The surface
A sheet whose height is Σ sin(nx)/n. Terms arrive one at a time, each smaller than the last, and every one leaves structure in the surface that never flattens back out.
terms: —
A swarm console for the Swarms API
An autonomous agent that proves its mathematics before it speaks.
Swarmonic is its console. Describe a roster — up to two thousand agents across fourteen coordination architectures — and it builds the exact Swarms API request, prices it at the published rate, and attaches the working to every number it reports.
It holds no key and sends nothing. You take the request and run it yourself.
CA
FzJMBMvospBKbWtbZn6Pz2PiSzGYsgbE9BtWbGnoswrm
Not affiliated with The Swarm Corporation. Every figure on this page is either quoted from the Swarms documentation or computed here, and the two are labelled differently.
01 — the thesis
Σ 1/n diverges.
Slowly. Inevitably.
The harmonic series grows without bound, but it takes about en terms to reach n. No single term matters. The sum is unstoppable.
A swarm is built the same way. One agent reads a filing, one checks a number, one argues the other side, one writes it down. No single agent is the answer. The answer is what the sum converges to — and unlike the series, the swarm is allowed to close.
Theorem (the swarm thesis). Let f(t) be one small, verifiable action taken by one agent per tick. Then ∫ f dt grows without bound.
Proof. Left as an exercise for the single agent. ∎
02 — the explainer, computed live
Why a series that never finishes
is the point.
Four panels. Nothing below is a picture of a calculation — it is the calculation, running now at your frame rate. Watch the second one climb out of the frame, and the third one refuse to.
It never closes
Rings stacked at heights 1, 1/2, 1/3, and on. The gaps shrink toward nothing and the stack still climbs out of the top of the frame, because Σ 1/n has no ceiling to reach.
terms: —
This one closes
The identical construction, one exponent different. The rings crowd against π²/6 and stop. Euler proved that ceiling exists in 1735. Panels 02 and 03 are the same object under two rules, which is the entire argument.
partial sum: —
The swarm
The same construction with agents in place of terms. Each new agent adds less than the last, and the roster still fills. The last panel is the reason the first three are worth reading.
agents: —
03 — the machine
Fourteen architectures.
One request each.
A swarm_type is the only thing that decides how a roster behaves. Same agents, same task, different topology, different answer. These descriptions are the strings the live API returns for each type — they are quoted, not written by us.
Two values that are rejected
"auto" and "SpreadSheetSwarm" are checked explicitly by the API and refused with 400, even though older documentation references them. GraphWorkflow and BatchedGridWorkflow are not swarm_type values at all — each is a separate, premium-gated endpoint.
04 — the loop
One request.
Six things happen.
Between your POST and your bill the API runs a fixed sequence: it refuses what it cannot bill, builds the roster from one field, coordinates it, forces the output into one shape, and charges by the token. Every stage below is a fact from the completions reference.
05 — the desk
Name the roster.
The request writes itself.
Compose a swarm here and take the exact request away — JSON, curl, Python, or JavaScript. The desk is a generator, not a client: it holds no key, sends nothing, and says so on every panel rather than in a footnote.
Nothing is sent
This page holds no API key and issues no requests. Paste the block above into your own terminal or notebook with SWARMS_API_KEY set. Your first call needs a funded account — billable endpoints require a credit balance strictly above $1.00.
06 — the reference
The shapes,
field by field.
Transcribed from the OpenAPI schema and the completions reference. Where the documentation contradicts itself, this page says so instead of picking a side quietly.
07 — what a swarm costs
The bill is linear.
The work is not.
Every number here is published pricing. The estimator below multiplies them; it does not guess a token count for you, because a guess dressed as a quote is the thing this page is against.
| item | price | charged |
|---|---|---|
| Input tokens | $6.50 | per 1M tokens |
| Output tokens | $18.50 | per 1M tokens |
| Agent | $0.01 | per agent, swarms & workflows |
| Image processing | $0.25 | per image supplied |
| MCP call | $0.10 | when mcp_url is set |
| Exa search | $0.04 | per search execution |
| Web scraper | $0.15 | per scrape execution |
| Night discount | −50% | tokens only, 8pm–6am PT, swarm completions only |
The discount is decided when the run finishes
A job that starts at 5:55am and ends at 6:02am Pacific is billed at the daytime rate. The per-agent fee is never discounted — it is added after the discount block. So the ceiling on your savings is exactly 50% of tokens and 0% of agents, no matter how large the fan-out.
08 — verification
It proves before it speaks.
Two layers. Below, a prover written in your browser's own integer arithmetic re-derives five claims every time this page loads — exact rationals, explicit tail bounds, no floating-point hand-waving. Every row is recomputed, not remembered.
qed :: boot prover
running
recomputing…
09 — the solver
Ask it something.
It answers with a proof.
The prover above settles five fixed claims. This is the same machinery pointed at whatever you type: a harmonic number to thirty digits, or the exact point at which the series passes a threshold — each one returned with the interval it was decided on, how wide that interval is, and which engine produced it. Nothing leaves the tab.
the harmonic number
n up to 19 digits
the first crossing
T up to 100
every crossing to 20, computed in this tab
| threshold | smallest n | H(n) | engine | time |
|---|
computing…
10 — telemetry
Live numbers.
Or an honest absence of them.
A dashboard that shows a frozen figure dressed as a live one is worse than a dashboard that shows nothing. This section makes no network calls by design, so every reading below is absent — and it says which endpoint would fill it, and with what.
Why there is no live panel
Reading any of the rows above needs a funded key in the x-api-key header, and a page that ships a key ships it to everyone who opens the page. A browser is the wrong place for one. Run these where the key already lives:
curl -s https://api.swarms.world/health
curl -s https://api.swarms.world/v1/account/credits -H "x-api-key: $SWARMS_API_KEY"
curl -s https://api.swarms.world/v1/rate/limits -H "x-api-key: $SWARMS_API_KEY"
curl -s https://api.swarms.world/v1/account/metrics/summary -H "x-api-key: $SWARMS_API_KEY"GET /health is the one endpoint that needs no key, and the only reading this page could have taken honestly. It returns {"status":"ok"} — a fact we verified once, by hand, and have not dressed up as a live one.
Start with one agent.
The sum does the rest.
Honest scope, stated once and plainly
Swarmonic is an independent page about a documented API. It is not affiliated with, endorsed by, or operated by The Swarm Corporation, and it does not run an agent, hold a key, or custody anything. The API facts are transcribed from docs.swarms.ai and the published OpenAPI schema, and verified against them; the mathematics is recomputed here on every load; the opinions are ours. Where the documentation contradicts itself, the contradiction is shown rather than resolved.