TFPT scientific source excerpt Original repository path: experiments/theory-contracts/native-arc-charge-source-interface-20261026/PROJECTOR-STATE-RIGIDITY.txt Original file SHA-256: 2e14e8875c263d71bd6bda61d005461cb718bdf30fdf34a6312db53167cd80c6 Inclusive original lines: 1-173 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.PROJECTOR-STATE.01 000002 | 26 October 2026. Full source coherence before primitive four-field fitting. 000003 | 000004 | 1. Corrected decisive question and original inputs 000005 | The last continuation correctly retained an explicit free conditional source 000006 | for its geometric averaging theorem. Is free quantization an independent 000007 | requirement AFTER a positive CAR source has the complete geometric projector 000008 | as its actual two-point covariance? No: the CAR state is then unique. 000009 | This sharpens the preceding four-field target. A mixed one-face covariance 000010 | does require higher/joint information. A FULL projector covariance does not. 000011 | Test success: positivity plus the actual full projector must determine all 000012 | ordered CAR words, without assuming Gaussianity or gauge invariance. 000013 | Abort the conclusion if only a marginal, a geometric projection without a 000014 | physical covariance identification, or an unrelated Euclidean inverse is given. 000015 | 000016 | Original source facts read and preserved: 000017 | - The same annulus Cauchy graph and its labelled faces, BOUNDARY-SOURCE.txt 000018 | in elliptic-moebius-source-bridge-20261026, lines201-239; W=1/2 and 000019 | B_Sigma=-i partial_s with NS lambda_n=2pi(n+1/2). 000020 | - The previous free/current and conformal continuations in this folder. 000021 | - Original v45 QFT source007, lines3-27 and115-146, defines the formal 000022 | admissible integral and higher vertices S_adm^(n). That schematic expansion 000023 | does not specify their actual CAR field insertions or compute a four-point. 000024 | - Original v45 QFT source003, lines2-41, gives a carrier-invariant Yukawa 000025 | trilinear type. Its coefficient and physical boundary insertion are not 000026 | obtained by identifying internal tensor indices with source coordinates. 000027 | - seam-collar-multifield-20261016/PROOF.txt, lines271-326: D_rel is not 000028 | automatically the physical kinetic K, its inverse is not the occupation 000029 | covariance, and P1 does not state the required Berezin measure/coupling. 000030 | These are source/type distinctions, not an assertion that every possible 000031 | fermion term is absent from the entire archive. 000032 | 000033 | 2. Exact CAR uniqueness theorem; no Gaussian premise 000034 | Let H be a complex Hilbert space with its charged CAR C* algebra and let 000035 | omega be ANY normalized positive state. Define its occupation covariance C 000036 | by omega(a*(f)a(g)) with the consistent inner-product convention. Positivity 000037 | and CAR imply 0<=C<=I. Suppose C=P=P*=P^2 on the WHOLE specified H. 000038 | In the GNS representation, for f in kerP, 000039 | ||a(f)Omega||^2=omega(a*(f)a(f))=0; 000040 | for f in ranP, 000041 | ||a*(f)Omega||^2=omega(a(f)a*(f))=||f||^2-=0. (1) 000042 | Thus the empty-particle annihilators and filled-hole annihilators annihilate 000043 | the cyclic vector. Split every smeared field using H=ranP direct_sum kerP. 000044 | The corresponding particle/hole operators obey CAR. Anticommuting their 000045 | annihilators to the right, and creators to the left, leaves only the scalar 000046 | CAR contractions in every finite word. All remaining terms have expectation0 000047 | by(1) and its adjoint. This determines every polynomial expectation uniquely. 000048 | Polynomials are norm dense in the CAR algebra, so the whole state is unique. 000049 | It is precisely the pure gauge-invariant quasifree/Fock state omega_P. 000050 | Odd and anomalous pair expectations vanish as CONSEQUENCES, not assumptions. 000051 | Every grouped neutral word has its Wick determinant with covariance P. 000052 | Araki1970, Lemma4.3, proves this arbitrary-state Fock uniqueness in self-dual 000053 | CAR form. The argument above is the charged CAR version, not a new general 000054 | CAR theorem. On infinite H it is uniqueness as a CAR state and its GNS 000055 | representation; no implementability in a separately fixed old Fock space 000056 | or global density matrix is asserted. 000057 | 000058 | 3. Apply to the ORIGINAL complete chiral annulus graph 000059 | For the actual NS eigenmode and r_n=exp(-W lambda_n)>0, its full projector is 000060 | P_n=1/(1+r_n^2) [[1,r_n],[r_n,r_n^2]]. (2) 000061 | On the complete two-face module its bounded functional calculus form is 000062 | P_A=[[I-C_A,X_A],[X_A,C_A]], 000063 | C_A=(I+exp(2W B_Sigma))^-1, 000064 | X_A=(2cosh(W B_Sigma))^-1=sqrt(C_A(I-C_A)). (3) 000065 | All functions commute and X_A^2=C_A(I-C_A), hence P_A^2=P_A=P_A*. 000066 | These expressions are bounded even though exp(-W B_Sigma) is unbounded. 000067 | The graph is understood as its closed Cauchy space, not an everywhere 000068 | defined bounded transfer operator. W=1/2 is the existing width, no fit. 000069 | The labelled second-face C_A is faithful with0, (4) 000086 | together with the two diagonal blocks and the actual joint CAR embedding. 000087 | It is NOT enough to call a geometric Calderon projector a covariance, to 000088 | purify a marginal by choice, or to identify the old Hardy state with C_A. 000089 | The former would insert the physical-state step that A/B must derive. 000090 | The map in(4) refers to the already given chiral scalar two-face source. 000091 | It is not an identification with the complete two-component Dirac vector 000092 | source, its normal Clifford action, OS admissible projector or physical time. 000093 | Existing full-Dirac vector-current cancellation results remain unchanged. 000094 | 000095 | 4. Exact positive countercontrols: why the missing coherence matters 000096 | For one actual mode put p=c_n, x=sqrt(p(1-p)). In the occupation basis 000097 | |00>,|10>,|01>,|11>, the geometric state is 000098 | |Psi_p>=sqrt(1-p)|10>+sqrt(p)|01>. 000099 | Its full covariance is [[1-p,x],[x,p]]=P_n, with p=r_n²/(1+r_n²). 000100 | Compare two other positive states: 000101 | rho_deph=(1-p)|10><10|+p|01><01|, 000102 | rho_prod=rho_left tensor rho_right with occupations(1-p,p). 000103 | All THREE have exactly the same complete individual-face states. 000104 | The last two even have the SAME full covariance diag(1-p,p), but 000105 | _deph=0, _prod=p(1-p)>0. (5) 000106 | Thus two-point data insufficiently determining general mixed states is 000107 | retained, not contradicted. The projector hypothesis is decisive. 000108 | For the geometric pure state =detP_n=0 and the coherence is x. 000109 | Here |omega(a_L* a_R)|²=p(1-p) saturates the two-point positivity bound. 000110 | For lambda_0=pi,W=1/2, the original mode values are 000111 | p=1/(1+exp(pi)), x=1/(2cosh(pi/2)). (6) 000112 | The controls are algebraic source distinctions on the existing mode, not 000113 | new proposed physical mixtures or chosen geometric distributions. 000114 | 000115 | 5. Quantitative finite-block certificate, without a Gaussian premise 000116 | Choose an actual FINITE P_A-reducing one-particle subspace, for example 000117 | any finite set of complete original NS two-face mode pairs. Its compressed 000118 | P remains a projection; an arbitrary spatial/one-face compression need not. 000119 | In a basis diagonalizing this P define the positive excitation counter 000120 | Q_P=sum_empty a_j* a_j+sum_filled a_j a_j*. 000121 | It has a unique zero vector Omega_P; all other eigenvalues are at least1. 000122 | Therefore Q_P>=I-rho_P, where rho_P=|Omega_P>=0, 000125 | 1-<=delta, 000126 | ||rho-rho_P||_1<=2sqrt(delta). (7) 000127 | The last step is the usual pure-target trace-distance inequality, also 000128 | obtained by purifying rho and comparing it with a target purification. 000129 | Hence every bounded word W IN THAT BLOCK satisfies 000130 | |omega(W)-omega_P(W)|<=2||W||sqrt(delta). (8) 000131 | No bounds on unmeasured modes, unsmeared fields or unbounded composites 000132 | follow. Exact delta0 on every finite reducing block gives the infinite 000133 | CAR theorem by density; no finite numerical sample proves that condition. 000134 | On the one-mode pair, write the real measured coherence as t*x with the 000135 | two target marginal occupations retained. Then 000136 | delta=4p(1-p)(1-t). (9) 000137 | Thus the missing coherent source information has an explicit bounded 000138 | error witness rather than a separately fitted four-point interaction. 000139 | 000140 | 6. Positive geometric mixtures and the actual remaining task 000141 | An orthogonal projection is an extreme positive contraction. If a positive 000142 | normalized ensemble of actual source states has mean covariance P_A, 000143 | then their covariances equal P_A almost surely: for a countable dense 000144 | set in kerP_A the nonnegative expectations vanish separately; repeat for 000145 | I-C on ranP_A. Each conditional source is then omega_PA by Section2. 000146 | Separability of the boundary Hilbert space makes the countable argument 000147 | valid. No conditional Gaussian premise is needed for this exact statement. 000148 | Signed or complex geometric integrands cannot be substituted for this 000149 | positive-ensemble argument. Their reconstructed total state may still 000150 | obey Section2 if it is positive and has the actual full covariance P_A. 000151 | The previous conformal averaging theorem remains valid in its own broader 000152 | algebraic weight scope; it is not replaced by an unjustified positive measure. 000153 | 000154 | The next original-source calculation is A/B, particularly(4), on the actual 000155 | marked closed real source carrier. P1 positivity on its OS space, internal 000156 | Yukawa invariants, a primitive spectral profile, and an arbitrarily selected 000157 | purification do not supply that joint boundary-field equation. If it fails, 000158 | retain its actual covariance/vertex rather than forcing the free target. 000159 | No P1 completion, 4D interaction or physical event/time selection is claimed. 000160 | 000161 | 7. Validation and primary literature 000162 | The checker independently constructs two-mode Jordan-Wigner CAR matrices, 000163 | checks(2)-(6) symbolically, constructs Q_P and its exact spectrum, and 000164 | tests the non-Gaussian marginal controls and coherence error identity. 000165 | The analytic proof in Section2 covers the infinite source; the finite 000166 | checker is not substituted for it. Normal and optimized checks are active. 000167 | Primary source: H.Araki, On Quasifree States of CAR and Bogoliubov 000168 | Automorphisms, Publ.RIMS6(1970/71)385-442, Lemma4.3 and Definition4.4. 000169 | https://ems.press/content/serial-article-files/41673?nt=1 000170 | This is an application of known CAR rigidity to the existing source, not a 000171 | new general theorem of nature or evidence for a programmed universe. 000172 | Verdict: EXACT_SCOPED_FULL_PROJECTOR_COVARIANCE_FORCES_ALL_CAR_WORDS; 000173 | MARGINALS_DO_NOT_FIX_CROSS_FACE_COHERENCE; ORIGINAL_P1_JOINT_SOURCE_OPEN.