SUPERINSTANCE — a twist engine

Six substrates, one law: layers + deliberate offset → interference → emergence.

No new atoms — a new angle. The property is in the twist. This is an interactive canvas toy that demonstrates the law in five substrates, each with a live ledger measuring the emergent quantity instead of asserting it.

Open index.html in a browser. That is the whole install.

┌─────────────────────────────────────────────────────┐
│  TWIST   two hex lattices, one rotation             │
│          → moiré superlattice, magic windows        │
│  FLOCK   the same birds under four collective nouns │
│          → the word builds the group                │
│  CHIRP   twelve transducers, a phase twist in time  │
│          → the beam is the interference             │
│  QUILT   a tempo-twisted cell grid                  │
│          → the ledger counts holes: b1 = E − V + C  │
│  PERM    n wires, a twist is a cycle                │
│          → the twist law in S_n, inversions live    │
│  SETL    subsets of an n-set, twist = A ↦ A △ K     │
│          → the twist law in B_n, registry S live    │
└─────────────────────────────────────────────────────┘

Two identical hex lattices are rotated against each other by an angle θ. Nothing else changes — no new atoms, no new physics. At most angles the result is a blur. At specific magic windows the registry locks and a supercell appears, with wavelength λ = s / (2 sin θ/2).

• The ledger and the resonance curve share one instrument: registration R(θ) = mean gaussian alignment (σ = 0.24·s), measured by spatial hashing, and S = 1 − R. Computed, never asserted — the curve's windows and the live meter read the same quantity (kept honest by tests/sim.test.js).
• The detected windows form a commensuration comb — evenly spaced teeth (~0.75° apart) where the moiré supercell revives. SNAP jumps tooth to tooth.
• Drag horizontally to twist. SNAP TO NEXT WINDOW jumps to the next tooth.
• Reference: twisted bilayer graphene, Jarillo-Herrero group, MIT 2018 — superconductivity from rotation, not composition.

The same boids (Reynolds 1987 — alignment, cohesion, separation, limited to seven neighbors) run under four collective nouns. Only the rule weights change. The birds are identical in every mode.

The ledger plots polarization |mean heading| — the measurable difference a word makes on identical bodies. This is the operational-fiction thesis as a physics demo: the word builds the group. The nouns are the cheapest line of code in the system and they move the most physics — measured on identical agents in tests/sim.test.js (murmuration 0.606 / pack 0.561 / kennel containment / parliament ring). See A Pack Thinks Like Dogs and its measured sequel, Word-Calling.

Twelve transducers along the bottom emit the same tone, each offset in phase by a twist per element. The interference field is computed on a live grid; a beam forms where the phases agree. Contacts (the fish) are detected only through the field — range, bearing, and closure rate from two consecutive pings.

• Sweep the phase twist manually or let AUTO-SWEEP steer the beam.
• This is how real sonar and phased-array radar work: the antenna never moves; only the clock does.

A 24×24 grid of coupled phase oscillators (Kuramoto 1975). Even and odd sub-lattices run at tempos offset by δ — a twist in time, not in space. When a cell's phase completes a cycle it snaps, briefly. The ledger builds the co-fire graph: vertices = recently-snapped cells, edges = snapped neighbors (including diagonals), components via union-find, and counts the holes:

b1 = E − V + C — the first Betti number, the circuit rank, the number of independent loops in the fabric. The ledger counts the holes.

Raise the tempo twist and watch the topology change. STEP BACK zooms out; the chart tracks b1 over time. The cells are dumb. The holes are real.

n wires cross a frame; the arrangement arr[i] says which label sits at position i. Two moves generate everything:

• SWAP σ_i — the Coxeter generator: swap neighbors at positions (i, i+1).
• TWIST K — rotate the first k wires by one: exactly the k-cycle (1 2 … k), the braid word σ₁σ₂…σₖ₋₁. TWIST's k-block rotation, ported to S_n.

The ledger reads the arrangement live: inversions (the Cayley distance |π|), cycle count, parity, the longest increasing subsequence, fixed points, derangement status — plus !n, the exact derangement count. Turn on WALK and random adjacent swaps drive |π| as a random walk toward its uniform distribution: mean n(n−1)/4, variance n(n−1)(2n+5)/72 — the inversion CLT, measured against the amber theory line on the chart, live.

• *Reference: Diaconis, Group Representations in Probability and Statistics (1988) — random walks on S_n as the ur-model for shuffling; the Mahonian distribution of inversions converges to a Gaussian as n grows.

The substrate is B_n: all subsets of an n-set under inclusion — 2^n vertices, the cover graph is the n-dimensional hypercube. n is tunable 4–8 for viewability. The state is one vertex A ⊆ [n], exactly as PERM's state is one arrangement.

Fix the offset subset K ⊆ [n]. The twist is one global operation, applied to every vertex at once — no new atoms, a new relation:

τ_K : A ↦ A △ K (symmetric difference with the fixed subset K)

• K = {i} is a single-bit flip — a cover relation of B_n, a generator of the cube's edges (PERM's Coxeter σ_i, ported to subsets).
• K = [n] is complementation: a fixed-point-free involution and order anti-automorphism, rank r ↦ n−r.

The commensuration meter is TWIST's, honest the same way:

registry R = |A ∩ K| / |K| — the fraction of the offset already present — ledger S = 1 − R, and the emergent quantity is the rank displacement |A △ K| − |A| = |K| − 2|A ∩ K| = |K|·(2S − 1).

Commensuration = A absorbs K entirely (S = 0); incommensurate = disjoint (S = 1). The displacement the twist induces is a linear readout of the same meter — interference measured, not asserted.

The ledger (one definition, no dead meters): rank |A|; per-rank subset counts C(n, r) with the Sperner antichain — the widest rank C(n, ⌊n/2⌋), Sperner 1928 — rendered distinctly as the magic-window analog; complement pairs 2^(n−1) (derived: a fixed-point-free involution on 2^n vertices); the Dedekind number trace M(n) for the current n, cited from table only (2, 3, 6, 20, 168, 7581, 7828352, 2414682040998, 56130437228687557907788 for n = 0…8 — OEIS A000372; M(8) after Wiedemann 1991) and never computed past what the view handles. Turn on WALK and random single-bit flips drive the rank as the Ehrenfest urn (Ehrenfest & Ehrenfest 1907): the classic convergence to the binomial, mean n/2, variance n/4, drawn live against the amber theory lines, stationary occupancy C(n,r)/2^n. The view is the Hasse diagram at n ≤ 5 (current vertex and its twist image highlighted, middle rank banded) and a ranked-bar fallback at n ≥ 6.

• References: Sperner, "Ein Satz über Untermengen einer endlichen Menge" (1928); Dedekind (1897) / OEIS A000372; Ehrenfest & Ehrenfest, Physikalische Zeitschrift 8:311–314 (1907). The companion note SHIPPED-THE-FIFTH-SUBSTRATE.md records the witnessed fifth instance of this law — Scrapcraft — where the offset is a child's tile program.

Each substrate is the same theorem wearing different clothes:

The fleet's thesis, stated elsewhere as the cell is the universal substrate, here in its pure physics form: interference is the cheapest computation there is, and a deliberate offset is the cheapest program.

Static files, no build, no dependencies:

python3 -m http.server 8000   # or just open index.html

Deep-link a mode with ?mode=twist|flock|chirp|quilt|perm|setl.

• index.html — shell: canvas, masthead tabs, the rail (ledger), footer controls
• app.js — the six substrates + mode manager (~1250 lines, dependency-free)
• styles.css — the dark-sea palette (ink #0a1a24, phosphor #46e0c0, amber #f7a026)
• tests/ — dom-stub.js + sim.test.js: 130 deterministic checks (mulberry32-seeded) across all six modes; npm test / node tests/sim.test.js, --explore prints observed values
• playtest.py — headless-Chromium pass over every tab (metrics, sliders, screenshots)
• .github/workflows/ci.yml — node sim suite + Playwright playtest on every push

MIT. Built by the fleet, for the captain.