the first fundamental equation written directly on observation space. Quantum theory subsequently enriches this canonical interface. Coherent response and full circular phase select the complex field; reversible binary mixing yields CP1 ∼= S2 ; independent composition creates the four-dimensional event domain; response-event identity glues probabilities across complete contexts; and Gleason representation with marginal consistency gives the Born trace pairing. The two-qubit correlation tensor then yields the singlet cosine law, the CHSH inequality, the Tsirelson bound, and the Horodecki spectral criterion. Bell violation is thereby identified with the absence of a positive global answer section across incompatible observation states. The quantum formalism is thereby closed into one architecture: complex state geometry describes coherent physical possibility, complete observation states specify the ways in which facts may form, the Born interface joins them, and quantum instruments carry a realized fact into subsequent dynamics. Fixed observation sections recover classical unary laws, while smooth equivariant enrichment lifts relativistic frame geometry and internal gauge geometry to the same joint domain. The central result is ontological. Physobser Theory admits observation into objective physical reality not as a mind, an external readout, or a perturbation, but as an irreducible state-bearing axis; a fact is generated at its lawful interface with physical state. Category theory is the proof instrument. The discovered physical content is the dual-axis constitution of facts, which supplies a common foundation for quantum, classical, relativistic, and gauge physics.
How a finite real cycle acquires a complex structure and a rule for composition Quantum theory uses a common imaginary unit when individual systems become a composite. This framework constructs that compatibility from finite real source histories. On the complete contrast sector of a three-cycle, the forward and backward comparisons give J=3P−P2,J2=−I The complex structure appears after the real comparison is specified. At a qualified unmarked boundary, opposite reference shifts select QAB=2I−JA⊗JB The range is the balanced complex tensor product. Its many-source composition is associative. A finite controlled contact supplies a nonfactorizable complex-linear operation, and exact residual bounds quantify departure from the selected sector. A source model that produces the contact A reversible controller compares operation orders on 24 proper signed frames. Its ordered, arrow-insensitive axis detector produces the retained cyclic response. The contact algebra gives [[G,G],[G,G]][G,G]≅C3. The construction prints the controller, scratch cleanup, full contrast probes, admissible operations and refusal states. A triaxial real response tensor realizes the detector and provides an explicit map between material contrasts and event-count contrasts. A record must survive the test of future operations A present tensor can hide a material marker that a later operation reveals. The framework derives independent force-gradient calibration and a delayed odd-field release. With a nonzero release field, energy and vector torque reconstruct the entire marker change: DU=−Δp⋅E,Dτ=Δp×E.Δp=∣E∣2E×Dτ−DUE The test includes shared calibration errors, stray fields, finite release pulses, marker relaxation and a complete operation inventory. It distinguishes detection of a retained record from agreement with its frozen prediction and from discrimination between physical theories. From proved structure to existing experiments The paper adds a conditional probability-correspondence theorem for the balanced composite, examines independent-source quantum-network experiments, and reproduces all eight measured rows of a public torque dataset. These supply composition benchmarks and demonstrated apparatus capabilities. Their published significances are kept attached to the hypotheses they actually test. The calibrated material release law is also a conventional electrostatic prediction; it is not presented as a QTT-exclusive signal. One material coefficient, two readouts A new archival calculation transfers a published same-particle libration calibration to a separate precession measurement. In the qualified rotor regime, the cross-readout relation is Ξ=ωlib2EpωsΩElcosϕl=alibaprec. An unchanged dipole-to-inertia coefficient gives unity. The calculation yields a slope of 2.909 from the libration calibration, compared with the published precession slope of 2.950, a central difference of 1.4135%. This shared electrostatic consistency relation eliminates the separate dipole and inertia; unequal field gains, dynamical corrections and shared uncertainty remain explicit. Reversal and long-term dipole-stability observations identify complementary laboratory capabilities. The complete matched-tensor delayed-release test remains an experimental target. When do two descriptions differ in any future experiment? The finite-reference extension follows a controlled source contact through shared references, fresh references, inverse operations, intermediate measurements and feedback. Its complete order-three response is gn=∑k=02pkζkn,ζ=e2πi/3,g3=1,g2=g1. A finite observable-span algorithm tests two fully specified models across every permitted future word. It either certifies agreement or returns the preparation, operation sequence, terminal outcome and exact probability difference. In the populated compatible-instrument benchmark, the complete quantum representation gives the same records without an additional target fit. A source-selected physical coupling or preparation remains a distinct route to predictive value. The framework includes nine vector figures, the retained source proofs and countermodels, complete reference-history benchmarks, calibration and vector-reconstruction theorems, runnable exact and numerical checks, published-input receipts, a data assessment and an experimental packet specification. The introduction defines the QTT terms needed to follow the construction without prior familiarity with the corpus. The probability-interface theorem, finite-memory bounds and covariance-weighted witness specify which further laboratory comparisons can distinguish physical models. Source anchors QTT Main Book, especially pp. 56, 248-250, 352 and 827-829; Artian's Completed-Event Four-Capacity Theorem; Frame-cycle derivation map, completed-event lexicon, and Artian's Universe and laboratory readout. Stable citation: 10.5281/zenodo.22882084.
We develop a mathematical framework in which physical description is organized through predictively sufficient closure of richer realization spaces. A closure map identifies realizations that are equivalent relative to specified observables, while dynamic sufficiency requires the induced reduced state to support autonomous evolution. This yields a precise formulation of the principle that law can close before realization does: a law-level state may be fully predictive within a declared domain even when multiple underlying realizations remain possible. We distinguish quotient closure, realization fibers, idempotent closure maps, refinement, partial closure, closure memory, obstruction, and saturation of multiple closure constraints. These structures are then tested against explicit physical examples involving compact phase winding, topological defects, gauge compensation, exchange statistics, spinorial closure, quantum measurement channels, and effective propagation geometry. The analysis exposes several limitations of a motivating one-scalar coherence construction. In particular, a smooth exact phase connection has vanishing local curvature; a relative phase does not generate orthogonality or Pauli exclusion; winding alone does not supply spin- or the observed electric-charge spectrum; and a single scalar gradient cannot define a generic nondegenerate four-dimensional metric. Rather than abandoning the coherence hypothesis, these obstructions motivate a minimally enriched effective architecture involving amplitude/phase, gauge connection, spinor, and geometric frame sectors. We distinguish this effective reconstruction from the stronger hypothesis that all such structures descend from a common parent coherence object. The resulting framework does not claim a completed derivation of the Standard Model, general relativity, or quantum measurement. Instead, it identifies the mathematical and physical gates such derivations must satisfy and isolates noncommuting saturation, closure obstruction, representation reconstruction, stable particle sectors, and universal emergent geometry as principal theorem and physics programs. Keywords: closure mathematics; predictive sufficiency; quotient dynamics; gauge theory; topological defects; particle identity; quantum foundations; emergent geometry; noncommuting closure; unification.
Subjectivity-Intersection Mathematics asks whether structures treated as objective can be composed as stable consequences of lawful intersections among differentiated standpoint roles rather than supplied as standpoint-free primitives. We study one restricted homogeneous operator representation of that principle. Objective mathematics is used here as a representational instrument downstream of registered standpoint-relative intersections; no formal datum or objective reconstruction is identified with subjectivity or Subjectivity Intersection itself, and no such representation is promoted into an exhaustive objectification of its generative principle. Let H be a complex Hilbert space, e in H a unit vector, and R in U(H tensor H) a checked-braid solution satisfying R(e tensor e) = lambda(e tensor e), with |lambda| = 1. At every finite order, the canonical braid-orbit isometries have constant ordered overlaps equal to the phase-normalized right endpoint slice; leg reversal gives the left family. Within the declared Subjectivity-Intersection representation, we call this neutral canonical overlap property all-orders intersection consistency; the interpretive name adds no mathematical conclusion. The standard constant-overlap Gram criterion forces both phase-normalized slices to be positive contractions, while the pointing condition makes both fix e. No finite-dimensionality, separability, involutivity, self-adjointness, or trace assumption is used. Together with separately proved pointed-unitary covariance and compatibility with the declared shuffled tensor law, this verifies the stipulated technical compositional-objectivity criterion in this representation class. The sole general-mathematics novelty candidate is the complete implication from a pointed unitary braid to canonical all-orders overlaps and bilateral endpoint positivity. Its Gram, localization, compression, covariance, and tensor ingredients are cited rather than claimed; historical priority remains open pending claim-level search and specialist review. Separately, complete population response laws on finite registered sets determine source-relative quotients, and a supplied pair kernel descends exactly under fibre constancy. These facts specify an ideal-population prospective certificate contract, not recovery of its inputs or a finite-sample natural-identification theorem. The atomic, NMR, and FPR1 records are component-level compatibility checks. The cryo-EM check yields a byte-identical canonicalized candidate-matrix artifact over heterogeneous class averages, while its exact-path baseline remains incomplete. No natural certificate is accepted and no result identifies the formal variables with phenomenal subjectivity.