This paper studies normalized probability states on pairs of parity-related null channels. The total weights of the two channels define an imbalance $\beta=w_{+}-w_{-}$, while the Bhattacharyya affinity between a state and its parity reverse defines a dimensionless clock functional $\chi$. For an arbitrary internal channel coordinate, the exact factorization\[\chi=\sqrt{1-\beta^{2}}\,\mathcal{A}(R_{+},R_{-})\]is established, where $R_{\pm}$ are the normalized internal profiles of the two channels and $\mathcal{A}(R_{+},R_{-})$ is their Bhattacharyya affinity. Consequently,\[\chi\leq\sqrt{1-\beta^{2}},\]with equality, away from the completely polarized endpoints, if and only if the two channels have the same normalized internal profile and differ only in their total weights. After identifying $\beta=v/c$, the saturation value is the special-relativistic proper-time factor. Rapidity becomes one half of the channel log-odds ratio, and reciprocal exponential reweighting furnishes a representation of collinear Lorentz boosts with additive rapidity and Einstein velocity composition. The analysis further shows that the ordinary directional first moment is profile-dependent, whereas the Lorentz clock factor is universal across the full common-profile saturation manifold. Rotational covariance constrains but does not uniquely determine the shared profile; maximum entropy supplies a canonical exponential family. Off saturation, the clock deficit is exactly controlled by the Hellinger separation of the two internal profiles and is quadratic for small profile mismatch. As an operational illustration, a pure $\ell=3$ directional moment is coupled to the gauge-invariant electric-octupole curvature\[\mathcal{E}_{ijk}=D_{\langle i}D_jE_{k\rangle}.\]In a linear-response completion, this produces a one-sided clock shift quadratic in $\mathcal{E}_{ijk}$ while leaving normalization and the directional first moment unchanged at leading order. The magnitude depends on an undetermined matter--directional Wilson coefficient, so the construction defines a controlled null test rather than a parameter-free forecast. The construction therefore distinguishes the empirically established Lorentz sector from a broader null-directional state space without, by itself, asserting that the additional states are physically realized.
This paper studies normalized probability states on pairs of parity-related null channels. The total weights of the two channels define an imbalance β=w+−w−, while the Bhattacharyya affinity between a state and its parity reverse defines a dimensionless clock functional χ. For an arbitrary internal channel coordinate, the exact factorization[\chi=\sqrt{1-\beta^{2}},\mathcal{A}(R_{+},R_{-})]is established, where R± are the normalized internal profiles of the two channels and A(R+,R−) is their Bhattacharyya affinity. Consequently,[\chi\leq\sqrt{1-\beta^{2}},]with equality, away from the completely polarized endpoints, if and only if the two channels have the same normalized internal profile and differ only in their total weights. After identifying β=v/c, the saturation value is the special-relativistic proper-time factor. Rapidity becomes one half of the channel log-odds ratio, and reciprocal exponential reweighting furnishes a representation of collinear Lorentz boosts with additive rapidity and Einstein velocity composition. The analysis further shows that the ordinary directional first moment is profile-dependent, whereas the Lorentz clock factor is universal across the full common-profile saturation manifold. Rotational covariance constrains but does not uniquely determine the shared profile; maximum entropy supplies a canonical exponential family. Off saturation, the clock deficit is exactly controlled by the Hellinger separation of the two internal profiles and is quadratic for small profile mismatch. As an operational illustration, a pure ℓ=3 directional moment is coupled to the gauge-invariant electric-octupole curvature[\mathcal{E}{ijk}=D{\langle i}D_jE_{k\rangle}.]In a linear-response completion, this produces a one-sided clock shift quadratic in Eijk while leaving normalization and the directional first moment unchanged at leading order. The magnitude depends on an undetermined matter--directional Wilson coefficient, so the construction defines a controlled null test rather than a parameter-free forecast. The construction therefore distinguishes the empirically established Lorentz sector from a broader null-directional state space without, by itself, asserting that the additional states are physically realized.
Null-Directional Kinematics (NDK) begins with a statistical ensemble of invariant-speed propagation directions rather than massive particle trajectories as primitive. Previous work showed that a small set of compositional axioms uniquely selects the Bhattacharyya coefficient as a statistical clock affinity and that its binary reversal sector reproduces the Lorentz clock factor. A companion closure result further showed that this statistical clock law, together with the inertial and null-direction structural inputs of NDK, closes the kinematic sector to the proper orthochronous Lorentz group. A subsequent construction connected the local geometry of the same statistical structure to nonrelativistic Schrödinger dynamics. The present paper asks a stronger question. Can the clock sector itself constrain a relativistic quantum dynamics? It is shown that the invariant proper occupation associated with the null-direction current is exactly the extensive form of the same Bhattacharyya overlap that defines the NDK clock. Within the local, derivative-free class of Lorentz-scalar transport actions that are extensive in occupation, this proper occupation fixes the free clock action up to the normalization of the inertial mass. Its exact quadratic auxiliary representation introduces a positive scalar density q that classically coincides with the proper occupation. The clock action then fixes the phase block of the local (q,S) geometry, while compatibility with the inherited density-phase two-form uniquely fixes the reciprocal Fisher density block within the stated local symmetry class. The natural complex coordinate Ψ=qeiS/κ and the linear Klein-Gordon equation are not assumed separately. Within that class, they follow from the inherited density–phase geometry. The quantum correction enters through the relationn=qmM,where M is the local invariant mass parameter, relating proper occupation to amplitude density while leaving the kinematic NDK clock ratio unchanged. Its nonrelativistic limit recovers the previously obtained Madelung-Fisher structure, while its first relativistic correction has the standard scalar mass-velocity coefficient. With standard external electromagnetic minimal coupling, the theory also reproduces the exact Klein-Gordon Coulomb spectrum. The Klein-Gordon equation, Fisher geometry, and auxiliary-field methods are established results and are not claimed as new. The proposed foundations result is the chain connecting a uniquely selected null-direction clock overlap to a relativistic scalar quantum dynamics. In this construction, the relativistic clock does more than share a statistical representation with quantum mechanics. It helps determine the form of the relativistic quantum transport theory.
A scalar clock rate is sought for comparing a probability distribution with its reversed counterpart. Four axioms are imposed: normalization at coincidence, symmetry under exchange, channelwise additivity, and multiplicative composition for statistically independent subchannels. These conditions uniquely determine the clock functional to be the Bhattacharyya coefficient. In the binary parity-opposed sector, the theorem immediately yields: dτ/dt=2w+w− If the signed first moment of the two-channel distribution is identified with the dimensionless drift β=v/c=w+−w−, then: dτ/dt=1−v2/c2 so the Lorentz factor appears as a corollary of the uniqueness theorem. The argument is self-contained and does not assume Minkowski geometry at the outset.
This paper asks what must be added to the one normalized relational interval and selected finite-dimensional carriers inherited from the first two Complementarity-First Time papers before those carriers function as quantum clocks, support endpoint comparisons, and share a universal matter coupling. The analysis first separates four parameter types: raw relation magnitude ρ, normalized relational interval ℓ, supplied local proper interval τ, and a shared comparison parameter s. In the declared order-unit placement class, the raw interval and raw generator compensators cancel exactly, leaving phase evolution per ℓ. For any one chosen parameter, a selected state-readout statistic is minimally phase-readable exactly when its grouped Bohr-frequency expansion has nonzero weight in at least one nonzero mode. A declared endpoint observable factors into signal transport and local clock calibration after the signal law, path, observers, and transitions are supplied. Complete labeled gap transfer is affine on each connected spectral graph. On a supplied common semiclassical path, a complete mass-energy operator is sufficient for common internal spectral scaling at the retained weak-field order; it does not derive universal trajectories or a unique matter action. An explicitly typed worldline-reparameterization-covariant, complement-even scalar model gives species-dependent clock rates, proving that the inherited one-time geometry and complement parity do not force universality. A two-level commuting perturbation has an internal affine blind spot, whereas an addressed three-level sector supports an anchor-free spectral-shape contrast. The prospective multiclock architecture therefore has rank three, with independent vector z = (Δ_ST, Δ_01, N_3)^T. It remains at STOP-SOURCE. No second physical time dimension, universal matter action, unique signal action, executed experiment, numerical redshift prediction, absolute second, numerical ℏ, objective chance law, total-carrier completeness, or thermodynamic arrow is claimed.
A companion letter established that four simple axioms on statistical comparison uniquely select the Bhattacharyya coefficient as a clock functional, and that its binary reversal sector yields the Lorentz time-dilation factor without presupposing a spacetime metric. The present paper asks whether that uniqueness structure can be extended from kinematics to dynamics. Five linked results are established. First, the continuum parity-reversal extension of the clock functional on the local celestial sphere S2 yields, in the weak-anisotropy dipole sector, a locally Minkowskian quadratic norm. Second, on the resulting spatial slices, a local action written in current form and constrained by probability conservation forces the transport current to be potential, j=ρ∇S/m. Third, an intrinsic second-order density geometry, taken as the local continuum descendant of the unique monotone Fisher-Rao metric identified by Čencov, fixes the density-gradient term uniquely up to overall scale as the Fisher information. Fourth, the resulting Euler-Lagrange system is exactly the Madelung form of quantum mechanics. Finally, by the standard substitution ψ=ρeiS/ℏ, the Schrödinger equation follows. The central result is not merely that Fisher information reproduces the Schrödinger equation, but that the Lorentz clock factor and the Schrödinger equation arise from sibling uniqueness structures on the same probability geometry: Bhattacharyya affinity and Fisher-Rao metric. The only undetermined dimensional constants are c and ℏ.
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.
This paper develops a framework-relative ontology of determinacy. Its core question is not what the world is made of, but what must minimally be accounted for if a stable deter- minate identity is internally self-representing. The primitive and functional core of Version 3.8 is retained: Being, the possibility of distinction, and domain-relative realizability define the sparse primitive basis; coherent self-representation then requires three non-equivalent func- tional questions—Support, Internal Encoding, and Correspondence. Explicit finite witness pairs establish uniform non-determination of each role from the other two in the unrestricted baseline class, while stronger canonical or equivariant subclasses remain open. Version 4.1.1 is a corrective consolidation of Version 4.1 rather than a new upstream on- tology step. It retains the Version 4.0 bridge revision and the Version 4.1 constructional layer. That layer adds the reverse evaluative direction Physics ⇒ Mathematics ⇒ Ontology and records where downstream structures constrain, but do not retroactively constitute, the ontol- ogy. Several previous bridge claims are weakened or rejected. Symmetry classification, tensor factorization, entanglement of a state, and path-dependent dynamics are kept as four distinct mathematical layers. A conditional expectation is treated as a localization map rather than as actualization; a branch instrument defines possible conditional events but does not select one realized branch; and a projection onto admissible spectral sectors is not called conservative dynamics without an independently derived dilation. The causal and propagation bridge is sharpened in two directions. First, a previous can- didate definition of a null relation by vanishing accumulated interval volume is proved degen- erate: every comparable pair admits a chain of links, so that criterion vanishes throughout the causal order. Link ensembles remain useful near-null approximants, but the null sector is therefore a limiting or measure-relative object rather than a pair-local primitive in the de- clared causal-set model. Second, the long-range free-chain calculation at 𝛼 = 2 is separated into two non-commuting limits. The infinite-range spectral symbol has maximal group veloc- ity 𝜋𝐽𝑎, whereas the envelope of finite-cutoff maxima tends to 2𝐽𝑎 Si(𝜋). These are spectral group-velocity statements; a causal signal cone still requires separate Lieb–Robinson-type hy- potheses. Isotropy and a unique physical Lorentz cone remain open. The defect bridge is updated with the completed 𝐾 = 512 data layer of the frozen ratio campaign. All 903 prescribed values of 𝜃 are present with eight branches each; 897 aggregate points are admissible and six lie in a short rejected interval. Neither √ 2 nor 𝜙 is a local or global minimum. The original preregistered primary ratio test nevertheless remains non- evaluable because its 𝐾 = 64 training gate is universally degenerate. Thus 𝐾 = 512 completes a characterization block; it does not retrospectively repair the primary estimator. Provenance registration and cross-platform verification remain pending. Version 4.0 turned Test 28A into an explicit reconstruction audit, and Version 4.1 retains that audit unchanged. For the standard three one-body extractions on ℋ = (C 3 ) ⊗3 , the combined linear map has exact rank 25 and kernel dimension 704. The hidden space splits as 192 two-body plus 512 fully traceless three-body directions; even the full 𝑆3-invariant class retains an exact 156-dimensional hidden sector. The preregistered continuation identifies an exact energy-selected fixed point, the totally antisymmetric state |𝜀⟩, while holonomy and variance objectives yield set-valued argmin correspondences. The historical non-triviality gate is therefore preserved but shown not to be a universal criterion, and no retroactive 𝑉28.4 verdict is declared. A prospectively frozen two-local QPH family provides an exact computer-assisted existence witness: its cycle-phase differential has rank 704/704 on the marginal kernel for the primary family and two preregistered replications, so marginals plus relational cycle phases are locally in- formationally complete in the fixed regularity stratum. Canonical modular-mismatch selectors are then audited rather than assumed. Identical additive qutrit references and every single pure mismatch under exact 𝑆3 protection obey strict capacity/rank no-go bounds. On the protected 156-dimensional domain a physically generated mixed mismatch has sufficient raw capacity but standard phase-only QPH remains incomplete. Exact cumulant analysis explains why order three is the natural next layer: the 𝑆3-hidden sector decomposes as 156 = 36 + 120, and the third connected cumulant is a complete affine coordinate on fibres of fixed proper marginals. A historical preregistered cubic-moment completion is retained only as a delicate T-NUM wit- ness, not as a theorem premise. This is the declared stopping point of the arity/cubic line: selecting one specific cubic observable would require a new independent physical law. Test 28B remains suspended. Version 4.0 added one further macro-level correction, retained in Version 4.1. Causal and conformal structure, transport geometry, stable matter sectors, and quantum correlations are treated as parallel downstream representations of one substrate model, not as a serial matter– space–time chain. A proposed identification of the matter admissibility projector Πadm with spacetime volume is rejected in its literal form: the projector is matter-branch specific, need not be nested under generative updates, and has no canonical finite trace in a type-III con- tinuum. In its place a conditional causal–transport conformal closure theorem is proved. If a causal reconstruction supplies a Lorentzian conformal class and an independent transport limit supplies a positive spatial metric on the same spacelike distribution, the conformal factor is fixed pointwise exactly when the two induced spatial metrics are conformally compatible. A scale-free generalized-eigenvalue defect provides a direct falsifier. The corresponding clock– cone–matter gate is also corrected by conformal weight: frequencies and dimensionful spectral scales have lapse weight one under a pure lapse deformation, whereas spectral ratios, localiza- tion fractions, and dimensionless stability-window coordinates have weight zero. A Tolman– thermal-time comparison is retained only as a relative lapse-profile gate; absolute time units remain open. Version 4.1 introduced a controlled common-substrate realization. Version 4.1.1 corrects its scope: parity alone excludes only an odd or linear mass/source identification with the same signed coordinate; an even law such as 𝑀 ∝ 𝑑 2 is not excluded by parity. On the declared graph Klein–Gordon sector, charge conjugation reverses Noether/gauge-current orientation while leaving mode energy and localization intensity unchanged. A grounded-graph Green- kernel theorem fixes the spatial uniformity of the cross-term sign, while the Coulomb sign itself is conditional on realizing a non-dynamical 𝐴0-type constraint sector; the same kernel in a dynamical scalar sector gives the opposite sign. A separate no-go proves that matter Dirichlet energy alone cannot select a nontrivial interior geometry. A state-selected positive geometry Hessian yields the conditional linear response 𝑦 = 𝐾 −1 𝐸 𝜏 , so weak 𝐶-even stress sources add while 𝐶-odd currents can cancel. The principal new bridge is a conditional stationary common causal–transport–matter con- struction. If one protected center supplies a common base, the transport branch supplies a positive spatial metric ℎ, and several modular/physical clocks agree on one relative lapse 𝑁, then 𝑔CTM = −𝑐 2 *𝑁2𝑑𝑇2 + ℎ has one timelike sign, predicts the same null front for all canonically normalized matter sectors, and fixes its stationary volume element without an additional scalar measure. Different masses and opposite 𝑈(1) representation weights remain lower-order data and do not alter the characteristic cone. The independently reconstructed HSMI conformal class remains a held-out test rather than an input to this construction. A second independent Version 4.1 result supplies a conditional physical law selecting com- plex rank two. A nonzero-Chern line cannot be closed under conjugation at rank one on 𝑆 2 , whereas 𝐿 ⊕ 𝐿¯ is the minimal conjugation-closed carrier; independently, ClC 2 ≃ 𝑀2(C) has minimal irreducible complex module dimension two. The existing odd-Chern defect-link con struction realizes this pair. If its Clifford algebra is physically admissible, the local carrier has 𝑀2(C) qubit kinematics. This is not an assertion that opposite electric charges form a coherent qubit, nor that every particle is a qubit. The resulting Version 4.1.1 is a constrained foundational proposal with a frozen primitive core, stronger positive and negative bridge results, a first common stationary geometry can- didate, an independently motivated minimal rank-two carrier, and a reproducibility-oriented ledger for every step from ontological determinacy toward physics. It also makes the external spin-sign, soft-graviton, equivalence-principle, and mass-budget inputs explicit without treat- ing them as evidence for the ontology.