TFPT scientific source excerpt Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/local_energy_eigenpacket_review.json Original file SHA-256: f911f9d9237c8056bb4f8add917682d02f43c0bca40e1da520942bb6c84e150f Inclusive original lines: 1-170 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 | { 000002 | "created_at_utc": "2026-10-18T15:38:07.152168+00:00", 000003 | "mode": "Independent read-only analytic/source review; no new model or suite run", 000004 | "verdict": "LEMMA_VALID_UNDER_EXPLICIT_FULL_NET_HYPOTHESES", 000005 | "lemma": { 000006 | "assumptions": [ 000007 | "H is self-adjoint, H>=0, H Omega=0 in the full given CAR representation.", 000008 | "alpha_t(A)=exp(+itH) A exp(-itH) is the chosen mathematical Heisenberg convention.", 000009 | "A is bounded and lies in F(I). There exist epsilon>0 and a proper larger interval J with alpha_t(A) in F(J) for every |t|0 of this same H. Zero vector is allowed and already implies A0 by separation." 000012 | ], 000013 | "proof_steps": [ 000014 | "For |t|alpha_t(A)", 000025 | "a finite-dimensional target eigenspace" 000026 | ] 000027 | }, 000028 | "primary_support": { 000029 | "authors": "Marcel Bischoff and Yoh Tanimoto", 000030 | "paper": "Construction of wedge-local nets of observables through Longo-Witten endomorphisms. II", 000031 | "url": "https://arxiv.org/abs/1111.1671", 000032 | "pdf": "https://arxiv.org/pdf/1111.1671", 000033 | "locations": [ 000034 | "Section2.1 property(7), PDFpage4: local Reeh-Schlieder property for Fermi nets", 000035 | "Section3.2, PDFpages15–16: C(K), standard subspaces, complex free fermion net and positive-energy Fock representation" 000036 | ], 000037 | "scope": "Supplies cyclic/separating local vacuum structure for the chiral free complex Fermi net. No Minkowski4D, physical P1 or source-selection inference." 000038 | }, 000039 | "quarter_phase_transfer": { 000040 | "issue": "The standard NS conformal-vacuum theorem should not simply be relabeled as a quarter-twisted Möbius vacuum theorem.", 000041 | "explicit_support_preserving_argument": "On a proper chart, multiplication by exp(-i(alpha-1/2)theta) maps the original C_alpha kernel to the NS kernel and preserves test-function support. The CAR/state isomorphism transfers regional cyclicity/separation. Endpoint transition and the transformed time generator remain part of the dictionary; no globally periodic gauge equivalence is asserted.", 000042 | "time_locality": "The actual H_alpha rotates the original circle, with its retained phase/bundle transition; the extra phase is an internal gauge action and does not enlarge support. Its fold transports regions to their actual multi-arc preimages. A proper base interval can also be embedded in a larger proper base interval under sufficiently small source-time translations." 000043 | }, 000044 | "same_source_H32_check": { 000045 | "full_source_formula": "H_alpha=sum_j (j+alpha)(n_j-1_(j<0)), with occupied sea j<=-1 and0=0 energies j+alpha>0; hole j<=-1 energies -(j+alpha)>0. Thus H_alpha>=0 and Omega has energy0.", 000046 | "normalization": "The source text writes U(t)=exp(-it K_alpha), where H_alpha=8 K_alpha. Its five native inputs have K_alpha4 and H_alpha32, not two different choices of dynamics.", 000047 | "construction_inspected": "source_packets() maps a word entry with weighted degree n+1 to original density(v,8(n+1),channel=k). series_response(...,grade4) retains exactly words whose sum of(n+1)=4. Each bilinear shifts occupied index by8(n+1) and preserves every residue charge. Thus every term has original cover excitation32 and all residue charges0, independently of coefficients b.", 000048 | "energy_function_caveat": "car.energy(bits) computes sum_j j Delta n_j. It equals the full H_alpha only on total charge0 because H_alpha=car.energy+alpha Q. The constructed packet has Q_k0 for each channel, so this equality is legitimate here. Do not use that finite formula as a proof that the alpha Q term can be discarded on arbitrary charged states.", 000049 | "reported_owner_small_call": { 000050 | "quarters": [ 000051 | "1/4", 000052 | "3/4" 000053 | ], 000054 | "support_sizes_per_five_states": [ 000055 | 694, 000056 | 710, 000057 | 711, 000058 | 713, 000059 | 688 000060 | ], 000061 | "all_source_energies": 32, 000062 | "all_channel_charges": [ 000063 | 0, 000064 | 0, 000065 | 0, 000066 | 0, 000067 | 0, 000068 | 0, 000069 | 0, 000070 | 0 000071 | ], 000072 | "Gram_rank": 5, 000073 | "execution_attribution": "Reported direct source_packets call by verification_split; independently checked the algebraic weighted-degree/density construction here, not repeated the call." 000074 | }, 000075 | "H_quartic_warning": "The compressed equality K|packet=4I5 (or H|packet=32I5) is not the net time dynamics. The proof uses the full H_alpha and its support-preserving automorphisms, not a finite quartic block or a newly postulated finite source." 000076 | }, 000077 | "limits": [ 000078 | "Exact preparation by one bounded operator in a proper local algebra is excluded only for this strict positive-energy eigenspace target.", 000079 | "This is not a bound on arbitrarily accurate local approximations, preparation probability, norm/cost tradeoffs or general finite-energy packets.", 000080 | "No statement follows about every local physical preparation, different admissible outputs, source/state changes or infinite/global protocols.", 000081 | "No automatic extension to unbounded point fields, affiliated operators or distributions without domain/control arguments.", 000082 | "A bounded global rank-one operator |psi_E>0, here E32. Support in the actual finite R_b4 packet is sufficient because that packet is contained in H32." 000146 | ], 000147 | "proof": [ 000148 | "Set v_j=A_j Omega, P=1_{E}(H), Q=I-P. The support assumption gives Q sigma Q=0.", 000149 | "For a trace-class outcome, normality/monotone convergence of the positive partial sums gives Tr(Q sigma)=sum_j ||Q v_j||^2=0. Every summand is nonnegative, hence Q v_j=0 for every j.", 000150 | "Equivalently, even without using a trace, for every xi in Ran Q, normal vector functionals give 0==sum_j ||^2. Positivity forces every v_j orthogonal to Ran Q. This avoids an unjustified exchange of a nonnormal limit and trace.", 000151 | "Thus each A_j Omega lies in the exact E>0 eigenspace. The proved local eigenpacket lemma yields A_j=0 individually, including the zero-vector case by separation.", 000152 | "Consequently sigma=0 and its success probability is0. Renormalization cannot produce a pure or mixed exact E32 state with nonzero outcome probability." 000153 | ], 000154 | "packet_metric_precision": "If a projector onto the five-state R_b4 packet is used, it means the orthogonal projector in the actual full CAR Hilbert-space metric, not a raw coordinate cut assuming Gram I. Only the inclusion Ran(P_packet) subset H32 is needed.", 000155 | "projection_in_the_proof_is_not_a_preparation": "The global spectral projector P_E and Q are mathematical support tests. No claim that they themselves belong to the local algebra or are physically available filters is made.", 000156 | "normal_convergence_precision": "A normal trace-nonincreasing Kraus outcome has trace-norm convergence of its positive vacuum-output partial sums, sufficient for the displayed norm-square identity. Bare unspecified convergence is not substituted for normality. The vector-functional proof also establishes the conclusion for positive ultraweak limits where a finite trace was not assumed.", 000157 | "luders_case": { 000158 | "statement": "Applies to a proper-local original Lüders projection P in F(I), with one Kraus operator A=P, or to a finite/countable same-local Kraus outcome of the stated type.", 000159 | "independent_consistency_check": "For any orthogonal projection P, =||P Omega||^2. If P Omega lies entirely at strictly positive energy, it is orthogonal to Omega, so P Omega=0. Local separation then gives P0. This elementary special case agrees with the general Kraus result." 000160 | }, 000161 | "scope_limits": [ 000162 | "The conclusion allows pure or mixed outputs and finitely/countably many local Kraus terms; positivity prevents cancellations outside the target spectral subspace.", 000163 | "It is not a theorem about every abstract normal AQFT CP map: an arbitrary CP map is not assumed to possess bounded Kraus operators inside this same local algebra.", 000164 | "No conclusion is drawn about approximate target preparation, quantitative success/energy/norm tradeoffs, different initial states, altered Hamiltonians or enlarged/global preparation procedures.", 000165 | "No extension to unbounded field insertions, uncontrolled infinite histories or a physically derived P1 preparation menu is asserted.", 000166 | "The result uses the full H_alpha=8K_alpha, not only the compressed quartic matrix. Exact H32 support is essential." 000167 | ], 000168 | "canonical_mutations": "None; only this existing work-area report was appended." 000169 | } 000170 | }