Λ-model — A Quartic Dispersion Framework

Author: Dimitar Kretski ORCID: 0000-0001-5108-2243 Affiliation: Center for Hydro- and Aerodynamics, Bulgarian Academy of Sciences, Varna, Bulgaria

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What this repository is for

This repository implements a one-parameter dispersion relation,

$$ \omega(k) = c,k,\sqrt{1 + \Lambda k^2}, $$

together with a validated numerical PDE solver for its wave equation, and a fitting tool that tests whether a real, measured $(k,\omega)$ dataset is consistent with this form.

What a researcher can actually do with it, in five minutes:

1. Test your own dispersion data. Run lambda_experimental_validator.py --omega your_omega.csv --k your_k.csv on any $(k,\omega)$ pairs you have — no domain assumptions required. You get a fitted $\Lambda$, its standard error, $R^2$, and an explicit significance test against $\Lambda=0$.
2. Simulate propagation under this dispersion. The 2D spectral PDE solver (wave_equation_2D_solver.py) is validated to machine precision in space and $O(1/N)$ in time (see Numerical status below) — use it directly if you need to propagate a field under a given $\Lambda$. The full leapfrog time-integration path (not just the spectral-exact shortcut) is separately validated for spatial convergence — see paper3_h_convergence_test.py and the status table.
3. If you work with BEC: the mapping $\Lambda=\xi^2/4$ is physically established (exact match to the Bogoliubov dispersion relation, not an analogy). Plug in your healing length and sound speed and test it against your own Bragg-spectroscopy data directly.
4. If you work with Dirac materials or photonic dispersion: the repository is explicit about which mappings are not currently supported, so you don't waste time chasing a claim that doesn't hold up (see Status table below). The fitting infrastructure still works on your data — you supply the physical mapping.
5. Extend it. Adding a new domain mapping is one function (lambda_from_<domain>(...)) plus a CLI entry; the fitting, error reporting, and significance-testing machinery is already there and does not need to be reimplemented.

Status of results — read before citing

Component	Status
Numerical PDE solver (spatial, spectral-exact shortcut)	✅ Exact (each Fourier mode assigned its analytic $\omega(k)$; no discretization in this path)
Numerical PDE solver (spatial, full leapfrog integration)	✅ Convergence order 2.0 confirmed between N=64→128 (genuinely discretized Laplacian², not a shortcut). ⚠️ N<64 unreliable (19% error at N=32). ⚠️ N>128 untested (CFL cost scales as N⁴); do not assume convergence beyond this without re-testing.
Numerical PDE solver (temporal)	✅ Validated $O(1/N)$ convergence
Discrete biharmonic operator $D_n^4$	✅ Validated $O(h^2)$ convergence
$\Lambda$-recovery from noisy data	✅ Validated, sub-percent accuracy (analytic $\omega(k)$ input; see above for recovery from actual simulated/integrated fields)
Fermi-LAT bound	✅ Audited: $\Lambda &lt; 1.4421\times10^{-53},\mathrm{m^2}$ (corrected from a $3/2$ normalization error)
EHT sensitivity ceiling	⚠️ Validated as a sensitivity estimate, not a fit to real M87* data
GW forecast	⚠️ Illustrative projection, not a fitted LIGO/Virgo/KAGRA constraint. Superseded by the matched-filter pipeline below.
GW matched-filter pipeline (Stages 1–4)	✅ Validated architecture — Stage 4 (coherent H1+L1) passes off-source null trials and injection/recovery on real strain. ⚠️ Physical inference currently blocked by a dominant waveform-model (PN-order) systematic: the GW150914 exploratory point estimate shifts from Λ=+6.57 (0PN) to Λ=−1.27 (1PN), a 65× larger swing than either statistical error bar. No physical result is claimed. See paper3/gw/matched_filter/STAGE4_FINAL_STATUS.md. Stage 5 (cross-waveform-family injection/recovery with validated waveform models) is required before any further real-event interpretation.
BEC mapping ($\Lambda=\xi^2/4$)	✅ Physically established (exact Bogoliubov coefficient match)
Dirac mapping ($\Lambda=(1-\eta^2)v_F^2/4$)	❌ Speculative, not supported by the cited literature (Fu 2009 describes an anisotropic $k^6$ effect, not this isotropic $k^4$ form)
Photonic mapping ($\Lambda=\beta_4^k/c^2$)	⚠️ Dimensionally correct and properly derived, but $\beta_4^k$ is not yet connected to any standard measurable dispersion coefficient

Any $\Lambda$ value extracted from simulated (not purely analytic) data must state the grid resolution used. Results from N<64 spatial grids should not be trusted; see paper3_h_convergence_test.py.

See paper3/paper3_final.tex for the full derivations and the honesty statement on what is proved, audited, forecast, or retracted.

Repository structure

Lambda-model/
├── README.md
├── LICENSE
├── requirements.txt
│
├── paper3/
│   ├── wave_equation_2D_solver.py       # Core PDE solver, omega(k)=ck*sqrt(1+Lambda k^2)
│   ├── paper3_Dn4_test.py               # Validates the discrete quartic operator D_n^4
│   ├── paper3_grid_convergence.py       # Temporal convergence + Lambda-recovery tests (analytic omega input)
│   ├── paper3_h_convergence_test.py     # Spatial h->h/2->h/4 convergence, REAL leapfrog PDE integration
│   ├── lambda_experimental_validator.py # Fits Lambda from (k, omega) data; domain mappings
│   ├── paper3_final.tex / .pdf          # Full writeup, including the honesty audit
│   ├── dispersion/                      # Extended validation module (fiber structural test, real-data pipeline)
│   ├── gw/                              # Gravitational-wave dispersion tests
│   │   ├── gwosc_chirp_dispersion_test.py       # legacy/invalidated — Hilbert extraction, 51.9σ bias
│   │   ├── gwosc_injection_recovery_test.py     # diagnostic — proved pipeline bias (Λ=0→Λ_fit=-1.91)
│   │   ├── gwosc_extraction_ablation.py         # diagnostic — localized bias to Hilbert-transform math
│   │   ├── gwosc_frequency_extraction_comparison.py  # diagnostic — tested 5 extraction methods
│   │   ├── gwosc_zero_crossing_injection_recovery_v2.py  # diagnostic — zero-crossing also unstable
│   │   └── matched_filter/              # VALIDATED phase-domain approach (Stages 1-4)
│   │       ├── waveform.py              # GR + Lambda phase-domain waveform model
│   │       ├── likelihood.py            # matched-filter likelihood, grid search
│   │       ├── synthetic_injection.py   # frequency-domain injection generator
│   │       ├── recovery_test.py         # Stage 1 — synthetic noise — PASS
│   │       ├── stage2_real_noise_recovery.py       # Stage 2 — real PSD, Gaussian noise — PASS
│   │       ├── stage3_real_strain_validation.py    # Stage 3 — real H1 strain — PASS (A+B,C); GW150914 exploratory
│   │       ├── stage4_coherent_h1l1_validation.py  # Stage 4 — coherent H1+L1 — PASS (4C,4D); PN-systematic found
│   │       ├── STAGE3_FINAL_STATUS.md   # Full Stage 3 audit trail
│   │       └── STAGE4_FINAL_STATUS.md   # Full Stage 4 audit trail
│   └── figures/                         # Generated plots
│
└── examples/
    ├── example_BEC.py                   # Lambda = xi^2/4 — established mapping
    ├── example_dirac.py                 # SPECULATIVE — raises a runtime warning
    └── example_photonic.py              # Corrected derivation; not yet measurable

Quick start

pip install -r requirements.txt

# Validate the numerical solver against the exact dispersion relation
python paper3/wave_equation_2D_solver.py

# Check the discrete biharmonic operator and grid/temporal convergence
python paper3/paper3_Dn4_test.py
python paper3/paper3_grid_convergence.py

# Spatial convergence test using REAL PDE time-integration (not the
# spectral-exact shortcut) -- confirms Lambda extracted from an actual
# simulated field converges as h -> h/2 -> h/4
python paper3/paper3_h_convergence_test.py

# Test whether YOUR data is consistent with the Lambda-model
python paper3/lambda_experimental_validator.py --omega data_omega.csv --k data_k.csv

# Gravitational-wave matched-filter pipeline (Stages 1-4, phase-domain
# approach validated after legacy time-domain methods were found biased)
python paper3/gw/matched_filter/recovery_test.py                    # Stage 1: synthetic
python paper3/gw/matched_filter/stage2_real_noise_recovery.py       # Stage 2: real PSD
python paper3/gw/matched_filter/stage3_real_strain_validation.py --h1 <H1.hdf5> --event GW150914
python paper3/gw/matched_filter/stage4_coherent_h1l1_validation.py --h1 <H1.hdf5> --l1 <L1.hdf5> --event GW150914

# Worked examples (each prints its own validity status)
python examples/example_BEC.py         # established mapping
python examples/example_dirac.py       # speculative — reads its own warning
python examples/example_photonic.py    # corrected, not yet measurable

What researchers in each field can do

Group	What you can do
Anyone with $(k,\omega)$ data	Run lambda_experimental_validator.py directly — no domain assumptions needed, just numbers.
Theorists	Inspect the covariant derivation in paper3_final.tex, verify the wave equation, propose new metric mappings.
BEC experimentalists	Supply your healing length $\xi$ and sound speed $c_s$; test $\Lambda=\xi^2/4$ against your measured dispersion. This mapping is physically established, not speculative.
Dirac-material / ARPES groups	The fitting tool works on your data, but do not treat $\Lambda=(1-\eta^2)v_F^2/4$ as a prediction of this framework — see the audit in paper3_final.tex §5.2 and the runtime warning in lambda_from_dirac(). If you can derive a literature-supported quartic (not $k^6$) correction for your material, we would like to hear about it.
Photonic-crystal / fiber-optics groups	$\Lambda=\beta_4^k/c^2$ is now correctly derived from the model, but $\beta_4^k$ is not yet connected to a standard telecom $\beta_4$ or any other directly-measured quantity — see paper3_final.tex §5.3 for what's missing. Contributions on this specific gap are welcome.
Numerical relativity / computational physics reviewers	Run paper3_h_convergence_test.py to independently verify the spatial convergence claim. The test integrates the actual fourth-order wave PDE via explicit leapfrog with a genuinely discretized biharmonic operator (not the spectral-exact shortcut used elsewhere in the repo) — this is the appropriate test for scrutinizing whether $\Lambda$ is a numerical artifact of grid spacing.

Honesty statement

This is a falsifiable framework, not a confirmed law of nature. A single value of $\Lambda$ that is simultaneously consistent with Fermi-LAT, EHT, GW, and BEC data would be evidence of a shared underlying mechanism. Inconsistent values would falsify the universal-$\Lambda$ hypothesis while leaving each domain-specific dispersion law intact.

This repository also documents its own audit trail: two of the three originally-proposed condensed-matter mappings (Dirac materials, photonic crystals) did not survive scrutiny in their original form and are reported here as negative/incomplete results rather than removed silently. The spatial-convergence claim underwent the same process: an earlier version of this repository asserted grid convergence based only on a spectral-exact shortcut and a temporal-resolution test, neither of which could have detected a genuine spatial-discretization problem had one existed. paper3_h_convergence_test.py closes that specific gap by testing the real leapfrog-integrated PDE instead. We consider negative and corrected results as important to publish as positive ones.

Paper 3 — Dispersion Validation

Paper 3 is currently treated as a constrained numerical and experimental-validation study rather than as a claim that every dispersive platform realizes the same Lambda model.

The central model is omega^2 = c^2 k^2 (1 + 2 Lambda k^2)

with a linear low-k branch and a quartic correction.

Current validation status

Component	Status
BEC Lambda mapping	Primary physical mapping
Lambda dispersion fitting	Implemented
Synthetic Lambda recovery	PASS
Spatial (h) convergence, real PDE	PASS, N=64-128 range only (see above)
Dimensional audit	Implemented
Fiber structural test	PASS for supplied demonstration regime
Fiber -> Lambda identification	REFUSED
Dirac independent Lambda validation	Not established
Full 2D QNM solver	Diagnosed but not yet physically complete

Fiber result

The supplied fiber event-horizon demonstration data give p = 3.99870888

for Omega ~ |Delta k|^p.

The local log-log slope is approximately 4 across the tested range.

This is consistent with a pure quartic regime and is structurally different from the Lambda-model low-k structure.

Therefore the repository does not interpret this result as an independent measurement of Lambda.

Dimensional safeguard

The experimental fiber quantity beta4 has units s^4 / m.

Consequently, beta4 / c^2

has units s^6 / m^3,

not m^2.

The validator therefore refuses the identification Lambda = beta4 / (4 c^2)

for this quantity.

This refusal is intentional and is part of the validation methodology.

Reproducibility

The Paper 3 dispersion tools can be run directly from the repository: python paper3/dispersion/lambda_experimental_validator.py

and python paper3/dispersion/fiber_event_horizon_structural_test.py

For the spatial-convergence audit specifically: python paper3/paper3_h_convergence_test.py

When no experimental input is supplied, the Lambda validator runs a synthetic self-test. Synthetic results must not be interpreted as experimental evidence.

The fiber validator currently uses a demonstration dataset and clearly labels it as such.

Real experimental data should replace the demonstration arrays before making experimental claims.

Citing this work

If you use this framework, please cite:

• Kretski, D. (2026). A Hamiltonian Oscillator Extension of Wave Propagation in Schwarzschild Spacetime. Zenodo. https://doi.org/10.5281/zenodo.22018715 (Paper 1, submitted to CQG, ref. CQG-117140)
• Kretski, D. (2026). A Hamiltonian Dispersion Framework for Kerr Photon Rings, Frequency-Dependent Shadow Sensitivity, Superradiance, and Eikonal Quasinormal Modes. Zenodo. https://doi.org/10.5281/zenodo.22051427 (Paper 2)
• Kretski, D. (2026). A Universal Quartic Dispersion Framework: Numerical Validation, Multi-Messenger Time-of-Flight Bounds, and Condensed-Matter Analog Mappings. (Paper 3, paper3/paper3_final.tex in this repository)

License

MIT License (see LICENSE). Free to use, modify, and redistribute with attribution.

Contact

Dimitar Kretski — Center for Hydro- and Aerodynamics, BAS, Varna, Bulgaria ORCID: 0000-0001-5108-2243