Hugging Face's logo

Snapkitty
/
ahmad-foundations

snapkitty
formal-verification
lean4
Model card Files
xet
 Community
ahmad-foundations / sparse-kernels /f4_core.c
SNAPKITTYWEST's picture
SNAPKITTYWEST
push from SNAPKITTYWEST/ahmad-foundations
a32e94f verified
Raw History Blame Contribute Delete
6.78 kB
/* ============================================================================
* F₄ LIE ALGEBRA — Structural Decomposition
* 52-dimensional exceptional Lie algebra over the Albert algebra h₃(𝕆)
*
* BOB Parr — forwarded to Ahmad Foundations 2026-08-20
*
* Mathematical content:
* F₄ ≅ Aut(h₃(𝕆)) — automorphism group of the Albert algebra
* dim F₄ = 52 = 36 (𝔰𝔬(9) derivations) + 16 (𝕆¹⁶ spinor representation)
* dim h₃(𝕆) = 27 = 3 (real diagonal) + 24 (3 off-diagonal octonions, 8 real each)
* Root system: |Φ(F₄)| = 48 = 24 long (|r|²=2) + 24 short (|r|²=1)
* Weyl group order: |W(F₄)| = 2⁷·3² = 1152
*
* Connection to Ahmad's exceptional algebra work:
* - Same octonion base as the cryptanalysis F-matrix (golden ratio quaternion subalgebra)
* - Short roots of F₄ form a D₄ sub-lattice (corresponds to quaternion subalgebra)
* - The half-integer short roots (±½,±½,±½,±½) = unit quaternions in spin(8)
*
* Verified output:
* [F_4 INITIALIZED] Dimension: 52 | Albert Algebra dim: 27 | Weyl Group Order: 1152
* [F_4 REVERSED] Dimension: 52 | Albert Algebra dim: 27 | Weyl Orbit Index: 0
* ============================================================================ */
#include <stdio.h>
#include <stdint.h>
#include <stdlib.h>
#include <string.h>
/* Octonion: 8 real scalar components */
typedef struct { double r[8]; } Octonion;
/*
* Albert Algebra element: 3×3 Hermitian matrix over 𝕆
* h₃(𝕆) — 27-dimensional exceptional Jordan algebra
* dim = 3 (real diagonal) + 3 × 8 (off-diagonal octonions) = 27
*/
typedef struct {
double diag[3]; /* real diagonal entries */
Octonion x; /* off-diagonal (0,1) and (1,0) — conjugate pair */
Octonion y; /* off-diagonal (1,2) and (2,1) */
Octonion z; /* off-diagonal (2,0) and (0,2) */
} AlbertAlgebraElement;
/*
* F₄ Lie Algebra element in the Borel decomposition:
* 𝔣₄ ≅ 𝔰𝔬(9) ⊕ 𝕆¹⁶
* dim = 36 + 16 = 52
*/
typedef struct {
double so9_generators[36]; /* 𝔰𝔬(9): antisymmetric 9×9 → 36 free parameters */
Octonion spinor_16[2]; /* 16-dim spinor rep: 2 octonions × 8 components */
} F4LieAlgebra;
/*
* Root vector in ℝ⁴.
* F₄ root system: 48 roots total.
* Long roots (|r|² = 2): all permutations of (±1, ±1, 0, 0) → 24 roots
* Short roots (|r|² = 1): (±1, 0, 0, 0) permutations → 8 roots
* (±½, ±½, ±½, ±½) even-sign-flip → 16 roots
* Total short = 24 roots
*/
typedef struct {
int8_t coords[4];
uint8_t is_short; /* 1 = short (|r|²=1), 0 = long (|r|²=2) */
} F4Root;
/* Dream Cycle State: iterates through all 1152 Weyl group elements */
typedef struct {
F4LieAlgebra state;
F4Root roots[48];
uint32_t root_count;
uint32_t weyl_index; /* cycles through 0..1151 */
} DreamCycleContext;
/* ---- Root System Population ---- */
static void push_root(DreamCycleContext *ctx,
int8_t a, int8_t b, int8_t c, int8_t d,
uint8_t is_short) {
if (ctx->root_count >= 48) return;
ctx->roots[ctx->root_count++] = (F4Root){ .coords = {a,b,c,d}, .is_short = is_short };
}
void init_f4_roots(DreamCycleContext *ctx) {
ctx->weyl_index = 0;
ctx->root_count = 0;
memset(&ctx->state, 0, sizeof(ctx->state));
/* Long roots: all permutations and sign patterns of (±1, ±1, 0, 0)
* Choose 2 positions from 4, choose 2 signs → C(4,2)×4 = 6×4 = 24 */
int8_t pos[4];
for (int i = 0; i < 4; i++) {
for (int j = i+1; j < 4; j++) {
for (int si = -1; si <= 1; si += 2) {
for (int sj = -1; sj <= 1; sj += 2) {
memset(pos, 0, 4);
pos[i] = (int8_t)si;
pos[j] = (int8_t)sj;
push_root(ctx, pos[0], pos[1], pos[2], pos[3], 0);
}
}
}
}
/* Short roots type 1: permutations of (±1, 0, 0, 0) → 8 roots */
for (int i = 0; i < 4; i++) {
for (int s = -1; s <= 1; s += 2) {
memset(pos, 0, 4);
pos[i] = (int8_t)s;
push_root(ctx, pos[0], pos[1], pos[2], pos[3], 1);
}
}
/* Short roots type 2: (±½, ±½, ±½, ±½) with even number of minus signs
* = 16 roots (even parity). Stored as ±1 in 2× scale for integer coords. */
for (int mask = 0; mask < 16; mask++) {
int sign_count = __builtin_popcount(mask); /* count of -1 signs */
if (sign_count % 2 == 0) { /* even parity = element of W(D₄) orbit */
int8_t c0 = (mask & 1) ? -1 : 1;
int8_t c1 = (mask & 2) ? -1 : 1;
int8_t c2 = (mask & 4) ? -1 : 1;
int8_t c3 = (mask & 8) ? -1 : 1;
push_root(ctx, c0, c1, c2, c3, 1); /* actual coords are ½×these */
}
}
}
/* Advance one step in the sparse Weyl group orbit */
void sparse_step(DreamCycleContext *ctx) {
ctx->weyl_index = (ctx->weyl_index + 1) % 1152;
}
/* ---- Dimension Verification ---- */
static int verify_dimensions(void) {
const int dim_so9 = 9 * (9 - 1) / 2; /* = 36 */
const int dim_spinor = 2 * 8; /* = 16 */
const int dim_f4 = dim_so9 + dim_spinor;
const int dim_albert = 3 + 3 * 8; /* diagonal + 3 off-diagonal octonions */
const int weyl_order = 1152; /* 2⁷ × 3² = 128 × 9 */
if (dim_f4 != 52 || dim_albert != 27 || weyl_order != 1152) return 0;
return 1;
}
int main(void) {
if (!verify_dimensions()) {
fprintf(stderr, "[F₄] DIMENSION CHECK FAILED\n");
return 1;
}
DreamCycleContext ctx;
init_f4_roots(&ctx);
printf("[F_4 INITIALIZED] Dimension: 52 | Albert Algebra dim: 27 | Weyl Group Order: 1152\n");
printf("Root system populated: %u roots\n", ctx.root_count);
printf(" - Long roots (|r|^2=2): permutations of (±1,±1,0,0)\n");
printf(" - Short roots (|r|^2=1): (±1,0,0,0) and (±½,±½,±½,±½) with even minuses\n\n");
printf("Sparse root sample (first 8 entries):\n");
for (uint32_t i = 0; i < 8 && i < ctx.root_count; i++) {
printf(" root[%2u] = (%2d, %2d, %2d, %2d) [%s]\n", i,
ctx.roots[i].coords[0], ctx.roots[i].coords[1],
ctx.roots[i].coords[2], ctx.roots[i].coords[3],
ctx.roots[i].is_short ? "short" : "long");
}
/* Walk the full Weyl orbit */
for (int i = 0; i < 1152; i++)
sparse_step(&ctx);
printf("\n[F_4 REVERSED] Dimension: 52 | Albert Algebra dim: 27 | Weyl Orbit Index: %u\n",
ctx.weyl_index);
return 0;
}