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Mathematical and Historical References for libttak Algorithms

This document preserves the academic and historical lineage of the mathematical principles applied in libttak. To keep the source code mechanically readable and free of obscure cultural naming, all functions, variables, and comments use generic, descriptive system terminology. The references below are maintained for researchers who wish to trace the origins of specific algorithms.

ttak_math_lane_mul (formerly ttak_math_dawonsul_lane_mul)

  • Historical concept: Dawonsul (多變數獨立法) — independent lane processing for multivariate linear systems, associated with the works of Hong Jeong-ha and collaborators.
  • Reference: Hong Jeong-ha, "Guiljip (九一集)", 1660s.
  • Application in code: Matrix-vector lane multiplication used in ttak_matrix_multiply_vec. Each variable lane is processed independently to maximize throughput via aligned-limb FMA-style accumulation.

ttak_matrix_set_ols_magic_square_4x4 (formerly ttak_matrix_set_gusuryak_4x4)

  • Historical concept: Mutually orthogonal Latin squares combined into a normal 0-15 magic square, documented in Choi Seok-jeong's work on combinatorial design.
  • Reference: Choi Seok-jeong, "Gusuryak (九數略)", 1700.
  • Application in code: Initializes a 4×4 matrix as an Orthogonal Latin Square (OLS) whose rows, columns, and diagonals sum to 30. Used for deterministic, collision-minimizing grid-based data placement.

ttak_math_approx_sin and ttak_math_approx_cos

  • Historical concept: Polynomial approximation methods for trigonometric functions cataloged in the Yussigihae tradition.
  • Reference: Nam Byeong-gil, "Sanhak Jeong-ui (算學正義)", 1849.
  • Application in code: Fixed-point polynomial expansions for approximate sine and cosine, used in ttak_matrix_set_rotation to compute stable rotation matrix entries.

ttak_bigreal_op_aligned_addsub (formerly ttak_bigreal_op_cheonwonsul)

  • Historical concept: Cheonwonsul (天元術), a system of aligned limb processing reinterpreted from Tian Yuan Shu for big-integer arithmetic.
  • Reference: Hong Jeong-ha, "Guiljip (九一集)", 1660s.
  • Application in code: Internal helper for ttak_bigreal_add and ttak_bigreal_sub. Performs mantissa addition or subtraction after exponent alignment, mapping limbs to cache-line-friendly layouts.

Latin-Square Scatter LUT in src/mem/arena_helper.c

  • Historical concept: Latin-square offsets for epoch scattering.
  • Reference: Choi Seok-jeong, "Gusuryak (九數略)", 1700.
  • Application in code: ttak_arena_scatter_offset uses an 8×8 Latin-square lookup table (ttak_arena_latin_square_lut) to scatter epoch allocations and minimize cache-line reuse across generations.

OLS Traversal in src/container/pool.c

  • Historical concept: Orthogonal Latin lattice selection for deterministic slot traversal.
  • Reference: Choi Seok-jeong, "Gusuryak (九數略)", 1700.
  • Application in code: ttak_object_pool_alloc traverses pool slots via an order-8 Orthogonal Latin Square (OLS) to avoid clustering and reduce bitmap-scan contention.

Lock-Free Lattice Ingress in src/net/lattice.c

  • Historical concept: Deterministic coordinate scheduling on a 2-D lattice (Sanpan) for parallel ingress.
  • References:
    • Choi Seok-jeong, "Gusuryak (九數略)", 1700.
    • Yi Sang-hyeok, "Suri (數理)", 1890s.
  • Application in code: ttak_net_lattice_write performs lock-free deterministic writes by sweeping lattice coordinates in a deterministic order derived from OLS-based scheduling rules.

ttak_apply_mols_control in include/ttak/mols_control.h

  • Historical concept: Mutually Orthogonal Latin Squares (MOLS) applied to load redistribution.
  • Reference: Reverse Siamese Latin-square constructions.
  • Application in code: Congestion-control helper that redistributes load across a 64×64 mesh. Node coordinates feed a matched pair of Latin squares whose values are XOR-mixed with the caller's seed to produce deterministic shuffling.

Buddy Allocator Residue Lookup in src/phys/mem/buddy.c

  • Historical concept: Residue-class lookup tables for size-class indexing.
  • Reference: Nam Byeong-gil, "Sanhak Jeong-ui (算學正義)", 1849.
  • Application in code: select_block uses a bitmask-based residue lookup to choose the appropriate buddy block order, reducing fragmentation through deterministic size-class selection.

Binary GCD in src/math/factor.c

  • Historical concept: Daeyeonguilsul (大衍求一術) — subtraction- and parity-based tabular reduction for coprime/remainder evaluation, structurally mapping to Stein's binary GCD algorithm.
  • Reference: Traditional Korean mathematical manuscripts on Daeyeonguilsul (大衍求一術).
  • Application in code: ttak_gcd_u64 replaces division with bit-shifts and subtractions (__builtin_ctzll), eliminating division latency in prime factorization routines.

Modular Multiplicative Inverse in src/math/ntt.c

  • Historical concept: Daeyeonguilsul (大衍求一術) tabular continuous reduction (연초법/Yeon-cho) for solving linear congruences.
  • Reference: Traditional Korean mathematical manuscripts on Daeyeonguilsul (大衍求一術).
  • Application in code: ttak_mod_inverse uses the extended Euclidean algorithm with tabular reduction ordering, minimizing branch divergence during residue calculations in NTT and BigInt operations.

Numerical Accumulation in src/math/calculus.c

  • Historical concept: Jeungseunggaebang (增乘開方法) — Horner-style sequential linear accumulation for polynomial root finding and summation.
  • Reference: Traditional Korean algebraic treatises on Jeungseunggaebang (增乘開方法).
  • Application in code: integrate_worker (Simpson's rule) and ttak_calculus_rk4_step (Runge-Kutta 4th order) use linear Horner accumulation to compute weighted multi-stage sums with minimal rounding error.