Validation: Soft matter (controlled test)
Frasch (2026e). Isolating the antisymmetric sector of a nonreciprocal colloidal model — kinetic unjamming, transient arrested coarsening, and irreversibility without coarse-grained circulation. Submitted to arXiv 9 September 2026; code and data at github.com/martinfrasch/nonrecip-modularity.
The claim
The four validations above are observational or computational. This one is an intervention. The hypothesis under test is a directed-graph extension of NWAP, formulated for this study and not published elsewhere: a nonreciprocal coupling can be split into a symmetric sector that selects structure by energy minimisation and an antisymmetric sector that adds circulation on top without disturbing it. Three consequences follow. The antisymmetric sector should leave equal-time structure untouched (solenoidality); its signature should appear as circulation in probability currents; and, carried over from the biology paper, it should leave a sustained modularity excess in the contact network. A fourth, weaker prediction is that cluster statistics should be controlled by the antisymmetric-to-symmetric ratio.
Method
The agent-based colloid model of Hara et al. (Phys. Rev. Lett. 137, 068302, 2026): size-asymmetric colloids driven by electrohydrodynamic flows, which exert unequal forces on each other. A reciprocity mixing parameter χ scales the antisymmetric part of the pair force while leaving the symmetric part bit-for-bit unchanged; χ = 0 restores Newton's third law exactly and χ = 1 recovers the published law. Particles, packing, noise stream and symmetric attraction are held fixed, so any difference between χ values is attributable to the antisymmetric sector alone. Four predictions were pre-registered with pass/fail thresholds before any χ run. 113 blocked-design runs at five system sizes from 10³ to 2.2 × 10⁴ particles, extended by continuations, committor relaunches, fluctuation–response twins and small-timestep calibrations.
The result
The mechanism the hypothesis assigned to the antisymmetric sector fails on all three counts, and the structural control holds.
- Solenoidality — refuted. Scaling the antisymmetric sector alone changes the cluster count by an order of magnitude, and the added drift demonstrably does not preserve the reciprocal stationary density. The sector restructures rather than circulating on top of a structure it leaves alone.
- Circulation — refuted in every projection examined. The dynamics are irreversible, with a resolved excess dissipation, yet no circulation is detectable in any coarse observable: signed-area rates in observable planes, the cluster-size coordinate, an all-pairs lag test. The antisymmetric pair law is itself predominantly gradient on the contact graph.
- Modularity excess — refuted with inverted sign. Newman modularity is degenerate on these two-dimensional contact networks, and its excess over a degree-preserving null is largest in the reciprocal reference and decreases with nonreciprocity.
- Structural control by the ratio — confirmed, non-monotonically. Weak nonreciprocity unjams the reciprocal gel (fivefold fewer clusters), strong nonreciprocity fragments it (two orders of magnitude more). No proposed functional predicts the dip; a balance of kinetic rates describes it.
What the antisymmetric sector measurably does: it unjams a kinetically arrested reciprocal gel; it sustains a fragment population whose channel pairs — shedding and reabsorption, fission and fusion, nucleation and dissolution — are each approximately balanced; the fragmented state is a long-lived transient whose condensation time scales linearly with system size; the selected scale of 8–10 particles is a critical-size crossover with a horizon-converged committor of one half; and its dissipation switches on across the same structural crossover at which the symmetric forces stop cancelling it (87–91 % of the unopposed contribution is cancelled above the crossover, all of it within error below). On single-particle coordinates one effective temperature, equal to the bath's, is shared by both species over a short-lag window that shrinks with drive.
What this domain adds to the programme
This is the discriminating test that the future-work page asked for: a laboratory-style perturbation with a single knob, in which a prediction could fail cleanly. It did. The decomposition survives as an instrument — constructing the split is what made every measurement possible, and the sector it isolates has real, measurable consequences — but the mechanism assigned to it does not, and the modularity-excess signature of the biology paper does not transfer to two-dimensional contact networks at this density (a diagnosis specific to those networks; it says nothing about the bipartite metabolic networks of Frasch 2026d, where the excess is measured against a different null on a different topology).
The programme learns three things. First, "some variational principle applies" forbids nothing; what makes a tier claim falsifiable is that each tier carries a measurable signature, and measuring one honestly is harder than the taxonomy suggests — two of three commonly invoked near-equilibrium diagnostics were defeated by their own construction. Second, whatever variational account applies to this system covers at most its dynamics at fixed structure within a short-lag window; structure selection itself is a kinetic balance that no functional tested here reproduces. That balance is the concrete thing a functional for structure selection would have to reproduce, and it is the sharpest target the programme now has. Third, the framework's author tested his own hypothesis and reports its failure; that is the register the programme intends to keep.
Code & data
- GitHub: martinfrasch/nonrecip-modularity — model, all measurements, per-run summaries, the pre-registered protocol and the manuscript source.
- Pre-registration:
paper/EXPERIMENT.mdin the repository, thresholds committed before any χ run; outcomes tabulated in the paper's Section 5.1. - arXiv: submitted 9 September 2026; identifier to follow.