{ "abstract": "v5 update (October 2026) — corrections. (1) Paper 2 is replaced by draft v0.5, which corrects computational errors in v0.4: Table 1 (impact-parameter columns), Section 7 and Eq. 20 (orbital frequency from the full dispersive Hamiltonian; correctly normalised Lyapunov exponent; retrograde ring added). The prograde correction is δΩ_ph/Ω_ph = +0.657 Λ̂E², not −0.219 Λ as stated in the v3 paragraph below, and eikonal corrections grow as l³. The photon-ring coefficients C_pro and C_ret, the chromatic E² scaling and the superradiant-threshold result are unchanged. (2) Paper 1: the quartic term of the scalar action (Eq. 7) is quartic in the field and does not produce the stated linear k⁴ dispersion (it drops out at linear order and has mass dimension 8, not 6). A linear dimension-6 candidate with the Laplacian of the preferred slices reproduces ω² = c²k²(1 + Λk²) and, in the eikonal limit, the Paper 2 Hamiltonian; a revised Paper 1 is not yet included. With the present Boyer–Lindquist/ZAMO foliation, the horizon boundary conditions for quasinormal modes are not fixed by the model, so the QNM results remain eikonal estimates. Details and check scripts: github.com/Kretski/Lambda-model (notes/: A0_LINEAR_ACTION, A1_KERR_CHECKS, A2a_SCHWARZSCHILD_BOUNDARY, EIKONAL_D_LAMBDA, THEORY_STATUS_GW). The CQG submission mentioned in the v2 paragraph (CQG-117140) was not accepted.\n\nComplete covariant derivation of a one-parameter Hamiltonian extension of\n\nmassless wave propagation in Schwarzschild spacetime. The Hamiltonian\n\nH = c k α(r) √(1 + Λk²) is derived from a scalar field action with quartic\n\nlocal-frame correction. Includes closed-form analytical solutions, numerical\n\nverification to machine precision, and EFT classification via dimension-6\n\nSME operators.\n\nv2 update (August 21, 2026): Added full manuscript PDF (v0.4, 11 pages, two\n\nproved theorems), companion code files for Theorems 1–2 and einbein\n\nfalsification test, and key figures from the Kerr extension (photon ring,\n\nchromatic shadow, Lyapunov analysis). Submitted to Classical and Quantum\n\nGravity (ref: CQG-117140).\n\nv3 update (August 21, 2026): Added Paper 2 draft — Kerr spacetime extension\n\nwith four theoretical results: (1) Photon-ring correction: δb = C(a*, ring)\n\n· Λ̂ · E², with C_pro = −0.6230 and C_ret = +11.495 (dimensionless, a*=0.9).\n\nThe E² scaling is proved analytically from the vanishing of g^μν k_μ k_ν at\n\nthe GR photon ring. (2) Chromatic shadow: the black hole shadow size acquires\n\na frequency-dependent shift, δθ ∝ Λ̂ · E² / D. The corresponding sensitivity\n\nscale satisfies Λ_max ∝ ω⁻² (exact algebraic scaling, confirmed numerically\n\nto slope −2.0000). (3) Superradiance: the kinematic threshold ω = mΩ_H is\n\nunchanged by the quartic dispersion; proof via the ZAMO 4-velocity shows\n\nn^μ k_μ → 0 at threshold, hence k⁴_loc → 0 identically. The amplification\n\nfactor away from threshold is not addressed here. (4) Eikonal QNM\n\ncontinuation: the classical photon-orbit/QNM correspondence\n\nω_QNM ≈ Ω_ph(Λ)·l − i(n+½)·λ_Lyap(Λ) is used as an effective eikonal\n\napproximation, with δΩ_ph/Ω_GR ≈ −0.219·Λ̂ in the Λ→0 expansion. The Λ→0\n\nlimit reproduces known Schwarzschild and Kerr photon-orbit benchmarks\n\n(Bardeen 1973) to numerical precision (Schwarzschild error < 2×10⁻⁷).\n\nv4 update (August 2026): Revised Paper 2 draft correcting several\n\npresentation issues identified in internal review prior to journal\n\nsubmission: (i) corrected the ZAMO 4-velocity / angular-velocity formula\n\nused in the superradiance proof; (ii) introduced explicit dimensionless\n\nnormalization Λ̂ = Λ/(GM/c²)² for all quoted coefficients, removing the\n\nearlier ambiguity between geometric-unit coefficients and SI-unit Λ;\n\n(iii) reworded the abstract and Table 2 to distinguish theoretical\n\nsensitivity estimates from observational constraints — no value of Λ has\n\nbeen measured or fitted in this work; (iv) separated the A3.3 fixed-\n\ntolerance scaling result (Λ_max ∝ ω⁻², the primary theoretical result)\n\nfrom the A3.4 resolution-scaled toy-model illustration (slope ≈ −2.495,\n\nnot a physical prediction); (v) softened the eikonal-QNM correspondence\n\nlanguage to reflect that it is used as an effective approximation, with\n\na first-principles dispersive wave-equation derivation deferred to a\n\nplanned Paper 3; (vi) disambiguated \"validated against GR\" throughout to\n\nmean specifically that the Λ→0 limit reproduces known benchmarks, not\n\nthat the Λ≠0 predictions have been independently confirmed. A new\n\ncompanion feasibility test (A3.3 TeV analysis) is also included, showing\n\nthat the theoretical E² chromatic-shift scaling is confirmed numerically\n\n(measured slope −2.000000 against expected −2.000000) while clarifying\n\nthat high photon energy alone does not provide spatial resolution: a\n\ndirect observational constraint on Λ requires the energy-dependent\n\nphoton-ring displacement to be spatially resolved, which current TeV\n\ninstruments do not provide model-independently.\n\nAll Λ→0 results are validated against known closed-form Schwarzschild and\n\nKerr benchmarks. The Λ≠0 sector consists of theoretical predictions and\n\nsensitivity estimates; no observational bound on Λ is claimed in this\n\nversion. Submitted to Classical and Quantum Gravity (Paper 1 ref:\n\nCQG-117140).", "abstract_shown": true, "affiliations": [], "article_type": null, "arxiv": null, "author_total": 1, "authors": [ { "name": "Dimitar Kretski", "orcid": "0000-0001-5108-2243" } ], "bibtex": "https://research.draconitic.com/paper/2we4re18nbh31.bib", "cited_by": 0, "doi": "10.5281/zenodo.23459797", "fulltext_url": null, "id": "2we4re18nbh31", "issns": [], "landing_url": "https://zenodo.org/records/23459797", "language": null, "licence": "CC-BY-4.0", "licence_label": "CC BY 4.0", "licence_source": "article", "licence_url": "https://creativecommons.org/licenses/by/4.0/", "metadata_licence": null, "native_subjects": [], "openalex": null, "pdf_url": "https://zenodo.org/api/records/23459797/files/paper2_draft_v05.pdf/content", "pmcid": null, "pmid": null, "published": "2026-10-18", "publisher": null, "ris": "https://research.draconitic.com/paper/2we4re18nbh31.ris", "sources": [ { "first_seen": "2026-11-01T12:20:53Z", "native_id": "23459797", "source": "zenodo", "url": "https://zenodo.org/records/23459797" } ], "subjects": [], "title": "A Hamiltonian Oscillator Extension of Wave Propagation in Schwarzschild Spacetime: Dispersion Index, Radial Wavenumber Dynamics, Covariant Derivation, and Observational Constraints on a Single Parameter", "title_text": "A Hamiltonian Oscillator Extension of Wave Propagation in Schwarzschild Spacetime: Dispersion Index, Radial Wavenumber Dynamics, Covariant Derivation, and Observational Constraints on a Single Parameter", "type": "preprint", "updated_at": "2026-11-01T12:20:53Z", "url": "https://research.draconitic.com/paper/2we4re18nbh31", "venue": "Zenodo", "versions": [] }