# Winding, not duration: geometric phase as a process clock and decision trigger

*Working title:* **Winding, not duration: geometric phase as a process clock and decision trigger in a classical belief system**

Author: [withheld in the repository copy]. Contact: [withheld].
Status: **DRAFT — not submitted. Luke submits by hand, and only if he chooses to.**
Revision: 2026-11-02 second sweep (related work, venues).

| | |
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| Venue (dated door) | Chaos (AIP Publishing), Special Topic "In Memory of Professor Erik Bollt: Celebrating Contributions to Nonlinear Dynamics, Chaos, and Complex Systems"; guest editors Belykh, di Bernardo, Porfiri, Porter, Sun; topics include Information Theory, Causal Inference, Data-Driven Dynamical Systems (Koopman and data-driven dynamics confirmed in scope), Network Dynamics. The only Chaos special topic open when the AIP list was re-read on 2026-11-02 |
| CFP | https://pubs.aip.org/cfp/pages/ebolt2026 |
| Deadline | **2027-01-31 (no time zone stated on the page; re-read 2026-11-02, unchanged)** |
| Format | Chaos article with a non-specialist Lead Paragraph; Fast Track ≤5,750 words (regular length not stated on the page read) |
| Review | Not stated on the pages read; AIP standard assumed (UNVERIFIED) |
| Fee / barrier | "AIP Publishing does not require page or color charges" (subscription route costs nothing; the arXiv preprint stays open); Author Select OA $3,800; waivers case-by-case for financial need. 2025: 31 days to first decision, 107 to acceptance; JIF 3.3 |
| Home | The Broadcast's paper section: `docs/papers/geometric-phase-process-time.md`, served at `/chartroom/geometric-phase-process-time` with its hash |

**Other doors**

| Venue | Deadline | URL | Fee / barrier |
|---|---|---|---|
| Physical Review Research | Rolling | https://journals.aps.org/prresearch/about ; https://journals.aps.org/authors/apcs | Fully OA; APC $2,910 (2026), CC BY; "APS does not currently provide other discounts or waivers", so no free route for an unaffiliated author in a high-income country; single-anonymized; ORCID required; handling of an empty affiliation UNVERIFIED |
| Physical Review E (Statistical physics; "Computational Physics, Machine Learning, and AI") | Rolling | https://journals.aps.org/pre/about | Subscription route, no charge listed; OA $2,910; single-anonymized; 2025: 6 days to first decision, 126 to acceptance |
| Entropy (MDPI) | Rolling | https://doaj.org/api/search/journals/issn:1099-4300 (mdpi.com returned 403) | APC up to CHF 2,600; a waiver policy exists but was unread; special issues UNVERIFIED |
| arXiv cond-mat.stat-mech (cross-list nlin.AO, cs.LG) | Rolling; 14:00 ET cutoff | https://info.arxiv.org/help/endorsement.html | A `cond-mat` endorsement covers cond-mat.stat-mech; cs.LG needs a `cs` endorsement; whether `nlin` is its own domain is not stated in the note (UNVERIFIED). Since 2026-01-21 automatic endorsement needs an academic email **and** prior arXiv authorship in the domain, so a first-time unaffiliated author goes by personal endorsement; "arXiv staff cannot waive endorsement requirements" |

## Lead paragraph (Chaos requires one)

A clock counts seconds whatever happens. A process has another time: how far its state has turned. We time a belief system by the solid angle its state sweeps on a sphere, its geometric phase, and we show on replayed data from a public game that a decision read from that angle alone is reached about six times sooner than a control (at least four times, as pinned), while at a fixed clock budget the two processes decide oppositely. The angle refuses to report when the path has not closed. We also report where the idea failed.

## Abstract (draft)

Clocks count seconds. A process has another time: the geometry its state path sweeps. We time a classical belief system, a density matrix over a small graph of qubit clusters simulated on a CPU, by the geometric phase its Bloch paths accumulate, not by the wall clock. A Berry tape (our name for the running record) integrates the solid angle of each closed walk on the Bloch sphere by L'Huilier's formula over the path polyline and stamps learning by that angle. Three honesty rules follow from the geometry. Winding on an open walk is NaN, never zero; holonomy refuses an open trajectory; loop invariants are emitted only at a certified return, and the absence of the key is the refusal. A closed equatorial loop reports W = 1 and Γ = π, a pole-hugging loop Γ → 0, and the same loop at three speeds agrees to 1e-9: traversal speed does not change the invariant. On replayed geometry from a public game whose ripeness mechanic is the Aharonov–Anandan phase, a harvest gate reading only accumulated phase opens at t = 300 for a driven loop and at t = 1800 for a pole-hugging control (≥4× pinned, ~6× measured), and at a fixed budget the two processes choose oppositely. Two negatives follow. Berry-phase rate failed a pre-registered test as a market regime detector in two disjoint years (AUC 0.4333 and 0.4322), and no winding gate yet holds live decision authority. That is owed and stated as dark.

## Contributions

1. **Process time as holonomy.** The Berry tape stamps the field's learning journey by accumulated solid angle of closed Bloch walks (L'Huilier's formula per slice). We place it exactly: a discrete Bargmann-invariant, Mukunda–Simon kinematic phase on the spin-½ sphere, used as a clock.
2. **Closure honesty as a refusal law.** Winding on an open walk is NaN, holonomy refuses open trajectories, loop invariants are emitted only at certified returns, and the absence of the key is the refusal. This departs from the literature's default of geodesic closure (Samuel–Bhandari) and is defended by the unravelling ambiguity (Pawlus–Sjöqvist).
3. **Speed invariance.** The same closed loop at three speeds reports identical winding and holonomy to 1e-9; a closed equatorial loop reports W = 1, Γ = π; a pole-hugging loop Γ → 0.
4. **The decision result on foreign geometry.** A harvest gate reading only accumulated γ opens at t = 300 for a driven loop and t = 1800 for a pole-hugging control (≥4× pinned, ~6× measured); at a fixed budget the two processes yield opposite choices. Because the game's world never sits still, the honest contrast is winding rate.
5. **Three clocks kept apart.** Compute time (dt, φ-clock, Berry tape), membrane cadence (`ingest_hold_s`), and in-universe time (drivers, opt-in); the hold is inert when unset.
6. **Two negatives and one refusal.** Berry-phase rate as a market regime detector: AUC 0.4333 (2025) and 0.4322 (2024), direction consistent, pre-registered filter never cleared. Live decision authority: DARK. Tape persistence across save/load: not claimed.

## Outline (about 7,000 words)

1. **Lead paragraph and introduction** (900 words). Business time counts trades, not seconds; thermodynamic length bounds how fast a protocol may run; the Hannay angle is gained per cycle however long the cycle took. We take the step none of these takes: read the geometric quantity as the time variable and let duration drop out. We say up front that the gate has never acted live.
2. **Geometry on the Bloch sphere** (900). The belief is a density matrix over clusters of qubits; each qubit's Bloch vector has a direction on the sphere. A closed walk of that direction encloses a solid angle, computed by L'Huilier's formula over the polyline and summed as the tape. For pure states this is the Aharonov–Anandan phase. For mixed states we claim only the kinematic solid angle of the direction, and we point to the open-system phase literature for what a physical phase would require.
3. **Closure honesty** (700). Three rules and the eleven tests behind them. A walk that has not returned has no loop, so its winding is NaN and its holonomy is refused; an absent key is the refusal. We contrast this with closing the path by a geodesic and explain why a belief system should refuse rather than close: an unclosed walk has no unique phase, and a typed zero would be a claim nobody measured.
4. **Three clocks** (500). Compute time, membrane cadence and in-universe time are kept untangled by design. The tape belongs to compute time. Cadence is a zero-order hold across one gap and is inert when unset. Confusing any two of these is how a clock is smuggled back in.
5. **Speed does not move the invariant** (400). The speed-limit literature proves duration ≥ geometric length over cost. We hold the geometry fixed and vary speed: the invariant does not move to 1e-9. That pin licenses reading geometry as time.
6. **The gate and the replay result** (900). Two tapes are built from the game's native solid-angle integrator over real playtester geometry, on the same machinery and clock. A gate opens when accumulated γ crosses a threshold. The driven loop opens at t = 300, the pole-hugging control at t = 1800. At a fixed clock budget between those times one process has decided and the other has not, so they choose oppositely. We state the pinned factor (≥4×) and the measured one (~6×) separately and explain the gap as sampling.
7. **The public instance** (500). In SpaceWheat a biome is a density matrix evolved exactly, and ripeness is the signed solid angle a closed Bloch walk banks, ripe at 2π, which the game's own ledger grades exact as the Aharonov–Anandan phase of the actual path. The S³ lift gives γ = Ω/2 = π at Ω = 2π with closure at 4π, and Gauss linking refuses open lifts. Zeno pinning in the game is the stopped clock: a watched state does not turn, so its process time stands still.
8. **What failed, and what is owed** (600). Berry-phase rate as a trending-versus-choppy market detector: pooled window AUC 0.4333 (2025) and 0.4322 (2024), direction consistent, pre-registered filter not cleared. No winding gate makes a consequential production decision; that is DARK. The tape does not survive save/load as part of the continuation gap; not claimed.
9. **Related work and the negative space** (900). Below.
10. **Threats; what this paper is not** (500). Below.

## Related work and the negative space

**The geometric-phase canon.** Berry (1984), Proc. R. Soc. Lond. A: the phase for a cycled spin is the solid angle swept (the solid-angle statement is from memory of the paper; the record is verified). Simon (1983), PRL: holonomy. Aharonov and Anandan (1987), PRL: the phase depends only on the closed curve in projective Hilbert space. Samuel and Bhandari (1988), PRL: open paths closed by a geodesic (content from memory). Mukunda and Simon (1993), Ann. Phys.: the kinematic, gauge-invariant definition via Bargmann invariants, whose discretization is the polygon-on-the-sphere computation.

**Classical and stochastic geometric phases.** Hannay (1985), J. Phys. A: angle holonomy per adiabatic cycle, independent of duration. Sinitsyn and Nemenman (2007), EPL, and Sinitsyn and Nemenman (2007), PRL: pumped flux in stochastic kinetics has a Berry-phase term. Sinitsyn (2009), J. Phys. A: the topical review. Rahav, Horowitz, Jarzynski (2008), PRL: directed flow and a no-pumping theorem.

**Open systems.** Sarandy and Lidar (2006), PRA: geometric phases for convolutionless master equations. Pawlus and Sjöqvist (2010), PRA: trajectory phases depend on the unravelling, the published argument for refusing a value.

**Geometry and time.** Crooks (2007), PRL, and Sivak and Crooks (2012), PRL: thermodynamic length. Ito (2018), PRL; Ito and Dechant (2020), PRX; Shiraishi, Funo, Saito (2018), PRL; Nicholson et al. (2020), Nature Physics: speed limits, in which geometry bounds duration. Clark (1973), Econometrica, and Ané and Geman (2000), J. Finance: business time. Wan et al. (2021), NeurIPS (venue verified 2026-11-02 via OpenAlex), and Li and Arora (2020), ICLR (venue from memory; UNVERIFIED after registry check 2026-11-02): angular update as the intrinsic clock of training.

**Measurement rate and the stopped clock.** Misra and Sudarshan (1977), J. Math. Phys., and Facchi and Pascazio (2008), J. Phys. A: Zeno dynamics. Erez et al. (2008), Nature: the measurement interval alone decides heating or cooling. Still, Sivak, Bell, Crooks (2012), PRL: non-predictive memory is dissipation.

**Thin frontier.** Du (2025), arXiv:2509.10900: geometric phase of stochastic oscillators. Zhang, Yang, Baptista, Zhang, Zhu, Lin, Grebogi (2026), arXiv:2607.15472: a machine-learned "universal dynamical clock" (uniform rotation through an equant-induced coordinate) from whose phase "a classical-mechanics counterpart of the Berry geometric phase emerges"; the clock produces a phase, it does not read accumulated holonomy as time (abstract read 2026-11-02; body not read). Magnot (2025), arXiv:2509.10536: group-valued holonomies along cycles of a Boltzmann-machine graph, learning framed as curvature flattening (abstract only).

**Negative space.** Stochastic pumping yields a geometric flux per cycle, not a clock, and nobody in that line triggers a decision on accumulated phase. Thermodynamic length and the speed limits make geometry bound duration; we invert the use and read geometry as time. Business time replaces seconds with event counts; the tape is business time whose ticks are solid angle. Angular-update clocks measure training progress as angle on a sphere of weights, not holonomy of a state path, and gate nothing. Samuel–Bhandari and Mukunda–Simon define open-path phases by closure; we refuse them, with Pawlus–Sjöqvist as the warrant. The observation that two processes choose oppositely at a fixed budget has no counterpart found. No source applies a Berry or Hannay phase to the dynamics of a learning system; the three 2025–2026 preprints are the nearest and are unreviewed. This sentence is not yet earned: the decisive searches ("holonomy as clock", "Berry phase in RL", "solid angle as trigger", "intrinsic time in RL") were NOT RUN on 2026-11-02 for budget and must be run before it is printed.

**Second sweep (2026-11-02).** The geometric-phase searches that ran were on radar, antenna arrays, neural training dynamics and pedagogy; the four that decide the gap sentence above ("accumulated geometric phase or holonomy as an intrinsic clock in a stochastic system", "Berry phase in a reinforcement-learning agent's decision", "solid angle on the Bloch sphere as a trigger in a learning system", "event-driven intrinsic time in reinforcement learning") were NOT RUN because the session's search budget closed, so the verdict on this paper's central claim is unchanged on evidence, not unchanged after search. Run them before the gap sentence is printed. What the sweep did find: the nearest neighbour is Zhang et al. (2026), arXiv:2607.15472, where a learned universal dynamical clock yields a classical counterpart of the Berry phase, the opposite direction from reading holonomy as the clock. In signal processing, geometric phase appears only as hardware and phasing: Bhandari (2000), arXiv:astro-ph/0003396, for IAU Symposium 199, where the phase between two differently polarised antennas has a geometric part that is discontinuous at singular points of parameter space, so a geometrically phased array can look in two directions at two wavelengths; and US patent 7,714,782, "Phased arrays exploiting geometry phase", for direction-of-arrival and beamforming. Every radar hit was a Pancharatnam–Berry metasurface for radar-cross-section reduction, not signal processing; the time–frequency and analytic-signal query did not run. For the public SpaceWheat instance (§7 of the outline), the pedagogy neighbours are Cavinato, Barbieri, Giliberti (2018), Eur. J. Phys. 39(5), DOI 10.1088/1361-6404/aacfcc, the vector potential and gauge invariance at high school, with its classroom precursor Barbieri and Giliberti (2015), Proc. GIREP-MPTL 2014, pp. 165–174; Huang and Zhao (2024), Eur. J. Phys. 45, DOI 10.1088/1361-6404/ad1397, a pedagogical introduction to honeycomb lattice gauge theory (given names not retrieved from Crossref); and Egri, Fodor, Hoelbling, Katz, Nogradi, Szabo (2007), Comput. Phys. Commun. 177, 631–639, DOI 10.1016/j.cpc.2007.06.005, "Lattice QCD as a video game", which is GPU computing, not pedagogy. No game that teaches gauge theory or a geometric phase was found, and no physics-education study of gauge-invariance misconceptions was located (a Hoyos–Pocoví record on "invariance" misconceptions surfaced as a snippet only; UNVERIFIED). Venues: the Chaos Erik Bollt Special Topic is the only Chaos special topic open as of 2026-11-02, deadline 2027-01-31 with no time zone stated, five guest editors confirmed; arXiv's endorsement rule changed on 2026-01-21 (automatic endorsement needs an academic email and prior arXiv authorship in the domain, else personal endorsement, no staff waiver); OpenReview moderation (up to two weeks on a personal email) does not touch this paper's doors, which are AIP, APS and MDPI. Verdicts: geometric phase as process clock or decision gate, unchanged on evidence with the decisive searches owed; closure honesty, unchanged, not searched; geometric phase in learning dynamics, narrowed to three unreviewed preprints (Du 2025, Magnot 2025, Zhang et al. 2026), none applying a phase to a learning trajectory; gauge pedagogy through a game, unchanged, with the three neighbours above.

## Evidence each claim rests on

| Claim | Pointer | Tier |
|---|---|---|
| Winding is NaN on an open walk; holonomy refuses open trajectories; closed equatorial loop W = 1, Γ = π; pole-hugging loop Γ → 0; same loop at three speeds identical to 1e-9; loop invariants emitted only at certified returns; absence of the key is the refusal | DARK: needs public evidence (a public repro or measurement) before it can be claimed | DARK |
| Two tapes off the game's native solid-angle integrator (L'Huilier over the path polyline); a harvest gate reading only accumulated γ opens at t = 300 (driven) vs t = 1800 (pole-hugging control), ≥4× pinned, ~6× measured; at a fixed budget the two processes yield opposite choices; the honest contrast is winding rate | DARK: needs public evidence (a public repro or measurement) before it can be claimed | DARK |
| Live decision authority (a winding gate making a consequential production decision) | — | DARK (owed) |
| Berry-phase rate as a trending-vs-choppy regime detector: pooled window AUC 0.4333 (2025) / 0.4322 (2024); direction consistent (higher rate in choppy windows); never clears the pre-registered filter | DARK: needs public evidence (a public repro or measurement) before it can be claimed | DARK |
| Berry-tape persistence across save/load | — | NOT CLAIMED (part of the continuation gap) |
| Three times kept untangled: compute time (dt, φ-clock, Berry tape), membrane cadence (`ingest_hold_s`), in-universe time (drivers, opt-in); hold inert when unset | DARK: needs public evidence (a public repro or measurement) before it can be claimed | DARK |
| SpaceWheat: ripeness = a closed Bloch walk banks signed solid angle (L'Huilier per slice), ripe at 2π; the Aharonov–Anandan phase of the actual path | SpaceWheat `BerryPhaseRegister.gd`; `docs/FOR_PHYSICISTS.md` | exact (owner's grade) |
| SpaceWheat: holonomy of the S³ lift γ = Ω/2 = π at Ω = 2π, closure at 4π; Gauss linking refuses open lifts (NaN) | `KnotRegister.gd`, `GaugeField.gd`, `BridgeRegister.gd`; `docs/design/GAUGE_CAMPAIGN.md` | exact / faithful / suggestive per ledger |
| SpaceWheat: each biome is a density matrix evolved exactly by U = exp(−iH·dt); purity conserved to machine precision; Zeno pinning among the authored channels | `Core/QuantumSubstrate/`, `native/src/`; `tools/zeno_assay.py` | exact within the model |
| The engine reads real SpaceWheat playtester tapes; a flat tape is reported as an honest null (the session that caught a game bug) | DARK: needs public evidence (a public repro or measurement) before it can be claimed | DARK |
| SpaceWheat public revision and releases | repo `8e67bcc` (2026-10-31); v0.2.0 (2026-06-17, Linux), v0.1.7-alpha (2026-10-19, Windows) | repo record |
| Learning outcomes, playtest counts, engagement for the game | — | DARK |

## DARK (do not type a number)

- Live decision authority for any winding gate.
- Berry-tape persistence across save/load (continuation gap; not claimed).
- Any use of the market Berry-rate signal after its filter; the window count is in RESULTS_S2.md and is not re-typed here.
- Learning outcomes or player data for the ripeness mechanic.
- Cross-platform determinism of the solid-angle integrator.
- Biome and channel counts in the game drift between documents; re-derive with `tools/atlas_plates.py` before quoting any.

## Threats to validity

- Replay, not live: the decision result is on recorded geometry, and the gate has never acted.
- One geometry from one game; the pole-hugging control is constructed, so the contrast is by design. The paper must say what a non-constructed control would be.
- The factor is 4× as pinned and 6× as measured; the gap is sampling, and the paper must show it.
- For mixed states the tape is a kinematic solid angle of the Bloch direction, not a measured interference phase; the open-system literature shows the physical phase depends on the unravelling.
- Discretization: L'Huilier over a polyline under-resolves fast turns; the 1e-9 speed pin is on a synthetic loop.
- The market negative is two years under a pre-registered filter; it is a negative for that use, not evidence against the clock.
- One author; the referees are the owner's CI tests.

## What this paper is not

- Not a quantum hardware or quantum advantage paper; everything runs on a CPU.
- Not a measurement of a physical Berry phase in an experiment.
- Not a stochastic-pump result, and not a thermodynamic bound.
- Not a trading signal; the market result is a negative.
- Not a learning-outcomes study of the game.
- Not a claim of live decision authority, and not a claim that the tape persists.
- Not a paper about, or citing, any earlier deployment the engine descends from.

## Luke's steps, if he wants to submit

1. ORCID (free; APS requires it for the corresponding author, IOP for the submitting author).
2. arXiv: register with organization "Independent"; start a cond-mat.stat-mech submission to trigger the endorsement email; find endorsers via "Which authors of this paper are endorsers?" on the abstract pages of Sinitsyn (2009), Crooks (2007) or Ito (2018). A `cs` endorsement is needed only for the cs.LG cross-list. Since 2026-01-21 the automatic path needs an academic email and prior arXiv authorship in the domain, so expect the personal-endorsement path; staff cannot waive it. Moderators may reclassify, so choose the primary by method.
3. Have a seat expand this scaffold into the Chaos template with the non-specialist Lead Paragraph; choose the subscription route (no charge). If Physical Review Research is preferred, budget $2,910 or use Physical Review E's subscription route instead.
4. Re-run [private, withheld] and [private, withheld] on a fresh checkout; record the SpaceWheat revision whose tapes were replayed (`8e67bcc`) and the engine commit SHA; regenerate `docs/figures/berry_loop_trace.png`. (DARK: needs public evidence (a public repro or measurement) before it can be claimed.)
5. Add a figure of accumulated γ against clock time for both tapes with the gate threshold and the two opening times marked; add the market negative as a table with its window count from RESULTS_S2.md.
6. Submit to the Chaos Special Topic before 2027-01-31; no time zone is stated, so treat it as end of day UTC-12 to be safe. Log the submission in the authority record when there is a receipt.