# The Loop Is in the Medium **Rate-Dependent Hysteresis in a Kuramoto Model Whose Coupling Remembers** Diego Rincón Independent Unpublished working paper — 2026-07-20 --- ## Abstract In the standard Kuramoto model the coupling $K$ is a control parameter set from outside, and for a unimodal frequency distribution the synchronisation transition is continuous: sweeping $K$ up and back down retraces the same curve. The `/field/medium` instrument on this site does something different — it derives coupling from the state of a connective medium, $K = K_{\max} \cdot Q \cdot (1 - D)$, where hydration $Q$ and densification $D$ have their own dynamics and $D$ builds slowly and takes work to reverse. $D$ is a memory term, so the medium's coupling depends on what has been done to it rather than only on the present drive. This paper measures the consequence. In a paired design across 16 seeds, sweeping shear up and back down produces a hysteresis loop in the memory arm that a memoryless control does not show: $t = 25.3$ where the sweep outruns the memory's relaxation, decaying monotonically and becoming statistically indistinguishable from zero in the quasi-static limit ($t = 0.88$, 9 of 16 seeds). Across a $5 \times 5$ parameter sweep the loop is positive in all 25 cells and its width correlates with the build-to-break ratio of densification at $r = 0.823$ — a dose-response on the parameter that constitutes the memory. The transition remains second-order: the memory lags it without sharpening it. What this does **not** establish is that the result is specific to this framework; rate-dependent hysteresis is the generic signature of any slow auxiliary variable, and that limit is stated rather than argued around. ## 1. The model Standard Kuramoto treats $K$ as given. Here it is derived: $$K = K_{\max} \cdot Q \cdot (1 - D)$$ $Q$ is hydration — free water against water-binding capacity. $D$ is densification — the aggregation that makes ground substance viscous and sticky when it is left still. Shear pumps hydration back and breaks aggregation; stillness drains hydration and lets densification accumulate. Densification relaxes on a timescale $\tau_D \approx 175$ integration steps at the parameters used here. The distinguishing feature is that $D$ builds slowly and takes work to reverse. The medium's coupling therefore depends on its history, not only on the shear currently applied. The framework calls this DC4 — *path dependence: cost depends on the route taken, not only the distance*. Stated that way it is a posited condition. The question here is whether it is a measurable one. ## 2. A first attempt that answered the wrong question The first experiment held each sweep point for 4000 steps before measuring, so the control arm would be honest. That is roughly 23 $\tau_D$. The memory variable equilibrated at every point, the loop was erased by construction, and the result was a clean null. This is recorded because the null is misleading and was nearly believed. It answered *is there a loop in the quasi-static limit*, where the answer is trivially no for any relaxational system. The design and the hypothesis were incompatible: a memory effect can only appear when the sweep is fast relative to the memory. That is a design error, not a finding, and any citation of it is a citation of nothing. ## 3. The measurement Sweeping shear up and back down at a range of hold times, in a paired design — the same natural frequencies and the same initial phases in both arms, so memory versus control is compared within seed rather than across — over 16 seeds. Loop area is the enclosed area between the up-sweep and down-sweep order parameter curves. | hold (steps) | $\div \tau_D$ | memory | control | difference | $t$ | seeds with diff > 0 | |---|---|---|---|---|---|---| | 150 | 0.85 | 0.155 ± 0.021 | 0.061 ± 0.024 | 0.094 ± 0.015 | **25.30** | 16/16 | | 300 | 1.71 | 0.071 ± 0.009 | 0.028 ± 0.012 | 0.044 ± 0.008 | **20.61** | 16/16 | | 600 | 3.42 | 0.028 ± 0.009 | 0.012 ± 0.010 | 0.016 ± 0.009 | **7.12** | 15/16 | | 1200 | 6.84 | 0.010 ± 0.004 | 0.004 ± 0.005 | 0.006 ± 0.004 | **6.36** | 15/16 | | 2400 | 13.68 | 0.002 ± 0.005 | 0.001 ± 0.005 | 0.001 ± 0.005 | 0.88 | 9/16 | $|t| > 2.13$ is $p < 0.05$ at 15 df, two-tailed. The shape is the whole result. The memory arm loops more than the control wherever the sweep outruns relaxation, the excess decays monotonically as the sweep slows, and at 13.68 $\tau_D$ it is indistinguishable from zero — 9 of 16 seeds is a coin flip. A loop that persisted quasi-statically would indicate something other than memory; a loop that never appeared would indicate no memory at all. This is the signature in between. **On the value of pairing.** An unpaired replication across 24 seeds gives ratios of 2.2–2.9× at settled rates, with enough seed-to-seed variance to argue about. A single seed gave 7.9×, which is a lucky draw and does not survive. The paired design removes the variance that made those numbers unstable: the same comparison, correctly conditioned, moves from arguable to $t = 25.3$. ## 4. Dose-response If the loop is produced by densification memory, its width should scale with how much memory there is. Sweeping the build rate $a_D$ and break rate $b_D$ over a $5 \times 5$ grid, paired, 6 seeds per cell, at a fixed sweep of 1.71 $\tau_D$: - The loop is **positive in all 25 cells**, ranging $38.1$ to $104.1 \times 10^{-3}$. - $\text{corr}(\text{build/break ratio}, \text{loop width}) = \mathbf{0.823}$. - Widest at $a_D = 0.55$, $b_D = 0.08$ — build/break ratio 6.88. - Narrowest at $a_D = 0.10$, $b_D = 0.15$ — ratio 0.67. An artefact of sweeping too fast would not track the build-to-break ratio; it would track the sweep rate alone, which is held constant here. The effect scaling with the parameter that constitutes the memory is the strongest single piece of evidence that the memory is what produces it. ## 5. What kind of transition Transition steepness in the widest-loop cell: memory 9.92, control 11.66, ratio 0.85. In the narrowest: memory 7.79, control 7.71, ratio 1.01. A steepness ratio near unity means the memory **lags** the transition without sharpening it. This is a hysteretic second-order transition, not the explosive first-order synchronisation seen in Kuramoto variants with frequency-degree correlation or inertia. The distinction matters: explosive synchronisation produces hysteresis by a different mechanism — bistability — and would show a steepness ratio well above 1. ## 6. Limits, and the one that matters most **Rate-dependent hysteresis is generic.** Any system with a slow auxiliary variable coupled to a control parameter will show a loop that grows as the sweep outruns the slow variable and vanishes when it does not. That is textbook, and it is what this measures. Nothing here shows the effect is peculiar to the displacement framework, to this medium model, or to connective tissue. What the result licenses is narrow: *in this model, path dependence is present, measurable, and scales with the memory that produces it.* It does not license the framework over any account that also posits a slow variable — and this is the third time in one day that a framework claim has turned out to be true in the company of every rival. That pattern is the finding, and it is not a comfortable one. **Other limits.** - One frequency distribution (unimodal), $N = 200$, one integration scheme, one definition of loop area. - The steepness result is two cells, not a fit across the grid. - $\tau_D \approx 175$ steps is measured at the default parameters and shifts across the $5\times5$ sweep, so "1.71 $\tau_D$" is nominal in §4. - The medium model is a caricature of ground substance. Nothing here is a claim about connective tissue, and the mapping from $Q$ and $D$ to hydration and densification is by construction rather than by measurement against any real material. - No comparison against an alternative memory implementation. Whether the specific functional form $K = K_{\max} Q(1-D)$ matters, or any slow variable in the coupling would do, is untested — and §6's first paragraph suggests the latter. ## Provenance Reproducible from this repository: ``` python3 research/run-experiment.py hysteretic-kuramoto/ensemble.ts # §3 python3 research/run-experiment.py hysteretic-kuramoto/phase-diagram.ts # §4, §5 python3 research/run-experiment.py hysteretic-kuramoto/rate-sweep.ts # single-seed python3 research/run-experiment.py hysteretic-kuramoto/experiment.ts # §2, the void null python3 research/hysteretic-kuramoto/replicate.py 24 # unpaired ``` Recorded output for each is committed under `research/results/`. All four experiments were written before this paper and, until 2026-07-20, had never been run. The verdict line in `rate-sweep.ts` still reports "not supported"; its criterion is unsatisfiable as written and is analysed in `research/hysteretic-kuramoto-note.md`. It has deliberately not been changed — rewriting a pass criterion until a result passes is not a repair.