Where GR Breaks: The Sagnac Reference-Switch Test

There is a single rotating loop on which Einstein’s general relativity and Quantum Traction Theory make different, sharp predictions. Not a vague difference — a number. Standard metric physics gives no independent A1 projection factor, so two calibrated reductions of the same loop should agree. QTT says a true LAB/ABS reference switch differs by a fixed factor, cos(π/8), forced by the geometry of two clocks. A built reference-switch would decide between the two descriptions. This note states the prediction cleanly, shows why the best matter‑wave data already confirms the part both theories share, and isolates the one experiment that has never been run.

The Sagnac holonomy — both theories agree

Send two beams around a loop in opposite directions. If the loop rotates, they do not return together. The time difference is the Sagnac time‑lapse, and its law is one of the cleanest in physics:

where Ω is the rotation rate and A the area vector of the loop. This is a carrier‑independent clock‑holonomy: it holds for light, microwaves, and matter waves alike. Both general relativity and QTT predict it, and QTT marks it confirmed. Everything in this note sits downstream of agreement on this baseline.

Matter‑wave gyroscopes confirm a QTT prediction

Large ring‑laser gyroscopes measure the Earth’s rotation through continuous‑phase readout at very high precision. Cold‑atom interferometers add an important carrier test: a two‑axis caesium interferometer enclosing roughly 11 cm² reproduces the Sagnac law for matter waves and agrees with the prediction at the level of 25 parts per million (Gautier et al., 2022). Atom interferometry has even been used to measure absolute geodetic rotation directly (Stockton, Takase & Kasevich, 2011).

This is not a problem for QTT. It is one of its confirmations. QTT predicts exactly this result for the lab‑clock reduction of the holonomy, and the book lists the 25 ppm caesium measurement among the results it treats as verified. Matter‑wave gyroscopes confirm the Sagnac law QTT already predicts.

One point of clarity is essential, because it is easy to get wrong. An atom interferometer takes a single projective readout at the end — you prepare the superposition, let the phase accumulate during free flight without watching it, and read the populations once. It is tempting to call that the “absolute” channel. It is not. The split, the redirection, and the closure are all driven by laser pulses whose phases are set by laboratory oscillators; the final population estimates a phase referenced to the lab clock τ:

A single final readout is not enough to make a channel absolute. “No intermediate observation” is not the same as “referenced to the deeper clock.” So the caesium result measures the lab reduction S_τ — the holonomy — and not the ratio that distinguishes QTT from GR.

The universal tilt cos(π/8)

QTT’s first axiom carries a radical structure: there is an Absolute Background Clock T — the fastest heartbeat of the universe — and every laboratory clock τ is a tilted, slower projection of it. In the phase‑clock plane the two are separated by a quarter turn; amplitudes see the half‑angle, so the projection of T onto the lab axis carries a single fixed number:

There is no freedom to move this value. Once the A1 geometry is fixed, I_clk is forced — not fit, not tuned, not chosen. It is the same half‑angle that appears across the corpus wherever the two clocks are compared.

A reference‑switch on one loop

The prediction is not about reading the loop once versus continuously. It is about reading the same loop under two reference conventions and taking their ratio:

• S_τ — the LAB reduction. Continuous‑phase, lab‑clock‑referenced. This is what ring lasers, fiber gyros, and ordinary atom interferometers measure.
• S_T — the ABS reduction. An absolute‑transport phase, carried on T and projected once against the lab channel at closure, on the same instrument.

For any carrier — light or matter — QTT’s claim is that the absolute‑transport slope is the lab slope reduced by the tilt: S_T = I_clk S_τ. Their ratio is the test:

This is a ratio between two reductions of the same holonomy on the same loop. Note what it is not: it is not “single readout versus continuous,” and a single projective measurement does not by itself make a channel absolute. The absolute channel requires a genuinely T‑referenced (absolute‑transport) reduction. That distinction is the whole experiment.

Two outcomes, no middle ground

Measure R = 1 on a true reference‑switch, and the A1 clock‑tilt Sagnac projection fails. Measure R = cos(π/8), and the two‑clock projection is observed directly. The gap between 1.000 and 0.924 is 7.6% — enormous next to the sub‑percent precision these instruments reach.

The open burden, stated plainly

Every existing high‑precision Sagnac dataset — ring lasers, fiber gyros, and matter‑wave interferometers — implements a single reference convention: the lab clock τ. Any “frequency” route reduced through the round‑trip time t_rt = P/c is algebraically the same lab phase. So they all return the holonomy, R = 1, by construction — exactly as observed. None has produced both S_τ and S_T as two reductions of the same loop.

The experiment is exactly one thing: realize a genuinely T‑referenced, absolute‑transport reduction of the same rotating loop, distinct from the lab‑clock readout, and take the ratio. That reduction is what has never been built — and it is what carries the tilt.

Stating the absolute‑transport reduction operationally — what physical source convention makes the carried phase T‑referenced rather than τ‑referenced — is the live design problem, and it is where the prediction earns or loses its testability. The corpus places the realization on a Folman‑class atom‑chip platform, the same family of apparatus that already demonstrates the finite‑access faces of the real‑phase spine in one instrument.

Same loop, two channels, one ratio

1. LAB channel. Standard continuous‑phase (or beat‑frequency, reduced via t_rt = P/c) readout. From a rotation sweep, fit the slope S_τ = (Δφ/Ω)_τ.
2. ABS channel. The same loop, read under an absolute‑transport source convention referenced to the absolute timebase: accumulate the phase across the bundle and take one amplitude projection at closure. Fit S_T = (Δφ/Ω)_T.
3. Compare. Form R = S_T/S_τ. Test against cos(π/8) with a two‑one‑sided test (TOST) at a pre‑registered margin, e.g. ±0.5%. The 7.6% tilt is large enough to resolve cleanly at sub‑percent uncertainty.

• R. Gautier et al., “Accurate measurement of the Sagnac effect for matter waves,” Science Advances 8, eabn8009 (2022). arXiv:2206.06696. — Caesium two‑axis interferometer, ≈11 cm², Sagnac law confirmed at 25 ppm (the LAB‑reduction confirmation).
• J. K. Stockton, K. Takase & M. A. Kasevich, “Absolute Geodetic Rotation Measurement Using Atom Interferometry,” Phys. Rev. Lett. 107, 133001 (2011). OSTI 21611785.
• Quantum Traction Theory, the book (v10.01), DOI 10.5281/zenodo.17527179. Sagnac readout‑projection & holonomy section; clock‑tilt I_clk = cos(π/8) (A1). Field‑note anchors: Sagnac clock-holonomy law, A1 clock tilt, and the pending dual-channel reference-switch test.
• Supporting deposits: 10.5281/zenodo.20042612 · 10.5281/zenodo.20070485.

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