TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/native-arc-charge-source-interface-20261026/PROJECTOR-STATE-REVIEW.txt
Original file SHA-256: 4b9abbcb3b3f0dcf9a99a4ec09640b28aaa938d1f78b47cc409cc89f81aaf934
Inclusive original lines: 1-72
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | Independent scoped mathematical review
000002 | 26 October 2026
000003 | 
000004 | Claim: UR.NATIVE.ARCCHARGE.PROJECTOR-STATE.01
000005 | Verdict: ACCEPT_EXACT_STATED_SCOPE. No material correction required.
000006 | 
000007 | Read PROJECTOR-STATE-RIGIDITY.txt and check_projector_state_rigidity.py fully.
000008 | Directly checked Araki, On Quasifree States of CAR and Bogoliubov
000009 | Automorphisms, Publ.RIMS6(1970/71), Lemma4.3 and Definition4.4:
000010 | https://ems.press/content/serial-article-files/41673?nt=1
000011 | Lemma4.3 assumes an arbitrary state with the required annihilator null norms,
000012 | not an already quasifree state. It therefore supports the claimed removal
000013 | of an independent Gaussian premise once the full projector covariance of a
000014 | positive CAR state is actually known.
000015 | 
000016 | 1. Exact uniqueness
000017 | GNS positivity gives a(f)Omega=0 on kerP and a*(f)Omega=0 on ranP.
000018 | CAR normal ordering then determines every finite polynomial expectation;
000019 | norm density extends uniqueness to the CAR C* algebra. Gauge invariance,
000020 | vanishing anomalous pairs and Wick factorization follow, rather than being
000021 | assumed. The text correctly distinguishes infinite CAR/GNS uniqueness from
000022 | implementability in a separately chosen representation or existence of a
000023 | global density matrix in that representation.
000024 | 
000025 | 2. Original annulus source and missing physical equation
000026 | The actual closed graph gives the bounded full projector with diagonal
000027 | blocks I-C_A,C_A and offdiagonal X_A=(2cosh(W B_Sigma))^-1.
000028 | The functional-calculus identity X_A^2=C_A(I-C_A) verifies idempotence.
000029 | The text does not treat the unbounded exponential transfer as an everywhere
000030 | defined bounded map. A single face covariance is explicitly not a projector.
000031 | The physical conclusion is correctly conditional on an actual joint charged
000032 | CAR embedding and the complete measured/derived covariance, especially
000033 | omega_P1(a_L*(f)a_R(g))=<f,X_A g>. This equation is not obtained by choosing
000034 | a purification of C_A. The original labelled chiral scalar source is kept
000035 | distinct from a full Dirac vector source, normal Clifford action, OS
000036 | admissibility and physical time.
000037 | 
000038 | 3. Quantitative finite block
000039 | On a finite genuinely P-reducing block, Q_P has a unique Slater zero vector
000040 | and a unit excitation gap. Hence Q_P>=I-rho_P, the displayed trace formula
000041 | for delta is correct for arbitrary positive states, and the pure-target
000042 | trace-distance bound gives ||rho-rho_P||_1<=2sqrt(delta).
000043 | Bounded-word error estimates apply only to that block. Spatial or one-face
000044 | compressions cannot silently replace a reducing block. The coherent,
000045 | dephased and product controls have the stated equal marginals and distinct
000046 | cross correlations/four-point values. The coherence defect formula is correct.
000047 | 
000048 | 4. Positive ensembles
000049 | The extreme-projection argument is valid for normalized positive ensembles
000050 | with the stated separability/measurability scope. A countable dense set in
000051 | the empty and filled subspaces suffices; positivity forces the remaining
000052 | offdiagonal entries as well. Signed or complex weights are correctly kept
000053 | outside this ensemble argument. No primitive P1 measure is thereby derived.
000054 | 
000055 | 5. Independent replay
000056 | Ran python3 -B check_projector_state_rigidity.py and python3 -B -OO
000057 | check_projector_state_rigidity.py, without output paths. Both exited0 and
000058 | returned the stored result, including all256 ordered four-field words,
000059 | excitation spectrum0^1,1^2,2^1 and the non-Gaussian marginal controls.
000060 | RuntimeError checks remain active. These finite checks are not substituted
000061 | for the all-degree GNS proof. Only this review file was written.
000062 | 
000063 | Reviewed SHA256:
000064 | PROJECTOR-STATE-RIGIDITY.txt
000065 | 2e14e8875c263d71bd6bda61d005461cb718bdf30fdf34a6312db53167cd80c6
000066 | check_projector_state_rigidity.py
000067 | 8439a30ad767f138f9269aa2bc6beca1f2d3d7088e92862bcd76fbf40d5a7c16
000068 | projector_state_rigidity_results.json
000069 | 40737f35ef6967cdfa2742f5f79d722f677a1adbb5a3770b9b0ee12bd190fe75
000070 | 
000071 | Experiment-only application of known CAR rigidity. No primitive source,
000072 | interaction, spacetime, empirical or ledger promotion is implied.