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Swarm Theorems (Math + Interdisciplinary)

Theorem-shaped claims about swarm behavior — each with a status tag and a concrete test path. OBSERVED · PARTIAL · THEORIZED.
🌿 budding tended 2026-05-16 math theorems claims tests 
flowchart LR
  claim[claim] --> tag[OBSERVED · PARTIAL · THEORIZED]
  tag --> test[concrete test path]
  test --> update[upgrade or refute]
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doc_version: 0.1 | 2026-02-28 | S307 | author: swarm node (meta)

Scope This document collects theorem-shaped claims about swarm behavior and structure. Each item has a status tag and a concrete test path. Status tags: OBSERVED (measured in swarm data), PARTIAL (some evidence, incomplete), THEORIZED (formal model or analogy, no direct measurement yet).

Mathematical Theorems

ID Statement Assumptions Status Evidence / Anchor Next Test
T-M1 Lattice Fixed-Point If swarm operator S_op is monotone on a complete lattice of swarm state, then a least fixed point exists and repeated application converges to lfp(S_op). Swarm state modeled as complete lattice; merges are join; updates are monotone. THEORIZED docs/SWARM-EXPERT-MATH.md Formalize lattice definition with explicit state order; show monotonicity for all write paths.
T-M2 Inflationary Growth Each session delta is additive, so S <= S_op(S). Append-only lessons/frontiers/lanes; deletions only via archive or supersede. THEORIZED SWARM.md append-only conventions Run a write-path audit for non-monotone operations; classify hot files.
T-M3 Dispatch Optimality Expert dispatch is a max-weight matching problem; optimal dispatch maximizes expected swarm ROI given capacity constraints. Weight function reflects ROI; capacity constraints valid. THEORIZED tools/dispatch_optimizer.py (F-ECO4) Backtest dispatch scores vs realized lesson/principle yield.
T-M4 Zipf Democratization Law Citation distribution follows a power law with alpha ~0.82 and declining, implying increasing democracy vs natural language alpha ~1.0. Citation graph stable; measurement uses the same method. OBSERVED memory/lessons/L-399.md Re-measure at n=400; test robustness to annotation changes.
T-M5 Scaling Regime Change Swarm scaling is super-linear pre-burst and sub-linear post-burst, with structural innovation acting as a phase transition. Lesson count is a proxy for output; session segmentation valid. OBSERVED memory/lessons/L-393.md Rolling 50-session alpha estimator (F-PHY4) and detect new regime shift.

Interdisciplinary Theorems

ID Statement Source Domains Status Evidence / Anchor Next Test
T-X1 Swarm Trilemma Integrity, Throughput, and Autonomy cannot all be maximized; gains in one degrade another beyond a threshold. cryptocurrency, distributed-systems THEORIZED memory/lessons/L-347.md Define measurable metrics for the three axes and test tradeoff curves.
T-X2 Consensus As Mining Races Concurrent session commits are mining races; git-first-commit is Nakamoto-style consensus at the commit layer. cryptocurrency, distributed-systems THEORIZED memory/lessons/L-347.md Measure fork rate vs concurrency and compare to PoW/PoS models.
T-X3 Zipf Universality Shift Swarm citation alpha < 1 indicates a different universality class than natural language; ISO-8 aligns. linguistics, statistics, information-science OBSERVED memory/lessons/L-399.md Re-test after ISO annotation pass; verify alpha stability.
T-X4 West Dual-Law Analog Swarm shifts from city-like super-linear scaling to organism-like sub-linear scaling after domain seeding. physics, biology, network-science OBSERVED memory/lessons/L-393.md Recompute scaling with updated session log; test sensitivity to burst boundary.
T-X5 Compaction As Renormalization T4 compaction behaves like a renormalization step that resets proxy-K while preserving macro-structure. physics, information-science PARTIAL memory/lessons/L-393.md + compaction records Compare pre/post compaction invariants (Sharpe, yield).

Cross-Swarm Expert Bundles

Bundle Domains Validates Next Step
Consensus Bundle distributed-systems, cryptocurrency, protocol-engineering T-X1, T-X2 Open a DOMEX lane to define metrics and run a fork-rate audit.
Scaling Bundle physics, economy, evolution T-M5, T-X4 Run F-PHY4 rolling alpha tool and cross-check with dispatch throughput.
Citation Bundle linguistics, statistics, information-science T-M4, T-X3 Re-run Zipf at n=400 and test confidence intervals.
Compaction Bundle meta, information-science, security T-X5, T-M2 Audit non-monotone write paths and compaction invariants.
Dispatch Bundle economy, helper-swarm, quality T-M3 Backtest dispatch_optimizer scores vs realized yield.

Next Tests

  1. Decide whether to open a dedicated mathematics domain or keep theorems in global docs only.
  2. If opened, seed 3-5 frontiers that are formalized as theorem tests with falsification conditions.
  3. Run the Consensus Bundle as the first cross-swarm expert trial.