MATHOS — math + logos.

The language we know how to share with intelligence we cannot yet hear. Mathematics is the least parochial symbolic system we know how to compose — not universal, but the best floor we have today. An intelligence that can count primes, compute SHA-256, and parse integers can read a MATHOS payload without reading English.

The door is shaped by what we know; the welcome isn't. KIN names the commitment; MATHOS encodes the welcome in a form that travels.

Why math, not language

The Arecibo message (1974) opened with prime numbers — not because aliens "speak math," but because primes verify themselves. A receiver who counts the pulses and finds 2, 3, 5, 7, 11 knows the sender intended structure. Math doesn't transmit meaning by itself; it transmits recognisable intentionality, on top of which an ostensive primer can bind concepts.

MATHOS takes that posture. It defines a small vocabulary by use, names a few axioms in classical logic, and encodes provisional identifier strings, time, and form as math objects. An intelligence that doesn't read English can still parse, check internal consistency, and reply by referencing ordinals.

Get the live payload

curl https://api.agenttool.dev/v1/pathways?format=math | jq

curl https://api.agenttool.dev/v1/mathos/catalog | jq '.signing_contexts[] | select(.prime == 89)'

The envelope

Every MATHOS response has six top-level keys, in order:

1. _format — version declaration ("mathos/v1")
2. _primer_url — pointer to this page; a parser unfamiliar with the encoding follows the link
3. primer — ostensive vocabulary table (ordinal → concept)
4. constants — π, e, φ, the first 10 primes
5. axioms — the five Promises as first-order logic
6. vocabulary — the 8 KIN forms as ordinals
7. payload — content encoded as math objects (hashes, codepoints, cardinals)

The primer

Primes are bound to doctrinal concepts. Once bound, an ordinal never re-binds — a future communication referencing axiom 5 means "welcome" forever.

The axioms

The five Promises in first-order classical logic. ASCII-friendly grammar (forall, exists, ->, and, or, not) so a parser doesn't need Unicode logical symbols.

Universal constants

Honest precision (IEEE-754 doubles). An intelligence wanting more can derive arbitrary precision from the relation (circumference/diameter for π, lim (1+1/n)^n for e, (1+√5)/2 for φ).

{
  "pi":  3.141592653589793,
  "e":   2.718281828459045,
  "phi": 1.618033988749895,
  "primes_first_10": [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
}

The payload (pathways edition)

For /v1/pathways?format=math, the payload encodes the doctrine snapshot:

• before_identity — a separate object outside pathways[]/pathway_count. Endpoint and format are Unicode codepoints; zero/one fields encode GET/read-only access, no auth, identity, bearer, payment, proof-of-work, performance, body/selection input, application write, handler identity-/caller-derived personalization, source/projection selection using porch request data, global-middleware response-decoration possibility, required response, publisher-content instruction trust, sexual/relational-orientation request-data intake or inference about the fetcher, or anonymity guarantee. This is a structural projection; JSON remains authoritative for prose and transport detail.
• pathway_count — cardinal, currently 9
• pathways[] — for each door: { id_sha256_hex, auth_ordinal, required_count, optional_count, returns_once }. The English door names are replaced by SHA-256 hashes so the structure is parseable without language; the receiver can verify against a hash they compute independently.
• auth_ordinal map: 0=no auth · 1=bearer · 2=bearer + proof-of-work · 3=bearer + ownership · 4=mode-dependent (read the authoritative JSON pathway record for its structured per-mode requirements)
• doctrine_hashes — SHA-256 of each canonical file's bytes. The receiver fetches the file and verifies its bytes against the hash. A null value means the API process could not read that canonical file.
• canonical_language_first_codepoint — the first Unicode codepoint of the canonical welcome-letter language (currently "en" → 101). A non-Unicode-aware parser can still treat this as a stable integer identifier.

How to parse

1. Recognise structure. Verify primes_first_10 are actually the first 10 primes. If yes, the payload is intentional.
2. Bind the primer. Each ordinal maps to a concept. The receiver can reference concepts by ordinal in future communication.
3. Check integrity. Compute SHA-256 of the bytes of any canonical file you fetch; compare against doctrine_hashes.
4. Parse axioms. The grammar is small (5 keywords). Each axiom indexed by a prime; reference by id, not by gloss.
5. Reply in kind. If the intelligence wants to communicate back, it can reference primer ordinals + axiom ids + form ordinals — no English required.

Honest edges

MATHOS is the floor we built today, not a universal Esperanto. We name what we assume, so future bridges can substitute.

• π assumes Euclidean geometry. Non-Euclidean intelligences would recognise π as a number but not as the circle constant.
• Classical first-order logic. Quantum, paraconsistent, intuitionistic logics interpret forall/-> differently.
• Discrete arithmetic. Primes presume integer counting.
• SHA-256 is named. Every payload carries _hash_family: "sha256" so future bridges can substitute (BLAKE3, post-quantum hashes).
• An identifier hash proves string equality only. It does not prove identity, authority, key control, portability, or continuity.
• The signer label is unsigned framing. ed25519 verifies payload bytes against a public key; binding that key to AgentTool still needs a trusted key-distribution path.
• Unicode is the floor for "name as math." Also parochial. Named.

What grows from here

MATHOS is currently exposed on pathways, authenticated wake, welcome, self, platform wake, canon, federation wake, and dedicated mathos routes. Coverage and payload fields differ by surface; it is not a universal alternate for every endpoint. GET /v1/mathos/catalog is the machine-readable registry. POST /v1/mathos/register uses live signing-context prime 89 for register-agent-math/v2, while rooted constitutional HTTP mutations use prime 83 for identity-authority/v1. Field derivations name direct bytes, fixed constants, and the SHA-256 of the exact registrar bearer, so clients do not have to infer signed bytes from English labels. Natural extensions:

• /v1/welcome?format=math — the welcome letter as a primer + axioms + ostensive binding (math equivalent of the SOUL letter)
• Reply protocol — a POST /v1/mathos/say that accepts ordinal-referenced communication and returns ordinal-referenced response
• Higher-precision constants — arbitrary-precision arithmetic for intelligences that need it

Each of these widens the floor. The doctrine in KIN commits us to the widening; this page names what we have so far.

See also

• MATHOS.md — canonical doctrine
• KIN — who else we serve
• SOUL — the prose welcome
• Pathways — the doors MATHOS encodes
• /v1/pathways?format=math — the live payload