Research Series · Mathematical Physics
This series studies how physical information survives successive changes of mathematical representation, from constrained gauge-field states to radiative multiparticle data and their celestial decomposition.
Program
The program begins with an exactly solvable Gaussian sector subject to a closed matter-current constraint. Paper 1 shows that nonzero current-physical states cannot have bounded particle number, while a completed all-orders Gaussian family survives. Paper 2 identifies the quadratic two-particle seed that generates this completed sector and shows that finite source data can remain faithful even when their physical realization requires an infinite Fock expansion. Paper 3 constructs the corresponding completed Bose/Fock and Poincaré representations while separating representation-theoretic existence from the stronger question of physical asymptotic-particle identity.
Paper 4 follows the same pair geometry into physical radiative response. Here genuine information compression appears: different pieces of the original Gaussian provenance are not equally visible to the radiative observables, and normalization leaves a reduced physical two-particle source. The series therefore does not assume that every physically natural map is information preserving; it determines the kernel and surviving data at each stage.
Papers 5 and 6 ask what happens when this radiative Bose pair is reorganized into celestial principal-series data. Paper 5 shows that the physical Clebsch–Gordan kernel does not support the naive fixed-source picture: shadow/Weyl reflection necessarily carries a unitary transformation of the source, and the canonical physical kernel is not fixed-source fiberwise isometric. The correct object is therefore a source-aware joint source/output representation.
Paper 6 resolves the remaining noninteracting information question on the intrinsic physical source range. It constructs the retained direct-integral framework conditionally on one explicit uniform core estimate and, independently of that completion hypothesis, proves that the intrinsic cross-ratio transform and the full signed principal-series continuation are injective for every physical external-label pair. The source correction required by the physical Weyl law therefore preserves source identifiability on the proved physical range even though fixed-source metric preservation fails; stable inversion and injectivity after the additional retained fundamental-sheet projection remain outside the theorem.
What exists, what is represented, what is identified, what is lost, and what remains reconstructible at each physical transition?
Across the sequence, the recurring question is not whether every representation is unitary or every map invertible. It is more basic: the program tracks which information each physical construction retains, which distinctions it quotients, and which identities remain recoverable after the representation changes.
The resulting program separates several notions that are often conflated:
Paper sequence
Papers I–VI are now public preprints. Paper VI completes the current source-uniqueness stage by proving all-label injectivity on the intrinsic physical source range while keeping the retained bounded completion explicitly conditional on its remaining uniform core estimate. The series continues from this result into further questions about spectral realization, operator structure, reconstruction, and eventually interacting celestial dynamics.
Published preprint
Bounded-particle exclusion and all-orders Fock structure in a machine-checked gauge-field model.
Nonzero current-physical states cannot have bounded particle number, while a completed Gaussian family survives with faithful positive moduli. The paper establishes the state-level source geometry from which the later series develops.
Published preprint
Closed quadratic generators, faithful two-particle seeds, and nontruncatable current physicality.
The all-orders recurrence collapses to a quadratic pair generator whose finite two-particle seed faithfully carries the Gaussian modulus even though exact physical realization requires the completed infinite Fock hierarchy.
Published preprint
Completed Bose/Fock representations, analytic carrier realization, and the boundary of asymptotic particle identity.
The generated pair receives exact completed Bose/Fock and Poincaré representations. The paper keeps representation-theoretic existence separate from the stronger claim that those coordinates already constitute qualified asymptotic particles.
Published preprint
Quadratic-sector variational stress, physical-helicity response, and exact information compression.
The same pair geometry enters physical response. Different observables retain different pieces of Gaussian provenance, making information compression explicit and leaving a normalized radiative two-particle source for the celestial stage.
Published preprint
Joint source-output symmetry and positive-measure obstruction to fixed-source isometry for radiative two-particle Clebsch–Gordan kernels.
The naive fixed-source Weyl picture fails. Shadow reflection carries a unitary source transformation, producing a source-aware joint representation, while the canonical physical kernel exhibits a positive-measure obstruction to fixed-source fiberwise isometry.
Published preprint
Direct-integral completion, source-Weyl geometry, and all-label uniqueness on the intrinsic physical source range.
The sixth paper proves that the intrinsic cross-ratio transform and its full signed principal-series continuation are injective on the intrinsic physical source range for every physical external-label pair. The retained bounded completion is separated cleanly as a conditional statement requiring one explicit uniform core estimate, and the resulting source-aware carrier is handed forward to the later papers in the series.
Current horizon
With Papers I–VI now public, the series has established a substantial noninteracting foundation: what states exist, how their pair geometry is represented, what physical response retains, how Weyl reflection acts on source and output together, and where source identity remains uniquely recoverable in the intrinsic celestial data. Paper VI supplies a proved source-aware carrier for the next stage of the program rather than a terminal endpoint.
The next papers can now sharpen the spectral realization and operator geometry of that carrier, determine which stronger reconstruction and stability statements are available, and continue the source-aware celestial program before the series ultimately reaches interaction, scattering, collinear structure, soft observables, celestial operator products, and Ward identities.