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Text
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json
Languages:
English
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Tags:
mathematics
harmonic-analysis
inverse-problems
arithmetic-matroids
flat-tori
finite-jet-reconstruction
License:
| // Independently implemented exact finite audit. No floating point or Python. | |
| // Build: g++ -std=c++20 -O2 cpp/exact_atlas.cpp -o exact_atlas | |
| // Run: ./exact_atlas results/exact_atlas.csv > results/cpp_report.json | |
| using Row=std::array<int,2>; | |
| using Mat=std::array<Row,3>; | |
| long long checks=0,states=0,energy_states=0; | |
| void check(bool b){++checks;if(!b)throw std::runtime_error("exact certificate failed at check "+std::to_string(checks));} | |
| int det(Row a,Row b){return a[0]*b[1]-a[1]*b[0];} | |
| int mod(int a,int q){int b=a%q;return b<0?b+q:b;} | |
| std::vector<int> supports(const Mat&a){ | |
| std::set<int>s; | |
| for(const auto&r:a){int bits=0;for(int i=0;i<3;++i)if(det(r,a[i]))bits|=1<<i;if(bits)s.insert(bits);} | |
| return {s.begin(),s.end()}; | |
| } | |
| std::pair<int,int> cogirth(const std::vector<int>&s,std::array<int,3>w){ | |
| int v=100000000,count=0; | |
| for(int mask:s){int sum=0;for(int i=0;i<3;++i)if(mask&(1<<i))sum+=w[i]; | |
| if(sum<v){v=sum;count=1;}else if(sum==v)++count;} | |
| return {v,count}; | |
| } | |
| int sine_energy(int q,int r){ | |
| if(q==2){int a[]={0,4};return a[r];} | |
| if(q==3){int a[]={0,3,3};return a[r];} | |
| if(q==4){int a[]={0,2,4,2};return a[r];} | |
| if(q==6){int a[]={0,1,3,4,3,1};return a[r];} | |
| throw std::runtime_error("unsupported exact alphabet"); | |
| } | |
| std::pair<int,int> minimum(const Mat&a,std::array<int,3>w,int q){ | |
| int low=100000000,count=0; | |
| for(int x=0;x<q;++x)for(int y=0;y<q;++y){if(x==0&&y==0)continue; | |
| int v=0;for(int i=0;i<3;++i)v+=w[i]*sine_energy(q,mod(a[i][0]*x+a[i][1]*y,q));++energy_states; | |
| if(v<low){low=v;count=1;}else if(v==low)++count;} | |
| return {low,count}; | |
| } | |
| int main(int argc,char**argv){try{ | |
| std::ofstream csv;if(argc>1)csv.open(argv[1]); | |
| if(argc>1&&!csv)throw std::runtime_error("cannot open CSV output"); | |
| if(csv)csv<<"a00,a01,a10,a11,a20,a21,Q,safe,unimodular,cogirth_unit,exact_squared_distance_unit\n"; | |
| std::vector<Row>rows;for(int x=-2;x<=2;++x)for(int y=-2;y<=2;++y)if(x||y)rows.push_back({x,y}); | |
| int candidates=0,faithful=0,uni=0,safe_count=0,unsafe_count=0,unsafe_exact=0; | |
| for(int i=0;i<int(rows.size());++i)for(int j=i+1;j<int(rows.size());++j)for(int k=j+1;k<int(rows.size());++k){ | |
| ++candidates;Mat a={rows[i],rows[j],rows[k]}; | |
| std::array<int,3>dets={det(a[0],a[1]),det(a[0],a[2]),det(a[1],a[2])}; | |
| int g=0;bool u=true;for(int z:dets){g=std::gcd(g,std::abs(z));if(std::abs(z)>1)u=false;} | |
| if(g!=1) { continue; } | |
| ++faithful; if(u) { ++uni; } | |
| auto supp=supports(a); check(!supp.empty()); | |
| for(int q=2;q<=11;++q){ | |
| bool safe=true;for(int z:dets)if(z&&std::gcd(std::abs(z),q)!=1)safe=false; | |
| if(safe)++safe_count;else ++unsafe_count; | |
| bool alias=false; | |
| for(int x=0;x<q;++x)for(int y=0;y<q;++y){if(x==0&&y==0)continue; | |
| ++states;int zero=0;for(int r=0;r<3;++r)if(mod(a[r][0]*x+a[r][1]*y,q)==0)zero|=1<<r; | |
| check(zero!=7); // Faithfulness on every composite alphabet too. | |
| for(int r=0;r<3;++r)for(int s=r+1;s<3;++s)if((zero&(1<<r))&&(zero&(1<<s))&&det(a[r],a[s]))alias=true; | |
| } | |
| check(safe==!alias); | |
| int exact=-1; | |
| if(q==2||q==3||q==4||q==6){ | |
| int c=sine_energy(q,1); | |
| for(std::array<int,3>w:std::vector<std::array<int,3>>{{1,1,1},{1,2,3},{3,1,2}}){ | |
| auto lam=cogirth(supp,w),answer=minimum(a,w,q); | |
| if(w==std::array<int,3>{1,1,1})exact=answer.first; | |
| if(safe)check(answer.first>=c*lam.first); | |
| if(u){check(answer.first==c*lam.first);check(answer.second==(q==2?1:2)*lam.second);} | |
| } | |
| if(!safe){ | |
| bool done=false; | |
| for(int r=0;r<3&&!done;++r)for(int s=r+1;s<3&&!done;++s){ | |
| int z=det(a[r],a[s]);if(!z||std::gcd(std::abs(z),q)==1)continue; | |
| std::array<int,3>w={1,1,1};w[r]=w[s]=4*q*q; | |
| auto lam=cogirth(supp,w),ans=minimum(a,w,q); | |
| check(ans.first<c*lam.first);done=true;++unsafe_exact; | |
| }check(done); | |
| } | |
| } | |
| if(csv){for(auto row:a)csv<<row[0]<<','<<row[1]<<',';csv<<q<<','<<safe<<','<<u<<','<<cogirth(supp,{1,1,1}).first<<','<<exact<<'\n';} | |
| } | |
| } | |
| check(candidates==2024); | |
| std::cout<<"{\n \"status\": \"passed\",\n \"arithmetic\": \"integer only\",\n \"candidate_matrices\": "<<candidates<<",\n \"primitive_matrices\": "<<faithful<<",\n \"unimodular_matrices\": "<<uni<<",\n \"matrix_modulus_cases\": "<<safe_count+unsafe_count<<",\n \"safe_cases\": "<<safe_count<<",\n \"unsafe_cases\": "<<unsafe_count<<",\n \"modular_configurations\": "<<states<<",\n \"exact_energy_configurations\": "<<energy_states<<",\n \"constructive_failure_checks\": "<<unsafe_exact<<",\n \"assertions\": "<<checks<<"\n}\n"; | |
| }catch(const std::exception&e){std::cerr<<e.what()<<'\n';return 1;}return 0;} | |