Now using two specific constructions from the October 25 OpenAI mathematics rather than an isotropic analogy: the seven-coordinate conserved Friedrichs wave-current tensor and the direction-dependent Appendix D Fourier disk operator, evaluated for finite plane-wave sums. Both retain interference. We compare them to Fourier magnitude, coherent field intensity and a three-delay virtual sensor.
This controlled comparison holds frequency components and coefficient magnitudes fixed. One waveform has phase 0; the other has the selected relative phase. That makes the Fourier power spectrum *identical by construction*. The original Scalar Slider3 synthetic system remains accessible from the back link.
We reproduce the actual finite-mode source, symmetric flux matrices, vector fields and full seven-coordinate Gram tensor. The displayed curve is its physical T-time component at a fixed auxiliary point, not a post-hoc arbitrary polarization.
We use the manuscript's e(W), J(W) and A⁰TT source on a fixed Q=0 fiber, separating zero and nonzero Fourier modes before the Dirichlet solve. The graph shows a local potential-coupled energy for that fiber slice, not the theorem's final integrated functional.
No lifted current or auxiliary PDE. Just the squared modulus of the complex wave field. It can also see phase interference when supplied the complex field.
A phase-space measurement of the real signal at one fixed virtual sensor: u(t), u(t−δ), u(t−2δ). This uses vastly fewer observations than a full-field Fourier decomposition.
| Representation | Relative difference, selected phase | Detections / 64 phase trials | Input available |
|---|---|---|---|
| Computing 64 seeded tests… | |||
The detection threshold is a numerical nonzero test (change > 10⁻⁸), without added measurement noise. Not a classifier accuracy, inferential p-value or equal-cost comparison. All inputs are synthetic and known. No representation is credited with new information if ordinary complex intensity or three delays also detect the same change.
These two components are implemented using the actual manuscript equations. What is not reproduced: the full spatial current regularization Jₕ, passive Q cutoff, integration over the entire complex fiber, the quaternionic cofactor energy estimate, the global local-smoothing bound, or a proof that any physical cavity can be queried more efficiently.
For the complete current, conservation and positive semidefiniteness are tested numerically on multiple coherent phase configurations. For the planar operator, disk boundary values, coefficient dependence on direction, and Fourier equation residuals are tested. The two examples use different finite mode scales suitable for their respective coordinate charts; they are not a single coupled reproduction of the complete proof.
The question this page can answer: Can the paper-derived quantities distinguish coherent waves that have the same conventional power spectrum? Yes. Are they uniquely sensitive to such changes, or proven better than phase space, ordinary coherent intensity or much cheaper features? No. Those comparisons need a genuinely matched sensor/compute budget and a downstream prediction task.