TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/native-arc-charge-source-interface-20261026/HALF-PERIOD-GNS-REVIEW.txt
Original file SHA-256: 3b8e6e7430e88f13ca12cc68ead3f9d677a418863688e95ed2d7bbfcb74c333b
Inclusive original lines: 1-73
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 | Independent scoped mathematical review
000002 | 26 October 2026
000003 | 
000004 | Claim: UR.NATIVE.ARCCHARGE.HALF-PERIOD-SEWING.01
000005 | Verdict: ACCEPT_EXACT_STATED_SCOPE. No material correction required.
000006 | 
000007 | Read HALF-PERIOD-GNS-SEWING.txt and check_half_period_gns_sewing.py fully.
000008 | The actual elliptic Green convention, annulus source and previous projector
000009 | rigidity were reviewed in the preceding independent checks. The reused
000010 | interaction was checked directly against seam-collar-multifield-20261016/
000011 | PROOF.txt Section12.4, including its explicitly diagnostic, non-P1 status.
000012 | 
000013 | 1. Actual midpoint kernel and field type
000014 | The existing S(t)=exp(-tB)(1+exp(-2WB))^-1 gives S(W)=X_A exactly by
000015 | bounded functional calculus. The reflected negative branch has the stated
000016 | sign. This identifies a kernel already present in the given free source.
000017 | It does not identify imaginary-time evaluations as independent CAR copies;
000018 | the displayed anticommutator and Hilbert-adjoint obstruction are correct.
000019 | 
000020 | 2. Graded HS fields, indices and GNS identification
000021 | For Gamma X=Pi X Pi, Omega=sqrt(rho), the partner b_j=-Gamma R_(a_j)
000022 | has adjoint b_j*=Gamma R_(a_j*). Both families satisfy CAR and mutually
000023 | anticommute. The trace identities in(3) are correct in the explicitly fixed
000024 | index convention C_ij=Tr(rho a_i* a_j); no unexplained transpose is needed.
000025 | The opposite cross matrix is the same Hermitian positive Gram matrix M.
000026 | For the actual free Gibbs density, mode traces give sqrt(c_j(1-c_j)) and
000027 | spectator traces1. The rejected L_Pi R_a has the exhibited spectator defects.
000028 | The resulting full projector covariance supplies all words by the previous
000029 | arbitrary-positive-state theorem. The GNS map is isometric on the cyclic
000030 | polynomial domain and onto HS(K): the original full left CAR algebra is
000031 | already cyclic there for the faithful normal density. This uses normal
000032 | cyclic closure, not a bounded inverse-density assertion.
000033 | 
000034 | 3. General source criterion
000035 | The HS Cauchy-Schwarz inequality, positivity of M_jj, and its equality
000036 | condition sqrt(rho)a_j=q_j a_j sqrt(rho) are correct. On a full finite CAR
000037 | algebra, equality for every mode of a covariance eigenbasis forces the
000038 | product quasifree density by scalar commutant and normalization.
000039 | The infinite alternative correctly uses all-mode saturation and the existing
000040 | normal HS realization to obtain a full projector covariance, then invokes
000041 | projector-state uniqueness. It does not extrapolate the finite proof using
000042 | an uncontrolled sigma^-1/2. Standard doubling or purity alone is not
000043 | mistaken for the needed saturation condition.
000044 | 
000045 | 4. Original interaction control and physical scope
000046 | The two-mode interacting weights, occupations, cross values, positive
000047 | saturation deficits, spectator-dependent half-density factors and four-point
000048 | defect agree with the stated original control. That control is not promoted
000049 | to a derived local interaction or source selection. The source/Sewing map
000050 | from actual P1 remains separate from the constructed abstract partner.
000051 | No identification of OS reflection with Tomita conjugation, physical time,
000052 | full Dirac vector source with the chiral scalar source, or primitive measure
000053 | is inferred. The constructive all-mode result is for the given free source.
000054 | 
000055 | 5. Independent replay
000056 | Ran python3 -B check_half_period_gns_sewing.py and python3 -B -OO
000057 | check_half_period_gns_sewing.py without output paths. Both exited0 with
000058 | the same stored result. This includes512 double-CAR matrix-unit checks,
000059 | 36 grouped four-field checks, the wrong parity control and the reused
000060 | interaction control. Rational simplifications in JSON are equivalent to
000061 | the common-denominator fractions in the proof. Checks remain active under
000062 | -OO. Only this review file was written.
000063 | 
000064 | Reviewed SHA256:
000065 | HALF-PERIOD-GNS-SEWING.txt
000066 | 663a31ee8a7067f6aabeb0be0ccff8b8af48f0e1da127eb8378258c134ceb033
000067 | check_half_period_gns_sewing.py
000068 | c26c50125ef471e5f6f48675fc96c8a4971e97aebb5893e5086f063de4904215
000069 | half_period_gns_sewing_results.json
000070 | 529fc875d0aa3352592a38eabca41b770f9bda9f26bc0b29c214ba6d396cb856
000071 | 
000072 | Experiment-only review. No physical P1 sewing, source measure, interaction,
000073 | empirical, spacetime or ledger promotion is implied.