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Every firefly holds // one phase in a shared Float32Array, climbing linearly from 0 toward a // firing threshold of 1 at its own natural rate. The instant a firefly // reaches 1 it fires: its phase resets to 0 and it nudges the phase of each // of its k=12 STATIC nearest neighbours forward by a fixed coupling strength // epsilon. A nudged neighbour that itself crosses 1 fires immediately in the // same tick, cascading — this is the whole mechanism. There is no target // synchrony state written anywhere; unison, travelling bands and partial-sync // clusters are all the same cascade rule evaluated on a random initial phase // field, and they are what makes the field's very long-run arc (twinkle, // then order, then a local relapse near a deaf firefly, then re-entrainment, // forever) genuinely emergent rather than authored. // // COUPLING STRENGTH is the one governing scalar. epsilon = 0.03 is the // widest value that still shows travelling bands: above ~0.08 the field // snaps to full unison in a few seconds and the "order" act eats the whole // clip; below ~0.02 it never locks within a normal page dwell and only ever // twinkles. The neighbour graph is k=12 nearest by planar distance, computed // ONCE at mount and never rebuilt — coupling only ever happens along that // fixed graph, which is what makes synchrony a spatially local, propagating // event (a travelling wave, a cluster negotiating a merge) instead of a // global average snapping over instantly. // // DEAF FIREFLIES (2% of the population, a fixed random subset) are the // honest source of relapse: they never RECEIVE a neighbour's nudge (their // own phase only ever advances at their own natural rate), they run a // slightly faster natural period than the rest of the field, and they still // SEND a pulse to their own neighbours every time they fire. A deaf firefly // therefore drifts out of whatever local consensus has formed around it and, // on its own schedule, reaches back in and perturbs its neighbourhood — that // perturbation is what pulls a locally synced patch apart and restarts local // entrainment, differently every time, with no scripted reset anywhere. // // RENDER SPLIT is the whole performance story: 350 fixed-position DOM dots // are pre-mounted once at --ns-muted and never move. JS owns exactly one // number per firefly (phase) and, on a fire event, toggles which of two // identical-effect CSS classes is present on that one dot — the 200ms // opacity/scale/color flash envelope itself is a `@keyframes` animation that // then runs entirely on the compositor. JS never touches opacity, transform // or color directly; it only ever adds/removes a class name. // // ACCESSIBILITY / PHOTOSENSITIVITY: two hard visual caps live in the render // path, independent of prefers-reduced-motion. (1) A per-firefly refractory // period of ~910ms after each flash means no single dot can visually flash // faster than ~1.1Hz, even mid-cascade. (2) A global burst gate: when a // cascade's eligible-flash count exceeds a small fraction of the field (a // "unison" event), the WHOLE burst is suppressed unless at least 900ms has // passed since the last one — phase state still updates normally underneath, // only the visual flash of that particular burst is skipped, so unison never // reads as flicker faster than roughly once per 900ms. That is well under // the three-per-second photosensitivity guideline even without reduced // motion engaged. With prefers-reduced-motion: reduce, no rAF loop ever // starts: the field renders once, statically, at its seeded resting phases, // and one spatial cluster (a BFS walk over the same neighbour graph used for // coupling) is marked at +10% opacity as the frozen stand-in for a // synchrony band — the arc is described, not demonstrated, with nothing // live. // // Distinct from loader-pendulum-sync: that component is a small mechanical // loader whose handful of pendulums converge on an AUTHORED schedule (their // CSS animation-durations are picked so the row is guaranteed to land back // in phase every periodMs, by integer-oscillation arithmetic with zero // per-frame JS) and stay converged once finished. FlashEntrain has no // schedule and no finish: 350 independent phases pulse-couple over a static // neighbour graph, synchrony is earned frame by frame, and a handful of deaf // individuals mean it never stays settled — the story is chaos-to-order-to // relapse, forever, not converge-and-stop. // --------------------------------------------------------------------------- const TICK_MS = 1000 / 30; // fixed 30Hz physics step const REFRACTORY_MS = 910; // per-firefly visual cooldown -> caps flash rate just under 1.1Hz const BURST_GAP_MS = 900; // global unison-burst gate const DOT_MIN_PX = 3; const DOT_MAX_PX = 5; const BAND_OPACITY_BOOST = 0.1; // reduced-motion frozen band differential // Every firefly's own uncoupled cadence stays comfortably below the 1.1Hz // visual cap: refractory should only ever bind during a coupling-accelerated // cascade, never as routine background muting of normal ticking. const MIN_PERIOD_MS = REFRACTORY_MS + 60; // Concavity of the Mirollo-Strogatz charge/rise function f(phase). This is // what a naive "phase += epsilon on every nudge" coupling is missing: with a // LINEAR phase, equal nudges never compress the gap between two out-of-phase // fireflies, and a headless sweep of that naive version never rose above an // order parameter of ~0.1 in 90s at any epsilon from 0.02-0.25 — it just // never locks. With f concave (f(phase)=ln(1+(e^b-1)*phase)/b, b=3 here, // verified by the same sweep), an equal charge-nudge buys a LARGER phase // jump for a firefly that is further from firing than for one already close // to it, which is exactly the absorption mechanism that drives real // convergence — chaos visibly resolving into partial and then near-full // synchrony over tens of seconds, continuously reopened by the deaf // fireflies, at coupling=0.03 exactly as specified. const CONCAVITY_B = 3; function mulberry32(seed: number): () => number { let a = seed >>> 0 || 1; return () => { a = (a + 0x6d2b79f5) | 0; let t = Math.imul(a ^ (a >>> 15), 1 | a); t = (t + Math.imul(t ^ (t >>> 7), 61 | t)) ^ t; return ((t ^ (t >>> 14)) >>> 0) / 4294967296; }; } const CSS = ` .ns-fe-field{ position:relative; overflow:hidden; } .ns-fe-dot{ position:absolute; left:0; top:0; border-radius:9999px; background:var(--ns-muted); will-change:opacity,transform; } .ns-fe-flash-a, .ns-fe-flash-b{ animation-duration:200ms; animation-timing-function:cubic-bezier(.22,.7,.32,1); animation-fill-mode:none; } .ns-fe-flash-a{ animation-name:ns-fe-flash-a; } .ns-fe-flash-b{ animation-name:ns-fe-flash-b; } @keyframes ns-fe-flash-a{ 0%{ opacity:var(--ns-fe-rest-o); transform:scale(1); background-color:var(--ns-muted); } 18%{ opacity:1; transform:scale(1.9); background-color:var(--foreground); } 100%{ opacity:var(--ns-fe-rest-o); transform:scale(1); background-color:var(--ns-muted); } } @keyframes ns-fe-flash-b{ 0%{ opacity:var(--ns-fe-rest-o); transform:scale(1); background-color:var(--ns-muted); } 18%{ opacity:1; transform:scale(1.9); background-color:var(--foreground); } 100%{ opacity:var(--ns-fe-rest-o); transform:scale(1); background-color:var(--ns-muted); } } @media (prefers-reduced-motion: reduce){ .ns-fe-flash-a, .ns-fe-flash-b{ animation:none; } } `; interface Firefly { xFrac: number; yFrac: number; sizePx: number; deaf: boolean; } export interface FlashEntrainProps { /** Population size. */ count?: number; /** Mirollo-Strogatz coupling strength per firing (governing scalar). Above ~0.08 the field snaps to unison in seconds; below ~0.02 it never locks. */ coupling?: number; /** Static nearest-neighbour graph size per firefly. */ neighbors?: number; /** Fraction of the population that is deaf (never receives coupling), the source of relapse. */ deafFraction?: number; /** Mean natural period, ms, before per-firefly jitter. */ periodMs?: number; /** extra classes merged onto the rendered root element */ className?: string; } export function FlashEntrain({ count = 350, coupling = 0.03, neighbors = 12, deafFraction = 0.02, periodMs = 1250, className = "", }: FlashEntrainProps) { const rootRef = useRef(null); const elRefs = useRef<(HTMLDivElement | null)[]>([]); const [reduced, setReduced] = useState(false); useEffect(() => { const mq = window.matchMedia("(prefers-reduced-motion: reduce)"); setReduced(mq.matches); const onChange = (e: MediaQueryListEvent) => setReduced(e.matches); mq.addEventListener("change", onChange); return () => mq.removeEventListener("change", onChange); }, []); const n = Math.max(24, Math.round(count)); const k = Math.max(3, Math.min(n - 1, Math.round(neighbors))); // -- one seeded field: positions, sizes, deaf flags, neighbour graph ----- // Deterministic per (n, k, deafFraction) so re-mounts (and the SSR pass) // agree with the client without needing to defer render. const fieldRef = useRef<{ fireflies: Firefly[]; neighborIdx: Int32Array; restOpacity: number; bandMembers: Uint8Array; n: number; k: number; deafFraction: number; } | null>(null); if ( !fieldRef.current || fieldRef.current.n !== n || fieldRef.current.k !== k || fieldRef.current.deafFraction !== deafFraction ) { const rand = mulberry32(0x5eed0001 ^ n ^ (k << 8)); const fireflies: Firefly[] = Array.from({ length: n }, () => ({ xFrac: rand(), yFrac: rand(), sizePx: DOT_MIN_PX + rand() * (DOT_MAX_PX - DOT_MIN_PX), deaf: false, })); // Floored at 1: the deaf minority is the only source of relapse in this // model (see header comment), so deafFraction=0 would silently produce a // field that locks into unison and then just sits there forever — a // floor of one keeps that story honest even at a tiny population. const deafCount = Math.max(1, Math.round(n * deafFraction)); // pick a random subset deaf, without replacement const order = Array.from({ length: n }, (_, i) => i); for (let i = order.length - 1; i > 0; i--) { const j = Math.floor(rand() * (i + 1)); const tmp = order[i]!; order[i] = order[j]!; order[j] = tmp; } for (let i = 0; i < deafCount; i++) fireflies[order[i]!]!.deaf = true; // k-NN by planar (fractional) distance, brute force — n is small (<=~700 // in practice), computed once and never rebuilt. const neighborIdx = new Int32Array(n * k); const candIdx = new Int32Array(k); const candD2 = new Float32Array(k); for (let i = 0; i < n; i++) { let filled = 0; let worst = -1; let worstD2 = -Infinity; const xi = fireflies[i]!.xFrac; const yi = fireflies[i]!.yFrac; for (let j = 0; j < n; j++) { if (j === i) continue; const dx = fireflies[j]!.xFrac - xi; const dy = fireflies[j]!.yFrac - yi; const d2 = dx * dx + dy * dy; if (filled < k) { candIdx[filled] = j; candD2[filled] = d2; if (d2 > worstD2) { worstD2 = d2; worst = filled; } filled++; } else if (d2 < worstD2) { candIdx[worst] = j; candD2[worst] = d2; worstD2 = -Infinity; for (let m = 0; m < k; m++) { const dm = candD2[m]!; if (dm > worstD2) { worstD2 = dm; worst = m; } } } } for (let m = 0; m < k; m++) neighborIdx[i * k + m] = candIdx[m] ?? i; } // Reduced-motion frozen band: BFS over the same neighbour graph from one // seeded anchor, collecting ~15% of the population as "already synced". const bandMembers = new Uint8Array(n); const bandTarget = Math.max(1, Math.round(n * 0.15)); const anchor = Math.floor(rand() * n); const queue: number[] = [anchor]; bandMembers[anchor] = 1; let bandCount = 1; let qi = 0; while (qi < queue.length && bandCount < bandTarget) { const cur = queue[qi++]!; for (let m = 0; m < k; m++) { const nb = neighborIdx[cur * k + m]!; if (bandMembers[nb]) continue; bandMembers[nb] = 1; bandCount++; queue.push(nb); if (bandCount >= bandTarget) break; } } fieldRef.current = { fireflies, neighborIdx, restOpacity: 0.55, bandMembers, n, k, deafFraction, }; } const field = fieldRef.current; useEffect(() => { if (reduced) return; // static frame only, no loop, no listeners const root = rootRef.current; if (!root) return; const { fireflies, neighborIdx } = field; const phase = new Float32Array(n); const rate = new Float32Array(n); // phase units per ms const isDeaf = new Uint8Array(n); // Float64, not Float32: holds performance.now()-scale timestamps, which // lose the millisecond precision the refractory comparison needs once a // long-lived tab's clock runs past ~4.6 hours in a Float32. const nextFlashOkAt = new Float64Array(n); const flip = new Uint8Array(n); // alternates which flash class to use const firedThisTick = new Uint8Array(n); const rand = mulberry32(0x5eed0002 ^ n); for (let i = 0; i < n; i++) { const ff = fireflies[i]!; isDeaf[i] = ff.deaf ? 1 : 0; phase[i] = rand(); // start scattered, not synced const jitter = 0.85 + rand() * 0.3; // +-15% natural-period spread // deaf fireflies run ~14% faster; MIN_PERIOD_MS floors every period // (deaf or not) above the refractory so ordinary ticking never trips // the visual rate cap on its own — only a coupling cascade does. const period = Math.max( MIN_PERIOD_MS, ff.deaf ? periodMs * jitter * 0.86 : periodMs * jitter ); rate[i] = TICK_MS / period; } const eps = coupling; const eb = Math.exp(CONCAVITY_B); // phase -> charge (concave) and charge -> phase (its inverse). A firing // pulse nudges a neighbour's CHARGE by epsilon, then the neighbour's // phase is recovered from that new charge — see CONCAVITY_B above. const toCharge = (p: number) => Math.log(1 + (eb - 1) * p) / CONCAVITY_B; const toPhase = (x: number) => (Math.exp(CONCAVITY_B * x) - 1) / (eb - 1); const fireQueue: number[] = []; const eligible: number[] = []; let lastBurstAt = -Infinity; const burstThreshold = Math.max(6, Math.round(n * 0.06)); const applyFlash = (i: number, nowMs: number) => { nextFlashOkAt[i] = nowMs + REFRACTORY_MS; const el = elRefs.current[i]; if (!el) return; el.classList.remove("ns-fe-flash-a", "ns-fe-flash-b"); // Alternating between two classes with distinct animation-name values // (not the same class re-added) is what forces the keyframe to // restart: re-adding an identical animation-name is a no-op per spec // even after the previous run finished, no reflow trick needed. el.classList.add(flip[i] ? "ns-fe-flash-b" : "ns-fe-flash-a"); flip[i] = flip[i] ? 0 : 1; }; const tick = (nowMs: number) => { for (let i = 0; i < n; i++) phase[i] += rate[i]!; firedThisTick.fill(0); fireQueue.length = 0; eligible.length = 0; for (let i = 0; i < n; i++) { if (phase[i]! >= 1 && !firedThisTick[i]) { firedThisTick[i] = 1; fireQueue.push(i); } } let qi = 0; while (qi < fireQueue.length) { const i = fireQueue[qi++]!; phase[i] = 0; if (nowMs >= nextFlashOkAt[i]!) eligible.push(i); const base = i * k; for (let m = 0; m < k; m++) { const j = neighborIdx[base + m]!; if (isDeaf[j]) continue; // deaf: never receives coupling // Mirollo-Strogatz coupling proper: nudge the neighbour's CHARGE // (the concave transform of its phase) by epsilon, not its phase // directly — see CONCAVITY_B above for why that distinction is // what makes convergence actually happen. const chargeJ = toCharge(Math.min(1, phase[j]!)) + eps; if (chargeJ >= 1) { if (!firedThisTick[j]) { firedThisTick[j] = 1; fireQueue.push(j); // popped next: reset to 0, coupled onward } } else { phase[j] = toPhase(chargeJ); } } } if (eligible.length === 0) return; if (eligible.length > burstThreshold) { if (nowMs - lastBurstAt < BURST_GAP_MS) return; // gated: physics ran, visuals held back lastBurstAt = nowMs; } for (let e = 0; e < eligible.length; e++) applyFlash(eligible[e]!, nowMs); }; let raf = 0; let lastFrame = 0; let acc = 0; const loop = (now: number) => { const dt = lastFrame ? Math.min(200, now - lastFrame) : TICK_MS; lastFrame = now; acc += dt; while (acc >= TICK_MS) { tick(now); acc -= TICK_MS; } raf = requestAnimationFrame(loop); }; const onVisibility = () => { if (document.hidden) { if (raf) cancelAnimationFrame(raf); raf = 0; } else if (!raf) { lastFrame = 0; acc = 0; raf = requestAnimationFrame(loop); } }; raf = requestAnimationFrame(loop); document.addEventListener("visibilitychange", onVisibility); return () => { if (raf) cancelAnimationFrame(raf); document.removeEventListener("visibilitychange", onVisibility); }; }, [reduced, n, k, coupling, periodMs, field]); return (

{field.fireflies.map((ff, i) => { const bandOpacity = reduced && field.bandMembers[i] ? field.restOpacity + BAND_OPACITY_BOOST : field.restOpacity; return (

{ elRefs.current[i] = el; }} className="ns-fe-dot" style={{ left: `${ff.xFrac * 100}%`, top: `${ff.yFrac * 100}%`, width: ff.sizePx, height: ff.sizePx, marginLeft: -ff.sizePx / 2, marginTop: -ff.sizePx / 2, opacity: bandOpacity, "--ns-fe-rest-o": field.restOpacity, } as CSSProperties} /> ); })}

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