TFPT scientific source excerpt Original repository path: experiments/theory-contracts/older-paper-source-recovery-20261025/POSITIVE-HALF-REVIEW.txt Original file SHA-256: 7c0b671b3606bb1ce376e163e4573a1c9e2dfeb0a87d86189ae8bdd4332c8a0b Inclusive original lines: 1-178 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: POSITIVE-HALF HEAT TRANSFER 000002 | ================================================ 000003 | 000004 | 25 October 2026. Exact analytic addendum to the recovered old circle source. 000005 | No Atlas, index, paper or other contract file was changed. 000006 | 000007 | Verdict 000008 | ------- 000009 | 000010 | PROVED as a direct consequence of the already fixed circle pair 000011 | 000012 | H=L2(S1), 000013 | B_alpha=-i partial_theta on D_alpha, 000014 | u=M_z, 0=0, and 000042 | 000043 | Pi=P_+-uP_+u*=|psi_0>0, 000046 | 000047 | T_t^+=P_+ exp(-t|B_alpha|)P_+, 000048 | R_t^+=exp(t alpha)T_t^+, 000049 | s=exp(-t). 000050 | 000051 | On P_+H their eigenvalues are 000052 | 000053 | T_t^+: exp[-t(n+alpha)], n=0,1,2,..., 000054 | R_t^+: 1,s,s^2,s^3,... . (1) 000055 | 000056 | On the complementary subspace (1-P_+)H both compressed operators vanish. 000057 | Therefore, as operators on the whole H, 000058 | 000059 | ||R_t^+-Pi||_1=sum_(n>=1)s^n=s/(1-s), (2) 000060 | 000061 | and the operator-norm error is s. The infinite zero block adds no trace norm 000062 | and no nonzero Fredholm eigenvalues. Thus (2) is exact on H, not only after 000063 | silently replacing H by P_+H. 000064 | 000065 | 2. Fredholm determinant and the null block 000066 | ------------------------------------------ 000067 | 000068 | For every t>0, T_t^+ and R_t^+ are positive trace-class operators. If z is a 000069 | Fredholm parameter, then 000070 | 000071 | det_H(I-zT_t^+) 000072 | =det_(P_+H)(I-zT_t^+|_(P_+H)). (3) 000073 | 000074 | Indeed the complementary zero eigenvalues of T_t^+ contribute factors one 000075 | to I-zT_t^+. They do not create extra zeros or divergent multiplicities in 000076 | the Fredholm determinant. 000077 | 000078 | Equation (2), together with continuity of the Fredholm determinant in trace 000079 | norm, gives 000080 | 000081 | det(I-zR_t^+) -> det(I-zPi)=1-z (4) 000082 | 000083 | for fixed z. Hence the same canonical projector obtained from the positive 000084 | Toeplitz defect is also the ground-mode limit of the positive heat 000085 | compression. 000086 | 000087 | 3. Quarter products and the origin of the two gradings 000088 | ------------------------------------------------------ 000089 | 000090 | Now set alpha=1/4 and 000091 | 000092 | q=exp(-t/4). 000093 | 000094 | Since exp[-t(n+1/4)]=q^(4n+1), 000095 | 000096 | D_+(q):=det(I-T_t^+) 000097 | =product_(m>=0)(1-q^(4m+1)). (5) 000098 | 000099 | The negative spectral half has eigenvalues 000100 | 000101 | |n+1/4|=m+3/4, n=-m-1, 000102 | 000103 | and therefore 000104 | 000105 | D_-(q)=product_(m>=0)(1-q^(4m+3)). (6) 000106 | 000107 | The full circle determinant factors exactly as 000108 | 000109 | D_full(q)=D_+(q)D_-(q) 000110 | =product_(k>=0)(1-q^(2k+1)). (7) 000111 | 000112 | Thus the congruence-one and congruence-three products are the positive and 000113 | negative spectral halves of the same quarter circle. The factor 1-q^3, which 000114 | caused the q^4 correction in the preceding full-circle review, is absent from 000115 | D_+. 000116 | 000117 | 4. First positive-half correction 000118 | --------------------------------- 000119 | 000120 | Let b=5/4. Compare 000121 | 000122 | Phi_+(q)=q D_+(q)^(-b) 000123 | 000124 | with its rank-one truncation 000125 | 000126 | Phi_1(q)=q(1-q)^(-b). 000127 | 000128 | The first omitted positive-half factor is 1-q^5. Hence 000129 | 000130 | D_+(q)^(-b) 000131 | =(1-q)^(-b)(1+bq^5+O(q^9)), 000132 | 000133 | and consequently 000134 | 000135 | Phi_+(q)-Phi_1(q)=bq^6+O(q^7) 000136 | =(5/4)q^6+O(q^7). (8) 000137 | 000138 | An exact formal expansion confirms equality of all coefficients through q^5 000139 | and gives at q^6 000140 | 000141 | [q^6]Phi_+=24163/8192, 000142 | [q^6]Phi_1=13923/8192, 000143 | [q^6](Phi_+-Phi_1)=5/4. (9) 000144 | 000145 | This does not modify the previously proved full-circle formula 000146 | 000147 | Phi_full(q)-Phi_1(q)=(5/4)q^4+O(q^5), (10) 000148 | 000149 | because (10) retains D_- and its leading factor 1-q^3. 000150 | 000151 | 5. Exact scope 000152 | -------------- 000153 | 000154 | The positive result is intrinsic to the existing circle operators: 000155 | 000156 | * P_+ is the canonical positive spectral projector of B_alpha; 000157 | * Pi is both its Toeplitz defect and the trace-norm ground limit of R_t^+; 000158 | * the zero block on (1-P_+)H is harmless for trace norms and Fredholm 000159 | determinants; 000160 | * the modulo-four products and the q^6 correction follow exactly from the 000161 | quarter spectrum. 000162 | 000163 | The first missing physical identification is still 000164 | 000165 | P_adm on the historical EM Hilbert space = P_+ on this circle sector, 000166 | 000167 | implemented by a state, trace and operator preserving intertwiner. Without 000168 | that map, D_+(q) is the canonical positive-half circle determinant, not the 000169 | original admissible EM determinant. Identifying its q with 000170 | delta_top exp(-2 alpha_EM), importing b=5/4, or treating exp(-t|B|) as charged 000171 | CAR or physical OS time are further independent steps. 000172 | 000173 | Exact check 000174 | ----------- 000175 | 000176 | The spectrum, factorization and rational series coefficients through q^9 000177 | were recomputed independently. The positive-half q^6 coefficient difference 000178 | is exactly 5/4. Status: PASS.