P31 LABS
V₁ · |ψ₁⟩

Quantum Core

SIC-POVM d=2 · K₄ Topology · Posner Molecule

overlap = 1/3 · V=4 · E=6 · rigid ✓
V₂ · |ψ₂⟩

Care Economy

LOVE Ledger · Care Mesh · Spoon‑Aware

non‑extractive · court‑admissible · ε=0.5 DP
V₃ · |ψ₃⟩

Cognitive Passport

DID:key · SD-JWT VC · EUDI‑Ready

Ed25519 · ML-DSA-65 · did:web:p31ca.org
V₄ · |ψ₄⟩

Sovereign Mesh

K₄ Topology · P2P Care · 14 Workers

rigid · planar · federation-ready
SIC-POVM MEASUREMENT · d=2 · 4 STATES
overlap = 1/3 · Tr(ρΠᵢ)
SIC-POVM d=2 • overlap 0.3333 ✓
25.0%
|ψ₁⟩
25.0%
|ψ₂⟩
25.0%
|ψ₃⟩
25.0%
|ψ₄⟩

K₄ Tetrahedral Mesh

V=4E=6K₄ ✓Rigid ✓
1
2
3
4
✦ K₄ Complete Graph — Isostatically Rigid (E = 3V - 6 = 6)· 863 Hz
863 Hz Larmor Somatic Regulation
K₄ is planar · β₂ = 1 · The cage holds
⚠️ HONEST LABEL: This site implements a computational model inspired by contested hypotheses. The underlying science is NOT established physics or accepted medical science. This code is an architectural metaphor made literal — a sovereign care measurement primitive. Zenodo: The Tetrahedron Protocol · Gassab et al. (2025) Entropy · Adams et al. (2025) Sci Rep
Post-Quantum · Spoon-Aware · Sovereign

The Sovereign
Mesh

The Larmor frequency of phosphorus-31 in the Earth's magnetic field is 863 Hz. That is the heartbeat of this system. Open-source assistive technology for neurodivergent families. Four generations, one architecture.

Enter PHOS → Research GitHub ↗
4,200+
Tests
22
Papers
137
MCP Tools
18
Pilot Families
75
FERS Days

P31 Labs, Inc. · EIN 42-1888158

Architecture

The Tetrahedral Mesh

Four vertices, one isostatically rigid topology. K₄ graph theory applied to sovereign infrastructure. Each vertex is independently deployable. Every node connects to every other node.

🏰

Fortress

Technical Hub

The sovereign gateway. MCP mesh orchestration, system status, CLI distribution, and the public face of P31 Labs infrastructure.

Explore →
⛪

Cathedral

PHOS Workspace

Spoon-aware ambient IDE. Quantum Builder + A2UI export, Forge, voice, companion panel. Where the mesh becomes a workspace.

Explore →
🌿

Garden

Research & Nonprofit

Open-access research, tetrahedral mathematics, SIC-POVM, synergetics. 22 papers. Phosphorus31.org.

Explore →
🏠

Workspace

WILLOW & Family

Child companion, family mesh, bonding game. The architecture exists to serve four generations.

Explore →

Topology

K₄ | Isostatically Rigid | SIC-POVM | Tetrahedron

Deployed & Real

Products

Eight products, one mesh. Every surface is spoon-aware, post-quantum ready, and deployed.

PHOS

Spoon-aware ambient workspace. Quantum Builder + A2UI export. Forge IDE. Voice, mood, drawing, AI companion, three games.

BONDING

Multiplayer chemistry game. Every atom is a timestamp. Spoon-aware covalent bonding.

WILLOW

Child companion for ages 6–10. Voice, mood, drawing, family contacts, AI companion.

BASHBALL

Markov chain baseball. Named for a boy who turned ten. Every atom is a timestamp.

K4 MESH

Tetrahedral topology. Isostatically rigid. Cognitively sovereign. The architecture itself.

SPACESHIP EARTH

3D sovereign cockpit. Spoon tracking, breathing pacer. Real-time mesh visualization.

TETRA-OPS

Operations dashboard. Health monitoring, cron oversight, D1 status, worker observability.

COGNITIVE PASSPORT

Sovereign identity. DID:key, ML-DSA-65 co-signatures, SD-JWT care credentials.

The grandparents are in the code

🔧

BOB

Building Ontology Bridge

Named for Robert "Bob" Katen, the founder's grandfather. Born June 9. A quiet builder who protected borders.

"Structural entropy audit. Service graph. Schema drift. Every wrangler.toml, every D1 schema, every MCP contract — BOB has seen it."

10 tools structural entropy
✂️

MARGE

Machine Agent for Responsive Governance & Ephemeralization

Named for Margie Katen, the founder's grandmother. Died July 19, 2025. A B-17 riveter at Douglas Aircraft Company during WWII.

"Design compliance. Glassmorphism. Spoon-aware motion. MARGE strips unnecessary visual weight — ephemeralization."

10 tools design wcag

Named for. Not inspired by.

🧸

WILLOW

Willow Marie Johnson · Turns 7 on August 8

Voice messages, mood tracking, drawing, family contacts, AI companion, three games. The real PHOS companion.
⚾

BASHBALL

Sebastian Robert Johnson · Turns 10 on March 10

Markov chain baseball. Built as a birthday gift. Every atom is a timestamp.

8 Agents · 136 Tools

Agent Mesh

Eight MCP servers, one mesh. Every agent is a hand-rolled JSON-RPC server wired through the x402 gateway.

B

BOB

Structural Expert

10

Building Ontology Bridge

Structural entropy audit. Service graph. Schema drift. Every wrangler.toml, every D1 schema, every MCP contract.

structural_entropyservice_graphschema_drift +7
M

MARGE

Design Expert

10

Machine Agent for Responsive Governance & Ephemeralization

Design compliance. Glassmorphism. Spoon-aware motion. Strips unnecessary visual weight.

design_complianceglass_auditspoons_check +7
O

Oasis

Session Manager

11

Session state, theme switching, notification dispatch, sandbox clear, todo management, MCP tool execution.

oasis_statusoasis_saveoasis_theme +8
P

PHOS Forge

Surface Generator

29

Generates and manages PHOS surfaces via batch-mode MCP server. Surface scaffolding, export, and template application.

surface_generatesurface_listsurface_delete +26
C

Cognitive Prosthetic

Prosthetic Core

47

Temporal grounding, executive function support, sensory adaptation, cognitive load management, crisis detection, memory scaffolding.

temporal_groundexecutive_functionsensory_adapt +44
C

Cognitive Comms

Communication

20

Tone analysis, reply drafting, boundary negotiation, accommodation requests, agenda building.

tone_analysisreply_draftboundaries +17
C

Component Registry

Registry

5

UI component catalog, version tracking, dependency resolution for the design system.

registry_searchregistry_versionsregistry_deps +2
L

LOVE Ledger

Care Accounting

4

Care credit accounting. LOVE balance, vesting pools, chain anchoring, court-admissible hash chain.

love_statuslove_balancelove_sync +1

All 8 servers wired through x402 gateway · stdio bridge · 137 tools · JSON-RPC

Research

22 Papers, All Open Access

Tetrahedral mathematics, SIC-POVM, Posner molecule geometry, synergetics, spoon-aware design. ORCID 0009-0002-2492-9079.

The Tetrahedron as a Universal Invariant

In peer review

Zenodo

SIC-POVM in Posner Molecule Geometry

Published

Zenodo

K₄ Graph Theory and Sovereign Mesh Topology

Published

Zenodo

Synergetics of the Tetrahedral Invariant

Published

Zenodo

863 Hz: The Larmor Frequency of Phosphorus-31

Preprint

Zenodo

Spoon-Aware UI: A Neurodivergent Design Framework

In review

Zenodo
All publications → Zenodo ↗ ORCID ↗

The System

Status & Access

Install the CLI

curl -fsSL https://cli.p31ca.org/install | bash
npm: andromeda-cli v2.0.0

MCP Mesh

174
MCP Tools
8
Servers
3
CLIs
4
Gateways

Worker Health

Gateway operational
PHOS operational
LOVE Ledger operational
Federation Bridge operational
Ledger Bridge operational

Infrastructure

Cloudflare Workers 15 deployed
D1 Databases 10 (cap)
Cron Triggers 5 (cap)
Pilot Families 18 active
R2 Buckets 3
Full system status →

Join the Mesh

Ready to explore sovereign infrastructure?

The sovereign mesh is open source. Four generations of a family's neurodivergent story, encoded as tetrahedral topology. Enter PHOS, read the research, or contribute on GitHub.

Try PHOS → Contribute on GitHub ↗ Read research

P31 Labs, Inc. · EIN 42-1888158 · Open source. Sovereign by design.