Oscillation as a constitutive substrate for perception

Is oscillation functional, or decorative? Does a perpetually-circling system of coupled networks — a non-settling “perceiving” system — represent distinctions a saturated, static network of equal capacity cannot?

Pre-registered · multi-phase Stuart–Landau triad + trained GRU triad central hypothesis refuted adversarial metric harnesses · multi-seed perceptome program

The minimal substrate is a cyclic triad A→B→C→A whose non-transitive (rock–paper–scissors) coupling forbids a common fixed point and forces a rotating limit cycle — a 120° three-phase splay wave. The program asked, in pre-registered phases, what this collective oscillator does. Each answer forced the next question: from memory, to temporal regulation, to the deepest one — what is the irreducibly-active ingredient of a living oscillation?

Headline. A trainable triad self-organizes a coupling-maintained rotating limit cycle and self-sustains it with its training loss switched off. It is not a memory store — the coupling that holds the cycle suppresses capacity, refuting the founding hypothesis. It is a phase regulator whose temporal code is Hopf-universal and coupling-independent, with coupling acting as a pure robustness knob — a universality that is emergent in the trained network. And the one invariant across the whole program: an oscillation's existence is cheap, but its directed rotation — the breaking of time-reversal symmetry — is the load-bearing, irreducibly-active ingredient.

λ* = 0.10

Hopf onset, analytic = numerical to 2×10⁻¹⁵

5 / 5

GRU seeds self-organize & self-sustain the splay

CV ~3×10⁻⁵

λ-invariance of the normalized temporal code

0.999

trained-GRU iPRC shape-match to the universal waveform

1 · The motor exists, and it is trainable

The cyclic Stuart–Landau triad undergoes a clean supercritical Hopf bifurcation: below the coupling threshold it collapses to consensus, above it a stable 120° splay limit cycle is born. The same regime is then shown to be trainable in a real network.

Stuart–Landau lambda-scan: Hopf bifurcation into a 120° three-phase limit cycle — **A clean Hopf bifurcation into the splay cycle.** Below λ* = 0.10 the triad collapses (grey); above it a stable limit cycle that is simultaneously a 120° three-phase lock (green ring). The amplitude follows √(λ−λ*) to r² = 1.000 (supercritical), the frequency is structurally flat at ω/2π ≈ 0.159, and the twin finite-time exponent stays at zero (no chaos). The analytic threshold matches the numerical eigenvalue to 2×10⁻¹⁵.

GRU triad training loss and held-out generalization across five seeds — **A trained GRU triad hosts the regime.** A shared-weight GRU triad trained on a phase-antagonistic + amplitude-pinning loss over its own autonomous unroll self-organizes the splay cycle and self-sustains it *with the loss switched off*, from held-out initial conditions — 5/5 seeds, 20/20 held-out ICs passing the full battery (bounded, sustained, spectral peak, non-chaotic, splay-not-consensus). Perturb or destroy one node and it re-locks to 120° in ~50–100 steps; isolate a node and it dies. A genuine coupling-maintained collective attractor, not three independent clocks.

What is earned here is mode-selection, attractor-formation, generalization, and coupling-maintained locking. Boundedness and the supercritical Hopf are partly baked into the normal form, and the loss prescribes the splay's geometry — stated plainly, not papered over.

2 · It is not a memory store — the founding hypothesis, refuted

The program began as a memory hypothesis: a non-settling oscillator should hold stimulus structure better than a static net of equal capacity. The data say the opposite.

Forgetting curves: the oscillator stores less than a static line-attractor — **The oscillator stores *less*, not more.** Stimulus-decodability across delays: a non-oscillating line-attractor of equal capacity (grey) holds the structure above the oscillator (blue) at every delay, and a dense noise sweep × 5 seeds shows monotonic forgetting with no stochastic-resonance peak. The strong coupling that re-locks the cycle is exactly what suppresses memory capacity — **coupledness and plasticity are antagonists**. This is the program's central, robust negative result.

3 · It is a phase regulator — coupling is a pure robustness knob on a Hopf-universal code

Pivoting from what it does not do (memory) to what it does (phase regulation) produced the program’s cleanest positive result — and a cross-substrate confirmation that it is not an artifact of the idealized model.

iPRC and entrainment vs coupling: the normalized temporal code is lambda-invariant — **Coupling tunes only robustness; the temporal code is invariant.** Across the whole coupling band the raw infinitesimal PRC diverges as 1/r (red), but the *amplitude-normalized* iPRC is flat (blue, PTP_norm ≈ 1.15, CV ~3×10⁻⁵) and the common-mode direction is exactly zero (grey). Coupling moves only the robustness axis — the transverse attraction rate, measured slope α ≈ −1.94ε (R² = 1.00) against the −2ε idealized line (centre). Entrainment-tongue width at fixed *relative* forcing is likewise invariant (right, blue); the apparent widening at fixed absolute forcing is removable 1/r amplitude geometry (red). Coupling and the temporal code are orthogonal knobs.

Trained GRU iPRC matches the Stuart–Landau Hopf-universal waveform — **The universality is *emergent* in the trained network.** The trained-GRU iPRC (blue, five seeds) inherits the Stuart–Landau Hopf-universal waveform (red) at shape-correlation 0.999, with normalized gain near the SL value (1.0–1.2 vs 1.15) and a small common-mode — so the universal temporal code is a property a trainable network *acquires*, not a designed-normal-form artifact.

4 · The clean code is a symmetric idealization — but the real object actively holds it

Everything above was proven in the symmetric triad. Breaking the symmetry (detuning the nodes) is the make-or-break test — and it separates the idealized model from the real one.

Per-node coherence under detuning: rigid vs retrained GRU, and the rescue gap — **On the real object the fragility is largely a *rigidity* artifact.** Under heterogeneity the clean symmetric code breaks (on the idealized Stuart–Landau equations it dies decisively — an order-parameter averaging artifact). But the rich 32-dim gated GRU absorbs the same detuning (left: rigid 1a in red holds through the soft band; retrained 1b in blue holds further), and re-training *actively* extends robustness — a multi-seed rescue gap that clears the pre-registered floor (right: +0.072 ± 0.020 at δ=0.20, +0.173 ± 0.057 at δ=0.30), with no node dominance and the order parameter intact: a genuine global coupling+readout retune, not a structural bypass. **The emergent collective level holds because the system actively holds it.**

5 · The deepest law — existence is cheap; rotation is irreducibly active

A “zoo” of dissection experiments asked what it actually takes to make the oscillation self-sustain without the prescriptive loss. One law held across all of them.

An oscillation's existence is cheap; its directed rotation is not. Self-sustaining motion can be born from conservation, from a per-step rotation target, or from a blind energy flow (van der Pol negative damping past a flow threshold). But every blind, passive route produces only a standing oscillation. The directed rotation — the chirality of the splay, which breaks time-reversal symmetry — is the load-bearing, irreducibly-active ingredient. Three routes produce it (a rotation-target prescription, angular-momentum conservation, an active non-reciprocal coupling) and all three are active / symmetry-breaking; none is a passive blind flow. A complementary minimal result sharpens it: on a dissipative map there is no smooth standing oscillation at all — any smooth oscillation requires a complex eigen-pair, i.e. rotation. Rotation is not decoration; it is the active core.

A second emergent structure from the same zoo: when the triad’s weights are allowed to change during operation, a clean two-layer stratification appears on its own — a fast oscillatory state and a slow latent weight-memory, decoupled, with the attractor robustly screening the slow store from the fast state (constant gap across freeze configurations, five seeds). A fast-perceptual / slow-infrastructural split that is robust and not bypassable by structure — a candidate primitive for a perceptual module.

6 · And the founding intuition, tested head-on

Do three coupled, non-settling nets solve a hard deceptive task better than one?

Coupled nets vs single net on a hard task: attribution ladder and long-horizon curves — **The advantage is ordinary cooperative training — and the non-settling chase *hurts*.** An attribution ladder with controls (24 seeds, long-horizon solve-curves on 8-bit parity) shows the "3 > 1" gain is mundane *cooperative joint-training* — a committee covering each other's errors — and that the non-transitive prediction-chase, the specific non-settling ingredient the hypothesis bet on, flatlines below even a single net and never settles. Replicated on a second task type (spectral-bias regression), where the non-settling conditions are the *only* ones that fail to escape a basin a plain net escapes.

What was found

The motor is real and trainable. SL Hopf at λ* = 0.10 (supercritical, r² = 1.000); a GRU triad self-organizes and self-sustains the 120° splay (5/5 seeds, 20/20 held-out ICs), coupling-maintained (re-locks after perturbation; dies in isolation).
(Negative.) It is not a memory store — it stores less than a static line-attractor of equal capacity; coupledness and plasticity are antagonists. The founding hypothesis is refuted.
It is a phase regulator with a Hopf-universal temporal code. The normalized iPRC is λ-invariant (CV ~3×10⁻⁵); coupling tunes only robustness (measured α ≈ −1.94ε, −2ε idealized). This universality is emergent in the trained GRU (shape-match 0.999), not a normal-form artifact.
The clean code is a symmetric idealization that breaks under detuning — but on the real trainable object the breakage is largely a rigidity artifact, and re-training actively holds the coupled structure (multi-seed rescue gap, genuine non-bypass adjustment).
The deepest invariant: an oscillation’s existence is cheap; its directed rotation — broken time-reversal symmetry — is the one irreducibly-active ingredient. On a dissipative map, smooth oscillation requires a complex eigen-pair.
An emergent fast/slow stratification — the attractor screens a slow weight-memory from the fast state — a candidate perceptual-module primitive.
The non-settling coupled dynamics buy robustness and structure, not the memory or optimization advantage hoped for — the prediction-chase is genuinely harmful; the committee gain is ordinary joint-training.

Phase	Question	Verdict
Motor (SL + GRU)	Does the regime exist and is it trainable?	GO
Memory	Stores more than a static net?	REFUTED
Phase regulation	Hopf-universal code, coupling = robustness knob?	GO (emergent in GRU)
Asymmetry (SL)	Does the clean code survive detuning?	NO-GO (averaging artifact)
Asymmetry (real object)	Does the trained net hold it?	YES — actively adapts
Existence vs rotation	What is irreducibly active?	rotation (time-reversal breaking)
Coupled-nets-vs-1	Does non-settling beat a single net?	no — chase hurts

What this means

The founding intuition — that a perpetually-circling, non-settling system perceives or remembers more than a static one — is wrong in its strong form, and the program shows precisely why: the coupling that sustains the cycle suppresses memory, and the non-settling dynamics buy no optimization advantage over ordinary cooperative training. But the dissection that refuted it surfaced something cleaner. The collective oscillator is a genuine, trainable, self-healing phase regulator carrying a substrate-independent (Hopf-universal) temporal code, and the single ingredient that makes any of it active rather than decorative is directed rotation — the breaking of time-reversal symmetry. Existence is cheap; chirality is the cost. That is a sharper and more portable claim than the one we set out to prove.

Reproducibility

Every phase freezes its decisive metric and pass/fail rule before seeing data; decisive metrics are validated against synthetic ground-truth harnesses with negative controls before they touch real data (this repeatedly caught masking artifacts and false-GOs in the program’s own frozen decisions); results are multi-seed with confidence intervals. Two substrates throughout — an analyzable Stuart–Landau triad and a trained GRU triad. Code (plain PyTorch + NumPy), pre-registrations, result JSONs, and figures are in the repository: github.com/mool32/oscillatory-perception.

Part of the perceptome program · pre-registration discipline per the AI-collaborative research methodology. Theodor Spiro · ORCID 0009-0004-5382-9346 · tspiro@vaika.org