Finite determinacy yields an allocation geometry and a causal cone but not quantummechanics: its axioms admit a classical model, so superposition and Born’s rule cannot followfrom them. This paper asks how far one may nonetheless go, and locates the remaining step.The minimal carrier of a continuous recombination is shown to be C, by classification of theconnected abelian Lie groups admitting the loop action by automorphisms, so the complexunit is not imported. Transport equivariance then forces contributions to combine by additionuniquely, while convex recombination is excluded for failing to close under composition. Thedependence of a record’s probability on the carrier is fixed at the square by an involutivesplitter, its normalisation following from involutivity rather than from any norm, with no otherexponent surviving repeated splitting. Phase-invariance of records and the density ceiling arederived. The chain then closes to an inner-product space, the amplitude being a unit vectorin L2 of the phase space; what is not obtained is the reduction to configuration space, and ofthe two polarisations available one rests on a primitive and the other on a theorem, neitherbeing forced. Several negative results are established unconditionally, among them that nopositive transport of a non-negative content can produce cancellation, and that no phase canbe built from a scalar or from any translation-invariant function of a cell.
Finite determinacy yields an allocation geometry and a causal cone but not quantummechanics: its axioms admit a classical model, so superposition and Born’s rule cannot followfrom them. This paper asks how far one may nonetheless go, and locates the remaining step.The minimal carrier of a continuous recombination is shown to be C, by classification of theconnected abelian Lie groups admitting the loop action by automorphisms, so the complexunit is not imported. Transport equivariance then forces contributions to combine by additionuniquely, while convex recombination is excluded for failing to close under composition. Thedependence of a record’s probability on the carrier is fixed at the square by an involutivesplitter, its normalisation following from involutivity rather than from any norm, with no otherexponent surviving repeated splitting. Phase-invariance of records and the density ceiling arederived. The chain then closes to an inner-product space, the amplitude being a unit vectorin L2 of the phase space; what is not obtained is the reduction to configuration space, and ofthe two polarisations available one rests on a primitive and the other on a theorem, neitherbeing forced. Several negative results are established unconditionally, among them that nopositive transport of a non-negative content can produce cancellation, and that no phase canbe built from a scalar or from any translation-invariant function of a cell.
A physical quantity is taken to have finitely many determinate digits rather than a realvalue partially resolved, the continuum being an idealisation the world does not honour.This paper develops the consequences by rank. An object is a family of related quantitiessharing one bounded joint depth — relatedness only is assumed, not conjugacy — and theexponential resolution law and a conversion cocycle follow, while parity separates closedfamilies from constrained ones. Reciprocal resolution, the antisymmetric form and a compatiblecomplex structure are each derived rather than posited, so what is usually takenas the canonical structure of a pair is here an output. A counting argument shows that norelation between pairs exists below rank two, so the single-pair theory is complete ratherthan deficient. The antisymmetric form is then assembled from a relation, which suppliescoupling, and from roles, which supply orientation and are primitive. A second and strictlystronger premise, a pointwise ceiling on record density, yields exact content-space results:an entropy floor, a characterisation of saturation as uniform product support, and a sharpvariance bound from confinement alone. Criticality is characterised as a crossover betweencompeting ceilings, first within a pair and then, at rank two and above, between them. Aclosing section collects results establishing what the construction provably cannot do, includingthat no scale-invariant datum can fix a scale-covariant normalisation. The axiomsare shown to admit a classical model, so quantum structure is not claimed to follow fromthem.
Finite Determinacy (FD) was developed from a representational question: whether theuse of real numbers for physical quantities should be read as implying arbitrarily precisephysical determination. Replacing that idealisation by objectively finite determinationleads to bounded shared resolution depth for related records. Close finite-informationand relational antecedents in quantum foundations were identified subsequently and areacknowledged here; they were not the route by which the present framework was developed,and no historical priority is claimed for those principles. This companion paper asks whatdynamical restrictions follow once the resulting finite-record geometry is combined withexplicit finite-rate premises.For self-containment the paper first reproduces the common foundation: bounded jointdepth and split-invariant refinement give exponential resolution and a non-zero two-registerwidth-product floor; a homogeneous, finitely additive area regularity promotes that scalarconstraint to a local area measure; and a primitive tick/readout role asymmetry suppliesorientation, after which the two-member sector is symplectic. The positive symmetric formused for compatible complex structure remains an independent structural input. Dynamicsthen adds a universal pace ceiling rmax, a saturated intra-pair exchange ceiling |dv/du| ≤ L,and an inter-pair causal conversion c∗. In the quantum branch, explicit coherent-recordand record-faithfulness assumptions lead through Weyl–Heisenberg representation theoryto finite-step unitary evolution and the Schrödinger law-form. In the gauge branch, recordgeneratedtransport and holonomy lead, under a minimal local reversible continuum closure,to Yang–Mills curvature-square dynamics. In gravity, locality, reversibility and isotropy givea minimal hyperbolic principal part, while an explicit spatially relational continuum closureleaves two transverse-traceless modes.The proposed contribution is therefore not a new finiteness or relationality principle,nor an exclusive derivation of familiar terminal equations. It is an independently developedfinite-geometric route from the physical interpretation of real-number representation to acommon pre-Hilbert, pre-gauge and pre-metric substrate, followed by sector-specific dynamicalconsistency conditions.
Finite observability is built in two stages. First, abstractly: a finite observer resolving realvaluedregisters carries a dimensionless ledger of distinctions, capped in total at λmax, withtwo coordinates — the capped total λ+ = λu + λv and the free split λ− = λv − λu. Second,the registers are reformulated as a conjugate pair (u, v) by positing the symplectic pairing,which dresses the capped total as a resolution cell of minimum area K = δu δv; a second,per-pair finiteness Luv ceilings the rate at which distinguishable change is exchanged betweenthe members, and marks the distinguished (null-diagonal) cell on the saturation locus. Theshared ledger makes the reciprocal cell a consequence, not a postulate: split-invariance forcesexponential resolution maps, and δu δv = K is the full-budget locus. From this we derive:noncommutativity at scale K; the imaginary unit as the saturated cell’s complex structure; aconstructed wave with Schr¨odinger-form evolution and a Born rule as Fubini–Study transitionprobability; one unconditional causal theorem (subluminal propagation equivalent to a nonnegativedispersion invariant); a forced first-order (Clifford) form; measurement decomposedinto the cell’s own motions, with one testable consequence — collapse of massive superpositionsabove a derived ceiling (∼ 1012–1014 amu) on the Di´osi–Penrose timescale via a conservative,non-radiating mechanism; the non-fixability of the scale; and composites as differentiated cellbundles. The physical reading — position and momentum, c, Dirac, gauge fields and gravity— is deferred entirely to a companion paper. The claim here is structural: the grammar of onefinite-observability conjugate pair, with no particular physical pair assumed.
Phase is forced by cancellation, before probability or physics enters the story. This paper continues a constructive sequence deriving only what is logically forced once incompatible closure criteria induce non-commuting stabilization operations. When distinct stabilization routes matter, a state-only description becomes lossy, but explicit route-tracking produces unbounded descriptive growth (“route explosion”). To summarize non-equivalent routes without erasing forced distinctions, the framework is compelled to introduce a contribution carrier and a combination rule. Regrouping invariance forces associativity; null contributions force an identity element; cancellation adequacy forces inverses, pushing the contribution space toward an abelian group (on the generated representational classes). Purely nonnegative scalar accumulation cannot support cancellation, so a phase-like orientation invariant appears prior to probability assignments, amplitudes, Hilbert space axioms, or physical interpretation. Finally, requiring descriptive boundedness under iterative route concatenation pressures the minimal orientation structure toward a compact (periodic) group. The forward pressure is then explicit: extract a nonnegative invariant magnitude from phase-bearing summaries in a way that is phase-invariant, additive over exclusive outcomes, and stable under composition—setting up the next paper’s derivation of a quadratic (Born-like) invariant.
Dynamique et transport au voisinage d'une transition de phase quantique en dimension deux Nous étudions le modèle O (N) relativiste, une généralisation quantique de la théorie φ⁴ utilisée en physique statistique pour étudier des transitions de phase. Ce modèle décrit certaines transitions de phase quantiques telles que la transition isolant de Mott-superfluide dans un gaz de bosons piégés dans un réseau optique ou la transition paramagnétique-antiferromagnétique dans un aimant. En deux dimensions d’espace, ces systèmes sont fortement corrélés près de la transition. Nous les étudions à l’aide du groupe de renormalisation non perturbatif, une formulation du groupe de renormalisation de Wilson. Nous nous intéressons aux propriétés universelles au voisinage de la transition de phase quantique à température nulle. Ainsi, nous déterminons les fonctions d’échelle universelles qui définissent la thermodynamique et démontrons que ces fonctions sont reliées à celles décrivant la force de Casimir critique dans un système classique tridimensionel. Ensuite, nous étudions le spectre d’excitation dans la phase ordonnée à température nulle. Pour N = 2 et 3, nous établissons l’existence d’un mode d’amplitude aussi appelé « mode de Higgs » par analogie avec le mécanisme de Higgs en physique des hautes énergies. Pour N = 1, nous montrons l’existence d’un état lié pour des dimensions proches de trois. Enfin, nous calculons la dépendance en fréquence de la conductivité à température nulle et confirmons son universalité, en particulier à la transition. Nous établissons que l’une des composantes du tenseur de conductivité dans la phase ordonnée est une quantité « superuniverselle », ne dépendant ni de la distance au point critique ni de N.
This paper studies the fixed-background many-body extension and the Wallstrom obstruction within the scalar density-phase framework of capacity geometry. Within the same local first-order homogeneous density-phase action class as the general scalar forcing theorem, the forcing argument extends immediately to declared configuration-space backgrounds and yields the standard spinless many-body Schrödinger equation without any factorisation ansatz. The bare density-phase equations do not force Wallstrom quantisation: a punctured-disc family gives explicit finite-action counterexamples, and the sharpened locallocking relation reduces that family only continuously, not integrally. The strongest continuum closure theorem is a domain-selection statement: on the intrinsic nonvanishing complex-field domain the forced action is the Schrödinger action, Wallstrom counterexamples are excluded, and circulation quantisation is automatic by degree theory. What remains open is whether the SGOC construction selects that single-valued complex domain rather than only the larger density-phase representation. A second contribution is a precise theorem map for the discrete carrier-transport approach. Native phase-sensitive SGOC data live on the Fourier/triad skeleton, while the discrete Wallstrom problem needs carrier data on a spatial triangulation of a nodal-complement component. The missing theorem is a genuine native-to-carrier transport theorem. On closed orientable surface carriers the remaining cases split into three transport regimes. Once a global carrier edge-phase field exists, the total-curvature scalar obstruction vanishes automatically by simplicial Stokes; once non-flat support is confined to a proper carrier,homology-killing enlargement and relative H^1 complement globalisation handle the rest of the downstream topology. Selector-side reductions are also consolidated. Before localisation, the remaining selector-side existence content is one universal H^2 transgression class. After localisation to a proper carrier, existence of a strong completion is automatic, and the only surviving noncanonical part is the ambient surface period torsor — an affine A^{2g}-torsor on genus g — with complement residues canonically separated off. Exchange sectors and spin are additional topological or multicomponent input, not outputs of the present scalar variational route. Keywords: - Schrödinger equation - Capacity geometry - Wallstrom obstruction - Many-body quantum mechanics - Madelung equations - Circulation quantisation - Domain selection - Carrier transport - SGOC programme Related identifiers (references cited in this paper): - 10.5281/zenodo.19156067 (cites) — Capacity Geometry: REG, Dirichlet Tension, and Quasi-Local Mass (SGOC Volume 2, Part 1) - 10.5281/zenodo.19156316 (cites) — The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge (Part 1A) - 10.5281/zenodo.19189901 (cites) — The Schrödinger Equation from Capacity Geometry: General Forced-Action Theorem on a Fixed Capacity Background - 10.5281/zenodo.19259771 (cites) — Radial Surcharge Inversion and Free-Boundary Host Selection - 10.5281/zenodo.19267944 (cites) — Free-Boundary Surface Selection for a First-Order Nonlinear Profile Equation in Three Dimensions Related identifiers (external references): - 10.1103/PhysRevA.49.1613 (cites) — Wallstrom, T. C., "Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations," Physical Review A 49(3) (1994), 1613–1617