diff --git a/research/p104-helper-detour-projection/.gitattributes b/research/p104-helper-detour-projection/.gitattributes new file mode 100644 index 0000000000..7467454cb8 --- /dev/null +++ b/research/p104-helper-detour-projection/.gitattributes @@ -0,0 +1 @@ +source/** -text diff --git a/research/p104-helper-detour-projection/NOTICE b/research/p104-helper-detour-projection/NOTICE new file mode 100644 index 0000000000..897ac4da05 --- /dev/null +++ b/research/p104-helper-detour-projection/NOTICE @@ -0,0 +1,5 @@ +New metric proof, canonical projection, source inventory and fee bounds: Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0. + +The exact PR5756 word is by Joel Pulikkan (GamingPuzzled), with Claude Opus 5.5 assistance. It retains PR5751 by Ali Romman with OpenAI Codex assistance and flyingviper PR5718. The original source NOTICE is retained unchanged. + +The native compiler is retained from research/p104-native-channel/vendor/clifford_exact.py and its predecessor line. The exact moment engine is by zxv with OpenAI Codex assistance. PR5772 supplies the previously published canonical Pauli image-space calculation; it is not claimed as new here. All original source pins and byte hashes are explicit. 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It includes noncanonical charts and +extra spectator axes. For any finite number of intermediate charts, a separate +sharp bound is attained in the tariff model by the two query endpoints. Longer +detours cannot increase this bound. + +On the exact PR5756 word, the checker inventories all **2534 helper queries and +559 independent helper targets**. At κ = .00311313795, even the optimistic gains +need at least **53 one-cut helpers**, or **33 arbitrary-path helpers**, to pay +one extra H1 native query. A single helper cannot pay that fee. This is a budget +for choosing future physical searches, not a new κ or a construction. + +## Contract and native metric + +The original helper has fixed endpoint charts A0 and B0. One paid query has +endpoints A and B. Assume these form a geodesic chain, with + +``` +u = d(A0,A), q = d(A,B) > 0, v = d(B,B0), +t = d(A0,B0) = u+q+v. +``` + +The source's nested canonical subspaces satisfy this condition. The function +`d(X,Y)` is the Gaussian rank of the relative Clifford map. If L_X and L_Y +are the images of the Z space, then +`d(X,Y) = n - dim(L_X intersect L_Y)`. It is a pseudometric: the intersection +codimension inequality gives its triangle inequality. Clifford phases and +rank-zero maps do not change it. The theorem grants all other costs for free, +so omitted scalar, normalization and ownership charges can only reduce the gain. + +Write `phi(r) = r^(1-C)` and `phi(0)=0`. A one-cut choice F replaces the helper +target by A0→F and F→B0, and replaces the paid query by A→F and F→B. It debits +both original jobs. The signed gain is + +``` +phi(d(A0,F)) + phi(d(F,B0)) - phi(t) +- phi(d(A,F)) - phi(d(F,B)) + phi(q). +``` + +These are proposed independent target credits. The theorem does not prove that +an arbitrary F permits their independent inputs or physical restoration. + +## One-cut optimum over every Clifford chart + +Put `x=d(A,F)`, `y=d(F,B)` and `h_r(z)=phi(z+r)-phi(z)`. The triangle inequality +and monotonicity give + +``` +phi(d(A0,F))-phi(x) + phi(d(F,B0))-phi(y) +<= h_u(x)+h_v(y). +``` + +Also `x+y >= q`. There are nonnegative x'≤x and y'≤y with x'+y'=q. Concavity +makes each h_r decreasing, so the right side is at most +`h_u(x')+h_v(q-x')`. For a power with exponent strictly between zero and one, +each h_r is convex. This follows from its second derivative for z>0 and +continuity at zero, or by integrating the convex function +`(1-C)*(z+s)^(-C)` over `0<=s<=r`. A convex function on `[0,q]` has a maximum +at an endpoint. Therefore the exact maximum gain is + +``` +max(phi(u)+phi(q+v), phi(u+q)+phi(v)) - phi(t). +``` + +It is attained by F=A or F=B in the tariff model because the source endpoints +are geodesic. F=A wins when u≥v; F=B wins when v≥u. To see this, subtract the +two endpoint scores to obtain `h_q(v)-h_q(u)`. Ties in the two scores do not +imply that only these two physical charts can tie. If u=v=0 every one-cut gain +is nonpositive. + +This proof uses only the native rank metric. It is not limited to canonical +binary charts or to dimension nine. Isometric spectator padding leaves the +four original distances unchanged. A new operation alphabet with a different +price, changed original endpoints or other shared jobs requires a new contract. + +## Canonical projection for every concave tariff + +There is a stronger structural statement for canonical subspaces and any +increasing concave phi with phi(0)=0. Suppose `A0<=A<=B<=B0` as subspaces. +For any canonical F, put `F'=A+(F intersect B)`. Then F' lies in `[A,B]`, and +its signed tariff is at least that of F. + +Use the previously published native image formula +`L(W)={(x,z):x in W, z+x in W_perp}` from PR5772. It gives +`d(X,Y)=dim X+dim Y-2 dim(X intersect Y)`. Put + +``` +x_i = dim(F intersect A_i), with A_i = A0,A,B,B0; +k = x_2-x_1; +e_1 = u-(x_1-x_0) >= 0; +e_3 = x_3-x_2 >= 0; +s = dim(A0)-x_0 + dim(F)-x_3 >= 0; +delta = e_1+e_3. +``` + +The paid ranks are `k+s+delta` and `q-k+s+delta`. The two target ranks exceed +these paid ranks by `u-2e_1` and `v-2e_3`. Monotonicity permits increasing those +increments to u and v. Decreasing the common arguments to k and q-k then +increases h_u and h_v by concavity. The resulting expression is exactly the +signed tariff at F'. This proves projection without a finite search. + +The endpoint maximum in the previous section needs the power hypothesis. +For the concave hinge `phi(r)=min(r,2)` and `(u,q,v)=(1,2,1)`, an interior cut +has score 2 and both endpoints have score 1. The verifier retains this +counterexample to prevent an incorrect extension to all concave tariffs. + +## Any finite detour path + +For intermediate charts F1,...,Fm, let the new target chain be +`A0,F1,...,Fm,B0` and the replacement paid chain be `A,F1,...,Fm,B`. +Every common interior edge cancels from the signed invoice. The gain is + +``` +phi(d(A0,F1))-phi(d(A,F1)) ++ phi(d(Fm,B0))-phi(d(Fm,B)) + phi(q)-phi(t). +``` + +Every increasing concave phi with phi(0)=0 is subadditive. The metric triangle +inequality bounds the first difference by phi(u) and the second by phi(v). +Hence every finite path obeys + +``` +gain <= phi(u)+phi(q)+phi(v)-phi(t). +``` + +For m≥2, the tariff bound is attained by F1=A and Fm=B. Additional interior +charts cannot raise it. This applies to arbitrary Clifford coframes; their +native ranks need not come from canonical subspaces. It does not apply to a +shared graph that credits more targets than the stated target path, to several +original queries charged as one, or to changed original helper endpoints. +Those are different invoices and must be checked separately. + +For source rank t≤9 and `0 2^26*N^5 avoiding all fixed denominators and rank witnesses" + }, + "outer": { + "functional_checks": { + "actual_new_BIT_stock_sublinear": "348599999999253/350000000000000", + "fixed_halo_explicit_metadata_branch": "26950000000001/350000000000", + "new_BIT_row_stock_sublinear_in_active_group": "1395799999997009/1400000000000000", + "new_semantic_group_guard": "3/1400000000000" + }, + "margins": { + "boundary_algebraic_matrix": "6101750387595641606867/1960000000000000000000000", + "fused_gaussian": "6101750387586924820601/1960000000000000000000000", + "global_coordinate_router": "2179196567/700000000000", + "group_bit_router": "3050875194493462410299/980000000000000000000000", + "group_scans": 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"theta_positive_polynomial_power": "2", + "union_header_fits_record": "3/1400000000000", + "union_tag_work_per_visit_fits_record": "3/1400000000000" + } + }, + "paid": { + "1": 41520, + "16": 416, + "2": 8172, + "24": 416, + "3": 8856, + "32": 416, + "4": 1674, + "40": 416, + "48": 416, + "5": 3132, + "56": 416, + "6": 108, + "64": 416, + "7": 270, + "72": 208 + }, + "paid_calls": 66852, + "previous_pr5718": "389127/125000000", + "productive_fields": 13390, + "query_shears": 15, + "rank_mass_surplus": 760, + "raw_ceiling": { + "derivative_upper": "-738596154004511222636095493817850924833979710424488126325902089/3138550867693340381917894711603833208051177722232017256448", + "interval": [ + "0", + "17/5000" + ] + }, + "raw_headroom": [ + "73/3125000000000", + "2337/100000000000000" + ], + "raw_root_interval": [ + "38914224667/12500000000000", + "311313797337/100000000000000" + ], + "relative_gain_percent": "4065/1037672", + "strictification": { + "alpha": { + "lower": "0.999999999981368230437503066447", + "upper": "0.999999999981368230437503066448" + }, + "cutoff": 16, + "delta": { + "lower": "0.944337257529929949513398128117", + "upper": "0.944337257529929949513398128118" + }, + "floor": 17, + "gamma": { + "lower": "0.975484709128690914856977434634", + "upper": "0.975484709128690914856977434635" + }, + "normalized_paid_upper": "1048576000976562501/1048576001073741824", + "power": 1024, + "reserve": "1/1000000000000", + "tail_block": 128, + "tail_decay": "125/128", + "tail_offset": "1/7", + "tested": "622627591/200000000000" + }, + "targets": { + "1": 432, + "145": 208, + "17": 416, + "2": 90, + "25": 416, + "33": 416, + "4": 1656, + "41": 416, + "49": 416, + "5": 126, + "57": 416, + "65": 416, + "7": 108, + "73": 208, + "8": 666, + "9": 6984 + }, + "whole_idle_jobs": 3120 + }, + "controls": { + "block_op_on_old_chart": "Scalar destination and coefficient", + "extra_target_credit": "Every productive helper has a native target", + "missing_block_move": "Scalar destination and coefficient", + "old_helper_start": "Scalar destination and coefficient", + "read_after_write": "Dependent operations keep their order", + "unpaid_query": "Complete field and call counts" + }, + "global_paid_delta": { + "1": 18, + "4": 18, + "5": -18 + }, + "helper_credit_delta": { + "1": -1, + "4": 1, + "5": 1, + "8": -1 + }, + "helper_finals": { + "405": [ + 477, + 1090 + ] + }, + "helper_starts": { + "741": [ + 477, + 1090 + ] + }, + "local": { + "endpoint_coefficients": 602176, + "helpers": { + "1": 24, + "2": 5, + "4": 92, + "5": 7, + "7": 6, + "8": 37, + "9": 388 + }, + "owned_fields": 9, + "owned_lifetimes": 11, + "paid": { + "1": 2220, + "2": 454, + "3": 492, + "4": 93, + "5": 174, + "6": 6, + "7": 15 + }, + "paid_rank_mass": 5987, + "peak_owned_fields": 2, + "public_inputs": 767, + "real_scalar_allowance": 224138 + }, + "local_paid_delta": { + "1": 1, + "4": 1, + "5": -1 + }, + "native": { + "distinct_maps": 2927, + "helper_normalizers": 1118, + "helpers": { + "1": 24, + "2": 5, + "4": 92, + "5": 7, + "7": 6, + "8": 37, + "9": 388 + }, + "main_adapters": 208, + "proof": "Every signed Pauli image and exact global phase; frame induction at every positive tensor width", + "recipe_payload_sha256": "b566ac8ebfb9df3d4c8a7be449c749283ec70dedbdaad7f943c264c692da544a", + "wrong_phase_rejected": true + }, + "physical": { + "central_square_columns": 53248, + "full_calls": 18, + "main_targets": { + "145": 208, + "17": 416, + "25": 416, + "33": 416, + "41": 416, + "49": 416, + "57": 416, + "65": 416, + "73": 208 + }, + "new_paid_jobs": 0, + "physical_contacts": 7488, + "ports": 104, + "route_payload_sha256": "7f4f0ca768ffbc4f93dcff575592c4c67411f46b4289e45b73f715a2ad47b142", + "tensor_rule": "Factorwise native maps; scalar signs stay outside the tensor power.", + "whole_idle_jobs": { + "16": 416, + "24": 416, + "32": 416, + "40": 416, + "48": 416, + "56": 416, + "64": 416, + "72": 208 + }, + "wrapper_sites": [ + 1, + 3, + 5, + 7, + 9, + 11, + 13, + 15, + 17, + 19, + 21, + 23, + 25, + 27, + 29, + 31 + ] + }, + "source": { + "pr5751": "8483f9a679d06c1e270141dbc686a7e76e7a7522" + }, + "status": "PASS_P104_R22_REFRAME_KAPPA_00311313795", + "transpose": { + "arithmetic": "Dense integer rows with binary denominators; reverse chronology", + "coefficients": 602176, + "columns": 776 + } +} diff --git a/research/p104-helper-detour-projection/source/WORD.json.gz b/research/p104-helper-detour-projection/source/WORD.json.gz new file mode 100644 index 0000000000..156ff6d627 Binary files /dev/null and b/research/p104-helper-detour-projection/source/WORD.json.gz differ diff --git a/research/p104-helper-detour-projection/source/clifford_exact.py b/research/p104-helper-detour-projection/source/clifford_exact.py new file mode 100644 index 0000000000..166841a037 --- /dev/null +++ b/research/p104-helper-detour-projection/source/clifford_exact.py @@ -0,0 +1,149 @@ +"""Small exact Gaussian-dyadic Clifford compiler, independent of gx internals. +Paulis use i^p X^x Z^z. State vectors use integer Gaussian numerators / 2^d. +The tableau is supplemented by a literal circuit, so global phases are kept. +""" +from dataclasses import dataclass + +def parity(x):return x.bit_count()&1 +def rank(rows): + p={} + for x in rows: + while x: + b=x.bit_length()-1 + if b not in p:p[b]=x;break + x^=p[b] + return len(p) +def transpose(rows,n):return tuple(sum(((rows[j]>>i)&1)<>j&1);a[j],a[k]=a[k],a[j] + for i in range(n): + if i!=j and a[i]>>j&1:a[i]^=a[j] + assert [r&((1<>n for r in a) +def basis_extend(cols,n): + out=[] + for x in list(cols)+[1<len(out):out.append(x) + assert len(out)==n + return tuple(out) +def nullspace(rows,n): + a=list(rows);p=[];k=0 + for j in range(n): + t=next((i for i in range(k,len(a)) if a[i]>>j&1),None) + if t is None:continue + a[t],a[k]=a[k],a[t] + for i in range(len(a)): + if i!=k and a[i]>>j&1:a[i]^=a[k] + p.append(j);k+=1 + return tuple((1<>j)&1)<>i&1:out=pmul(out,T[i]) + for i in range(n): + if z>>i&1:out=pmul(out,T[n+i]) + return out +def compose(A,B):return tuple(apply(A,p) for p in B) +def inverse(T): + n=len(T)//2;cols=[x|(z<>n;u,v,p=apply(T,(x,z,0));assert (u,v)==identity(n)[j][:2];out.append((x,z,(-p)%4)) + return tuple(out) +def qvalue(Q,alpha,x,n): + return (sum(alpha[i] for i in range(n) if x>>i&1)+2*sum(parity((Q[i]>>(i+1))&(x>>(i+1))) for i in range(n) if x>>i&1))%4 +def mul_i(a,p): + x,y=a;p%=4 + return [(x,y),(-y,x),(-x,-y),(y,-x)][p] +@dataclass +class Op: + n:int + tab:tuple + circuit:list + def then(self,b): + assert self.n==b.n + return Op(self.n,compose(b.tab,self.tab),self.circuit+b.circuit) + def adjoint(self): + circ=[] + for e in reversed(self.circuit): + if e[0]=='P':circ.append(('P',inv(e[1],self.n))) + elif e[0]=='D':circ.append(('D',e[1],tuple((-x)%4 for x in e[2]))) + elif e[0]=='C':circ.append(('C',e[1],-e[2])) + elif e[0]=='phase':circ.append(('phase',(-e[1])%4)) + else:circ.append(e) + return Op(self.n,inverse(self.tab),circ) + def zero_state(self,basis=0): + N=1<>i)&1 for i in range(n)) + assert all(alpha[i]%2==(Q[i]>>i)&1 for i in range(n)) + return Op(n,tuple([(1<>i&1 else 0,1<>i&1 else 0) for i in range(n)]),[('C',q,sgn)]) +def xgate(n,b):return Op(n,tuple([(1<>i)&1)) for i in range(n)]),[('X',b)]) +def fourier(n,axes=None): + out=ident(n) + for i in range(n) if axes is None else axes:out=out.then(cgate(n,1<len(selected):picks.append(j);selected.append(x) + assert len(picks)==r and all(x>>r==0 for x in Bcols) + Br=transpose(selected,r);Bi=inv(Br,r);Dr=transpose([Dcols[j]&((1< 0, 'positive interval denominator') + return mul(a, rounded(1/b[1], 1/b[0])) + + +@lru_cache(None) +def log_bounds(n): + """Exact atanh series, binary reduction; round only the final interval.""" + need(isinstance(n, int) and n >= 1, 'positive integer logarithm') + + def series(x): + u = (x-1)/(x+1) + term, total = u, Q(0) + for j in range(80): + total += 2*term/(2*j+1) + term *= u*u + tail = 2*term/(161*(1-u*u)) + return total, total+tail + + k = n.bit_length()-1 + a, b = series(Q(2)) + c, d = series(Q(n, 2**k)) + return rounded(k*a+c, k*b+d) + + +def exp_point(x): + """Outward fixed-point Taylor enclosure, with binary scaling/squaring.""" + need(abs(x) < 16, 'bounded exponential argument') + if x < 0: + a, b = exp_point(-x) + return rounded(1/b, 1/a) + divisions = 0 + while x > Q(1, 2): + x /= 2 + divisions += 1 + term = total = (Q(1), Q(1)) + for j in range(1, 65): + term = scale(term, x/j) + total = rounded(total[0]+term[0], total[1]+term[1]) + following = scale(term, x/65) + # Every remaining ratio is <= 1/132. Twice the next term is safe. + result = rounded(total[0], total[1]+2*following[1]) + for _ in range(divisions): + result = mul(result, result) + return result + + +def exp_interval(a): + return exp_point(a[0])[0], exp_point(a[1])[1] + + +@lru_cache(None) +def power(n, exponent): + return exp_interval(scale(log_bounds(n), exponent)) + + +def moment(histogram, exponent): + low = high = Q(0) + for n, count in histogram.items(): + a, b = power(n, exponent) + low += count*a + high += count*b + return rounded(low, high) + + +def display(a): + unit = 10**30 + + def fixed(n): + sign = '-' if n < 0 else '' + s = str(abs(n)).rjust(31, '0') + return sign+s[:-30]+'.'+s[-30:] + + return {'lower': fixed((a[0]*unit).__floor__()), + 'upper': fixed(-((-a[1]*unit).__floor__()))} + + +def encoded(x): + if isinstance(x, Q): + return str(x) + if isinstance(x, dict): + return {str(k): encoded(v) for k, v in x.items()} + if isinstance(x, (list, tuple)): + return [encoded(v) for v in x] + return x + + +def source_environment(path): + env = {'__file__': str(path), '__name__': 'pinned_source_oracle'} + exec(compile(path.read_bytes(), str(path), 'exec'), env) + return env + + +def authenticate(): + result = {} + for row in json.loads((HERE/'VENDOR.json').read_text())['files']: + path = HERE/'vendor'/Path(row['path']).relative_to( + 'research/higher-moment-strictification/vendor') + digest = sha256(path.read_bytes()).hexdigest() + need(digest == row['sha256'], 'changed source: '+row['path']) + raw = path.read_bytes() + need(sha1(b'blob '+str(len(raw)).encode()+b'\0'+raw).hexdigest() + == row['git_blob'], 'changed Git source blob') + result[row['path']] = digest + return result + + +def invoice_oracle(): + """Run #4426 unchanged, rebinding only its two file-location globals.""" + path = HERE/'vendor/invoice-review/verify.py' + env = source_environment(path) + env.update(ROOT=path.parent, PHASES='DATA-PHASES.json') + output = io.StringIO() + with redirect_stdout(output): + env['main']() + result = json.loads(output.getvalue()) + need(result['status'] == 'PASS_PR4416_INVOICE_REBUILD', 'source invoice oracle') + need({int(k): v for k, v in result['targets'].items()} == T + and {int(k): v for k, v in result['paid'].items()} == P, + 'exact literal-word invoice') + return result + + +def tail_certificate(v=150, cutoff=10, floor=15, u=Q(3, 10)): + t = 1-C + need(v >= 1 and 1 < cutoff < floor < 121, 'strict finite ranks') + need(t+u >= 1, 'tail moment exponent at least one') + target, paid = moment(T, t), moment(P, t) + alpha = divide(paid, target) + gamma = divide(moment(P, t+u), mul(power(cutoff, u), target)) + delta = divide((Q(sum(T.values())*floor),)*2, + mul(power(cutoff, C), target)) + limits = (1-Q(1, 10**8), Q(181, 200), Q(439, 500)) + for name, bounds, limit in zip(('alpha', 'gamma', 'delta'), + (alpha, gamma, delta), limits): + need(0 < bounds[0] <= bounds[1] < limit < 1, name+' certified limit') + a, g, d = limits + bound = 1/(1+v*(1-a))+2*g**v+2*d**v + need(bound < 1, 'complete charged normalized moment') + old_gamma = divide((Q(sum(k*n for k, n in P.items())),)*2, + mul(power(cutoff, C), target)) + old_lower = alpha[0]**v+2*old_gamma[0]**v + need(old_lower > 1, 'rank-one-tail estimate fails at this tensor power') + return {'tensor_power': v, 'cutoff_base': cutoff, 'floor_base': floor, + 'paid_moment_exponent': t+u, 'tail_offset_u': u, + 'tested_saving': C, 'alpha': display(alpha), 'gamma': display(gamma), + 'delta': display(delta), 'rational_upper_limits': limits, + 'normalized_paid_upper': bound, 'strict_moment_gap': display((1-bound,)*2), + 'old_rank_one_gamma': display(old_gamma), + 'old_rank_one_bound_lower': display((old_lower, old_lower)), + 'ambient': '121^'+str(v), 'minimum_target': str(floor)+'^'+str(v), + 'maximum_paid': str(cutoff)+'^'+str(v), + 'productive_fields_before_uniform_compiler': '7688^'+str(v), + 'rank_one_surplus': '131723^'+str(v)+'-131115^'+str(v), + 'materialized_product_or_uniform_word': False} + + +def uniform_rank(v): + t, p, theta = 1-C, 13, Q(6, 7) + gap = (C-KAPPA)/7 + uniform_saving = C-gap + need(10**7 < 15**6, 'theta upper bound') + need(2*(10**v)**7 <= (121**v)**6, 'shifted logarithm inequality') + eta = scale(exp_interval(scale(log_bounds(121), -v*C)), t) + need(eta[0] > theta**p, 'exit contraction below complement capacity') + base = exp_interval(scale(log_bounds(121), v*C)) + need(base[1] < 2**p, 'entire ordinary base range fits') + # Integer grid: gap*H is integral, avoiding enormous rational powers. + step = gap.denominator + need(gap.numerator == 1, 'chosen rational gap grid') + + def passes(j): + H = step*j + return (H+1)**p < 2**j + + lo, hi = 1, 2 + while not passes(hi): + lo, hi = hi, 2*hi + while hi-lo > 1: + mid = (lo+hi)//2 + if passes(mid): + hi = mid + else: + lo = mid + H = step*hi + need(H > 7*v and passes(hi) and not passes(hi-1), 'final integer comparison') + need(eta[1] < theta**(p-1), 'one smaller logarithmic power fails') + return {'t': t, 'uniform_saving': uniform_saving, 'gap': gap, + 'logarithm_power': p, 'theta_upper': theta, 'eta': display(eta), + 'eta_minus_theta_to_p': display((eta[0]-theta**p, eta[1]-theta**p)), + 'm_to_saving': display(base), 'H': H, 'N': '2^'+str(H), + 'integer_test': {'left_base': H+1, 'left_exponent': p, + 'right_base': 2, 'right_exponent': hi}, + 'grid_step': step, 'previous_grid_point_rejected': True, + 'smaller_logarithmic_power_rejected': True} + + +def power_majorant_comparison(v, uniform_saving): + """Retain #4416's older power proof, changing only the new unit's ranks.""" + t, z, h = 1-C, 1-uniform_saving, Q(1, 7) + z0 = (t+z)/2 + delta = z0-t + B, debt = (4+7*h)*v, 1+7*v*(1-z0) + ratio = (delta*B+debt)/(h*delta) + H0 = -((-ratio.numerator)//ratio.denominator) + ratio = H0*(1-z0)/(z-z0) + H1 = ratio.numerator//ratio.denominator+1 + need(H0 >= 7*v and H0*h*delta >= delta*B+debt, 'power majorant threshold') + need((H0-1)*h*delta < delta*B+debt, 'previous power threshold rejected') + need(H1*(z-z0) > H0*(1-z0) >= (H1-1)*(z-z0), + 'strict power-majorant comparison and previous-rank control') + return {'H0': H0, 'H1': H1, 'N': '2^'+str(H1), 'theta_upper': 1-h, + 'scope': 'Tail transformation alone, retaining the older power majorant.'} + + +def outer_ledger(uniform_saving): + d = (C-KAPPA)/7 + params = {'kappa': KAPPA, 'tested_saving': C, + 'complex_uniform_saving': uniform_saving, 'b': KAPPA+5*d, + 'a': KAPPA+4*d, 'q': KAPPA+3*d, 'bit_saving': Q('0.0027'), + 'epsilon': 1-d, 'beta': 1-3*d, 'r0': d/10, + 'stop': d, 'layout': d, 'd': d} + env = source_environment(HERE/'vendor/outer_endpoint.py') + row = env['ledger'](params) + need(len(row['strict_checks']) == 54 and len(row['functional_checks']) == 3, + 'complete first outer ledger') + need(all(v > 0 for group in ('strict_checks', 'functional_checks') + for v in row[group].values()), 'all outer predicates') + frozen = json.loads((HERE/'vendor/p7-RESULT.json').read_text())['endpoint']['outer'] + need(encoded(row) == frozen, 'unchanged full p7 endpoint ledger') + parsed = ast.parse((HERE/'vendor/check_ledger.py').read_bytes()) + function = next(n for n in parsed.body + if isinstance(n, ast.FunctionDef) and n.name == 'ledger') + constants = [n for n in parsed.body if isinstance(n, ast.Assign) + and any(isinstance(t, ast.Name) and t.id.startswith('KAPPA_') + for t in n.targets)] + eps, beta, q, a = (params[k] for k in ('epsilon', 'beta', 'q', 'a')) + grid = 10**10 + backoff = Q((eps*q*grid).__floor__(), grid)-KAPPA + need(backoff > 0, 'strict derived oracle grid backoff') + oracle = {'Q': Q, 'R0': params['r0'], 'ZETA': Q(1, 10**8), + 'SLACK': d, 'STOP': d, 'FLOOR': grid, 'BACKOFF': backoff, + 'EPS': eps, 'BETA': beta, 'THETA_POWER': Q(2), + 'HALO_POWER': 48, 'CORE_POWER': 128, 'STRASSEN_TAU': Q(45, 16), + 'b': params['b'], 'q': q, 'a': a, 'bit': params['bit_saving'], + 'online': Q('0.001730970419199163702738'), + 'old': Q('0.001730970419199163702738')} + code = ast.fix_missing_locations(ast.Module(body=constants+[function], type_ignores=[])) + exec(compile(code, 'pinned 63-predicate ledger', 'exec'), oracle) + second = oracle['ledger']('higher_moment_smaller_ranks') + need(second['kappa'] == KAPPA and second['margins'] == row['margins'], + 'all nine rows match both ledgers') + need(len(second['strict_checks']) == 63 and + all(v > 0 for v in second['strict_checks'].values()), 'all 63 legacy predicates') + controls = {} + for name, updates in [('closed_epsilon', {'epsilon': Q(1)}), + ('missing_layout', {'layout': Q(0)}), + ('unfunded_kappa', {'kappa': C}), + ('closed_uniform_gap', {'complex_uniform_saving': params['b']}), + ('insufficient_BIT', {'bit_saving': Q('0.0023')})]: + bad = env['ledger'](dict(params, **updates)) + failed = sorted(k for k, v in bad['strict_checks'].items() if v <= 0) + need(failed, 'outer control rejected: '+name) + controls[name] = failed + for name, options in [('old_per_axis_scan', {'old_scan': True}), + ('group_equals_dimension', {'beta_': eps})]: + bad = oracle['ledger']('reject_'+name, **options) + failed = sorted(k for k, v in bad['strict_checks'].items() if v <= 0) + need(failed, 'legacy control rejected: '+name) + controls[name] = failed + return {'ledger': row, 'legacy_63_checks': second['strict_checks'], + 'legacy_grid': grid, 'legacy_backoff': backoff, + 'minimum_margin_minus_kappa': min(row['margins'].values())-KAPPA, + 'rejected_controls': controls} + + +def finite_controls(): + def tensor(histogram): + out = Counter() + for k, n in histogram.items(): + for h, b in histogram.items(): + out[k*h] += n*b + return out + + def split(histogram, cutoff): + out = Counter() + for k, n in histogram.items(): + q, r = divmod(k, cutoff) + out[cutoff] += q*n + if r: + out[r] += n + return +out + + target, paid = tensor({2:1, 5:1}), tensor({1:2, 4:1}) + padded, completion = Counter(), Counter() + for k, n in target.items(): + padded[max(k, 9)] += n + if k < 9: + completion[9-k] += n + final = split(paid+completion, 4) + mass = lambda x: sum(k*n for k, n in x.items()) + need(mass(padded)-mass(final) == 7**2-6**2 == 13, 'exact paid padding mass') + need(min(padded) > max(final), 'exact toy separation') + need(final == Counter({1:5, 4:9}), 'literal toy expanded inventory') + # t=1/2, sigma=3/4, R=100, J=1 refutes a sublinear tail extension. + invalid = power(100, Q(1, 2))[1]+2*power(100, Q(3, 4))[1] + need(invalid < 100, 'sublinear upper-tail moment cannot bound splitting') + return {'toy_targets': padded, 'toy_paid': final, 'rank_one_surplus': 13, + 'sublinear_tail_counterexample': {'R': 100, 'J': 1, 't': '1/2', + 'sigma': '3/4', 'actual_charge': 100, + 'invalid_upper': display((invalid, invalid))}} + + +def verify(): + sources = authenticate() + invoice = invoice_oracle() + tail = tail_certificate() + rank = uniform_rank(tail['tensor_power']) + endpoint = outer_ledger(rank['uniform_saving']) + controls = finite_controls() + rejected = [] + for name, updates, expected in [ + ('too_short_product', {'v': 100}, 'complete charged normalized moment'), + ('sublinear_tail', {'u': C/2}, 'tail moment exponent at least one'), + ('lost_separation', {'floor': 10}, 'strict finite ranks'), + ]: + try: + tail_certificate(**updates) + except ValueError as error: + need(str(error) == expected, 'wrong rejection: '+name) + rejected.append({'mutation': name, 'failed_at': str(error)}) + else: + raise ValueError('accepted mutation: '+name) + old = json.loads((HERE/'vendor/p7-RESULT.json').read_text())['endpoint']['compiler'] + return encoded({'schema': 'higher-moment-native-strictification/1', + 'status': 'PASS_HIGHER_MOMENT_STRICTIFICATION', 'kappa': KAPPA, + 'invoice_oracle': invoice, 'strictification': tail, + 'uniform_rank': rank, 'outer': endpoint, + 'tail_only_power_majorant': power_majorant_comparison( + tail['tensor_power'], rank['uniform_saving']), + 'finite_controls': controls, 'rejected_mutations': rejected, + 'predecessor_H': old['H1'], + 'rank_exponent_reduction_factor': Q(old['H1'], rank['H']), + 'tensor_power_reduction_factor': Q(6240, tail['tensor_power']), + 'source_sha256': sources, + 'verifier_sha256': sha256(Path(__file__).read_bytes()).hexdigest(), + 'scope': 'Conditional charged transformation and finite compiler rank. Native all-column, physical, analytic, precision, storage, fixed-tape and recovery premises remain inherited. No new kappa, selected frontier or practical word.'}) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--out', type=Path) + args = parser.parse_args() + result = verify() + if args.out: + need(not args.out.exists(), 'new output path required') + args.out.write_text(json.dumps(result, indent=2, sort_keys=True)+'\n') + else: + need(json.loads((HERE/'RESULT.json').read_text()) == result, 'changed saved receipt') + print(json.dumps({'status': result['status'], 'tensor_power': result['strictification']['tensor_power'], + 'uniform_rank': result['uniform_rank']['N'], + 'strict_moment_gap': result['strictification']['strict_moment_gap'], + 'kappa': result['kappa']})) + + +if __name__ == '__main__': + main() diff --git a/research/p104-helper-detour-projection/source/source-local.py b/research/p104-helper-detour-projection/source/source-local.py new file mode 100644 index 0000000000..c181baa903 --- /dev/null +++ b/research/p104-helper-detour-projection/source/source-local.py @@ -0,0 +1,208 @@ +"""Check the complete P104 word with an independent helper inventory. + +Adapted from Ali Romman's PR5544/5545. David Arias supplied the word; +Cinna Davis supplied private reuse; Gareth Davidson supplied composition. +Prepared by Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. +""" +from collections import Counter +from fractions import Fraction as Q +from functools import lru_cache +from hashlib import sha256 +from pathlib import Path +import gzip +import importlib.util +import json +import sys + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] + +def need(ok, why): + if not ok: + raise ValueError(why) + +def read(path): + raw = path.read_bytes() + return json.loads(gzip.decompress(raw) if path.suffix == '.gz' else raw) + +def load(path, name): + spec = importlib.util.spec_from_file_location(name, path) + module = importlib.util.module_from_spec(spec) + sys.modules[name] = module + spec.loader.exec_module(module) + return module + +def add(dst, src, q): + for k, v in tuple(src.items()): + value = dst.get(k, 0) + q*v + if value: + dst[k] = value + else: + dst.pop(k, None) + +def span(basis): + out = {0} + for v in basis: + need(type(v) is int and 0 < v < 512 and v not in out, 'Independent binary basis') + out |= {x ^ v for x in tuple(out)} + return frozenset(out) + +def replay(w): + public = w['public'] + owned = w.get('owned_fields', 8) + need(type(owned) is int and 8 <= owned <= 9, 'Owned inventory bound') + need(type(public) is int and public == 767, 'Public inventory bound') + need(len(w['initial']) == len(w['final']) == public, 'Public boundary size') + need(len(w['program']) < 26000, 'Fixed operation bound') + frames = [span(x) for x in w['frames']] + rows = [{r:Q(1)} for r in range(public+owned)] + current = list(w['initial']) + [None]*owned + paid, scalar, live_owned, peak = Counter(), 0, set(), 0 + lifetimes = 0 + for e in w['program']: + op, r = e[:2] + need(0 <= r < public+owned, 'Role index') + if op == 'MOVE': + _, r, a, b, private = e + need(current[r] == a and (r in live_owned) == private, 'Transport source and ownership') + need(frames[b] < frames[a] if private else frames[a] < frames[b], 'Nested chronological transport') + paid[abs(len(w['frames'][b])-len(w['frames'][a]))] += 1 + current[r] = b + elif op == 'INIT': + need(r >= public and r not in live_owned, 'Owned initialization') + live_owned.add(r) + lifetimes += 1 + peak = max(peak, len(live_owned)) + rows[r] = {} + current[r] = e[2] + scalar += 2 + elif op == 'CLEAR': + need(r in live_owned and current[r] == e[2], 'Owned erasure at declared frame') + rows[r] = {} + current[r] = None + live_owned.remove(r) + scalar += 2 + else: + q, f = Q(e[-2]), e[-1] + need(q and current[r] == f, 'Scalar destination and coefficient') + n, d = abs(q.numerator), q.denominator + need(d & (d-1) == 0, 'Dyadic scalar coefficient') + need(abs(q)+1 < 2**16, 'Fixed scalar growth bound') + if op == 'SCALE': + need(n & (n-1) == 0, 'Dyadic unit SCALE') + rows[r] = {k:q*x for k,x in rows[r].items()} + scalar += 2*(abs(n.bit_length()-d.bit_length()) + int(q < 0)) + else: + s = e[2] + need(s != r and 0 <= s < public+owned and current[s] == f, 'Common scalar chart') + if op == 'COPY': + need(r in live_owned and not rows[r], 'COPY into cleared owned field') + rows[r] = {k:q*x for k,x in rows[s].items()} + else: + need(op == 'ADD', 'Scalar opcode') + add(rows[r], rows[s], q) + scalar += 2*(3+n.bit_length()-1+n.bit_count()+d.bit_length()-1) + need(not live_owned and 0 < peak <= 2 and current[:public] == w['final'], 'All public and owned exits') + live = set(w['live']) + for r,row in enumerate(rows): + expected = {r:Q(1)} if r < public else {} + if r-112 in live: + expected[r-112] = Q(1) + need(row == expected, 'All public and owned rows') + main = live | {r+112 for r in live} + helpers = Counter(len(w['frames'][w['final'][r]])-len(w['frames'][w['initial'][r]]) for r in range(public) if r not in main) + need(len(live) == len(w['ports']) == 104 and len(main) == 208 and min(helpers) > 0, 'Independent productive inputs') + return {'paid':dict(paid), 'helpers':dict(helpers), 'public_inputs':public, 'owned_fields':owned, 'owned_lifetimes':lifetimes, + 'peak_owned_fields':peak, 'real_scalar_allowance':scalar, 'endpoint_coefficients':(public+owned)*(public+owned), + 'paid_rank_mass':sum(r*count for r,count in paid.items())} + +def reverse_columns(w): + """Check the transpose with dense dyadic rows and the reverse chronology.""" + public = w['public'] + n = public+w.get('owned_fields', 8) + rows = [[int(i == j) for j in range(n)] for i in range(n)] + exponents = [0]*n + for e in reversed(w['program']): + op,r = e[:2] + if op == 'MOVE': + continue + if op in ('INIT','CLEAR'): + rows[r] = [0]*n + exponents[r] = 0 + continue + q = Q(e[-2]) + qe = q.denominator.bit_length()-1 + if op == 'SCALE': + rows[r] = [q.numerator*x for x in rows[r]] + exponents[r] += qe + else: + s = e[2] + level = max(exponents[s], exponents[r]+qe) + a, b = 1 << (level-exponents[s]), q.numerator << (level-exponents[r]-qe) + rows[s] = [a*x+b*y for x,y in zip(rows[s],rows[r])] + exponents[s] = level + if op == 'COPY': + rows[r] = [0]*n + exponents[r] = 0 + r = s + while exponents[r] and not any(x & 1 for x in rows[r]): + rows[r] = [x//2 for x in rows[r]] + exponents[r] -= 1 + for r,row in enumerate(rows): + expected = [0]*n + if r < public: + expected[r] = 1 << exponents[r] + if r in w['live']: + expected[r+112] = 1 << exponents[r] + need(row == expected, 'Independent dense transpose columns') + return {'columns':n, 'coefficients':n*n, 'arithmetic':'Dense integer rows with binary denominators; reverse chronology'} + +def native(w): + n = w['public'] + C = load(ROOT/'research/p104-native-channel/vendor/clifford_exact.py', 'p104_productive_clifford') + V = load(ROOT/'research/p104-native-channel/vendor/native_verify.py', 'p104_productive_gaussian') + @lru_cache(None) + def chart(i): + return C.canonical_frame(9, tuple(w['frames'][i])) + recipes = {} + def check(a,b): + if (a,b) not in recipes: + recipe,_ = C.lower_child(chart(a).adjoint().then(chart(b))) + expected = V.adjoint(V.frame(w['frames'][a],9)) + V.frame(w['frames'][b],9) + V.equal(V.recipe(recipe),expected,'Native chart change and exact global phase') + need(recipe['rank'] == abs(len(w['frames'][a])-len(w['frames'][b])), 'Native boundary rank') + recipes[a,b] = recipe + return recipes[a,b] + for e in w['program']: + if e[0] == 'MOVE': + check(e[2],e[3]) + main = set(w['live']) | {r+112 for r in w['live']} + helper_profile = Counter() + for r in range(n): + if r not in main: + helper_profile[check(w['initial'][r],w['final'][r])['rank']] += 1 + adapters = [] + for r,q in zip(w['live'],w['ports']): + need(w['frames'][w['initial'][r]] == [q], 'Literal central port') + need(span(w['frames'][w['final'][r+112]]) == frozenset(x for x in range(512) if not (x&q).bit_count()%2), 'Literal output hyperplane') + tests = [ + (C.cgate(9,q,1).adjoint().then(chart(w['initial'][r])), V.adjoint((('C',q,1),))+V.frame(w['frames'][w['initial'][r]],9)), + (chart(w['final'][r+112]).adjoint().then(C.cgate(9,q,-1).then(C.fourier(9))), V.adjoint(V.frame(w['frames'][w['final'][r+112]],9))+(('C',q,-1),)+V.fourier(9))] + for actual,expected in tests: + recipe,_ = C.lower_child(actual) + need(recipe['rank'] == 0, 'Free intrinsic main adapter') + V.equal(V.recipe(recipe),expected,'Main endpoint phase') + adapters.append(recipe) + bad = json.loads(json.dumps(next(iter(recipes.values())))) + bad['left']['phase'] = (bad['left']['phase']+1)%4 + a,b = next(iter(recipes)) + try: + V.equal(V.recipe(bad),V.adjoint(V.frame(w['frames'][a],9))+V.frame(w['frames'][b],9),'Wrong phase control') + except ValueError: + pass + else: + raise ValueError('Wrong native phase accepted') + payload = [[a,b,recipes[a,b]] for a,b in sorted(recipes)] + adapters + return {'distinct_maps':len(recipes),'main_adapters':len(adapters),'helper_normalizers':2*sum(helper_profile.values()), + 'helpers':dict(helper_profile),'recipe_payload_sha256':sha256(json.dumps(payload,sort_keys=True,separators=(',',':')).encode()).hexdigest(), + 'wrong_phase_rejected':True, 'proof':'Every signed Pauli image and exact global phase; frame induction at every positive tensor width'} diff --git a/research/p104-helper-detour-projection/verify.py b/research/p104-helper-detour-projection/verify.py new file mode 100644 index 0000000000..2c145422de --- /dev/null +++ b/research/p104-helper-detour-projection/verify.py @@ -0,0 +1,444 @@ +"""Check the helper-detour projection, path bound and exact source inventory. + +Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. +The retained native and interval engines keep their original notices. +""" +import argparse +from collections import Counter, defaultdict +from copy import deepcopy +from fractions import Fraction as Q +from hashlib import sha1, sha256 +from itertools import product +import gzip +import importlib.util +import json +from pathlib import Path +import sys +import tempfile + +HERE = Path(__file__).resolve().parent +sys.dont_write_bytecode = True + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def load(path, name): + spec = importlib.util.spec_from_file_location(name, path) + module = importlib.util.module_from_spec(spec) + sys.modules[name] = module + spec.loader.exec_module(module) + return module + + +def read(path): + raw = path.read_bytes() + return json.loads(gzip.decompress(raw) if path.suffix == '.gz' else raw) + + +def span(vectors): + out = frozenset({0}) + for v in vectors: + out |= frozenset(x ^ v for x in out) + return out + + +def basis(space): + out, have = [], frozenset({0}) + for v in sorted(space): + if v not in have: + out.append(v) + have = span(out) + need(have == space, 'Exact subspace basis') + return out + + +def dim(space): + n = len(space) + need(n > 0 and n & (n - 1) == 0, 'Binary subspace cardinality') + return n.bit_length() - 1 + + +def distance(a, b): + return dim(a) + dim(b) - 2 * dim(a & b) + + +def subspaces(n): + seen, todo = {frozenset({0})}, [frozenset({0})] + universe = frozenset(range(1 << n)) + while todo: + a = todo.pop() + for v in universe - a: + b = a | frozenset(x ^ v for x in a) + if b not in seen: + seen.add(b) + todo.append(b) + return sorted(seen, key=lambda s: (len(s), sorted(s))) + + +def compositions(total, count): + if count == 1: + yield (total,) + else: + for first in range(total + 1): + for tail in compositions(total - first, count - 1): + yield (first,) + tail + + +def signed(paid, targets): + out = Counter(r for r in targets if r) + out.subtract(r for r in paid if r) + return {r: v for r, v in out.items() if v} + + +def evaluate(coefficients, powers): + lo = hi = Q(0) + for r, count in coefficients.items(): + a, b = powers[r] + lo += count * (a if count > 0 else b) + hi += count * (b if count > 0 else a) + return lo, hi + + +def add(a, b): + return a[0] + b[0], a[1] + b[1] + + +def source_modules(): + pins = read(HERE / 'SOURCES.json') + for row in pins['direct_sources']: + raw = (HERE / row['local_path']).read_bytes() + need(sha256(raw).hexdigest() == row['sha256'] and + sha1(b'blob ' + str(len(raw)).encode() + b'\0' + raw).hexdigest() == row['git_blob'], + 'Exact source bytes: ' + row['local_path']) + return (load(HERE / 'source/clifford_exact.py', 'detour_native'), + load(HERE / 'source/moments.py', 'detour_moments')) + + +def finite_geometry(C): + spaces = subspaces(4) + need(len(spaces) == 67, 'All binary subspaces in dimension four') + ops = [C.canonical_frame(4, tuple(basis(s))) for s in spaces] + inverses = [op.adjoint() for op in ops] + for i, a in enumerate(spaces): + for j, b in enumerate(spaces): + actual = C.rank([x for x, z, p in inverses[i].then(ops[j]).tab[4:]]) + need(actual == distance(a, b), 'Actual native distance on every small pair') + small = subspaces(3) + flags = checks = 0 + for a0 in small: + for a in small: + if not a0 <= a: + continue + for b in small: + if not a < b: + continue + for b0 in small: + if not b <= b0: + continue + flags += 1 + for f in small: + projected = span(tuple(a) + tuple(f & b)) + need(a <= projected <= b, 'Projected chart lies in the query interval') + for h in range(1, 4): + cost = lambda x, y: min(distance(x, y), h) + old = cost(a0, f) + cost(f, b0) - cost(a, f) - cost(f, b) + new = cost(a0, projected) + cost(projected, b0) - cost(a, projected) - cost(projected, b) + need(old <= new, 'Every discrete concave hinge respects projection') + checks += 1 + # All two-qubit Lagrangians test the metric premise beyond canonical charts. + lagrangians = [] + def symp(x, y): + return (((x & 3) & (y >> 2)).bit_count() + ((y & 3) & (x >> 2)).bit_count()) % 2 + for s in subspaces(4): + if dim(s) == 2 and all(symp(x, y) == 0 for x in s for y in s): + lagrangians.append(s) + need(len(lagrangians) == 15, 'All two-qubit Lagrangians') + triangle_checks = 0 + for a, b, f in product(lagrangians, repeat=3): + af, bf, ab = (2 - dim(x & y) for x, y in ((a, f), (b, f), (a, b))) + need(af <= ab + bf, 'Lagrangian distance triangle inequality') + for h in (1, 2): + need(min(af, h) - min(bf, h) <= min(ab, h), 'Concave endpoint path bound') + triangle_checks += 1 + # Literal full sums check the cancellation for every path of up to three cuts. + a0, a, b, b0 = (span(v) for v in ([], [1], [1, 2], [1, 2, 4])) + paths = 0 + for length in (1, 2, 3): + for path in product(small, repeat=length): + for h in (1, 2, 3): + phi = lambda r: min(r, h) + targets = [a0] + list(path) + [b0] + paid = [a] + list(path) + [b] + gain = sum(phi(distance(x, y)) for x, y in zip(targets, targets[1:])) + gain -= sum(phi(distance(x, y)) for x, y in zip(paid, paid[1:])) + gain += phi(distance(a, b)) - phi(distance(a0, b0)) + bound = phi(distance(a0, a)) + phi(distance(b, b0)) + phi(distance(a, b)) - phi(distance(a0, b0)) + need(gain <= bound, 'Literal arbitrary-path upper bound') + paths += 1 + return dict(native_pairs=len(spaces)**2, binary_flags=flags, flag_hinge_checks=checks, + lagrangians=15, metric_hinge_checks=triangle_checks, literal_paths=paths) + + +def dimension_patterns(): + patterns = checks = 0 + for row in compositions(9, 10): + y, z = row[::2], row[1::2] + blocks = [x + w for x, w in zip(y, z)] + if blocks[2] == 0: + continue + a0, u, q, v, outside = blocks + a, b, b0 = a0 + u, a0 + u + q, a0 + u + q + v + x0, x1, x2, x3 = (sum(y[:i]) for i in range(1, 5)) + f = sum(y) + s, e1, e3, k = z[0] + y[4], z[1], y[3], y[2] + delta = e1 + e3 + target = [a0 + f - 2*x0, b0 + f - 2*x3] + paid = [a + f - 2*x1, b + f - 2*x2] + need(target == [u + k + s + e3 - e1, q - k + v + s + e1 - e3] + and paid == [k + s + delta, q - k + s + delta], 'Exact flag defect decomposition') + for h in range(1, 10): + phi = lambda r: min(r, h) + raw = sum(map(phi, target)) - sum(map(phi, paid)) + projected = phi(u+k) + phi(q-k+v) - phi(k) - phi(q-k) + need(raw <= projected, 'All rank-nine patterns and concave hinges') + checks += 1 + patterns += 1 + need(patterns == 37180, 'All feasible nine-dimensional flag intersection patterns') + return dict(patterns=patterns, hinge_checks=checks) + + +def power_endpoints(E): + count = 0 + for c in (Q('.00311313795'), Q('.0034'), Q('.01'), Q(1, 2), Q(9, 10)): + powers = {r: E.power(r, 1-c) for r in range(1, 10)} + for total in range(1, 10): + for u in range(total): + for q in range(1, total-u+1): + v = total-u-q + endpoint = 0 if u >= v else q + best = signed([endpoint, q-endpoint], [u+endpoint, q-endpoint+v]) + for k in range(q+1): + actual = signed([k, q-k], [u+k, q-k+v]) + gap = Counter(best) + gap.subtract(actual) + gap = {r: n for r, n in gap.items() if n} + need(not gap or evaluate(gap, powers)[0] > 0, 'Exact power endpoint optimum') + count += 1 + # This is a necessary restriction: an arbitrary concave tariff need not have an endpoint maximum. + phi = lambda r: min(r, 2) + endpoint = phi(1)+phi(3)-phi(0)-phi(2) + interior = 2*phi(2)-2*phi(1) + need(interior == 2 and endpoint == 1, 'Reject extension of endpoint theorem to every concave tariff') + return dict(power_comparisons=count, hinge_endpoint_counterexample=dict(u=1, q=2, v=1, endpoint=endpoint, interior=interior)) + + +def all_clifford_one_cut(E): + n = 3 + def symp(x, y): + return (((x & 7) & (y >> 3)).bit_count() + ((y & 7) & (x >> 3)).bit_count()) % 2 + lagrangians = [s for s in subspaces(6) if dim(s) == 3 + and all(symp(x, y) == 0 for x in basis(s) for y in basis(s))] + need(len(lagrangians) == 135, 'All three-qubit Lagrangian images') + def image(w): + return frozenset(x | (z << n) for x in w for z in range(1 << n) + if all(((x ^ z) & v).bit_count() % 2 == 0 for v in w)) + a0, a, b, b0 = [image(span(v)) for v in ([], [1], [1, 2], [1, 2, 4])] + metric = lambda x, y: n-dim(x & y) + need([metric(x, y) for x, y in ((a0, a), (a, b), (b, b0), (a0, b0))] == [1, 1, 1, 3], + 'Exact geodesic source boundary in the full Clifford domain') + checks = 0 + for c in (Q('.00311313795'), Q(1, 2), Q(9, 10)): + powers = {r: E.power(r, 1-c) for r in range(1, 4)} + upper = signed([], [1, 2]) + for f in lagrangians: + delta = Counter(upper) + delta.subtract(signed([metric(a, f), metric(f, b)], [metric(a0, f), metric(f, b0)])) + delta[1] -= 1 + delta = {r: m for r, m in delta.items() if m} + need(not delta or evaluate(delta, powers)[0] > 0, 'Power endpoint optimum for every Clifford image') + checks += 1 + return dict(lagrangians=135, exact_power_checks=checks) + + +def source_inventory(word, receipt): + frames = [span(v) for v in word['frames']] + need(all(dim(s) == len(v) for s, v in zip(frames, word['frames'])), 'Independent source frame bases') + public = word['public'] + main = set(word['live']) | {112+r for r in word['live']} + helper_ids = set(range(public)) - main + need(len(main) == 208 and public == 767 and len(helper_ids) == 559, + 'Exact source main and public interface') + need('main = live | {r+112 for r in live}' in (HERE/'source/source-local.py').read_text(), + 'Literal source main-field contract') + current = word['initial'][:] + [None]*word['owned_fields'] + paid, helpers, queries = Counter(), Counter(), [] + for h in sorted(helper_ids): + a0, b0 = frames[word['initial'][h]], frames[word['final'][h]] + need(a0 < b0, 'Positive nested source helper boundary') + helpers[distance(a0, b0)] += 1 + for pc, event in enumerate(word['program']): + op = event[0] + if op == 'MOVE': + _, h, a, b, private = event + need(current[h] == a and private == (h >= public), 'Actual source MOVE frame and ownership') + current[h] = b + r = distance(frames[a], frames[b]) + need(r > 0, 'Positive paid source rank') + paid[r] += 1 + if h in helper_ids: + a0, b0 = word['initial'][h], word['final'][h] + need(frames[a0] <= frames[a] < frames[b] <= frames[b0], 'Actual helper-query flag') + u, q, v = dim(frames[a])-dim(frames[a0]), r, dim(frames[b0])-dim(frames[b]) + queries.append(dict(pc=pc, helper=h, frames=[a0, a, b, b0], ranks=[u, q, v])) + elif op == 'INIT': + _, h, frame = event + need(h >= public and current[h] is None, 'Fresh owned frame lifetime') + current[h] = frame + elif op == 'CLEAR': + _, h, frame = event + need(h >= public and current[h] == frame, 'Owned frame cleanup') + current[h] = None + elif op in ('ADD', 'COPY'): + _, dst, src, value, frame = event + need(current[dst] == current[src] == frame, 'Actual scalar operand frames') + else: + need(op == 'SCALE' and current[event[1]] == event[3], 'Actual scalar scale frame') + need(current == word['final'] + [None]*word['owned_fields'], 'All source frame endpoints') + need(dict(paid) == {int(k): v for k, v in receipt['local']['paid'].items()}, 'All source paid ranks') + need(dict(helpers) == {int(k): v for k, v in receipt['local']['helpers'].items()}, 'All source target ranks') + wrong = Counter(distance(frames[word['initial'][h]], frames[word['final'][h]]) + for h in range(208, public)) + need(wrong != helpers, 'Reject a contiguous-main partition despite equal field counts') + need(len(queries) == 2534 and len({r['helper'] for r in queries}) == 559, 'Complete helper-query inventory') + return queries, paid, helpers + + +def source_controls(word, receipt): + controls = [] + bad = deepcopy(word) + bad['live'] = list(range(104)) + controls.append(('contiguous-main-role-assumption', bad, receipt)) + bad = deepcopy(word) + first = next(i for i, e in enumerate(bad['program']) if e[0] == 'MOVE') + bad['program'][first][2] = (bad['program'][first][2] + 1) % len(bad['frames']) + controls.append(('wrong-query-source-frame', bad, receipt)) + bad = deepcopy(receipt) + bad['local']['helpers']['9'] -= 1 + controls.append(('missing-original-helper-target', word, bad)) + rejected = [] + for name, candidate, expected in controls: + try: + source_inventory(candidate, expected) + except ValueError as error: + rejected.append(dict(control=name, rejection=str(error))) + else: + raise ValueError('Accepted invalid source control: ' + name) + return rejected + + +def bounds(queries, E, receipt): + groups = defaultdict(list) + for row in queries: + groups[row['helper']].append(row) + outputs = {} + for c in (Q('.00311313795'), Q('.0034'), Q('.01')): + powers = {r: E.power(r, 1-c) for r in range(1, 10)} + selected = {} + for mode in ('one_cut', 'arbitrary_path'): + choices = [] + for helper, rows in sorted(groups.items()): + candidates = {} + for row in rows: + u, q, v = row['ranks'] + if mode == 'one_cut': + parts = [u, q+v] if u >= v else [u+q, v] + else: + parts = [u, q, v] + parts = tuple(sorted(r for r in parts if r)) + candidates.setdefault(parts, row['pc']) + values = [(evaluate(signed([sum(parts)], parts), powers), parts, pc) + for parts, pc in candidates.items()] + values.sort(key=lambda x: (x[0][0], x[1]), reverse=True) + win, parts, pc = values[0] + for loss, other, other_pc in values[1:]: + need(win[0] > loss[1], 'Exact source profile ordering') + choices.append(dict(helper=helper, pc=pc, target_parts=parts, gain=win)) + total = (Q(0), Q(0)) + for x in choices: + total = add(total, x['gain']) + ordered = sorted(choices, key=lambda x: x['gain'][0], reverse=True) + for left, right in zip(ordered, ordered[1:]): + lc = signed([sum(left['target_parts'])], left['target_parts']) + rc = signed([sum(right['target_parts'])], right['target_parts']) + need(lc == rc or left['gain'][0] >= right['gain'][1], + 'Exact ordering for the minimum helper count') + prefix = (Q(0), Q(0)) + required = None + for i, x in enumerate(ordered, 1): + previous = prefix + prefix = add(prefix, x['gain']) + if prefix[0] >= 1: + need(previous[1] < 1, 'Exact minimum number of relaxed helpers for one H1 fee') + required = i + break + need(required is not None, 'Source envelope can fund a unit fee with enough helpers') + selected[mode] = dict(total_gain=E.display(total), minimum_helpers_for_one_H1=required, + maximum_gain=E.display(ordered[0]['gain']), zero_gain_helpers=sum(x['gain'] == (0, 0) for x in choices), + choices=[dict(helper=x['helper'], pc=x['pc'], target_parts=x['target_parts'], gain=E.display(x['gain'])) for x in choices]) + outputs[str(c)] = selected + # Exact universal cap at the largest declared exponent. Jensen bounds three parts by 3,3,3. + c = Q('.01') + p = {r: E.power(r, 1-c) for r in range(1, 10)} + cap = evaluate({3: 3, 9: -1}, p) + need(0 < cap[0] <= cap[1] < Q(1, 10), 'Every one-target detour gain is below one tenth of H1 through C=.01') + return outputs, dict(three_part_cap_at_one_percent=E.display(cap), ordinary_work_charged=False) + + +def run(): + C, E = source_modules() + geometry = finite_geometry(C) + patterns = dimension_patterns() + endpoints = power_endpoints(E) + clifford = all_clifford_one_cut(E) + word, receipt = read(HERE/'source/WORD.json.gz'), read(HERE/'source/RESULT.json') + queries, paid, helpers = source_inventory(word, receipt) + controls = source_controls(word, receipt) + budgets, cap = bounds(queries, E, receipt) + result = dict(status='PASS_HELPER_DETOUR_PROJECTION_AND_SHARP_PATH_BUDGETS', geometry=geometry, + dimension_patterns=patterns, endpoint_checks=endpoints, full_clifford_check=clifford, source_head='a1b338e5d9271b541f2a198f4fc19cefc565feed', + source_queries=len(queries), source_helpers=sum(helpers.values()), + source_paid=dict(sorted(paid.items())), source_targets=dict(sorted(helpers.items())), + full_query_zero_gain=sum(row['ranks'][0] == row['ranks'][2] == 0 for row in queries), + budgets=budgets, universal_cap=cap, source_controls=controls, + limits=['Optimistic recursive tariff bounds, not physical independent-target cuts.', + 'Choose at most one original query per helper; alternative cuts are not additive.', + 'Shared-query graphs, changed endpoints and extra ordinary/native fees require a new contract.', + 'Source frame and rank checks do not repeat its scalar endpoint, native-phase or compiler admission.']) + return result, dict(schema='p104-helper-query-flags/1', source_head=result['source_head'], queries=queries) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--write', action='store_true') + args = parser.parse_args() + with tempfile.TemporaryDirectory(prefix='detour-bytecode-') as cache: + sys.pycache_prefix = cache + result, queries = run() + for name, value in (('RESULT.json', result), ('QUERY-BOUNDS.json', queries)): + # JSON normalizes integer dictionary keys and tuples in the reproducible receipt. + normalized = json.loads(json.dumps(value)) + if args.write: + (HERE/name).write_text(json.dumps(normalized, sort_keys=True, indent=2)+'\n') + else: + need(read(HERE/name) == normalized, 'Exact retained receipt: '+name) + print(result['status']) + print('Geometry:', result['geometry']) + for c, rows in result['budgets'].items(): + print(c, {m: dict(gain=v['total_gain'], fee_helpers=v['minimum_helpers_for_one_H1']) for m, v in rows.items()}) + + +if __name__ == '__main__': + main()