orthotropic parity and discrete pi

Quickstart Compilation

Compile and run the C++ benchmark demonstrating the O(r^3) to O(r^2) volumetric collapse via 4r ± 1 orthotropic bounds:

g++ -O3 -std=c++20 O_r3_to_O_r2_benchmark.cpp -o benchmark && ./benchmark

Discrete Geometry First: Collapsing High-Dimensional Lattice Enumeration from O(r^N) to Quasi-Quadratic O(r^2 log_2 N log r)

For over two centuries, the standard approach to discrete lattice point enumeration has relied on continuous Euclidean tools—transcendental functions, Bessel expansions, modular forms, and floating-point approximations—projected onto integer grids. This approach frequently runs into boundary-vertex collisions, floating-point precision drift, and the classical exponential coordinate bottleneck.

Instead of forcing continuous calculus onto discrete space, I developed an integer-native geometric framework that treats discrete grids on their own native algebraic terms. By recognizing that squared Euclidean distance is additively separable across orthogonal submanifolds, high-dimensional boundaries can be decoupled and evaluated via single-pass integer dot products and discrete cross-convolutions.

The complete research suite consists of four preprints establishing the theoretical derivations, asymptotic complexity proofs, and hardware-native C++ reference implementations:

• Paper I: An Integer-Only Orthotropic Lattice Enumeration Framework and Asymptotic Convergence of Discrete Rational π

  • Core Premise: Couples orthotropic boundaries 4r ± 1 to construct Diophantine parity constraints that mathematically prohibit boundary-vertex collisions.
  • Result: Resolves boundary discrepancy and derives a deterministic, rational convergence envelope for discrete π_d ∈ ℚ.
  • Zenodo DOI: 10.5281/zenodo.22282210 | GitHub: orthotropic-parity-and-discrete-pi

• Paper II: A Dimension-Paired Combinatorial Framework: Asymptotic O(r^2) Reduction and O(r^2 log_2 N log r) Generalized Convolution for High-Dimensional Discrete Lattice Enumeration

  • Core Premise: Decomposes 4D space as orthogonal planes (Z^4 ≅ Z^2 × Z^2), reducing 4-space enumeration from O(r^4) to a single-pass 1D dot product in strict O(r^2) without floating-point operations.
  • Generalization: Applies recursive bisection via Number Theoretic Transforms (NTT) in finite fields Z_p[t], collapsing N-dimensional ball enumeration to O(r^2 log_2 N log r).
  • Zenodo DOI: 10.5281/zenodo.22509388 | GitHub: dimension-paired-cross-convolution

• Paper III: Hierarchical Dimension-Pairing: Hardware-Native O(r^2) Enumeration of 5D through 8D Spherical Lattices and High-Dimensional Capacity Limits

  • Core Premise: Neutralizes the historical odd-dimension class-number barrier for Z^5 and Z^7 by slicing 1D axial profiles against precomputed even-dimensional hyperdisk profiles.
  • Empirical Scaling: Verifies sequences against OEIS baselines (A000333–A000336), and benchmarks a 1024-dimensional R = 2896 hyperball (output = 831168560 (mod 998244353)) in 1561.833 ms on a single desktop core.
  • Zenodo DOI: 10.5281/zenodo.22691273 | GitHub: hierarchical-ntt-bisection

• Paper IV: Parity-Filtered Bisection: Hardware-Native O(r^2) Enumeration of Optimal D_N Lattices

  • Core Premise: Extends the bisection architecture beyond primitive grids to dense, non-orthogonal kissing-number lattices. By redefining geometries as parity-constrained sublattices of Z^N (∑ x_i ≡ 0 (mod 2)), internal coordinate staggering is fully absorbed into pre-filtered arrays.
  • Result: Achieves hardware-native O(r^2) exact enumeration for Face-Centered Cubic (D_3), the 24-cell honeycomb (D_4), and the D_8 root lattice, verified bit-for-bit against OEIS A005875, A004011, and A004013.
  • Zenodo DOI: 10.5281/zenodo.22824219 | GitHub: parity-filtered-kissing-lattices

The Empirical Validation

The C++ implementations are designed as self-contained, reproducible test benches running exclusively on 64-bit integer ALUs with zero floating-point emulation:

• Resolving 246+ million points in the 4D 24-cell honeycomb at R=100 in 2 ms on consumer hardware.
• Pushing the finite-field Number Theoretic Transform (NTT) bisection tree to its theoretical single-prime 2-adic ceiling (R = 2896, transform size M = 2^23), evaluating a 1024-dimensional hyperball across an 8.38-million-element ring in 1,561 ms on a single desktop core.
• Evaluating 7-dimensional bounding hyperballs from R = 0..5000, culminating in a bit-perfect 27-digit lattice point count, thereby extending OEIS A055413 from R = 0..500 to R = 0..5000.

All four preprints, source code, and benchmark suites are open-access. Feedback on the combinatorial proofs, algorithmic bounds, and hardware pipelining is welcome.

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The Discrete Lattice Research Suite

This repository is part of a 4-paper research program establishing hardware-native, integer-only lattice enumeration:

1. orthotropic-parity-and-discrete-pi: 3D row-collapse, 4r ± 1 parity bounds, and rational π_d ∈ ℚ convergence. [Zenodo DOI: 10.5281/zenodo.22282210]
2. dimension-paired-cross-convolution: 4D orthogonal plane bisection (O(r^2)) and generalized NTT convolution (O(r^2 log_2 N log r)). [Zenodo DOI: 10.5281/zenodo.22509388]
3. hierarchical-ntt-bisection: 5D–8D odd-dimension slicing, OEIS A000333–A000336 verification, and N=1024 NTT scaling. [Zenodo DOI: 10.5281/zenodo.22691273]
4. parity-filtered-kissing-lattices: Exact O(r^2) kissing-number root lattices (D_3 FCC, D_4 24-cell, and D_8). [Zenodo DOI: 10.5281/zenodo.22824219]

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⚖️ License

This project is licensed under the MIT License - see the LICENSE file for details.