Research ManuscriptStandalone LaTeX Package Complete (Pre-submission)v1.0.0 (September 2026)

Underdetermination and Negative Identifiability Theorems in Graph-Fiber Field Theories

Ivan PasevGlobal Institute of Logic & Cybernetics (GILC)

Abstract

We present a comprehensive mathematical analysis of structural underdetermination, parameter identifiability obstructions, and predictive boundaries in field theories formulated on discrete internal fibers. While discrete combinatorial structures (such as graph Laplacians) provide mathematically tractable regularization mechanisms and generate multi-scale mediator towers, we prove that finite combinatorial data alone cannot fix low-energy physical observables without extensive auxiliary Lagrangian choices. Specifically, we prove four foundational negative results: (1) Spectral Incompleteness Theorem: graph degree regularity bounds operator norms but fundamentally underdetermines Laplacian spectra, multiplicities, and spectral measures; (2) Pullback Universality Theorem: unrestricted pullback operators are surjective onto all positive atomic operators, demonstrating that unconstrained coupling maps possess zero predictive power; (3) Direct Path Incompleteness: bare combinatorial graph Laplacians cannot be mapped directly to atomic transition shifts in the absence of a derived matter-coupling Lagrangian and source state; and (4) Continuous Parameter Identifiability Obstruction: when a three-parameter spectral mediator model is constrained by two scalar transition frequencies, the Jacobian and Fisher information matrices are strictly rank-deficient (rank(F) ≤ 2 < 3), resulting in singular Fisher information and continuous parameter degeneracy. We formalize these boundaries as an epistemic firewall that strictly prevents retrospective parameter fitting from masquerading as prospective physical predictions.

Inspect Proof Map Cite Work

This scholarly research manuscript presents foundational mathematical proofs of inverse-problem rank deficiency, spectral underdetermination, and parameter identifiability obstructions in discrete graph-fiber field theories.

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1. Core Mathematical Results & Analytical Scope

This work establishes four foundational negative identifiability theorems (OBJ-PRG-NEG):

1. Degree Underdetermination (NEG-KP-R1-SPECTRAL-INCOMPLETE-01): Proof by counterexample ($Q_5$ vs $K_{5,5}$) that degree regularity ($d=5$) bounds operator norms ($\parallel {\mathrm{\Delta }}_d\parallel \le 10$) but underdetermines spectra, eigenvalue multiplicities, and spectral measures.
2. Pullback Universality (NEG-P2-PULLBACK-UNDERDET-01): Theorem proving that the mapping $B\mapsto B^{\ast }{K}_{\lambda }B$ is surjective onto all positive self-adjoint operators $W\ge 0$, establishing that unconstrained pullback maps possess zero predictive power.
3. Direct Path Incompleteness (NEG-P2-DIRECT-KP-PATH-01): Mathematical proof that combinatorial Laplacians $(V,E,{\mathrm{\Delta }}_G)$ require three independent auxiliary choices (matter coupling map $C$, source state $\eta$, and mass scales) to form a well-posed atomic shift functional.
4. Fisher Rank Deficiency (NEG-P2-C3-P4-UNDERDET-01): Proof that projecting 3 model parameters $(A,m_0,{\mathrm{\Lambda }}_K)$ onto 2 atomic observables yields a rank-deficient Jacobian ($\mathrm{rank}(J)\le 2<3$) and singular Fisher information ($det(F)=0,{\lambda }_{min}(F)=0,\kappa (F)=\mathrm{\infty }$).

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2. Reviewer Proof & Verification Crosswalk

Canonical Negative ID	Mathematical Proposition	Analytical Status	Empirical / Numerical Testbed	Primary Literature Anchor
NEG-KP-R1-SPECTRAL-INCOMPLETE-01	Graph Degree Underdetermination Theorem	Proven Analytically (Counterexample $Q_5$ vs $K_{5,5}$)	Graph Laplacian Spectrum Benchmark	Spectral Graph Theory
NEG-P2-PULLBACK-UNDERDET-01	Pullback Universality & Surjectivity Theorem	Proven Analytically ($B=K_{\lambda }^{-1/2}J{W}^{1/2}$)	Operator Square-Root Reconstruction	Operator Algebra
NEG-P2-DIRECT-KP-PATH-01	Combinatorial Mapping Incompleteness	Proven Analytically	Source Coupling Invariance Test	Metrology Foundations
NEG-P2-C3-P4-UNDERDET-01	Parameter-to-Observable Fisher Rank Deficiency	Proven Analytically ($\mathrm{rank}(F)\le 2<3$)	Permutation Null Test ($p_{\text{null}}=0.62$)	Aster et al. (2018), Gutenkunst (2007)

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3. Explicit Epistemic Boundaries & Conditions for Prospective Falsifiability

Epistemic Firewall: Zero Synthetic Physical Claims

Because current theory lacks a first-principles microscopic Lagrangian deriving $(A,m_0,{\mathrm{\Lambda }}_K,\eta )$, any match to atomic spectroscopy is an exact retrodiction ($\mathrm{DOF}=0$). It is strictly forbidden to claim prospective physical validation (ACTIVE_P4_SEALS = 0).

Necessary Conditions for Prospective Status

For discrete-fiber models to achieve prospective scientific standing, three milestones are required:

1. First-principles derivation of the source state $\eta$ from underlying gauge symmetry.
2. Derivation of the physical matter current coupling from a fundamental gauge action.
3. Acquisition of at least $N\ge 4$ independent isotope shift transitions to break the 1-dimensional degeneracy curve.

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4. Citation & Bibliographic Metadata

Standard Citation

Pasev, I. (2026). Underdetermination and Negative Identifiability Theorems in Graph-Fiber Field Theories. Research manuscript, version 1.0.0. https://ivanpasev.com/04-mathematics/manuscripts/identifiability-obstructions

BibTeX Record

@misc{Pasev2026NegativeIdentifiability,
  author       = {Ivan Pasev},
  title        = {Underdetermination and Negative Identifiability Theorems in Graph-Fiber Field Theories},
  year         = {2026},
  month        = {September},
  howpublished = {Research manuscript, version 1.0.0},
  institution  = {Global Institute of Logic & Cybernetics (GILC)},
  url          = {https://ivanpasev.com/04-mathematics/manuscripts/identifiability-obstructions}
}

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5. Canonical Continuations

Direction	Target Resource	Purpose
Mathematics Hub	Formal Mathematics Root →	9-tier status taxonomy and submission-grade manuscripts
Continuum Analysis	Manuscript B: FQFT Continuum Core →	Complementary positive variational Dirichlet analysis and Mosco convergence
Falsifiability Index	Falsifiability Index & Negative Boundaries →	Explicit failure conditions and parameter identifiability limits
Review Intake	SFR Review Portal & Critique Intake →	Five-gate critique protocol and formal referee inquiry gateway