A coarse description of a conservative amplitude dynamics is not determined by an energy compression alone. We develop a conditional account of refinement, unresolved contrasts, and effective preparation for finite weighted graphs and a supplied one-dimensional local chain. Norm conservation, covariance, and additive refinement permit different site measures. Repeated-value refinement closes exactly only when the measure-normalized external rates agree within each coarse block. When they do not, eliminating the complementary coordinates yields coherent memory and an initial-data forcing term, not stochastic noise or irreversible decoherence. A three-site example separates return to the coarse subspace at one time from autonomous coarse evolution. On a uniform pair-refined cycle, a contrast of vanishing norm retains finite energy feedback; a corrected nearest-neighbor generator has a fixed-band, finite-time second-order compressed estimate. Static elimination induces both stiffness and a generally nondiagonal metric. For uniformly positive nonuniform couplings, the normalized static graph has fourth-order low-energy eigenvalue accuracy, but its general prepared-state estimate is separately only first order. A contracting inverse series controls finite-range approximation, with additional costs for normalization and evolution. First-resolvent dressing improves an isolated low mode's state and phase ceiling to fourth order under quantitative spectral separation. The contribution is the source-bound combination of these conditional statements and obstructions, not a new general Schur, projection-memory, or subspace-perturbation mechanism. Geometry, measure, couplings, preparation, clock, action scale, and any empirical readout interpretation remain supplied.
A coarse description of a conservative amplitude dynamics is not determined by an energy compression alone. We develop a conditional account of refinement, unresolved contrasts, and effective preparation for finite weighted graphs and a supplied one-dimensional local chain. Norm conservation, covariance, and additive refinement permit different site measures. Repeated-value refinement closes exactly only when the measure-normalized external rates agree within each coarse block. When they do not, eliminating the complementary coordinates yields coherent memory and an initial-data forcing term, not stochastic noise or irreversible decoherence. A three-site example separates return to the coarse subspace at one time from autonomous coarse evolution. On a uniform pair-refined cycle, a contrast of vanishing norm retains finite energy feedback; a corrected nearest-neighbor generator has a fixed-band, finite-time second-order compressed estimate. Static elimination induces both stiffness and a generally nondiagonal metric. For uniformly positive nonuniform couplings, the normalized static graph has fourth-order low-energy eigenvalue accuracy, but its general prepared-state estimate is separately only first order. A contracting inverse series controls finite-range approximation, with additional costs for normalization and evolution. First-resolvent dressing improves an isolated low mode's state and phase ceiling to fourth order under quantitative spectral separation. The contribution is the source-bound combination of these conditional statements and obstructions, not a new general Schur, projection-memory, or subspace-perturbation mechanism. Geometry, measure, couplings, preparation, clock, action scale, and any empirical readout interpretation remain supplied.
A finite amplitude dynamics, a configuration-space propagator, and a mechanical action are different mathematical objects. We give a conditional one-dimensional bridge between them while retaining the assumptions that make each identification possible. Common-additive-shift invariance selects a relative two-mode quadratic energy; a supplied first-order kinetic term and action scale turn its graph Laplacian into reciprocal norm-preserving dynamics. The resulting amplitude action vanishes on solutions and therefore cannot simply be identified with the configuration action. A convergent ordered-jump expansion retains diagonal dwell phases, hopping factors, and time-simplex measure. On a supplied spatial lattice, inverse-square scaling gives second-order fixed-band convergence and strong convergence for each fixed square-integrable state, but not all-mode real-time operator-norm convergence. The raw return-kernel density has two distinct mesh-subsequence limits. A declared Gaussian resolution instead gives an explicit uniform lattice-to-continuum bound, separate finite-volume control, and locally uniform removal of resolution at nonzero time. With independently declared mass and action calibration, the limit has the free mechanical action and exact finite-slice normalization. For smooth positive variable couplings, midpoint edges and uniform site measure select a divergence-form operator. Different measures or orderings retain the same classical principal symbol while changing lower-order terms. The Liouville coordinate and amplitude transformation fixes a local correction, endpoint factors, and the internal measure; these must be carried together. Smooth-orbit lattice and state-level time-slicing estimates give controlled comparisons without asserting a raw variable-coefficient kernel limit. The contribution is the organized conditional bridge and its obstructions, not the invention of the standard component methods or a derivation of geometry, the action-scale value, physical couplings, or empirical probability from primitive complementarity.
We study when a reduced description preserves the response to an applied source. For systems defined by continuously differentiable state updates and observations a bounded surjective linear coarse map gives differentiable descent when the projected maps are constant on each fibre. Their common source response is then preserved on the invertible domain. For approximate descent an exact error identity separates defects in the update and source coupling from defects in the readout and direct term. Resolvent and contour estimates carry these errors and the displacement of the reference branch to visible poles and their residues at shifted critical roots. In a reciprocal Hilbert-space realisation an isolated finite-dimensional critical subspace with a positive crossing matrix has a positive-semidefinite finite-rank residue when the endpoint operator is self-adjoint and the complementary inverse is bounded. Source overlap determines its visible directions. Arithmetic and reflecting-heat examples separate stationary response preservation from memoryless dynamical closure. In the two-mode example pole displacement decays geometrically with the retained memory length. The residue error includes an additional factor of that length. For a matched vacuum covariance the residue is the zero-mass atom of its positive spectral measure. This identification uses the same physical projection and source pairing together with the continuation and subtraction conventions and endpoint matching.
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.
Complex multiplication, coherent addition, and quadratic readout are related ingredients of quantum calculation, but they answer different reconstruction questions. We separate them on a supplied reciprocal carrier. The centralizer of a complex structure on a real two-plane is the complex algebra; its invertible transports are nevertheless not closed under the addition needed for destructive interference at a selected output. A one-outcome dephaser and a reversible phase kick show that certainty of a public outcome does not imply nondisturbance. Within an explicitly assumed positive quadratic response class, calibrated zero-leakage comparisons on a connected tree select the all-ones Gram kernel. We prove quantitative stability estimates and retain the distinction between unconditional record-local operations, selective conditioning, and a controlled inverse interaction. A local normalized-weight ansatz, arbitrary anchored orthogonal refinement, and additive coarse-graining of recorded outcomes force weights proportional to quadratic size. Equal binary refinement alone does not: a positive continuous increasing log-periodic function supplies a counterexample, while binary and ternary refinement at every size, with continuity, recover linearity in size. Real rotations implement the requisite refinements and controlled tagging without full complex control. Fixed-complement preservation and reciprocal reversal admit different generator sectors; independently switchable real mixers on a connected graph plus one addressable local phase generate the unitary Lie algebra. A common scalar phase does not. Supplementary finite action–phase and history examples expose the remaining calibration assumptions. The contribution is an assumption-controlled synthesis of conditional selectors and obstructions, not the introduction of the standard component constructions. The carrier, comparison response, control resources, tensor interpretation, and empirical probability interface are not derived from primitive complementarity.
We investigate the information retained after eliminating hidden degrees of freedom from a finite-dimensional positive-definite Hermitian operator. A single energy-dependent elimination signature connects orthogonal residual geometry, linear innovation, Feshbach–Schur reduction, resolvent response, and dynamical memory. Its value at zero determines the minimum energy after elimination, while its negative first derivative is the Gram matrix of the full-state reconstruction. The main result solves an extremal problem over all positive realizations sharing the same signature value, first derivative, and known lower spectral threshold of the hidden block. We construct a minimal canonical realization that saturates a matrix-valued lower bound throughout the subthreshold energy interval. We determine the exact minima and suprema of the first indexed eigenvalues, allowing arbitrary finite hidden dimensions. All lower minima are attained simultaneously by one realization, while a single family approaches all upper suprema. Operator equality with the extremal bound at one energy enforces the entire extremal signature. The deviation from this bound controls the high-energy component of the static reconstruction and satisfies optimal comparisons across energies. The full signature identifies the active hidden realization up to unitary equivalence, whereas first-order data determine only its minimum required dimension. Noncommuting and three-mode examples demonstrate the benefit of retaining complete matrix information and distinguish reduced visible-state weight from destructive interference in a fixed spectral readout.
A coarse description of a dynamical system generally fails to evolve autonomously: evolving and coarse-graining in different orders gives different predictions, and the compressed propagator fails the semigroup law. We show that this path dependence is the observable boundary of missing state. For a finite continuous-time Markov chain with generator L, coarse channel C, and stochastic section R satisfying CR=I, define P=RC, Q=I−P, and K=CLR. The exact coarse propagator Tt=CetLR obeys a Volterra equation with memory kernel M(t)=CLQetQLQQLR. Every memory jet M(k)(0) is recursively recoverable from the short-time derivatives of Tt alone. The rank of the block Hankel operator H∞=(M(i+j)(0))i,j≥0 is exactly the smallest number of auxiliary linear state variables required to turn the coarse dynamics into an exact autonomous, time-homogeneous system. A minimal completion is constructible by Ho--Kalman factorization and is unique up to similarity. If d=min{k:M(k)(0)=0}, then every smooth classical f-divergence generically scales as t2d+4. Exact Markov examples realize exponents 4,6,8 and closure ranks 1,2,3. We also derive finite-sample singular-value certificates for hidden predictive coordinates, including a matrix-Bernstein bound with intrinsic-variance control. Finally, histories are identified when they induce the same future law under every admissible intervention, yielding the minimal counterfactual predictive state. The results turn coarse-graining into a constructive state-completion problem and separate ordinary linear realization from the harder requirement of a positive or Markov realization.