Enlarging observation space extends the available event families, but its effects on correlation bounds and dynamical spectra depend on different assumptions. Within the quantum representation of the physical--observation dual-axis structure (PODA), the real Gram geometry of the Born pairing yields the dimension-independent CHSH bound \(2\sqrt{2}\) and a two-dimensional saturating configuration. For a fixed \(d\)-dimensional maximally entangled state, arbitrary-rank binary projections give an even--odd dependence, whereas rank-one events yield \(2+(4\sqrt{2}-4)/d\). These bounds admit an experimental comparison at fixed preparation. We then introduce an independent joint field with a quadratic action and positive gradient energy along observation space. On the full \(\mathbb{CP}^1\) with its standard projective metric and Hopf connection, we derive the covariant eigenvalues and degeneracies for integer phase weight. In the minimal nonzero sector \(\lvert q \rvert=1\), the mass levels obey \(M_n^2=m_X^2+[2n(n+2)+1]m_S^2\), where \(m_X\) is an independent local mass parameter and \(m_S\) is determined by the lowest observation eigenvalue and the gradient coefficient. The two lowest distinct mass levels uniquely reconstruct both scales, giving \(m_X^2/m_S^2=(7-r)/(r-1)\) with \(r=M_1^2/M_0^2\in(1,7]\). Higher levels satisfy relations such as \(3M_2^2=8M_1^2-5M_0^2\) without further parameters. Exactly equal mass spacings occur if and only if \(m_X=m_S\). These spectral results depend on the specified action, full observation geometry, and phase representation. The scale ratio remains a dynamical parameter, while the rest-energy relation retains the form \(E_{0,n}=M_nc^2\).
Enlarging observation space extends the available event families, but its effects on correlation bounds and dynamical spectra depend on different assumptions. Within the quantum representation of the physical--observation dual-axis structure (PODA), the real Gram geometry of the Born pairing yields the dimension-independent CHSH bound (2\sqrt{2}) and a two-dimensional saturating configuration. For a fixed (d)-dimensional maximally entangled state, arbitrary-rank binary projections give an even--odd dependence, whereas rank-one events yield (2+(4\sqrt{2}-4)/d). These bounds admit an experimental comparison at fixed preparation. We then introduce an independent joint field with a quadratic action and positive gradient energy along observation space. On the full (\mathbb{CP}^1) with its standard projective metric and Hopf connection, we derive the covariant eigenvalues and degeneracies for integer phase weight. In the minimal nonzero sector (\lvert q \rvert=1), the mass levels obey (M_n^2=m_X^2+[2n(n+2)+1]m_S^2), where (m_X) is an independent local mass parameter and (m_S) is determined by the lowest observation eigenvalue and the gradient coefficient. The two lowest distinct mass levels uniquely reconstruct both scales, giving (m_X^2/m_S^2=(7-r)/(r-1)) with (r=M_1^2/M_0^2\in(1,7]). Higher levels satisfy relations such as (3M_2^2=8M_1^2-5M_0^2) without further parameters. Exactly equal mass spacings occur if and only if (m_X=m_S). These spectral results depend on the specified action, full observation geometry, and phase representation. The scale ratio remains a dynamical parameter, while the rest-energy relation retains the form (E_{0,n}=M_nc^2).
The physical--observation dual-axis (PODA) structure treats physical and observation configurations as distinct arguments of joint dynamics. We study the analytic conditions under which this structure admits rigorous path integrals and finite complex measures for sequential quantum histories. On compact smooth joint geometry, a specified covariant Hamiltonian has a Feynman--Kac representation. A uniform complex-time product estimate permits simultaneous removal of Abel damping and time slicing at arbitrary relative speeds. For fixed finite bounded-readout protocols, class operators converge strongly and the trace-class branch-pair operators converge in trace norm. Finite instruments with complete Kraus families also yield trace-norm convergence of their record states. Energy-form regularity gives quantitative error bounds. A completed cylinder-test space carries a consistent, strongly positive history functional. The stronger finite-measure question is controlled by covariant transport: calibrated rank-one readouts on a finite-dimensional fibre satisfy the sharp estimate ∥μγ∥TV≤e2cγ, where cγ is the off-diagonal covariant variation. In the specified common-action class, conserved energy and smooth, globally labelled rank-one readouts imply this regularity when the kinetic geometry is complete and the effective potential is bounded below, or when the chosen energy confines the trajectory to a compact set. In infinite dimension, finitely many readout sectors allow unbounded internal energies and bounded transitions. The construction extends to countable outcomes when the operator of block norms is bounded on ℓ2; nonatomic readouts admit a double-Dyson construction when an absolute integral kernel is bounded on L2. In these Dyson constructions, pure preparations with rank-one discrete readouts or scalar continuous fibres have exact variation ∥eTBv∥22. Whole-sector and spatial-domain cutoffs converge in variation, whereas nonatomic readout quantization converges weakly and remains separated in variation norm. Mixed curvature survives in the covariant constructions, and invariant quantum embeddings recover physical-sector histories. Unrestricted basis switching, a bounded-Hamiltonian countable example and free-particle position histories establish the limits of finite-measure existence; exact physical recovery transfers the last obstruction to the joint theory. Analytic two-axis models exhibit the resulting dynamics and verify the distinct convergence statements. A proposed two-mode trapped-ion realization links the magnetic oscillator's transport phase to independent spectral and driven-response tests. Exact finite-time Gaussian evolution shows that nearly complete wave-packet return can coexist with a substantial phase correction. The protocol therefore infers the coupling from finite-time interference and predicts the other observables without refitting, with calibration and uncertainty requirements explicit. These are model predictions, not measured data. Each construction is tied to its stated geometric, dynamical and readout hypotheses.
For finite-dimensional binary systems and their bipartite composites, this paper establishes a conditional representation chain from a physical–observation dual-axis structure to quantum Bell correlations. An observational fact is jointly generated by a complete physical state and a complete observation state. Additivity and associativity of coherent responses, two-sided cancellation, and a full circle group of scalar phase transport select C among finite-dimensional unital associative real response algebras. Reversible mixing then determines a Hermitian metric up to an overall positive factor. Normalized two-channel responses form S3, and quotienting by their common phase yields the rank-one observation-event space and, with an outcome label, the pure observation fiber S3/U(1)=CP1≃S2. The universal bilinear composite of two independent systems is C4. Probabilities are first assigned within complete measurement contexts. Events with the same response filter are operationally interchangeable, so the local assignments glue along the observation axis into a positive orthogonally additive measure on the full projection lattice. Gleason's theorem then gives the unique trace representation p(P∣x)=Tr(ρxP), while composite marginal consistency returns the trace pairing to each local binary fiber. Every two-qubit correlation therefore has the universal form Eρ(a,b)=aTTρb. Joint rotational invariance and perfect equal-axis anticorrelation uniquely fix Tρ=−I3, producing the singlet cosine law E=−a⋅b=−cosθ. Finally, the local commutators in the squared CHSH operator confine all realizable correlations to 22. Fine-measurement completeness supplies saturating directions, while sharp preparation and reversible covariance make the singlet accessible. Thus CP1, the Born pairing, the cosine correlation law, and the Tsirelson bound arise within one chain linking phase structure, independent composition, and cross-context probability consistency.
We calculate the formation and measurement of entanglement from an initially independent two-source preparation in physical--observation dual-axis (PODA) geometry. A common matter--field action determines the linear response kernel, and a Dirac reduction fixes the leading spin coupling. Excitation and return of the field generate an ordered exchange operator that creates a second joint amplitude. The component of state motion normal to the product-state manifold determines the initial loss of local purity. In a finite cavity, a polynomial embedding bounds the difference between the full evolution and an effective exchange description over the long preparation interval, with all physical mode families of the chosen background included. At g=8×10−4, the total-variation error in the joint readout distribution is below 7.56×10−5, and an independent bound establishes source entanglement before measurement. We then derive the effects of local transport, unread observation degrees of freedom, and finite Gaussian pointers on the correlations and conditional states. Exact reduction conditions determine when spin variables alone suffice for later predictions. A rotor model and a worked source-to-detector calculation connect these results to independent calibration. The quantum representation is conditional on stated operational assumptions. The numerical bounds apply to the linear-source Hamiltonian defined here; their extension to a microscopic apparatus requires estimates for the additional orbital and nonlinear field terms.
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.
A quantum phase reference is part of the state from which an observation record emerges. Within the physical-observation dual-axis structure, we describe such a reference by a compact quantum coordinate coupled to a finite-dimensional physical system. A covariant rotor action gives the joint evolution and expresses the readout's phase acceleration through ordered correlations and a covariant force. In a qubit model whose dynamics preserve two adjacent observation momenta, the first harmonic depends on four complex correlations. For fixed nondegenerate parameters, its full complex history requires eight real state variables. For phase increments alone, a common amplitude and phase factor can be removed, leaving three complex correlation ratios. We prove that these six real variables are locally minimal at generic states for every fixed nondegenerate parameter choice. Six suitably timed phase estimates and an initial reference recover the ratios by solving real linear equations; an error bound relates uncertainty in those estimates to the subsequent phase prediction. The role of the retained correlations is illustrated by preparations with equal marginals and equal initial phase, rate, and acceleration, but different future phases. We also describe how the state continues through a zero of the first harmonic and how finite integration windows affect the available information. The resulting theory connects a joint quantum action to phase prediction from finite records, subject to independent physical characterization of the phase inertia and coupling.
Physical--observation dual-axis (PODA) Theory holds that physical reality is realized jointly through physical state and observation state. A change of observation state can therefore change the realized physical fact even when the physical state is held fixed. We develop the phase law of this PODA structure. A coherent readout of the joint observation law gives a smooth scalar complex response g. Wherever this response is nonzero, differentiating its polar decomposition gives the observed phase rate and the one-form Ag=Im(dg/g). A transported phase with the same increments differs from the observed phase by a constant reference factor. We derive its differential equation and unique initial-value solution, and show that the equation is horizontal lifting for the unique response-compatible connection on the trivial principal U(1) phase bundle. We call the resulting equation c∗Ωg=0, the Phase Transport Fundamental Equation (PTFE), the first fundamental equation of observation space in PODA Theory. Its group-valued, covariant-derivative and continuous real-phase forms express one transport law: a path and an initial phase determine a unique horizontal lift. Integration gives finite phase comparison, composition along successive paths, and inverse transport between two observation states of the same physical source. Positive amplitude rescaling leaves the law unchanged; synchronized changes of response and fiber coordinates preserve its covariant form. The induced connection is flat and has trivial circular holonomy, although a continuously unwrapped phase may wind. We distinguish the full response G0, the phase-transport structure G1, and the task quotient G2. At the last level, maximal invariants describe the quotient, and a target can be recovered exactly when it is constant on each fiber of the quotient observation. Phase transport and the information retained after a task reduction thus follow from the same joint observation law.