TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/collective-native-marginal-origin-20261026/PROOF.txt
Original file SHA-256: 72ff4cac1b6e32d71cc1d4f0fb39fbf59754935f72455b7e5a3ba5ffce0b369e
Inclusive original lines: 21-91
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.

000021 | 1. The first anisotropic source moment is the fourth
000022 | Define M_r=(1/60)sum_a |z_a^tensor r><z_a^tensor r|.
000023 | For r=1,2,3 the actual source satisfies EXACTLY
000024 |     M_1=I4/4, M_2=Pi_sym2/10, M_3=Pi_sym3/20.
000025 | For r=4 it satisfies the PREEXISTING native parent identity
000026 |     40 M_4 = Pi_C + Pi_sym4.
000027 | The checker independently rebuilds M_r from the original roots with Gaussian
000028 | integer tensor products. No target5 enters M_r. It then extracts
000029 |     Pi_C = 40 M_4 - Pi_sym4
000030 | and compares with the literal code projector, with no change of frame.
000031 | Thus the fourth moment is the first departure from the uniform complex
000032 | projective moment in this source. This reuses, rather than claims discovery of,
000033 | the original moment parent. The uniform lower moments are not a claim that
000034 | all conceivable nonlinear recognition procedures fail below four copies.
000035 | 
000036 | 2. Actual parent and algebraic filter
000037 | H_nat=I-20M_4 has the exact orthogonal decomposition
000038 |     H_nat=0 Pi_C + (1/2)(Pi_sym4-Pi_C) + (I-Pi_sym4).
000039 | The multiplicities are5,30,221 and the normalized mathematical gap is1/2.
000040 | These identities are checked by exact projector actions, not eigenvalue fits.
000041 | On the code exp(-itH_nat) is the identity. The associated algebraic filter is
000042 |     (20M_4)^n=Pi_C+2^(-n)(Pi_sym4-Pi_C), n>=1.
000043 | Convergence requires a nonzero code component. This is NOT a derived physical
000044 | time step, deterministic preparation, thermal bath or allowed instrument.
000045 | A unitary evolution does not remove the other components merely from this gap.
000046 | 
000047 | 3. The whole-code state is invisible in all proper marginals
000048 | Set rho_C=Pi_C/5 and rho_S=Pi_sym4/35. For every subset A of size1,2 or3,
000049 |     Tr_(A complement)(rho_C)=Tr_(A complement)(rho_S).
000050 | In particular every three-register marginal is Pi_sym3/20. Partial tracing
000051 | gives the two-register Pi_sym2/10 and one-register I4/4 marginals.
000052 | The checker verifies all4+6+4 proper subsets exactly. It uses integer-scaled
000053 | projectors Pc24=24Pi_C and Ps24=24Pi_sym4, so tests partial traces of
000054 | 7Pc24-Ps24=0 with no floating tolerance. Globally the states differ strongly:
000055 |     (1/2)||rho_C-rho_S||_1=6/7.
000056 | This comparison concerns the TWO MIXED STATES, not all pure code states.
000057 | Individual three-register outputs can and do retain logical information.
000058 | 
000059 | 4. Exact locality obstruction, including frustrated Hamiltonians
000060 | Let H be ANY Hermitian operator on the SAME four registers, a sum of terms
000061 | each supported on at most three registers. No ancillas or new sites are allowed.
000062 | The marginal identity implies Tr(H rho_C)=Tr(H rho_S).
000063 | If ALL of C is a ground space with ground energy E0, the left side is E0.
000064 | Since H-E0 I is positive semidefinite, Tr((H-E0 I)rho_S)=0 implies
000065 | (H-E0 I) annihilates the support of rho_S, which is ALL Sym^4(C^4).
000066 | Therefore dim ground(H)>=35. Its exact ground space cannot be only C of dim5.
000067 | No assumption of commuting terms, group invariance, positive individual terms
000068 | or frustration freedom is used. The argument is the standard local-marginal
000069 | necessary condition, applied to the actual TFPT source:
000070 | J. Chen, Z. Ji, B. Zeng, D. L. Zhou, Phys. Rev. A86,022339(2012),
000071 | https://arxiv.org/abs/1110.6583 . No novelty priority is asserted.
000072 | The existing four-register parent ATTAINS the required interaction locality.
000073 | This proves necessity for this exact whole-ground-space task, not for preparing
000074 | one chosen state or realizing an effective interaction using extra degrees.
000075 | 
000076 | [OMITTED: non-scientific conversation or execution metadata]
000077 | [OMITTED: non-scientific conversation or execution metadata]
000078 | [OMITTED: non-scientific conversation or execution metadata]
000079 | [OMITTED: non-scientific conversation or execution metadata]
000080 | [OMITTED: non-scientific conversation or execution metadata]
000081 | [OMITTED: non-scientific conversation or execution metadata]
000082 | [OMITTED: non-scientific conversation or execution metadata]
000083 | [OMITTED: non-scientific conversation or execution metadata]
000084 | [OMITTED: non-scientific conversation or execution metadata]
000085 | [OMITTED: non-scientific conversation or execution metadata]
000086 | [OMITTED: non-scientific conversation or execution metadata]
000087 | [OMITTED: non-scientific conversation or execution metadata]
000088 | [OMITTED: non-scientific conversation or execution metadata]
000089 | [OMITTED: non-scientific conversation or execution metadata]
000090 | [OMITTED: non-scientific conversation or execution metadata]
000091 | [OMITTED: non-scientific conversation or execution metadata]