TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/graded_boundary_functor/old_source_embedding/REVIEW-CLAUDE-OLD-SOURCE.txt Original file SHA-256: df2871ace6df72300e56f7377368f21625f74441c8eab4f214ab0a3fea92f4dd Inclusive original lines: 1-200 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 REVIEW: THE ORIGINAL QUARTER SOURCE, ITS FOLD AND ITS EXACT SCALES 000002 | 29 October 2026. Analytic source review; no numerical inference, Atlas change, 000003 | new physical source or change of pinned files. 000004 | 000005 | VERDICT 000006 | 000007 | [OMITTED: non-scientific conversation or execution metadata] 000008 | [OMITTED: non-scientific conversation or execution metadata] 000009 | [OMITTED: non-scientific conversation or execution metadata] 000010 | [OMITTED: non-scientific conversation or execution metadata] 000011 | [OMITTED: non-scientific conversation or execution metadata] 000012 | [OMITTED: non-scientific conversation or execution metadata] 000013 | [OMITTED: non-scientific conversation or execution metadata] 000014 | [OMITTED: non-scientific conversation or execution metadata] 000015 | 000016 | The identification of the ENTIRE positive/negative-grade source with the 000017 | old same-orientation NS cover is stronger and does not follow from the 000018 | second spectral multiset. It needs the explicit support and winding map 000019 | in section 5. In particular the old cover winding is not the new scalar 000020 | multiplication by exp(i x) on the finite coefficient space. 000021 | 000022 | [OMITTED: non-scientific conversation or execution metadata] 000023 | [OMITTED: non-scientific conversation or execution metadata] 000024 | [OMITTED: non-scientific conversation or execution metadata] 000025 | [OMITTED: non-scientific conversation or execution metadata] 000026 | [OMITTED: non-scientific conversation or execution metadata] 000027 | [OMITTED: non-scientific conversation or execution metadata] 000028 | [OMITTED: non-scientific conversation or execution metadata] 000029 | [OMITTED: non-scientific conversation or execution metadata] 000030 | [OMITTED: non-scientific conversation or execution metadata] 000031 | [OMITTED: non-scientific conversation or execution metadata] 000032 | [OMITTED: non-scientific conversation or execution metadata] 000033 | [OMITTED: non-scientific conversation or execution metadata] 000034 | 000035 | 1. EXACT LOCAL SOURCE EMBEDDING, NOT A DIMENSION OR TEMPERATURE MATCH 000036 | 000037 | Let K_alpha=L2(S1,dx/(2pi)), A_alpha=-i partial_x+alpha on periodic H1, 000038 | with alpha=1/4 or 3/4. Use the ACTUAL supplied finite space F, its grading g, 000039 | the scalar connection A_alpha tensor I_F and the given selfadjoint odd Phi. 000040 | Take a constant unit vector e with g e=e and, initially, Phi=0. Define 000041 | V_e f=f tensor e. 000042 | Then V_e is an isometry onto a reducing one-particle subspace and, on the 000043 | full H1 domains, 000044 | h_0 V_e=V_e A_alpha, h_0=A_alpha g. 000045 | Consequently V_e intertwines every bounded Borel function and 000046 | exp(i t h_0)V_e=V_e exp(i t A_alpha) 000047 | with exactly the same QUARTER time parameter t. The support is preserved: 000048 | supp(V_e f)=supp(f). This is an actual local CAR embedding for all test 000049 | functions, not only a map of Fourier eigenvalues. 000050 | 000051 | There is a whole normal/two-face map as well. Put 000052 | V_e^normal(f_+,f_-)=(f_+ tensor e,f_- tensor e). 000053 | It intertwines B_0 with Z A_alpha, the new complex structures Ical, the 000054 | geometric conjugations C0=i0 N, and F=S tensor g with scalar S. Apply it 000055 | on both oriented faces, retaining C_faces=(C0,-C0). It then intertwines the 000056 | ENTIRE G_ell(B), including both diagonals and both crossing blocks, and 000057 | the product reflection face_swap tensor F. No original J_F equality is 000058 | asserted; C0 and that finite charge conjugation remain distinct operations. 000059 | 000060 | The selfdual CAR map c_scalar(u)->c_full(V_e^faces u) is therefore a unital 000061 | *-monomorphism preserving the actual graph state and all its words. The 000062 | graph-to-HS creation chart already proves the common reflection/time 000063 | diagram. In ordinary charged notation it restricts to 000064 | a_alpha(f)->a_h(V_e f), 000065 | with annihilators antilinear in f and creators linear. For example on face0, 000066 | c(E V_e f)=a_R^*(V_e f), 000067 | so the ordinary old annihilator is its adjoint slot, not an incorrectly 000068 | relabeled complex-linear annihilator. No kappa or inverse time is needed. 000069 | The corresponding cyclic GNS map is unitary ON THE CYCLIC IMAGE, not onto 000070 | the entire many-channel Hilbert space. 000071 | 000072 | The same whole construction holds whenever Phi e=0, even if Phi is nonzero 000073 | on other channels. For a varying smooth Phi(x), the sufficient condition 000074 | is Phi(x)e=0 for every x, with the scalar connection unchanged. Thus the 000075 | statement 'Phi!=0 always destroys the old source' would be too strong. 000076 | The constant-Phi iff result is reviewed in section 6 below. 000077 | 000078 | 2. GEOMETRIC TEMPERATURE: THE FULL OPERATOR AND LENGTH RESCALING 000079 | 000080 | The original source uses the explicit normalized maps 000081 | theta=2pi s, 000082 | W_0 f(theta)=exp(i theta/2)f(2theta), 000083 | W_1 f(theta)=exp(3i theta/2)f(2theta). 000084 | With normalized circle measures these W maps are isometries; no extra sqrt2 000085 | is inserted. Their ranges are the two reducing NS mode-parity subspaces. 000086 | The actual H1-domain identities are 000087 | B_cov W_alpha=2 W_alpha A_alpha, 000088 | B_Sigma W_alpha=4pi W_alpha A_alpha, 000089 | where the original s->theta unitary is understood in the second equation. 000090 | The old annulus has width W=1/2. It follows by FUNCTIONAL CALCULUS that 000091 | C_A W_alpha 000092 | =(1+exp(2W B_Sigma))^-1 W_alpha 000093 | =W_alpha(1+exp(4pi A_alpha))^-1. 000094 | The crossing factor simultaneously becomes 000095 | (2cosh(W B_Sigma))^-1 W_alpha 000096 | =W_alpha(2cosh(2pi A_alpha))^-1. 000097 | Thus the entire old two-face graph, not just its marginal occupation, has 000098 | the scalar Quarter graph at ell=2pi, beta=2ell=4pi. 000099 | 000100 | At the normal differential-operator level this is the coordinate change 000101 | r=4pi t_old, 0<=t_old<=1/2, 0<=r<=2pi, 000102 | partial_t_old+4pi A_alpha=4pi(partial_r+A_alpha). 000103 | If raw radial Lebesgue L2 norms are also transported, the unitary is 000104 | (U_rad f)(r)=(4pi)^(-1/2) f(r/(4pi)). 000105 | The normalized tangential boundary test maps are the W maps above. This 000106 | explains exactly why widths and generators rescale inversely; no fitted 000107 | temperature, one-particle amplitude or absolute physical energy is added. 000108 | 000109 | The real-time statements must keep these actual scales: 000110 | exp(i t B_cov)W_alpha=W_alpha exp(i (2t) A_alpha), 000111 | exp(i t B_Sigma)W_alpha=W_alpha exp(i (4pi t) A_alpha). 000112 | Hence the new SAME-t source diagram is the diagram for the old QUARTER 000113 | generator H_alpha. The old NS cover-angle parameter has a factor 2, and 000114 | the s-normalized parameter has a factor 4pi. Calling all three t while 000115 | retaining the same operators would be false already on one eigenmode. 000116 | These are actual positive rescalings, not inverse-time obstructions. 000117 | 000118 | 3. THE FULL CHARGE LAW FOLLOWS IN THIS GIVEN FREE CAR SOURCE 000119 | 000120 | On the original negative sea r<0 the positive excitation energy for A_alpha 000121 | is the actual sea-relative generator H_alpha. Filling q>0 lowest particles 000122 | costs sum_(r=0)^(q-1)(r+alpha); making -q holes gives the same formula 000123 | F_alpha(q)=q^2/2+(alpha-1/2)q. 000124 | The original Maya/partition correspondence gives oscillator degree N in 000125 | every charge sector, so on the finite-energy core and its closure 000126 | H_alpha=N+F_alpha(Q). 000127 | This is the all-level operator identity; a check only through E<=6 is not 000128 | its proof. Since the excitation partition product converges, the complete 000129 | normal free CAR state at this covariance is exactly 000130 | rho_alpha=exp(-4pi H_alpha)/Z_alpha. 000131 | The oscillator partition factor is independent of q. Therefore 000132 | Prob(Q=q)=exp(-4pi F_alpha(q))/sum_(m in Z)exp(-4pi F_alpha(m)). 000133 | This law and all joint field correlations are fixed in the FULL GIVEN CAR 000134 | graph source; they are not additional choices along V_e. 000135 | 000136 | DERIVATION.txt section 5 remains correct for its different, weaker premise: 000137 | KMS on the neutral current algebra alone does NOT determine the central 000138 | charge law. Supplying the full free CAR graph is the changed sufficient 000139 | premise that determines it here. The thermal charge distribution is not 000140 | the charge-zero vacuum and must not be replaced by one. 000141 | 000142 | 4. EXACT EIGHTFOLD: FIELDS, TEST NORMS, ENERGY AND DELTA 000143 | 000144 | Retain the actual original point-field formulas 000145 | eta_r(x)=psi((x+2pi r)/8)/sqrt8, 000146 | chi=U_alpha^* eta, 000147 | U_alpha(r,a)=exp(2pi i(a+alpha)r/8)/sqrt8, 000148 | mu_a=(a+alpha)/8, a=0,...,7. 000149 | Their underlying one-particle unitary sends the old mode r=8n+a to the 000150 | mode n of the a-th mu_a channel. In twisted-function coordinates this is 000151 | exactly the fold and Fourier matrix displayed above; a periodic gauge 000152 | coordinate includes the usual exp(-i mu_a x) factor on each channel. 000153 | Both factors sqrt8 matter: one is the continuous fold normalization and 000154 | the other normalizes the finite Fourier matrix. The resulting channel 000155 | CAR modes are the original a_(8n+a), with no new independent arc modes. 000156 | 000157 | On full Sobolev domains the unitary F_alpha obeys 000158 | F_alpha A_alpha F_alpha^*=8 direct_sum_a A_mu_a, 000159 | F_alpha P_-(A_alpha)F_alpha^*=direct_sum_a P_-(A_mu_a). 000160 | Thus the same sea Fock source satisfies 000161 | H_alpha=8 sum_a H_mu_a, 000162 | F_alpha exp(i t A_alpha)F_alpha^* 000163 | =direct_sum_a exp(i (8t) A_mu_a). 000164 | The folded Gibbs coefficient is beta_fold=8(4pi)=32pi, and its ACTUAL 000165 | occupation is (1+exp(32pi A_mu_a))^-1. A beta=4pi channel covariance with 000166 | unrescaled A_mu_a would be a different state, already in the first mode. 000167 | 000168 | The full-CAR charge vector p=(p_0,...,p_7) in Z^8 therefore has law 000169 | Prob(p) proportional to 000170 | exp[-32pi sum_a (p_a^2/2+(mu_a-1/2)p_a)]. 000171 | If a further fixed total charge or other lattice restriction is imposed, 000172 | its law is the corresponding CONDITIONAL law; unrestricted Z^8 must not 000173 | be silently retained after that additional restriction. 000174 | 000175 | The original reference difference is exactly 000176 | delta_a=mu_a-1/2, 000177 | as stated and derived by point splitting in CAR-OCTIC-DICTIONARY.txt. 000178 | It is transported without alteration by this embedding and fold. Delta 000179 | is the specified sea-to-conformal-reference constant, not the thermal mean 000180 | charge and not an output of KMS without that reference. The actual neutral 000181 | polynomial, its matrix A, old subtractions and derivative counterterms are 000182 | then transported by the SAME field map. In the arc chart the covariance is 000183 | still U_alpha diag(C_mu) U_alpha^*: the eight arcs remain correlated. 000184 | 000185 | Localization also retains its exact meaning: a proper fold interval has 000186 | eight actual preimage arcs. W_0/W_1 already pull a Quarter interval back 000187 | to two cover arcs. Neither operation proves single-point locality on the 000188 | unfolded original circle. Tensoring with constant e is local on the given 000189 | Quarter circle and commutes with this actual eightfold; those facts do not 000190 | remove either multiarc support map. 000191 | 000192 | 5. THE SECOND GRADE IS A COUNTERPROPAGATOR, NOT THE WHOLE OLD COVER TUPLE 000193 | 000194 | The free one-particle split is exactly 000195 | h_0=A_alpha tensor I_(F+) direct_sum (-A_alpha) tensor I_(F-). 000196 | On periodic scalar functions set 000197 | (R_sp f)(x)=exp(-i x) f(-x). 000198 | This unitary involution satisfies 000199 | (-A_alpha)R_sp=R_sp A_(1-alpha). 000200 | It proves the second spectral identification operatorially, but reflects