Maintenance-first ordering
Maintenance-first preserved sustained cycling but suppressed many moves that had been selected and were affordable before maintenance.
SUSTAINED CYCLE / WEAK TRANSPORTResearch Release / Sep 13, 2026
Natural Math: Fixed Local Rules Become More Reliable Under Proportional Scale
Reliability increased across the tested proportional scales without increasing local rule complexity; normalized throughput did not improve.
- Project
- Natural Math
- Status
- EXPERIMENTAL_RESULT
- Evidence class
- EARNED_WITH_TESTED_SCOPE
- Research area
- Local deterministic formation and proportional scale
- Authors
- James Allen Clow, Melissa Ellen Clow
- Release ID
- RR-2026-09-13-NM-SCALE
The result
Key result
For this frozen mechanism and tested x1-x3 scaling regime, larger proportional systems created more opportunities for locally required conditions to coincide. Transported-substrate reuse became reliable in all 64 tested seeds at x2 and x3, while normalized work per agent-tick declined.
x144/640.01324 work / agent-tick
x264/640.00970 work / agent-tick
x364/640.00867 work / agent-tick
| Scale | Grid | Agents | Ticks | Lifetime | Median work | Re-formations | Independent reuse | Transported reuse median | Transport-success seeds |
|---|---|---|---|---|---|---|---|---|---|
| x1 | 17x17 | 18 | 512 | 7 | 122 | 381.5 | 181.5 | 1 | 44/64 |
| x2 | 34x34 | 72 | 1024 | 14 | 715.5 | 2341.5 | 996.5 | 6 | 64/64 |
| x3 | 51x51 | 162 | 1536 | 21 | 2157 | 7679 | 2959 | 12 | 64/64 |
Failed preregistration preserved
The original transported-substrate gate remains failed
The frozen carry correction preregistered at least 48/64 transported-success seeds at x1. The executable result was 44/64, so cycle gate C6 failed and the full cycle remained failed.
- A later 256-seed extension produced 171/256 successes, or 66.8%, with a Wilson 95% interval of about 60.8%-72.3%.
- The four 64-seed blocks were 44/64, 45/64, 41/64, and 41/64.
- Late work remained 256/256, supporting a conditional small-scale transport boundary rather than run-to-run noise.
- The separate scale-test gate required at least 32/64 transported successes, so x1 passed that later scale-test gate without retroactively passing the original 48/64 carry gate.
What changed
What changed
- World side scaled as 17*s, agents as 18*s^2, horizon as 512*s, formation lifetime as 7*s, and material feed per tick as s^2.
- The number and duration of possible local causal histories increased with scale.
- Total cycling and transported-reuse reliability increased across the tested range.
What stayed frozen
What did not change
- Movement score, carry rule, trace rule, formation rule, maintenance priority, dissolution rule, work rule, local costs, relation count, and tie-break.
- Movement remained one edge per tick.
- Local formation, movement, maintenance, and work costs remained unchanged.
- Cargo capacity remained three units of ordinary conserved material.
- No global planner, global map, curiosity bonus, future-state representation, or broadcast knowledge was added.
Why it matters
Why it matters
- A capability can be opportunity-limited rather than rule-limited.
- Scale can change the space of admissible local causal histories even when the local rule does not change.
- Before adding reasoning complexity to an unreliable system, test topology, lifetime, resource availability, transport, sparse encounter rate, and the number of independent causal histories.
Causal and diagnostic audit
Negative and diagnostic results
Movement-first ordering
Reversing the order strengthened transport but destroyed the larger sustained-work cycle.
STRONGER TRANSPORT / CYCLE COLLAPSERejected arbitration variants
Immediate-energy arbitration, terminal-life arbitration, and commitment reserve before formation did not preserve the desired cycle and universalize transport together.
FAILEDInterpretation
Interpretation boundary
A fixed set of local deterministic rules can preserve useful cycling across proportional increases in system size, while some higher-order capabilities become more reliable because larger systems create more opportunities for locally required conditions to coincide.
Scale changes the space of possible local causal histories even when the local rule does not change.
RULE COMPLEXITY != OPPORTUNITY COMPLEXITY.
Limitations
Limitations
- Only scale factors x1, x2, and x3 were tested.
- World size, population, horizon, lifetime, and input schedule scaled together, so this release does not isolate which scaling dimension caused the reliability change.
- The result belongs to this deterministic integer-discrete mechanism and its declared conditions.
- The local-condition audits identify necessary and diagnostic conditions, not a complete causal decomposition of every failed seed.
Not established
What is not established
- Arbitrary-scale continuation or a universal scaling law.
- Improved normalized efficiency.
- Intelligence, life, consciousness, or biological reproduction.
- A general or philosophical theory of emergence.
- Retroactive passing of the original 44/64 preregistered gate.
Relation to current work
Canonical relation
This is a Natural Math mechanism-level experimental branch. It does not replace the Natural Math v5 frozen integer baseline and does not rewrite earlier Natural Math as Consequential Formation.
This result is being examined as a mechanism-level example for the broader Consequential Formation and Formative Field program: capability reliability may sometimes be limited by the availability of admissible local histories rather than by insufficient local reasoning complexity. This is interpretation, not evidence that artificial intelligence was demonstrated.
Source artifacts
The article is the entry point. These files are the record.
Evidence precedence: executable and raw receipt; frozen specification and preregistration; experiment report; handoff; interpretation.
Complete source package
original package
SHA-256 2b7f4c0b0ec2eb3ed8aa6ee25e149bf8e9ac10f44b54832e103833ef417ea1cfHandoff for Jim
handoff
SHA-256 b21232ae4184fa43a7b9f579fc9f4dec3f35824d05a8cce4132cc37fecd8abfeCarry correction evidence
executable receipts
SHA-256 58b540adbb2c6b385fd6c1e845a50d32153624efc3f64172848bfe1b357f33b9Carry stability and local-condition audit
audit receipts
SHA-256 50485202f1413588a8b61ec7157b775f4b96e0d4b0ec8a2795eafbd7171e189aProportional scale test
preregistered experiment
SHA-256 a0732b4ade249f22b4491ba32f9df8f27d3f4fde2c0fc0fa843a2fe1cbca85b9Formed-movement causal audit
diagnostic receipts
SHA-256 ba67af4b14741b95ae05c370fe7e1f3c6bf5d598294ee30631c59768cd37376aTick-order energy audit
accepted instrumentation
SHA-256 4b8f1997ceb9eaef83765019d042233db6e1a22a9b9f228c16e875b2b3a6cb3bSource package README
source orientation
SHA-256 503ccfa803504463415277f82765c22fe1dcc6dfeaae2ce0276f38c66419ccb4Original package checksum manifest
source manifest
SHA-256 893287337553cebda95afd3d12332d15321329e75d0c3bd9c93be5e8f8eeff43Next question
Next question
Which scaling dimension - topology, population, horizon, formation lifetime, input rate, or independent-history count - causally changes transported-reuse reliability under a newly frozen factorial or ablation design?