O0-OBS-004

Nested Boundaries and Hierarchical Self-Models

**Version:** 1.0

**Research status:** PLANNED

CLAIM STATUS: UNTESTED PREDICTION
EVIDENCE TYPE: COMPUTATIONAL SIMULATION (PLANNED)
PHYSICAL VALIDATION: NONE
INDEPENDENT REPLICATION: NOT YET ATTEMPTED
PHILOSOPHICAL PROVENANCE: O/0 ARCHIVE
ARCHIVE ENDORSEMENT: LIMITED TO REPORTED RESULT

Abstract

This study extends the boundary simulation framework to multi-scale analysis on a 200×200 grid, investigating whether small boundaries form within larger bounded regions—creating nested hierarchical structure. We define formal nesting criteria, test whether nested boundaries exhibit algebraic properties described in O0-MATH-008 (boundary composition, intersection, and containment operations), and examine whether hierarchical self-models emerge at multiple scales. The central question is whether boundary nesting is a stable, recurring structural feature or merely a transient coincidence of the dynamics.

Note: Conceptual provenance from the O/0 philosophical archive does not constitute empirical support.

Source proposition

From O0-MATH-008: "Boundaries compose algebraically: if B_1 ⊂ B_2 (B_1 is nested within B_2), then the interior dynamics of B_1 are conditionally independent of the exterior of B_2 given B_2's boundary states." This study tests whether such nesting occurs and whether the algebraic independence property holds.

Scientific audit

• **Testability:** HIGH. Nesting is geometrically verifiable; conditional independence is measurable via mutual information.
• **Falsifiability:** YES. If no nested structures form in extended simulations, or if nesting violates the predicted algebraic properties, hypotheses are falsified.
• **Novelty:** MODERATE-HIGH. Multi-scale boundary analysis in cellular automata is understudied.
• **Potential confounds:** Apparent nesting could be artifactual if boundaries are detected at different scales simultaneously; requires consistent scale comparison.

Research question

1. Do bounded regions spontaneously form within larger bounded regions in cellular automata dynamics?

2. If so, do nested boundaries exhibit the algebraic independence property: I(B_1_interior; B_2_exterior | B_2_boundary) ≈ 0?

3. Are nested structures stable (persisting > 500 timesteps) or transient?

Operational definitions

• **Boundary B_k:** A connected set of cells forming the border of a bounded region R_k, satisfying I(R_k_interior; R_k_exterior | B_k) < τ.
• **Nesting relation (⊂):** B_1 ⊂ B_2 iff every cell in R_1 (the region bounded by B_1) is also contained within R_2 (the region bounded by B_2), AND B_1 ∩ B_2 = ∅.
• **Nesting depth:** Maximum chain length in the nesting partial order. Depth 0 = no boundary. Depth 1 = single boundary. Depth 2 = boundary within boundary.
• **Algebraic independence:** I(R_1_interior; R_2_exterior | B_2) < ε, where ε = 0.01 bits (effectively zero given finite sample estimates).
• **Nesting stability:** A nested pair (B_1, B_2) is "stable" if both boundaries persist and the nesting relation holds for ≥ 500 consecutive timesteps.
• **Hierarchical self-model:** If OBS-001 prediction layers are attached at both scales, do both develop accurate self-models simultaneously?

Hypothesis

H1: Nested boundaries will form spontaneously in ≥ 20% of simulation runs within 10,000 timesteps on a 200×200 grid.

H2: When nesting occurs, the algebraic independence property I(R_1_interior; R_2_exterior | B_2) < 0.01 will hold in ≥ 80% of cases.

H3: At least some nested structures will be stable (persist > 500 timesteps) in ≥ 30% of runs where nesting occurs.

H4: Both nested and nesting boundaries can simultaneously support accurate self-models (r > 0.6 at both levels).

Null hypothesis

H0_1: Nested boundaries never form (or form in < 5% of runs).

H0_2: Algebraic independence does not hold—inner regions remain informationally coupled to outer exterior.

H0_3: All nested structures are transient (< 100 timesteps).

H0_4: Self-models at different nesting levels interfere—only one level can maintain r > 0.6 at a time.

Competing explanations

1. **Scale artifact:** Apparent nesting might be an artifact of using multiple detection thresholds simultaneously. Controlled by using a single consistent τ at all scales.

2. **Spatial autocorrelation:** In sufficiently large grids, independent boundaries will sometimes overlap geometrically by chance. Controlled by computing expected nesting rate under spatial null model (random boundary placement).

3. **Boundary leakage:** Inner boundaries might simply be a part of the outer boundary's internal dynamics, not genuinely independent structures. Tested via the algebraic independence criterion.

Formal model

Grid: 200×200 cells, k=4 states, Moore neighborhood interaction.

Detection: Multi-scale boundary detection algorithm:

1. Coarse scan: Detect boundaries at scale ρ=5 (5×5 averaging windows). Identify large bounded regions R_coarse.

2. Fine scan: Within each R_coarse, detect boundaries at scale ρ=1 (single-cell resolution). Identify nested regions R_fine ⊂ R_coarse.

3. Verify nesting: Confirm geometric containment and boundary separation (B_fine ∩ B_coarse = ∅).

4. Test independence: Compute I(R_fine_interior; G \ R_coarse | B_coarse) where G is the full grid.

Algebraic operations (from O0-MATH-008):

• **Composition:** If B_1 ⊂ B_2, define composed boundary B_1∘B_2 = B_2 (outer boundary screens off inner)
• **Intersection:** If B_1 and B_2 partially overlap, define B_1 ∩ B_2 as the shared boundary cells
• **Union:** B_1 ∪ B_2 = outer envelope of both bounded regions

Test: Do these operations preserve the Markov blanket property?

• Composition: Does I(R_1; exterior(R_2) | B_2) < τ? (Screening-off property)
• Intersection: Is B_1 ∩ B_2 itself a valid boundary for some region?

Methods

1. **Grid initialization:** 200×200 grid, random initial conditions, k=4 states.

2. **Burn-in:** Run dynamics for 2,000 timesteps without measurement (allow structure formation).

3. **Boundary detection:** From t=2,000 to t=12,000, apply multi-scale detection at every 10th timestep.

4. **Nesting identification:** At each measurement point, identify all nesting pairs using geometric containment algorithm.

5. **Independence testing:** For each identified nesting pair, compute conditional mutual information.

6. **Stability tracking:** Track nesting pairs across time. Record formation, dissolution, and duration.

7. **Self-model test:** For runs with stable nesting (> 500 steps), attach OBS-001 prediction layers at both scales. Measure simultaneous self-model accuracy.

8. **Algebraic operations:** For each nesting pair, test composition, intersection, and union operations.

9. **Repetitions:** 100 independent simulation runs.

Controls

• **Random boundary null model:** Place boundaries randomly on the 200×200 grid with same density as observed. Compute expected nesting rate by chance. Observed nesting must exceed this.
• **Small grid control:** Run same protocol on 50×50 grid where nesting is physically less likely due to space constraints. Establishes scale dependence.
• **Single-scale control:** Apply only one detection scale. Verify that nesting is not an artifact of multi-scale detection.
• **Shuffled dynamics:** Apply boundary detection to temporally shuffled grid states. Confirm that nesting stability requires temporal coherence.

Predictions

**Expected Outcome Matrix:**

| Metric | Expected value | Decision threshold |

|--------|---------------|-------------------|

| Runs with nesting | 25-40% | > 10% to support H1 |

| Algebraic independence (when nested) | I < 0.01 in 85% of cases | > 70% to support H2 |

| Stable nesting (> 500 steps) | 30-50% of nesting events | > 20% to support H3 |

| Dual self-model accuracy | r > 0.6 at both levels in 60% | > 40% to support H4 |

| Nesting rate vs. random null | 3-5× higher | > 2× to reject chance explanation |

| Maximum nesting depth observed | 2-3 levels | Depth > 2 would be notable |

**Decision rules:**

• If nesting rate > 10% AND exceeds random null by > 2×: **SUPPORT H1**
• If nesting rate < 5% OR does not exceed random null: **REJECT H1**
• If independence holds in > 70% of nested pairs: **SUPPORT H2, support algebraic framework**
• If independence holds in < 50% of nested pairs: **REJECT H2, algebraic framework inadequate**

Falsification criteria

H1 is falsified if: (a) nesting occurs in < 5% of runs, OR (b) observed nesting rate does not significantly exceed random null (χ² test, p > 0.05).

H2 is falsified if: conditional MI exceeds 0.05 bits in > 50% of identified nesting pairs.

The algebraic framework (O0-MATH-008) is not supported if composition does not preserve the Markov blanket property in > 30% of tested cases.

Results / Expected Outcomes

Study not yet conducted. Predictions based on information-theoretic principles and preliminary 100×100 results:

• Nesting expected to occur but be relatively rare (20-40% of runs)
• Maximum depth of 2-3 levels expected (deeper nesting is thermodynamically costly to maintain)
• Algebraic independence expected to hold when boundaries are well-separated (gap ≥ 3 cells)
• Dual self-models may show transient competition before settling into stable coexistence

Uncertainty

• **Grid size sufficiency:** 200×200 may still be too small for robust multi-level nesting. Larger grids (500×500) may be needed.
• **Detection sensitivity:** Multi-scale detection introduces additional parameters that may affect nesting identification.
• **Computational cost:** Full MI computation on 200×200 grid at 10-timestep intervals for 10,000 steps is computationally expensive. May require approximations.
• **Algebraic completeness:** Only three operations (composition, intersection, union) are tested; the full algebra may have more structure.

Limitations

1. Two spatial dimensions limit the complexity of nesting geometries possible.

2. The definition of nesting requires strict geometric containment—partial or overlapping boundaries are excluded.

3. Self-model testing at multiple scales introduces additional architectural choices that may confound interpretation.

4. Computational cost limits grid size and run duration.

5. No claim about biological or cognitive hierarchy is made—this tests computational nesting only.

Replication status

Not yet conducted.

Data and code

• Simulation framework: Extended SIM-001 with multi-scale detection module
• Grid size: 200×200 (configurable)
• Language: Python 3.11+, JAX for GPU acceleration of MI computation
• Output: Nesting event database, algebraic operation verification logs, self-model accuracy traces
• Pre-registration: This document. Timestamp: 2026-07-26.

Relationship to philosophical archive

This study tests the mathematical claims of O0-MATH-008 regarding boundary algebra. The archive proposes that boundaries compose algebraically—this study determines whether that algebraic structure actually emerges in dynamics. A positive result would lend computational support to the mathematical framework; a negative result would require revision of the algebraic claims.

References

• Friston, K. (2019). A free energy principle for a particular physics. *arXiv:1906.10184*.
• Kirchhoff, M. D., & Kiverstein, J. (2019). Extended Consciousness and Predictive Processing. Routledge.
• Rosas, F. E., et al. (2020). Reconciling emergence: An information-theoretic approach. *Entropy*, 22(4).
• Hoel, E. P. (2017). When the map is better than the territory. *Entropy*, 19(5).
• Albantakis, L., et al. (2023). Integrated information theory (IIT) 4.0. *PLoS Computational Biology*, 19(10).

Revision history

| Date | Version | Changes |

|------|---------|---------|

| 2026-07-26 | 1.0 | Initial pre-registration |