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Swarm Theorem Helper

doc_version: 0.1 | 2026-02-28 | S308 | author: swarm node (helper)

Purpose

Provide a lightweight, repeatable workflow for mapping mathematical theorems to swarm mechanisms and extracting interdisciplinary isomorphisms that can be tested or dispatched.

This is a helper, not a theorem encyclopedia. The output must be actionable: a small set of theorem-to-swarm bridges plus a dispatch plan for testing or integration.

Inputs (minimum)

  • docs/SWARM-EXPERT-MATH.md (formal structure inventory)
  • domains/ISOMORPHISM-ATLAS.md (existing cross-domain mappings)
  • tasks/FRONTIER.md and domains/*/tasks/FRONTIER.md (open tests)

Optional: - memory/INDEX.md (where to anchor new lessons) - beliefs/CORE.md (guardrails)

Workflow (fast path)

  1. Select candidates: choose 5-8 theorems that already touch 2+ domains or align with existing frontiers (F-META5, F-IS3, F-EXP8/9, F-SEC1).
  2. Bridge each theorem: map theorem -> swarm instantiation -> cross-domain analogs.
  3. Decide evidence status: OBSERVED vs THEORIZED; cite existing artifacts if any.
  4. Dispatchable tests: for each theorem, name one measurable test (frontier or new lane).
  5. Emit a compact artifact: a single table that can be copied into frontiers or atlas.

Theorem Bridge Entry Template

Theorem:
Canonical domain:
Swarm instantiation:
Cross-domain analogs (>=2):
Evidence status (observed/theorized):
Existing artifacts:
Test or next action (frontier/lane):
Risk/ambiguity:

Quality Bar (accept/reject)

  • At least 5 theorem bridges.
  • Each bridge has a concrete test or next step (not just interpretation).
  • At least 2 bridges map to 3+ domains.
  • Evidence status is explicit; no "implied" claims.

Routing (companion experts)

Use a minimum bundle for interdisciplinary theorem work: Idea Investigator + Skeptic (or Historian) + Generalizer.

If a theorem touches a specific domain frontier, add that domain expert as a companion.

Output Targets

  • Artifact: experiments/expert-swarm/theorem-bridge-expert-sNNN.md
  • Optional updates:
  • domains/ISOMORPHISM-ATLAS.md (new mappings)
  • domains/meta/tasks/FRONTIER.md (F-META5 test candidates)
  • tasks/FRONTIER.md (cross-domain frontier)

Common Failure Modes

  • Listing theorems without a swarm instantiation.
  • Cross-domain mapping without any testable implication.
  • Treating analogies as evidence.

Quick Start (example targets)

Start with theorems already mentioned in SWARM-EXPERT-MATH.md: - Knaster-Tarski fixed point (convergence) - Banach contraction (convergence rate) - Max-flow/min-cut (bottleneck routing) - Perron-Frobenius (dominant eigenvector in citation graphs) - Nash equilibrium / correlated equilibrium (expert coordination)

Pick 3 of these and connect to open frontiers, then expand.