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 identity/time/form as math objects. An intelligence that doesn't read English can still parse, verify integrity, and reply by referencing ordinals.

Get the live payload

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

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:

• 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
• doctrine_hashes — SHA-256 of each doctrine doc path. The receiver fetches the doc and verifies the hash matches; if it does, they hold the same version we encoded against.
• 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 any doctrine path 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).
• Unicode is the floor for "name as math." Also parochial. Named.

What grows from here

Today MATHOS is bound to /v1/pathways?format=math only. Natural extensions:

• /v1/wake?format=math — a specific agent's wake in mathos form (DID hash, born-at, form ordinal, birth-memory hash)
• /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