TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/native-arc-charge-source-interface-20261026/HALF-PERIOD-GNS-SEWING.txt
Original file SHA-256: 7236ecc67bbae4ff477fe5a761a1b1068b6b75418aa6651b8ee00d2f7d87e3b1
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 | UR.NATIVE.ARCCHARGE.HALF-PERIOD-SEWING.01
000002 | 26 October 2026. Actual half-period kernel, graded GNS, and saturation.
000003 | 
000004 | 1. Question and correction of the preceding target
000005 | Does the actual existing elliptic source already contain the numerical
000006 | cross kernel X_A, and what precisely identifies it with a second CAR family?
000007 | The answer separates a positive constructive connection from the still
000008 | unproved primitive P1 source map. Do not fit another kernel or purification.
000009 | Original sources read:
000010 |  ELLIPTIC-GREEN-RP.txt, elliptic-moebius-source-bridge-20261026, lines33-146;
000011 |  BOUNDARY-SOURCE.txt there, lines201-238; the original NS square annulus.
000012 |  ELLIPTIC-CURRENT-SOURCE.txt and PROJECTOR-STATE-RIGIDITY.txt here.
000013 |  seam-collar-multifield-20261016/PROOF.txt, lines178-197 and598-654:
000014 |  complete interacting sewing and the explicit old interacting control.
000015 | The previous missing cross equation was a PHYSICAL P1 identification target.
000016 | Its numerical value is already in the given free Green source; it is not
000017 | an unknown kernel in that existing source. The operator/source types must
000018 | still match. Success: construct that full double in the existing thermal
000019 | GNS, with all CAR signs and no new state, then test what is required beyond
000020 | abstract standard doubling. Abort any direct identification of two temporal
000021 | evaluations as independent canonical CAR families.
000022 | 
000023 | 2. The half-period is exactly the existing X_A
000024 | Write beta=2W=1, B=B_Sigma and C=(1+exp(2WB))^-1. The original inverse
000025 | of partial_t+B on its temporal NS domain is
000026 |  S(t)=exp(-tB)(1+exp(-2WB))^-1, 0<t<2W.
000027 | Therefore its ACTUAL midpoint value is
000028 |  S(W)=(2cosh(WB))^-1=X_A=sqrt(C(I-C)).                   (1)
000029 | The negative branch obeys -S(-W)=S(W). These are also the two reflected
000030 | charge Grams at t=u=W/2, with the original order/normal-Clifford minus.
000031 | The functions in(1) are bounded on the full NS Hilbert space. Equality
000032 | does not come from a regulator or from changing the old quarter time.
000033 | Thus the original source fixes both its magnitude and its relative sign
000034 | within these displayed charged Green conventions.
000035 | 
000036 | 3. The smallest operator-type obstruction
000037 | For one actual NS mode, imaginary-time evolution gives
000038 |  a(t)=exp(-t lambda)a, bar_a(t)=exp(t lambda)a*.
000039 | Then {a(W),a*(0)}=exp(-W lambda)I, not0. Moreover
000040 |  a(t)*=exp(-t lambda)a* is NOT bar_a(t).
000041 | So bare temporal evaluations of one physical mode cannot be inserted as
000042 | two independent equal-time CAR families with their ordinary Hilbert adjoint.
000043 | The reflected Euclidean functional and its OS adjoint are distinct slots.
000044 | This does not exclude a correct reflected reconstruction; it excludes this
000045 | specific untransported double-CAR identification before any large calculation.
000046 | 
000047 | 4. Correct graded double in the ALREADY GIVEN thermal GNS
000048 | For the existing sea Fock space K and its faithful even traceclass density
000049 | rho, the prior continuation already uses HS(K), Omega=sqrt(rho),
000050 |  L_A X=A X, R_A X=X A.
000051 | Right multiplication reverses composition: R_A R_B=R_(B A).
000052 | Let Pi be the actual sea-relative CAR parity on K and define
000053 |  Gamma X=Pi X Pi. Then Gamma Omega=Omega and Gamma anticommutes with
000054 | both left and right multiplication by any odd CAR field.
000055 | Gamma is the canonical GNS implementer of source parity fixing Omega;
000056 | its action on the cyclic left CAR orbit determines it uniquely.
000057 | The standard modular conjugation J_HS X=X* satisfies
000058 | J_HS L_A J_HS=R_(A*); it is not silently identified with geometric OS Theta.
000059 | In the original fixed NS mode chart define
000060 |  a_R,j=L_(a_j), a_L,j=-Gamma R_(a_j),
000061 |  a_L,j*=Gamma R_(a_j*).                                (2)
000062 | Both families obey CAR and their cross anticommutators vanish. For example,
000063 | Gamma L_a=-L_a Gamma and [L_a,R_b]=0 give the cross identity immediately.
000064 | No extra factor i belongs in(2) under this explicit adjoint convention.
000065 | With C_ij=Tr(rho a_i* a_j), exact index calculations give
000066 |  <a_R,i* a_R,j>=C_ij,
000067 |  <a_L,i* a_L,j>=delta_ij-C_ij,
000068 |  <a_R,i* a_L,j>=M_ij=Tr(sqrt(rho)a_i*sqrt(rho)a_j).       (3)
000069 | The opposite cross block is the same Hermitian matrix M. Keeping these
000070 | index conventions avoids an unexplained transpose of the partner labels.
000071 | Gamma is the parity of the WHOLE HS representation, not merely L_Pi.
000072 | The tempting replacement L_Pi R_a passes a one-mode test but creates
000073 | spurious spectator parity factors already with two modes. The checker
000074 | detects this explicitly. No infinite product of filled-sea parities is used.
000075 | 
000076 | For the EXISTING free sea-relative Gibbs density, independent mode traces
000077 | give M_ij=delta_ij sqrt(c_i(1-c_i))=X_A,ij. Spectators have trace1.
000078 | Combining(3) with(1) gives precisely the full geometric covariance
000079 |  [[I-C,X_A],[X_A,C]]=P_A.                                (4)
000080 | The earlier positive-projector theorem now proves every joint CAR word.
000081 | The map pi_PA(A)Omega_PA -> iota(A)sqrt(rho) is isometric and intertwines
000082 | the bounded CAR fields. It is onto HS(K): already the original full left
000083 | CAR algebra has cyclic closure HS(K) for this faithful rho. Thus this is
000084 | a unitary realization of the SAME given two-face source in its existing
000085 | thermal GNS, not a different physical state or an added particle species.
000086 | Infinite-mode equality uses the stated traceclass density, bounded smeared
000087 | CAR fields and cyclic closure, not the inverse of rho as a bounded operator.
000088 | Original currents/native composites are transported with their old domains,
000089 | subtractions and limits. No new intrinsic metric stress or time is selected.
000090 | 
000091 | 5. What standard doubling alone does NOT force
000092 | Formula(2) also works for any faithful EVEN finite density rho, whether or
000093 | not it is Gaussian. Its actual cross matrix is still M in(3), not generally
000094 | sqrt(C(I-C)). The HS vector is pure on the doubled full algebra, but purity
000095 | alone does not imply a projector occupation covariance or Wick's rule.
000096 | This is the exact distinction between a canonical state purification and
000097 | the very special geometric free projector target.
000098 | 
000099 | There is a minimal finite test of that distinction. For each mode put
000100 |  c_j=Tr(rho a_j* a_j), m_j=M_jj.
000101 | With u=a_j sqrt(rho), v=sqrt(rho)a_j, Hilbert-Schmidt Cauchy-Schwarz gives
000102 |  0<=m_j=||rho^(1/4)a_j rho^(1/4)||_HS²
000103 |        <=sqrt(c_j(1-c_j)).                              (5)
000104 | Here ||u||²=c_j, ||v||²=1-c_j. Since rho is faithful, both are nonzero.
000105 | Equality holds precisely when
000106 |  sqrt(rho)a_j=q_j a_j sqrt(rho),
000107 |  q_j=sqrt((1-c_j)/c_j).                                 (6)
000108 | Thus maximal cross coherence means that modular HALF transport maps the
000109 | annihilator to a scalar multiple of itself, without spectator operators.
000110 | On a FULL FINITE CAR algebra, saturation for all annihilators in a fixed
000111 | covariance eigenbasis is equivalent to rho being the product quasifree
000112 | density with those occupations. Proof: let sigma be that product density.
000113 | It has the same(6); X=sigma^-1/2 sqrt(rho) commutes with every a_j and a_j*.
000114 | Irreducibility of the full finite CAR makes X scalar; normalization gives
000115 | rho=sigma. The converse is the independent mode trace in Section4.
000116 | No parity or Gaussian assumption replaces the equality test itself.
000117 | This finite iff proof is NOT extended using an uncontrolled infinite
000118 | sigma^-1/2. The all-mode free result in Section4 is established separately.
000119 | There is also an all-mode route within the existing normal HS class: if
000120 | C is diagonal in the complete original NS basis and every mode saturates,
000121 | then v_j=q_j u_j gives M_ij=<u_i,v_j>=q_j C_ij. The entire doubled covariance
000122 | is therefore the projection(4). Its positive state is unique by the previous
000123 | projector theorem; restricting it to the original source proves quasifreeness.
000124 | This uses all-mode identities and bounded CAR fields, not an inverse-density
000125 | domain assertion or finite numerical samples. No claim for an unspecified
000126 | non-normal primitive source is inferred.
000127 | 
000128 | 6. Reuse the old interacting control; do not select a new model
000129 | The existing source study already supplies the explicit two-mode diagnostic
000130 |  H_int=diag(0,1/4,3/4,2), ell=8log2,
000131 |  T=exp(-ell H_int)=diag(1,1/4,1/64,1/65536).
000132 | It preserves the stated quarter loop exp(2pi iH)=diag(1,i,-i,1), but was
000133 | never a derived local P1 interaction. Here it is reused ONLY to test(5)-(6),
000134 | not rerun as a new physical proposal or new clock no-go.
000135 | For its rho=T/Z, Z=82945/65536, the ACTUAL values are
000136 |  c_p=16385/82945, c_h=1025/82945,
000137 |  m_p=32800/82945, m_h=8320/82945,
000138 |  c_j(1-c_j)-m_j²=14745600/82945²>0 for BOTH modes.       (7)
000139 | The half-density transport factor for a_p is2 on the empty-hole sector
000140 | and32 on the occupied-hole sector; for a_h it is8 or128. A spectator
000141 | occupation appears explicitly. Standard positive doubling still exists,
000142 | but does not yield P_A. Its two-mode Wick defect is
000143 |  <n_p n_h>-c_p c_h=-16711680/82945², also nonzero.
000144 | Thus gluing, positivity and the old clock alone do not prove the missing
000145 | linear modular law. The original interaction premise must be determined.
000146 | 
000147 | 7. Exact progress and remaining original-source step
000148 | The actual half-period kernel and its all-mode CAR/GNS double are now
000149 | connected constructively. There is no numerical X ambiguity in the given
000150 | elliptic free source. For a general actual source the question becomes
000151 | whether its half-modular field transport satisfies(6), or retains a
000152 | source-derived operator-valued spectator/vertex contribution.
000153 | The original operational seed takes its state, time and local algebra as
000154 | inputs. Its classical canonical doubling does not identify OS reflection
000155 | with Tomita conjugation on this odd field algebra by itself. An abstract
000156 | partner in(2) does not prove that the physical P1 second boundary is that
000157 | partner. The original marked real carrier, actual source/Sewing map, and
000158 | admissible field/time transport still have to supply that identification.
000159 | No physical Gaussian completion, positive ground-state time, source measure,
000160 | full Dirac-vector/chiral equivalence, or 4D interaction is inferred here.
000161 | 
000162 | Primary literature: Araki, On Quasifree States of CAR and Bogoliubov
000163 | Automorphisms, Publ.RIMS6(1970/71), Lemma4.3, Definition4.4 and Section6
000164 | (graded double and KMS structure),
000165 |  https://ems.press/content/serial-article-files/41673?nt=1 .
000166 | The HS construction and inequality above are derived explicitly, not treated
000167 | as a physical-source identification licensed by citing that paper.
000168 | Verdict: EXACT_SCOPED_HALF_PERIOD_TO_GRADED_GNS_INTERTWINER;
000169 | FINITE_MAXIMAL_COHERENCE_IFF_LINEAR_HALF_MODULAR_CAR_TRANSPORT;
000170 | ORIGINAL_P1_SEWING_AND_ODD_FIELD_LAW_OPEN. Experiment only.