From 33ec006ab679bf0b408965cc9dd39fc127061ca5 Mon Sep 17 00:00:00 2001 From: flyingviper <44173173+flyingviper@users.noreply.github.com> Date: Fri, 23 Oct 2026 18:35:47 +0200 Subject: [PATCH] Certify a joint productive-helper and native-map fusion family --- .../.gitattributes | 4 + .../COMBINED-WORD.json.gz | Bin 0 -> 207018 bytes .../COMMUTATION.json | 566 +++++++++++++ .../FLOW.json.gz | Bin 0 -> 75176 bytes .../README.md | 132 +++ .../RESULT.json | 163 ++++ .../SHARED-MAPS.json | 754 ++++++++++++++++++ .../SOURCES.json | 36 + .../TASK.json | 76 ++ .../VERIFY.json | 32 + .../WITNESS.json.gz | Bin 0 -> 21302 bytes .../check_fusion.py | 377 +++++++++ .../check_rewrite.py | 232 ++++++ .../check_shared.py | 197 +++++ .../verify.py | 99 +++ 15 files changed, 2668 insertions(+) create mode 100644 research/productive-helper-fusion-certificate/.gitattributes create mode 100644 research/productive-helper-fusion-certificate/COMBINED-WORD.json.gz create mode 100644 research/productive-helper-fusion-certificate/COMMUTATION.json create mode 100644 research/productive-helper-fusion-certificate/FLOW.json.gz create mode 100644 research/productive-helper-fusion-certificate/README.md create mode 100644 research/productive-helper-fusion-certificate/RESULT.json create mode 100644 research/productive-helper-fusion-certificate/SHARED-MAPS.json create mode 100644 research/productive-helper-fusion-certificate/SOURCES.json create mode 100644 research/productive-helper-fusion-certificate/TASK.json create mode 100644 research/productive-helper-fusion-certificate/VERIFY.json create mode 100644 research/productive-helper-fusion-certificate/WITNESS.json.gz create mode 100644 research/productive-helper-fusion-certificate/check_fusion.py create mode 100644 research/productive-helper-fusion-certificate/check_rewrite.py create mode 100644 research/productive-helper-fusion-certificate/check_shared.py create mode 100644 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z@%t~oe_R36U%Y<&{FnM+5KQ2|fBE`4mG2*)zP!Bt^6jrL#)k>;{^iBJB;)k!OAIf+ zzM2mzyoG5{f3YtgRDO$-TV75ZyuQ7>e6TOC@zs8Kn|#V_oq4en^!@a`?KeKWeE#zJ zz6-1zuHnezkhxE_U)}LQ4BG^PTzX_5MG@6&4=GU z#NU)r*e+3_wQ5lefaCk*YDqc`Q!8SFQ5PNR`mD%>mP3uct*Uw{q5!R z>o1?byuN)a4gI--YWB0 and not len(s)&(len(s)-1),'binary subspace cardinality') + return len(s).bit_length()-1 + +@lru_cache(None) +def base(s): + b=[];covered=frozenset({0}) + for v in sorted(s): + if v not in covered:b.append(v);covered=span(b) + need(covered==s,'binary subspace closure') + return tuple(b) + +@lru_cache(None) +def join(a,b):return span(base(a)+base(b)) + +def mv(g,x): + result=0 + for i,col in enumerate(g): + if x>>i&1:result^=col + return result + +@lru_cache(None) +def image(g,s):return frozenset(mv(g,x) for x in s) + +def inverse(g): + lookup={mv(g,x):x for x in range(512)} + need(len(lookup)==512,'native map invertible') + return tuple(lookup[1<>9 + need(x in s and all(((x^z)&u).bit_count()%2==0 for u in base(s)),'canonical zero-phase chart') + +def reconstruct_energy(witness,payload,proof_mode): + old,low,edges,fixed,maps,centers,climbs,initial,final,target=source_graph(payload,proof_mode) + proposals={int(i):chart for i,chart in witness['proposals'].items()} + expected={i for i in range(len(old)) if old[i]!=low[i] and i not in fixed} + need(set(proposals)==expected,'all and only distinct eligible least-frame proposals') + need(len(expected)==(1079 if target else 896),'declared variable inventory') + for i,chart in proposals.items():chart_is_canonical(chart,low[i]) + if not target:need(not expected.intersection(initial+final),'all helper and main endpoints fixed') + else: + need(witness['binding']['combined_word_payload_sha256']==SOURCE_SHA256,'helper witness source binding') + need(witness['retained_helper_exit_pins']==target['retained_helper_exit_pins'],'independent helper exit repins') + need(witness['adapter_demand_nodes_pinned']==target['adapter_demand_nodes_pinned'],'independent adapter pin inventory') + tables=[];paid=Counter(centers);lower_paid=Counter(centers);helpers=Counter();new_helpers=Counter() + linear=defaultdict(lambda:ZERO);pair=defaultdict(lambda:ZERO) + zero_changed=0;warps_changed=0 + for a,b,op,sign in edges: + if a==b and not op:continue + left=[old[a],low[a] if a in expected else old[a]] + right=[old[b],low[b] if b in expected else old[b]] + if op:left=[image(maps[op],s) for s in left] + ranks=[distance(left[i],right[j]) for i,j in [(0,0),(1,0),(0,1),(1,1)]] + need(min(ranks)>=0 and max(ranks)<=9,'physical rank range') + if sign<0: + need(min(ranks)>0,'all helper states strictly productive');helpers[ranks[0]]+=1;new_helpers[ranks[3]]+=1 + else: + if ranks[0]:paid[ranks[0]]+=1 + if ranks[3]:lower_paid[ranks[3]]+=1 + if a not in expected and b not in expected:continue + tables.append([a,b,op,sign,ranks]) + if sign>0 and not ranks[0]:zero_changed+=1 + if op:warps_changed+=1 + v00,v10,v01,v11=[atom(r,sign) for r in ranks] + if a==b: + linear[a]=add(linear[a],sub(v11,v00)) + else: + if a in expected:linear[a]=add(linear[a],sub(v10,v00)) + if b in expected:linear[b]=add(linear[b],sub(v01,v00)) + if a in expected and b in expected: + key=tuple(sorted((a,b)));pair[key]=add(pair[key],sub(sub(v11,v10),sub(v01,v00))) + need(tables==witness['tables'],'every ordered four-state rank table from raw events') + need(paid==climbs+centers,'entire source paid multiset') + need(paid==Counter({int(k):v for k,v in witness['baseline_paid'].items()}),'witness baseline invoice') + need(lower_paid==Counter({int(k):v for k,v in witness['proposal_paid' if target else 'lower_paid'].items()}),'witness least invoice') + need(helpers==Counter({4:28,7:7,8:47,9:326}),'all 408 helper rewards') + if target: + need(helpers==Counter({int(k):v for k,v in witness['baseline_helpers'].items()}),'witness baseline helper invoice') + need(new_helpers==Counter({int(k):v for k,v in witness['proposal_helpers'].items()}),'witness proposal helper invoice') + need(witness['zero_productivity_combinations']==[] and witness['all_binary_combinations_strictly_productive'],'declared strict productivity') + return linear,{k:v for k,v in pair.items() if v!=ZERO},{'source_nodes':len(old),'all_edges':len(edges),'orientation':'forward','proposals':len(expected),'affected_edges':len(tables),'old_zero_edges_affected_including_warps':zero_changed,'old_zero_paid_edges_affected':zero_changed-warps_changed,'native_warps_affected':warps_changed,'helper_endpoints_changed':len(expected.intersection(initial+final)),'proposal_helper_rewards':dict(new_helpers),'boundary_mode':proof_mode,'adapter_pins_after_nonidentity_map':target['adapter_pins_after_nonidentity_map'] if target else None,'retained_helper_exit_pins':target['retained_helper_exit_pins'] if target else None,'restored_frozen_least_differences':sum(old[i]!=low[i] for i in fixed),'helper_rewards':dict(helpers),'baseline_paid':dict(paid)} + +def graph_energy(d,linear,pair): + ids=d['ids'];need(ids==sorted(linear),'ordered graph variable IDs') + n=len(ids);S=n;T=n+1;sw={int(k):v for k,v in d['switch'].items()} + need(set(sw)==set(ids) and set(sw.values())<={0,1},'binary switch') + need(d['groups']==len(ids) and d['pair_terms']==len(pair),'reported graph inventory') + reachable=set(d['terminal_reachable']) + selected=[i for j,i in enumerate(ids) if sw[i]^(j not in reachable)] + need(selected==d['selected_groups'],'reported minimizing assignment') + # Expand cut arcs as a quadratic in original x, independently of the + # author's switched-submodular graph construction. + coeff=defaultdict(lambda:ZERO);coeff[()]=tuple(d['graph_constant']) + def multiply(a,b): + out=defaultdict(int) + for ma,ca in a.items(): + for mb,cb in b.items():out[tuple(sorted(set(ma+mb)))]+=ca*cb + return out + def y(i):return {():sw[ids[i]],(ids[i],):1-2*sw[ids[i]]} + def one_minus(p):return {k:(1 if k==() else 0)-v for k,v in {():0,**p}.items()} + for a,b,cap in d['source_arcs']: + need(0<=a=0 for k in range(1,10)):return 'concave' + mass=sum(r*poly[r] for r in range(1,10));need(mass>=0,'nonnegative at C=0') + for j in range(16): + l=Q(j,3200);h=Q(j+1,3200);bound=Q(0) + for r in range(1,10): + a=power_interval(r,h)[0];b=power_interval(r,l)[1] + coefficient=poly[r] + if not mass: + la,lb=log_interval(r);a*=la;b*=lb;coefficient=-coefficient + bound+=coefficient*(a if coefficient>=0 else b) + need(bound>0,'rational interval positivity') + return 'derivative' if not mass else 'value' + +def verify_flow(d): + n=len(d['ids'])+2;S=n-2;T=n-1;initial=[[] for _ in range(n)];arcs=[] + for a,b,c in d['source_arcs']: + ai,bi=len(initial[a]),len(initial[b]);c=tuple(c) + initial[a].append((b,c,bi));initial[b].append((a,ZERO,ai));arcs.append((a,ai,b,bi,c)) + final=d['final_residual'];need(len(final)==n,'residual node count') + for old,row in zip(initial,final): + need(len(old)==len(row),'residual adjacency shape') + for (ob,oc,oi),(b,c,i) in zip(old,row):need((ob,oi)==(b,i),'residual incidence and reverse pairing') + divergence=[ZERO]*n;polynomials=set();reachable=set(d['terminal_reachable']);need(S in reachable and T not in reachable,'separating cut') + cut=ZERO + for a,ai,b,bi,c in arcs: + residual=tuple(final[a][ai][1]);flow=tuple(final[b][bi][1]) + need(add(residual,flow)==c,'individual arc capacity conservation') + divergence[a]=add(divergence[a],flow);divergence[b]=sub(divergence[b],flow) + polynomials.update([c,residual,flow]) + if a in reachable and b not in reachable:cut=add(cut,c) + for a,row in enumerate(final): + for b,c,r in row: + if a in reachable and b not in reachable:need(tuple(c)==ZERO,'cut saturated identically') + need(all(v==ZERO for i,v in enumerate(divergence) if i not in (S,T)),'flow conservation at every variable') + need(add(divergence[S],divergence[T])==ZERO and divergence[S]==cut,'exact max-flow/min-cut equality') + need(add(cut,tuple(d['graph_constant']))==tuple(d['minimum_polynomial'])==ZERO,'identically zero energy minimum') + need(all(type(i)is int and 0<=ibarrier: + trace.append([ordinals[prior],ordinal]) + out[prior]=summed(out[prior],event) + if out[prior] is None:previous.pop((a,b)) + continue + previous[a,b]=index + if event[0]=='scalar' and event[1][0]=='add': + a,b=map(tuple,event[1][1:3]);writes[a]=index;reads[b]=index + else: + for role in roles(event):barriers[role]=index + if event[0]=='phase':phase=index + out.append(deepcopy(event));ordinals.append(ordinal) + return trace + + +def commute(add,event): + """Exact elementary-matrix criterion, with conservative other barriers. + + An ADD i<-j commutes with k<-l iff i!=l and j!=k. All other + operations crossing its two live roles are rejected. Center events affect + owned C only. Adapters are inert native markers in both retained engines. + Phase markers are conservative barriers, matching the credited rewrite. + """ + a,b=map(tuple,add[1][1:3]) + if event[0]=='phase':return False + if event[0]=='scalar' and event[1][0]=='add': + x,y=map(tuple,event[1][1:3]) + return a!=y and b!=x + if event[0] in ('scalar','warp'): + return not ({a,b}&roles(event)) + need(event[0] in ('center','adapter'),'Unknown crossed event') + return True + + +def trace_verify(program,trace): + raw=[deepcopy(e) for e in program if e[0]!='climb'] + prior_right=-1;crossed=cancelled=0 + for left,right in trace: + need(type(left) is int and type(right) is int and 0<=leftprior_right,'Trace must follow chronological incoming events') + prior_right=right + a,b=raw[left],raw[right] + need(a is not None and b is not None and live_add(a) and live_add(b), + 'Trace endpoints are surviving live ADDs') + need(a[1][1:3]==b[1][1:3],'Different elementary matrices') + need(a[1][1]!=a[1][2],'Self-ADD is not a nilpotent elementary generator') + for event in raw[left+1:right]: + if event is not None: + need(commute(b,event),'Noncommuting crossed instruction') + crossed+=1 + raw[left]=summed(a,b);raw[right]=None + cancelled+=raw[left] is None + return [e for e in raw if e is not None],{'merges':len(trace), + 'cancelled_pairs':cancelled,'crossed_instruction_checks':crossed} + + +def span(basis): + result={0} + for vector in basis:result|={x^vector for x in tuple(result)} + need(len(result)==1<>i&1:result^=c + return result + +def image(columns,s):return frozenset(mv(columns,v) for v in s) + +def inverse(columns): + lookup={mv(columns,v):v for v in range(1<=0,'consistent marked-line dimensions') + return k+l-2*intersection + +def projected_chart(chart): + need(len(chart)==9 and all(type(v)is int and 0>9))&u).bit_count()%2==0 for u in basis(s)),'canonical chart phase') + return s + +def chronological_nodes(payload): + c=payload['canonical'];frames=list(map(span,c['frames'])) + nodes=[];fixed=set();warps=[] + def new(value,frozen=False): + i=len(nodes);nodes.append(value) + if frozen:fixed.add(i) + return i + current=[new(frames[f],r<224) for r,f in enumerate(c['start'])] + for event_index,event in enumerate(payload['programs']['forward']): + kind=event[0] + if kind=='scalar': + gate,f=event[1:];refs=gate[1:3] if gate[0] in ('add','copy') else gate[1:2] + n=new(frames[f],any(typ!='L' for typ,r in refs)) + for typ,r in refs: + if typ=='L':current[r]=n + elif kind=='warp': + _,role,op,a,b=event;need(role[0]=='L','persistent native occurrence') + i=new(frames[a]);j=new(frames[b]);current[role[1]]=j + warps.append({'event':event_index,'op':op,'input':i,'output':j}) + elif kind=='adapter':fixed.add(current[event[2]]) + else:need(kind in ('climb','center','phase'),'known unchanged source event') + for r,f in enumerate(c['final']):new(frames[f],r<224) + return nodes,fixed,warps + +def verify(payload,witness): + raw=json.dumps(payload,sort_keys=True,separators=(',',':')).encode() + need(hashlib.sha256(raw).hexdigest()==SOURCE_SHA256,'pinned combined source') + nodes,base_fixed,warps=chronological_nodes(payload) + props={int(i):projected_chart(chart) for i,chart in witness['proposals'].items()} + need(all(0<=i0 else (y,x)) + else:need(da in (0,9),'only zero/full or line constraints outside late occurrence') + need(len(prescribed)<=1,'single normalized shared prelude line') + line=next(iter(prescribed)) if prescribed else None + if line: + lines[op]=list(line);need(mv(maps[op],line[0])==line[1],'incumbent shared line compatibility') + a,b=chosen['input'],chosen['output'];left=[nodes[a],props.get(a,nodes[a])];right=[nodes[b],props.get(b,nodes[b])] + bounds=[];attained=[] + for i,j in [(0,0),(1,0),(0,1),(1,1)]: + bound=lower_bound(left[i],right[j],line) + actual=distance(image(maps[op],left[i]),right[j]) + need(bound==actual,'original shared G attains each independent lower bound') + bounds.append(bound);attained.append(actual) + need(bounds[0]==0,'baseline late rank zero') + if a in props or b in props: + need(old_cost_tables[a,b,op,1]==bounds,'shared rank table matches full invoice certificate') + tables.append([a,b,op,bounds]) + need(len(selected)==72 and len(lines)==12,'all 72 late maps and 12 prelude lines') + need(selected==witness['selected_late_maps'],'selected occurrence inventory') + need(lines=={int(k):v for k,v in witness['prelude_lines'].items()},'prelude line inventory') + need(sorted(extra_fixed-base_fixed)==witness['frozen_earlier_warp_nodes'],'other occurrence frozen-node inventory') + need(witness['excluded_proposals']==[] and witness['changed_native_tables']==[],'no excluded choices or improved native table') + return {'status':'PASS_INDEPENDENT_SHARED_MAP_EXTENSION','source_sha256':SOURCE_SHA256, + 'native_occurrences':len(warps),'selected_late_maps':len(selected),'four_state_minima':4*len(selected), + 'prelude_lines':len(lines),'other_occurrence_dimensions':{str(k):v for k,v in sorted(profiles.items())}, + 'other_native_nodes_fixed':len(extra_fixed),'baseline_fixed_native_nodes':len(extra_fixed&base_fixed), + 'all_original_shared_maps_attain_minima':True,'changed_native_tables':0, + 'late_tables_sha256':hashlib.sha256(json.dumps(tables,separators=(',',':')).encode()).hexdigest(), + 'scope':'Geometric lower bound over all compatible shared G matrices for the declared original/least chart choices. All other native occurrence charts remain fixed; no physical or phase admission and no new kappa.'} + +def small_formula_audit(): + cases=maps_total=0 + for n in (1,2,3): + universe=range(1<>(v-1)&1) for mask in range(1<<((1<