Bulk Scaling Laws and the Parity Conjecture
Abstract
We discover that the bulk plateau values of the SLD central moment differences are NOT constant for r ≥ 4 — they scale linearly with system size n. Combined with the known linear scaling of cycle values, this yields exact formulas for the asymptotic cycle/bulk ratio. The simple parity conjecture (even→2, odd→3/2) is falsified: R₅ = 13/8, not 3/2.
1. The Discovery: Bulk Values Scale with n
Prior work assumed the bulk plateau value was constant in n for all moments. This is true for μ₂ and μ₃ but FALSE for μ₄ and μ₅.
μ₄ bulk plateau values
| n | Bulk Δμ₄ | In 32nds |
|---|---|---|
| 16 | -103/32 | -103 |
| 18 | -7/2 | -112 |
| 20 | -121/32 | -121 |
The sequence -103, -112, -121 increases by 9 per 2 units of n.
2. Corrected Asymptotic Ratios
Even moments (μ₄)
Odd moments (μ₅)
3. The Actual Ratio Sequence
| Moment | Asymptotic Ratio | Type |
|---|---|---|
| μ₂ | 2 (exact all n) | Additive |
| μ₃ | 3/2 (exact all n) | Sub-additive |
| μ₄ | 2 | Additive |
| μ₅ | 13/8 = 1.625 | Sub-additive |
| μ₆ | 3 (conjectured) | Super-additive |
4. Verified Structural Formulas
5. Physical Interpretation
The breakdown of simple parity means the interaction between two boundary removals is not captured by a simple even/odd rule. Instead, each moment order has its own characteristic ratio that encodes how boundary effects combine.
For the Laflamme-3T framework: the Ψ measure must be sensitive to the full moment hierarchy, not just the variance. A system that appears identical at the variance level may be structurally different at the kurtosis level or higher. Consciousness measures based only on second-order statistics miss the rich structure that emerges at higher orders.