TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/primitive_insertions_review/UV-INSERTION.txt Original file SHA-256: aca7b4ac6f9bedaacfa92b2168d88f6ad837f3a7d03005c03d8e71eeb51a0a1f Inclusive original lines: 1-128 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 | CONSTANT ORIGINAL ODD INSERTION: ALGEBRAIC RESPONSE AND SEA DOMAINS 000002 | 29 October 2026. Analytical appendix to ORIGINAL-INSERTION-REVIEW.txt. 000003 | No new source, scan, suite, original edit or Atlas mutation. 000004 | 000005 | RESULT 000006 | The algebraic CAR insertion does not automatically have a Fock generator on 000007 | the old sea domain. Nevertheless the full coupled one-particle evolution is 000008 | implementable in that same one-dimensional sea representation. The two facts 000009 | are compatible and must not be turned into a physical exclusion. 000010 | 000011 | 1. Exact existing class 000012 | Let K be the given finite internal coefficient space, gamma=gamma_F, 000013 | gamma*=gamma, gamma^2=1. On L2(S1,K), with period2pi and periodic H1 domain, 000014 | A=-i partial_theta+alpha, alpha=1/4 or3/4, 000015 | h0=A gamma, Phi=deltaPhi=Phi* constant, {gamma,Phi}=0, Phi!=0, 000016 | V=delta h=-i gamma Phi, h=h0+V. 000017 | These are precisely the existing finite-coupled chart and an actual constant 000018 | odd insertion type. V*=V, {gamma,V}=0, V^2=Phi^2. There is no spatial or normal 000019 | profile change in this calculation. Bounded V preserves the one-particle H1 000020 | domain. None of this alone is a statement about a many-body vacuum domain. 000021 | 000022 | Sources: graded_boundary_functor/FINITE-COUPLED.txt and 000023 | charge_transport/DERIVED-PAIRING.txt:17-30,83-95; the actual v252/v254 000024 | one-form is followed in original_insertions/PROOF.txt:25-51. The latter's 000025 | section4 already warns against differentiating a massive Fock Hamiltonian 000026 | on a fixed massless vacuum. The present appendix evaluates that precise 000027 | constant-insertion condition, rather than reopening the field chart. 000028 | 000029 | 2. Smallest obstruction: the insertion alone 000030 | For Fourier momentum p=n+alpha, n in Z, put s=sign(p). The old negative 000031 | projector is P0(p)=(I-s gamma)/2. Since V anticommutes with gamma, 000032 | [P0(p),V]=-s gamma V, 000033 | ||[P0(p),V]||_HS^2=Tr_K(Phi^2)>0. 000034 | Summing over all n gives infinity. The off-diagonal particle-hole block 000035 | has half this positive squared norm on each mode. Thus its formal pair 000036 | creation action on the old sea vacuum is not a square-summable two-excitation 000037 | vector. The normally ordered quadratic insertion is not supplied as an 000038 | operator with the old vacuum in its domain. Subtracting a scalar cannot 000039 | change this off-diagonal divergence. 000040 | 000041 | One can prove a stronger group statement here without inferring it from 000042 | the divergent infinitesimal norm alone. On every Fourier block, 000043 | exp(iu V)=cos(u |V|)+i V sin(u |V|)/|V|, 000044 | with the continuous value on ker V. Its off-diagonal block is nonzero and 000045 | has the same finite positive norm for infinitely many n whenever 000046 | 0<|u|=0. 000066 | Since sum_(n in Z)1/(n+alpha)^2 converges, 000067 | P(h)-P(h0) is Hilbert-Schmidt in this one-dimensional circle class. 000068 | No small-mass assumption is required. Zero internal singular values simply 000069 | give zero contribution. The selfdual conjugate test slots of the EXISTING 000070 | whole chart give the same HS criterion; they add no physical species. 000071 | 000072 | Consequently the pure quasifree sea representations are unitarily equivalent. 000073 | This does NOT say their vacuum vectors or their generators are identical. 000074 | The coupled sea is the state determined by P(h), not an independently fitted 000075 | new state. Its particle/hole Fock time has a positive excitation generator. 000076 | Transport through the representation equivalence gives a genuine strongly 000077 | continuous implementation of the full exp(it h) CAR evolution in the old 000078 | Fock Hilbert space, with its own transported domain and coupled ground vector. 000079 | 000080 | 4. Direct check of the full evolution 000081 | The same conclusion is visible without replacing the generator by V: 000082 | U_t(p)=cos(t E_p)+i(p gamma+V) E_p^-1 sin(t E_p), 000083 | [P0(p),U_t(p)]=i[P0(p),V] E_p^-1 sin(t E_p). 000084 | Hence 000085 | ||[P0,U_t]||_HS^2 000086 | <=Tr_K(Phi^2) sum_n 1/(n+alpha)^2 < infinity. 000087 | The bound and dominated convergence also give HS continuity at t=0. 000088 | The full kinetic evolution produces the inverse-momentum suppression absent 000089 | from the pure insertion group. A non-HS infinitesimal commutator therefore 000090 | cannot be used to reject the full exp(it h) evolution. 000091 | 000092 | Indeed the old vacuum need not lie even in the coupled energy form domain. 000093 | In this nonzero constant class its coupled excitation energy, obtained as 000094 | the increasing sum of the positive finite-mode energies, is 000095 | 1/2 sum_n Tr_K[(p^2+Phi^2)^(1/2)-|p|I]. 000096 | Its summand is asymptotic to Tr_K(Phi^2)/(4|p|), so the sum diverges. 000097 | The corresponding coupled Hamiltonian exists, but its old vacuum is not 000098 | in Dom(H_coupled^(1/2)), hence not in Dom(H_coupled). This is why writing 000099 | H_coupled=H0+an already defined mass operator on the entire old finite-particle 000100 | vacuum core is unjustified. This domain fact does not obstruct its unitary 000101 | time group or the pure-state representation equivalence above. 000102 | 000103 | 5. Source and theorem status 000104 | The Atlas was searched for this exact insertion/whole-evolution distinction. 000105 | Retrieved findings include 000106 | finding:graded-boundary-clifford-typing-transports-full-scalar-and-finite-coupled-source, 000107 | finding:actual-charge-j-transport-retains-full-dirac-and-distinct-pairings, 000108 | finding:full-dirac-cauchy-vector-cocycle-cancels-and-endpoint-domain-distinction. 000109 | Those records and the cited old domain warning are retained. The retrieved 000110 | sources do not contain this explicit constant-Phi Fourier HS calculation; 000111 | this is a targeted search statement, not a complete-corpus absence claim. 000112 | 000113 | Primary standard result: H. Araki, On Quasifree States of CAR and Bogoliubov 000114 | Automorphisms, Publ. RIMS6(1970/71),385-442, Theorem6 p430 (pure Fock states: 000115 | projector difference HS) and Theorem7 p432 (unitary implementability: 000116 | off-diagonal block HS). The source was checked directly: 000117 | https://ems.press/content/serial-article-files/41673 000118 | https://ems.press/journals/prims/articles/2511 000119 | These standard equivalence/implementation criteria are used; the Fourier 000120 | identities and bounds above are the direct application to the given chart. 000121 | 000122 | Scope: fixed finite internal K, one spatial circle, constant selfadjoint odd 000123 | Phi, the already specified sea/CAR quantization, and the common periodic H1 000124 | one-particle domain. No result selects Phi from P1, proves physical OS time, 000125 | excludes interactions or extrapolates this HS convergence to higher spatial 000126 | dimension. The algebraic source-response test in the original review remains 000127 | valid and must be checked on the original source side. It requires no 000128 | unproved inner insertion operator or new quantum state.