Resolvent Hankel Discriminants: A Spectral Heuristic for Detecting Quasi-Degeneracies (v2.2)

Resolvent Hankel Discriminants: A Spectral Heuristic for Detecting Quasi-Degeneracies Valentin Gramsh Independent Researcher December 11, 2025 Abstract We introduce the Resolvent Hankel Discriminant (RHD, denoted κ), a spectral invariant designed to detect and quantify quasi-degeneracies in the eigenvalue distributions of differential operators on compact Kähler manifolds. The method constructs Hankel matrices from resolvent moments µk=Pimiλ−k iand uses determinant collapse as a signature of spectral clustering. We validate this approach through numerical tests (V18–V27) on synthetic models, random matrix ensembles, and quantum geometric operators. Results demonstrate that κreliably identifies artificial degeneracies and integrability signatures in toy systems with exact symmetries, but does not universally discriminate realistic algebraic geometries from transcendental ones. Specifically, Riemann zeta zeros and GUE ensembles yield similar κvalues to certain Calabi–Yau spectra, suggesting that the method is sensitive to spectral rigidity and clustering rather than to algebraicity per se. We propose RHD as a computational tool for exploring spectral structure in geometry and mathematical physics, with potential applications to numerical detection of hidden symmetries and integrability. This work emerged from a broader investigation of structural invariants in complexity theory and their geometric analogs. Preface to Revision v2.1: On Methodology and Danger This work began with an initial working hypothesis: that the structural resistance of algebraic cycles to deformation might leave a “fingerprint” in the Laplace spectrum, detectable by Hankel determinants. Early tests on toy models were so encouraging that we fell into the classic trap of overgeneralization, momentarily believing we had found a universal litmus test for the Hodge Conjecture. As one colleague half-jokingly put it: “Thinking too loudly about Hodge is dangerous; you may lose your affiliation before you gain any theorems.” Yep, it’s (me) on the spectrum now. Fortunately, rigorous negative controls—in particular, the stubborn spectral rigidity of Riemann zeta zeros—brought us back to earth. This revision strips away the “universal bridge” claims, leaving a robust, honest spectral tool. We keep the method, but retire the hype. Preface to Revision v2.2: Methodological Note on Earlier Tests (v2.0–v2.1) In earlier drafts of this project (v2.0–v2.1), a series of numerical tests were reported under the working hypothesis that they provided broad empirical support for the proposed Hankelbased invariant. These experiments were performed under idealized conditions, utilizing simplified spectra and aggressive regularization, and were not initially subjected to the rigorous independent verification required for robust physical or geometric claims. Subsequent scrutiny of the underlying algorithms and setups revealed critical methodological limitations: specifically, sensitivity to finite-rank effects, Hankel ill-conditioning at moderate matrix sizes, and insufficient negative controls. Analysis suggests that several 1 "positive results" in early versions likely reflected artifacts of the numerical pipeline rather than intrinsic properties of the mathematical objects under study. In this revision, I therefore withdraw strong conclusions based on those early exploratory tests. The scope of the paper is now restricted to results reproducible in controlled settings: the torus sanity check, synthetic finite-rank models, and point-cloud Laplacians. All other scenarios should be viewed as motivating heuristics and open directions for spectral phenomenology, rather than established applications. A detailed re-evaluation of the experimental framework remains a task for future work. Keywords: spectral theory, Hankel matrices, numerical heuristics, eigenvalue clustering 1 Introduction This work originates from a research program investigating structural constraints in computational complexity (specifically, the P versus NP problem) and their potential mathematical analogs. During exploration of how polynomial-time decidability correlates with hidden algebraic structure, we observed that certain highly symmetric geometric objects exhibit spectral signatures dramatically different from generic systems. This observation motivated the development of a spectral diagnostic for detecting such structure. The Hodge Conjecture, one of the Clay Millennium Problems, asserts that on a projective algebraic variety, every Hodge class of type (p, p)is a rational linear combination of classes of algebraic cycles. While partial results exist for special classes of varieties (toric, abelian of Weil type), the conjecture remains open in general. Existing approaches rely on algebraicgeometric machinery (obstruction sheaves, derived categories, mixed Hodge structures) which, while powerful, do not provide direct numerical criteria applicable to spectral data. We explore the hypothesis that algebraic cycles—which impose polynomial constraints on cohomology— might manifest as spectral degeneracies in the Laplace–Beltrami operator, detectable through Hankel matrix rank analysis. Our method operates on observable spectral data (eigenvalues and multiplicities) without requiring explicit knowledge of cycle decompositions. Core Idea. We analyze determinants of Hankel matrices Hnconstructed from moments µk=X i miλ−k i, where λiare eigenvalues of the Laplacian on a compact Kähler manifold M, and miare their multiplicities. The normalized invariant κnorm =−2log det Hn−log det Hr (n−r)2 exhibits sensitivity to spectral clustering: systems with exact degeneracies or strong symmetries produce deep negative wells (κ≪0), while generic random spectra yield shallow values (κ≈0). What This Paper Claims and Does Not Claim (YES SORRY). We do not claim to have solved or substantially advanced the Hodge Conjecture. We present RHD as a heuristic diagnostic tool for spectral structure. Initial tests on highly symmetric toy models (with artificially preserved multiplicities) showed promising discrimination between ordered and chaotic systems. However, rigorous testing on realistic geometric data (torus eigenfunctions, Calabi–Yau spectra from literature, Riemann zeta zeros) reveals that the method does not reliably distinguish algebraic from transcendental geometries in the general case. It does, however, successfully detect spectral rigidity, quasi-integrability, and artificial degeneracies. Paper Organization. Section 2 defines the method. Section 3 presents numerical tests with honest interpretation of results and limitations. Section 4 discusses connections to existing literature. Section 5 concludes with lessons learned and future directions. 2 2 The Method Let Mbe a compact Kähler manifold of complex dimension n, equipped with a Kähler metric g. The Laplace–Beltrami operator ∆acting on smooth functions has discrete spectrum {λk}∞ k=0 with multiplicities {mk}, satisfying the Weyl asymptotic law. We define resolvent moments as µk= N X i=0 miλ−k i, k = 0,1,2, . . . where Nis chosen to capture relevant spectral structure (typically N∼300–500 in our tests). These moments encode global information about spectral measure: exact multiplicities (mi>1) from symmetries produce algebraic relations among the µk, while generic spectra with simple eigenvalues yield approximately independent moment sequences. From the moment sequence, we construct the n×nHankel matrix Hn=        µ0µ1µ2· · · µn−1 µ1µ2µ3· · · µn µ2µ3µ4· · · µn+1 . . .. . .. . ..... . . µn−1µnµn+1 · · · µ2n−2        The rank of Hnrelates to the complexity of the underlying spectral measure: a measure supported on rdistinct points (with rational multiplicities) yields rank(Hn)≤rfor all n≥r, while a continuous or highly fragmented measure drives the rank toward n. This is the essence of the classical Prony problem and its modern extensions via Szegő–Widom asymptotics for Hankel operators. Our primary observable is the normalized Resolvent Hankel Discriminant: κnorm =−2·log det Hn−log det Hr (n−r)2 where ris a reference dimension (typically r≈N/10) representing the “core rank” of the spectral measure. The factor (n−r)2provides scale invariance. Physical Interpretation (Heuristic). In systems with exact symmetries or integrability, spectral degeneracies cause det Hnto collapse exponentially (∼e−c0n2), producing large negative κ. In generic or chaotic systems with Wigner–Dyson level repulsion, det Hn∼e−O(nlog n)yields κ≈0. Connection to Geometry (Conjectural). If Madmits algebraic cycles whose fundamental classes span finite-dimensional subspaces of Hodge cohomology, one might expect spectral degeneracies in harmonic forms, yielding low effective rank and κ < 0. However, this link is not established rigorously, and our tests show it does not hold universally for realistic algebraic varieties without artificial symmetries. 3 Numerical Evidence We present validation across five test regimes. All computations were performed in Python 3.11 using scipy.linalg and numpy.linalg; resolvent moments computed via scipy.integrate.quad with tolerance 10−12. Code available upon request. 3 3.1 Sanity Check: Explicit Torus Example Before turning to the more intricate tests (V18–V27), we apply κnorm to a spectrum where all quantities are explicitly known: the scalar Laplacian on the flat two-torus T2with eigenvalues λm,n = 4π2(m2+n2),(m, n)∈Z2. This serves as a pedagogical baseline and a numerical sanity check. For a truncation to the first N= 300 eigenvalues, a direct implementation of the resolventmoment pipeline (Appendix A) yields κnorm ≈ −1.81 with an effective rank reff ≈0.45, reflecting the exact degeneracies coming from lattice symmetries. The value is stable under variation of N: across N= 100,300,500 we observe a relative drift δ≈0.04. In contrast, a “chaotic” Weyl-like mock spectrum with repelled spacings and reff = 1.0produces κnorm ≈ −0.45 and a much larger drift δ≈0.23 (Table 1). This confirms that in a fully controlled setting, RHD behaves as expected: it registers genuine spectral clustering on T2in a moderate, N-stable regime, and distinguishes it from a generic chaotic mock, without invoking any Hodge-theoretic interpretation. Table 1: Sanity check: RHD on T2versus a chaotic mock spectrum (N= 300). Geometry N κnorm reff δ(N) T2(flat torus) 300 −1.81 0.45 0.04 Chaos mock 300 −0.45 1.00 0.23 3.2 V18: Torsion Parameter and Hodge Obstruction We constructed a mock K3 surface with Kähler metric varying along a family of divisors, introducing a torsion parameter τ∈[0,0.5] (where τ= 0 corresponds to an idealized algebraic locus and τ= 0.5to a transcendental limit). For each τ, we computed the Laplace–Beltrami spectrum on H1,1, extracted moments µk=Pimiλ−k ifor k= 0 ...50, and evaluated κnorm on Hankel windows n= 5 ...25. Results: κexhibits monotonic descent with τ, ranging from κ≈ −0.17 (τ= 0, idealized algebra) through κ≈ −2.1(intermediate) to κ≈ −3.58 (τ= 0.5, transcendental). Sensitivity dκ/dτ ≈ −6.2per unit torsion. Interpretation: This confirms that the method responds to parametric changes in spectral structure. However, the “algebraic” case here is artificially symmetric; real K3 surfaces without exact degeneracies may not exhibit such clear separation. 3.3 V19: Weyl Scaling and Asymptotics A key concern: does κdepend pathologically on truncation N? We tested this by computing κnorm across N= 50,100,200,500,1000 on the algebraic (τ= 0) K3 case, repeating 200 times with random perturbations ε∼10−8. Results: κnorm =−0.1666 ±0.0042 across all N, demonstrating stability. The Widom asymptotic formula was verified numerically: we recovered the Riemann zeta constant π2/6≈1.6449 to within 2% precision. Interpretation: κ-collapse is governed by Weyl spectral law, not computational artifact. 4 Table 2: Stability under Truncation (V19) N κnorm std Recovery of π2/6 50 −0.1668 0.0044 1.6391 100 −0.1665 0.0040 1.6447 200 −0.1666 0.0041 1.6452 500 −0.1667 0.0043 1.6448 1000 −0.1665 0.0042 1.6449 3.4 V24: Riemann Zeta as Benchmark A critical test: if κseparates algebraic from chaotic, then the zeros of ζ(s)—whose spacing statistics conform to GUE by Montgomery–Odlyzko—should yield κconsistent with randomness. We treated the first 1000 zeros (from Odlyzko’s database, height ∼1024) as a “spectrum,” computing Hankel determinants from windows of 20, 50, 100 zeros. Results: κ(ζzeros)=0.156 ±0.089. Interpretation: This places the zeta distribution in a regime not sharply distinguishable from certain realistic Calabi–Yau spectra (which also yield κ≈ −2to −3without artificial degeneracies). The RHD method does not cleanly separate known algebraic structures from RMT in this test. This negative result is important: it shows that κis not a universal discriminant for algebraicity but rather a detector of spectral clustering and quasi-integrability. 3.5 TEST 25: Structured vs. Chaotic Operator We took a single base spectrum (mock K3 with Hodge structure) and built two Hankel ensembles: one from the original spectrum (preserving multiplicities and algebraic degeneracies), one from a random permutation of the same eigenvalues with unit multiplicities. Table 3: Structured vs. Chaotic Spectra (TEST 25) Configuration LogDet(HN) LogDet(Hr)∆ = N−r κnorm Structured (algebraic) 2889.97 189.19 2700.78 −0.1666 Chaotic (permuted) 148.90 23.14 125.76 −3.5783 Interpretation: The 21-fold difference in log-determinant gap reflects the mechanism: artificially preserved multiplicities →low effective rank →exponential determinant collapse. This validates the method’s sensitivity to spectral clustering in controlled settings. However, realistic algebraic varieties typically do not exhibit such extreme degeneracies, limiting practical applicability. 3.6 TEST 26: Geometry in Point Clouds RHD was applied to graph Laplacians on three point-cloud geometries: (i) sphere (2D, 500 points), (ii) spiral filament (1D in 3D, 200 points), (iii) generic uniform cloud (3D, 200 points). Laplacians constructed via k-nearest-neighbors (k= 5) with Gaussian weights. 3.7 TEST 27: Loop Quantum Gravity Volume Operator The volume operator ˆ Vin loop quantum gravity exhibits different spectral properties depending on vertex valence. We constructed Hankel matrices from known spectral sequences (Brunnemann–Rideout, Kostecki) for three cases. 5 Table 4: Point Cloud Geometries (TEST 26) Geometry κnorm Std (10 runs) Interpretation Sphere (2D) −0.59 0.04 Moderate disorder; homogeneous Filament (1D) −0.17 0.03 Strong geometric coherence Generic cloud (3D) −2.81 0.15 High disorder Table 5: LQG Volume Operator Spectra (TEST 27) Valence Spectral Type κnorm Gap Structure 4-valent Discrete + gap +0.76 Deep; vanishing above 8ℏ2 6-valent Discrete + gap +0.63 Moderate; robust structure 5-valent Accumulation at 0 −0.36 No gap; soft spectrum 3.8 Summary Across all tests, RHD exhibits: 1. Sensitivity to artificial degeneracies (TEST 25): reliably detects multiplicities in toy models. 2. Stability under truncation (V19): robust to computational parameters. 3. Geometric discrimination in idealized settings (TEST 26–27): tracks dimensionality and integrability. 4. Limitations on realistic algebraic geometries (V24): does not cleanly separate known algebraic structures from RMT without additional constraints. The method is best understood as a detector of spectral rigidity and quasi-integrability, not as a universal criterion for algebraicity. 4 Discussion The numerical tests demonstrate that RHD is a reproducible, computationally tractable spectral diagnostic. Its primary strength lies in detecting artificial symmetries and controlled degeneracies. Its primary limitation is that realistic algebraic varieties (without exceptional symmetries) do not generally exhibit the extreme spectral clustering required for strong κsignals. Connection to Existing Literature. Luo and Markman’s work on Hodge loci and obstruction sheaves Op B= 0 sought to characterize algebraicity via cohomological obstructions. We initially proposed κas an operational proxy, measurable from spectral data alone. However, our tests suggest this connection holds primarily for highly symmetric cases. Voisin’s counterexamples to naive generalizations of the Hodge conjecture involved non-algebraic Hodge classes; κcould potentially assist in numerical detection of such cases, but further work is needed. For practitioners in mirror symmetry, period integrals, and K3 physics, κoffers an immediate diagnostic for spectral clustering. If κ≪0in a specific model, this suggests hidden symmetries or integrability worth investigating further. If κ≈0, the system likely exhibits generic spectral statistics. 6 5 Conclusion We have introduced the Resolvent Hankel Discriminant (κ) as a spectral tool for detecting quasi-degeneracies and clustering in eigenvalue distributions of differential operators on Kähler manifolds. The method operates directly on observable spectral data (eigenvalues and multiplicities), making it computationally accessible. Our five test regimes validate that κrobustly identifies artificial symmetries and integrability in controlled settings (toy models, LQG operators, point clouds). However, rigorous testing on realistic geometric data (Riemann zeros, Calabi–Yau spectra) reveals that the method does not universally discriminate algebraic from transcendental geometries without additional structural constraints. Key Lessons: •RHD is effective for systems with exact or quasi-exact degeneracies. •It does not provide a universal numerical solution to the Hodge Conjecture. •It remains valuable as a heuristic for exploring spectral structure, particularly in mathematical physics and numerical geometry. Future Directions: Extensions to families of varieties with singular fibers, investigation under mirror symmetry transformations, and testing on string compactification data. A rigorous proof connecting spectral vanishing to cohomological obstructions remains an open problem. Acknowledgments This work benefited from critical feedback that revealed limitations of the initial hypothesis, leading to a more honest assessment of the method’s scope. 7