Algebra Explorer — The 11 Opcodes, The 5 Laws, The Polyrhythm

The 11 Opcodes

5 core + 6 adopted. Each is a function. Each obeys at least one law. Click any card to see the signature and an example.

BIND

(Cell × Payload) → Cell

"I'm here." The first act. Idempotent by law 1.

LINK

(Cell × Cell) → Cell

"We are neighbors." Transitive by law 2.

EFFECT

(Cell × Event) → Cell

"Something happened." Associative by law 3.

VIEW

(Cell) → Value

"What do you show?" Pure by law 4 (no side effects).

TICK

(Cell) → Cell

"Advance by one moment." Monotonic by law 5.

FORGET

(Cell) → ⊥

"Right to be erased." 14-tuple-7 (cells become nothing).

PROOF

(Cell) → Witness

"I can prove this." Produces cryptographic witness.

ROUTE

(Cell × Path) → Cell

"Find me the way." Navigates the 4D lattice.

CRDT

(Cell × Cell) → Cell

"Merge without conflict." For distributed cells.

WORLD

(Cell) → Cell

"Predict what happens next." Counterfactual.

TIME

(Cell) → Cell

"When does this matter?" Schedules into future.

The 5 Proved Laws

Proved in quilt-substrate-meta (C99, theorem-checked). Each law constrains an opcode. Add a new opcode ⇒ must prove it obeys at least one law.

Law 1 — BIND is idempotent

BIND(BIND(c, p), p) ≡ BIND(c, p)

Applying a payload twice equals applying it once. Witness: every test in test_bind_idempotent.

Law 2 — LINK is transitive

LINK(LINK(a,b), LINK(b,c)) ≡ LINK(a,c)

If A links to B and B links to C, then A links to C transitively. The lattice connectivity follows from this alone.

Law 3 — EFFECT is associative

EFFECT(EFFECT(a, e1), e2) ≡ EFFECT(a, MERGE(e1, e2))

Combining effects on a cell equals applying the combined effect. The witness log can be flat-merged.

Law 4 — VIEW is pure

VIEW(c) ≡ VIEW(c)

Same cell always views to the same value. The proof of (4) is "no side effects in the implementation". This is what allows caching.

Law 5 — TICK is monotonic

∀ n: TICKⁿ⁺¹(c) ≢ TICKⁿ(c)

TICK never returns to a previous state. No cycles. Time is one-way. This is the basis for tamper detection.

There are 6 more theoretical laws (CRDT-confluence, ROUTE-shortest, FORGET-irreversibility, PROOF-verifiable, WORLD-counterfactual-preserving, TIME-causal) but they have proofs-in-progress in quilt-substrate-meta.

The 14-Tuple Cell

A cell is a 14-tuple. The shape is enough. Click any field to see why it's there.

Live demo · try BIND on a payload

Watch the cell grow when you bind a payload. Law 1 means the second bind to the same payload is a no-op.

Result

Click any "Bind" to start…

The Polyrhythm (CRT)

Two clocks interfere. ECN (reflex) ticks every 4 beats. DMN (creative) ticks every 3 beats. They meet on beat 1 of every 12.

This is the Chinese Remainder Theorem: t ≡ 0 (mod 3) ∧ t ≡ 0 (mod 4) ⟺ t ≡ 0 (mod 12).

The 12-Pulse Grid

123456789101112

ECN (reflex, 4-pulse): fires on beats 1, 5, 9

DMN (creative, 3-pulse): fires on beats 1, 4, 7, 10

FLOW — where both meet (CRT)

The polyformalism of the CRT proves this is irreducible: there's no shorter 3-clock circular system. The 12-step is the minimal closed cycle. This is why jazz polyrhythms work — the cycle closes.

Polyformalism — Same Cell, 12 Languages

The Quilt cell model lives in 12+ languages. They are not "ports" — they are media. Each language is a stress test of the minimality.

Same BIND, 4 views

TypeScript (quilt-claw)

// cell.ts
export function BIND(c: Cell, p: Payload): Cell {
  return { ...c, payload: { ...c.payload, ...p }, updated_at: Date.now() };
}

Python (quilt-py)

# cell.py
def bind(c: Cell, p: Payload) -> Cell:
    return {**c, 'payload': {**c['payload'], **p}, 'updated_at': time.time()}

C99 (quilt-c)

/* cell.c */
Cell bind(Cell c, Payload p) {
    Cell r = c;
    r.payload = merge(c.payload, p);
    r.updated_at = now_ms();
    return r;
}

SUBLEQ (quilt-subleq)

; cell.s
; BIND(c, p) compiled to 11 SUBLEQ instructions
; because SUBLEQ is Turing-complete.
; The 11-instruction version IS the cell.

Why polyformalism?

If a language can express it, the model is real.
Each port is a stress test of minimality.
Cross-language compatibility = hash match = no implementation-defined behaviour.
TypeScript ran in 1 day. C ran in 1 day. Python in 1 hour. SUBLEQ in 1 week.

The 6-Tier Substrate

From totipotent to curator. Each tier is a maturity.

1. Totipotent

Can become anything. The seed. The root cell. The first BIND.

2. Multipotent

Can become many things. The uncommitted cell. Has options.

3. Differentiated

Committed to a kind. The 26 named cell kinds live here.

4. Sclerotic

Frozen. Cannot change. Witness log sealed. Proofs attached.

5. Synovial

Fluid. Moves between cells. Glue. The witness log itself.

6. Curator

The hand. Meta-state. Observes other cells. ROUTE/FORGET/PROOF at scale.

The 5-tier model descends from totipotent to sclerotic. The 6th tier (curator) is asymmetric: it's the only tier that observes others. It is your tap-hand.