Appendix E: Mathematical Symmetry Derivations

This document provides the mathematical justification and implementation details for the even_collars parameter in eq_caps.

Mathematical Derivations

The standard EQ algorithm determines the collar count through rounding:

\[ n_{\text{collars}} = \max\!\bigl(1,\;\operatorname{round}\bigl((\pi - 2\,c_{\text{polar}}) \,/\, \alpha_{\text{ideal}}\bigr)\bigr) \]

Forced Parity

When even_collars=True, we constrain \(n_{\text{collars}}\) to be even by rounding the half-count:

\[ n_{\text{half}} = \max\!\bigl(1,\;\operatorname{round}\bigl((\pi - 2\,c_{\text{polar}}) \,/\, (2\,\alpha_{\text{ideal}})\bigr)\bigr), \qquad n_{\text{collars}} = 2\,n_{\text{half}}. \]

This ensures that the boundary at \(k = n_{\text{collars}}/2\) sits exactly at:

\[ c_{\text{polar}} + \tfrac{n_{\text{collars}}}{2}\,\alpha_{\text{fit}} \;=\; c_{\text{polar}} + \tfrac{\pi - 2\,c_{\text{polar}}}{2} \;=\; \frac{\pi}{2}. \]

Architectural Optimizations

Cache Reuse Strategy

PyEQSP’s recursive partitioning uses a _private cache for \(S^{d-1}\) partitions. Hemispherical Symmetry: Partitions generated with even_collars=True are perfectly symmetric about the equator. This allows the Southern hemisphere regions to be calculated once and then mirrored, providing a 100% cache hit rate for the second half of the partition generation for \(S^3\).

Interface Symmetry

To maintain polymorphic compatibility, all property functions must accept even_collars, even if the property (like total area error) is parity-invariant. This ensures call-site stability for users while allowing internal mathematics to exploit symmetry when available. For foundational citations, see the Volume 2 References.

Audit Summary (PyEQSP 1.0b2 Beta)

  • Derivation Alignment: The even_collars algebraic derivation structurally mirrors the runtime calculation in eq_caps() (eqsp/partitions.py).

  • Cache Integrity: The \(n_{\text{collars}}/2\) cache boundary correctly mirrors collar \(k\) to \(n_{\text{collars}} - k + 1\) ensuring 100% cache utilization for \(S^3\) southern hemisphere regions.

  • Status: VERIFIED