TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/admissible_pair_20261028/HODGE-TIME-IDENTITY-TEST-20261102.txt Original file SHA-256: 8e6ee0625b6b38800b7a63d0bec9d7b0b54cbcc2de202e786f5648f7eab5ef00 Inclusive original lines: 1-151 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | HODGE CHANNEL AND A CONTINUOUS IDENTITY START 000002 | ============================================ 000003 | 000004 | 2 November 2026. Exact finite-dimensional consequence of the given Hodge 000005 | amplitudes. This is not an identification of TFPT physical time, a deduction 000006 | of an original measurement, or a no-go for a larger process with additional 000007 | amplitudes, registers, memory, or a distinguished finite-time event. 000008 | 000009 | Let H=Lambda^2 C^5, dim H=10, and C_i:H -> H* be the five original epsilon 000010 | amplitudes from hodge_complement_channel.py. An arbitrary unitary identification 000011 | H* -> H may be made before comparing them with an identity evolution. Since the 000012 | result below is uniform over all unitary target maps, that identification does 000013 | not affect the bound. 000014 | 000015 | Write |A>> for vectorization, >=tr(A*B), and 000016 | 000017 | P5 = (1/6) sum_i |C_i>><>< H*, Holder's inequality gives 000031 | 000032 | |tr(C(z)*U)| <= ||C(z)||_1 = 6||z||, 000033 | <> = (1/6) sum_i |tr(C_i*U)|^2 <= 6. (1) 000034 | 000035 | The second line also follows by maximizing over the unit coefficient vector 000036 | z. The bound is sharp: the checker extends the actual signed partial 000037 | permutation C_0 to a 10 by 10 unitary, attaining 6. Set omega_U=|U>><= 2(1 - 3/5) = 4/5 (2) 000048 | 000049 | for every U. In fact the exact normalized Choi trace distance sharpens this. 000050 | Let u=|U>>/sqrt(10), q= <= 3/5. There are four eigenvalues 1/5 000051 | on P5 perpendicular to its projection of u; the two remaining eigenvalues 000052 | have opposite sign for q<1 and give 000053 | 000054 | ||P5/5 - |u>= (4+2 sqrt(6))/5 = 1.779795897... . (3) 000056 | 000057 | The unitary completing C0 has q=3/5, so the MINIMUM OF THIS CHOI lower bound 000058 | over unitaries is attained. The actual diamond distance is nevertheless 2, 000059 | and this holds FOR EVERY U, by the following separate argument. 000060 | 000061 | For fixed U, let K be the set of complex five-vectors with coordinates 000062 | Tr(rho U* C_i), where rho runs over all density operators on H. K is a 000063 | compact convex subset of C^5. If zero were not in K, strict convex separation 000064 | would give a complex vector z such that Re Tr(rho U* C(z))>0 for every rho. 000065 | This would mean Re(U* C(z)) is strictly positive definite. However, C(z) has 000066 | a nonzero kernel for every z, so for a nonzero x in that kernel, 000067 | =0. Contradiction. Thus some density rho obeys 000068 | Tr(rho U* C_i)=0 simultaneously for all i. 000069 | 000070 | Here C^5 is treated as R^10: every real linear functional can be expressed as 000071 | Re sum_i z_i a_i after relabelling the arbitrary complex coefficients z_i. 000072 | Hence no unstated complex linear separation or conjugation convention is used. 000073 | 000074 | Purify rho=sum_j p_j |e_j>=sum_j sqrt(p_j)|e_j> tensor |j> on an untouched reference. 000076 | It has norm one, and for any A on the carrier, 000077 | =Tr(rho A). The inner product of the output for U 000078 | and each Hodge Kraus output is precisely Tr(rho U* C_i)/sqrt(3)=0. Hence the 000079 | pure unitary output is orthogonal to the support of the Hodge output, and the 000080 | states are perfectly distinguishable in one use. Both output states have 000081 | trace one because U and Phi are trace preserving. Consequently 000082 | 000083 | ||Phi - Ad_U||_diamond = 2 for every U:H -> H*. (4) 000084 | 000085 | This convex argument is not an original general distinguishability theorem; 000086 | compare the operator-subspace criterion for channel discrimination in the 000087 | primary papers https://arxiv.org/abs/1605.02294 and 000088 | https://arxiv.org/abs/0908.0119. Its present input is the specific rank-six 000089 | identity of the original epsilon amplitudes. 000090 | 000091 | Explicitly, in the original ordered duad basis (01,02,03,04,12,13,14,23, 000092 | 24,34), U equals C0 on the last six basis states and is identity on the 000093 | first four. On the input |01>, U outputs |01>, while C0|01>=C1|01>=0, 000094 | C2|01>=|34>, C3|01>=-|24>, C4|01>=|23>. The Hodge output is therefore 000095 | (|34><34|+|24><24|+|23><23|)/3, orthogonal to |01><01|. This is an 000096 | unassisted perfect distinguisher and proves distance 2 for that actual U. 000097 | 000098 | Combined with the already established R086 shared-TP theorem, any complete 000099 | globally TP operation whose every joint Kraus operator lies strictly in 000100 | span{C_i} tensor B(F,F') has precisely Phi as carrier marginal, including on 000101 | correlated inputs. Consequently such a family cannot be a finite-dimensional 000102 | continuous evolution with a unitary/identity carrier limit at t=0 for all 000103 | sufficiently small t. Finite-dimensional strong continuity gives the needed 000104 | channel norm continuity. This remains true for any unitary choice of the 000105 | carrier/dual identification, even if that choice varies with t. 000106 | 000107 | An explicit compatible mathematical embedding 000108 | --------------------------------------------- 000109 | 000110 | Fix any unitary U:H -> H*, and write Psi=Ad_(U*) o Phi, now a CPTP map on H. 000111 | For any separately chosen lambda>0 the bounded generator L=lambda(Psi-id) 000112 | defines the continuous CPTP semigroup 000113 | 000114 | E_t = exp(t L) = exp(-lambda t) sum_(k>=0) (lambda t)^k Psi^k/k!, 000115 | E_t = id + lambda t(Psi-id) + O(t^2). 000116 | 000117 | Every term is CPTP and the nonnegative Poisson weights sum to one. The k=0 000118 | Kraus operator exp(-lambda t/2)I is the unchanged carrier branch; the 000119 | other branches concatenate the actual original epsilon Kraus operators 000120 | U*C_i/sqrt(3). The checker verifies trace preservation for the one- and 000121 | two-jump terms. The jump intensity lambda, the Poisson schedule and the 000122 | identification U are mathematical choices here, not outputs of P1, S_adm, 000123 | the TFPT clock network or a physical time scale. 000124 | 000125 | Each nonzero C(z) has rank six, whereas U has rank ten. Thus U is outside 000126 | span{C_i}; equivalently I is outside span{U*C_i}. Further, this is not merely 000127 | an artifact of the displayed Kraus decomposition. If any Kraus decomposition 000128 | of E_t were entirely in that strict span, the original 25-product TP theorem 000129 | would force E_t=Psi. But E_t -> id as t -> 0, while (3) bounds Psi away from 000130 | id (indeed (4) gives distance 2). Moreover 000131 | ||E_t-id||_diamond <= 2(1-exp(-lambda t)) < 2 for EVERY finite t, so E_t 000132 | cannot equal Psi at ANY finite time. Hence at least one Kraus amplitude must 000133 | be outside the strict span for EVERY Kraus decomposition of E_t at EVERY 000134 | finite t. This conclusion says nothing about which extra amplitudes are 000135 | actually supplied by the original source or about the infinite-time limit. 000136 | 000137 | This is an actual compatibility condition on how the previously identified 000138 | E8 bracket operation can enter a source/time dictionary: it can be a 000139 | particular event or finite transition; alternatively, a full continuous 000140 | process needs other components, such as an identity/no-event amplitude, 000141 | outside this strict bracket-only Kraus span. We do not infer which of these 000142 | the original source chooses. A single sub-TP branch need not obey R086's 000143 | global-TP conclusion. Original reflection, two boundaries, dynamics, family, 000144 | Eta, reference, and the measured or discarded register must still be mapped. 000145 | 000146 | Reproduction from repository root: 000147 | python3 -B experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/admissible_pair_20261028/check_hodge_time_identity.py 000148 | 000149 | Output: PASS 14 exact checks, including polynomial identities in all complex 000150 | coefficients and a sharpness witness for (1). No Lean run, physical validation, 000151 | claim promotion, or construction of the full source is asserted.