TFPT scientific source excerpt Original repository path: experiments/theory-contracts/older-paper-source-recovery-20261025/HEAT-TRANSFER-IR-CLOSURE.txt Original file SHA-256: eef9329b3804c06de8d36e9313084ce9c34595f3c78476552615d006c4422a48 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 | OLDER CIRCLE SOURCE AND COMPLETE HEAT TRANSFER 000002 | ============================================== 000003 | Date: 2026-10-25 000004 | Verdict: PROVED for the originally supplied quarter-circle representative. 000005 | Application to the original electromagnetic source: CONDITIONAL on the 000006 | same admissible operator/trace and parameter identification stated below. 000007 | 000008 | Decision-relevant question 000009 | -------------------------- 000010 | The older papers contain a concrete scalar circle Dirac operator and later 000011 | introduce a full positive heat transfer. Can the later rank-one factor be 000012 | recovered from these existing objects without choosing a free vector? 000013 | 000014 | Answer: YES as a canonical ground-mode limit with an explicit error bound. 000015 | NO as an exact identity for the full heat operator at finite heat time. 000016 | The full quarter-circle determinant is an odd Euler product. It gives a 000017 | unique all-order continuation within this operator class and first differs 000018 | from the rank-one opening at fourth order in its lowest heat weight. 000019 | 000020 | 1. Original inputs and necessary orientation repair 000021 | ---------------------------------------------------- 000022 | The fullv21 geometric A3 passage supplies a one-dimensional tangential 000023 | operator, circle multiplication and twisted boundary domains: 000024 | 000025 | documents/archive--paper-latex--tfpt-theory-fullv21/sections/ 000026 | 001-section-introduction-the-problem-of-parameters.tex:230-251; 000027 | paper-source:1fe2ddfabc74ac0e94a4686d. 000028 | 000029 | Its printed D_Gamma=i partial_theta has the opposite sign to its printed 000030 | eigenvalues n+varphi/(2 pi). Fix the orientation explicitly by B=-D_Gamma. 000031 | Use a for the boundary twist, not for the electromagnetic coupling: 000032 | 000033 | H=L2([0,2 pi],dtheta/(2 pi)), 000034 | Dom(B_a)={f in H1:f(2 pi)=exp(2 pi i a)f(0)}, 000035 | B_a=-i partial_theta, 000036 | psi_(n,a)(theta)=exp(i(n+a)theta), n in Z. 000037 | 000038 | Then B_a psi_(n,a)=(n+a)psi_(n,a). The domain is self-adjoint and the 000039 | displayed basis is complete. Multiplication u=exp(i theta) preserves the 000040 | domain and gives the strong same-space covariance and Toeplitz index in 000041 | ORIGINAL-CIRCLE-REVIEW.txt. This is the old concrete gluing representative, 000042 | not a representative newly chosen for this investigation. 000043 | 000044 | The old heat-transfer formula appears in 000045 | 000046 | documents/archive--paper-latex--tfpt-31/sections/ 000047 | 014-section-downstream-cosmology-module-on-the-observation-c.tex:4-24, 000048 | 000049 | and in the later tfpt-42 FRW/transfer section. Those sources write 000050 | 000051 | U_Sigma=P_adm exp(-ell_Sigma |B_Sigma|)P_adm. 000052 | 000053 | The present calculation uses the ORIGINAL CIRCLE representation of B, 000054 | with no freely fitted potential or spectral weights. The papers do not 000055 | identify this scalar H with Ran(P_adm), or B_a with B_Sigma on that range. 000056 | That is the first additional physical/operator identification if this 000057 | calculation is used for the electromagnetic transfer. 000058 | 000059 | 2. The complete quarter-circle absolute spectrum 000060 | ------------------------------------------------ 000061 | For a=1/4 or a=3/4, 000062 | 000063 | sigma(|B_a|)={(2m+1)/4:m=0,1,2,...}, (1) 000064 | 000065 | and every eigenvalue is simple. The two branches have the same absolute 000066 | spectrum but different signed eigenvectors: 000067 | 000068 | a=1/4: lowest |B| line is span(psi_(0,a)); 000069 | a=3/4: lowest |B| line is span(psi_(-1,a)). 000070 | 000071 | Let P_gap=1_{ {1/4} }(|B_a|). It is a canonical rank-one projection of 000072 | the supplied operator. No unit vector is selected independently of B_a. 000073 | The two quarters cannot be distinguished by |B_a| alone. 000074 | 000075 | The positive Toeplitz defect Pi=P_+-uP_+u* always selects psi_(0,a) when 000076 | 00 set 000082 | 000083 | T_t=exp(-t|B_a|), 000084 | R_t=exp(t/4) T_t, 000085 | r=exp(-t/2). 000086 | 000087 | In the absolute-energy basis, T_t has eigenvalues exp[-t(2m+1)/4], 000088 | and R_t has eigenvalues 000089 | 000090 | 1,r,r^2,r^3,... . (2) 000091 | 000092 | All these are determined by the old given spectrum. Consequently 000093 | 000094 | ||R_t-P_gap||_1=sum_(m>=1)r^m=r/(1-r). (3) 000095 | 000096 | Thus R_t tends to P_gap in TRACE NORM as t tends to infinity. On this 000097 | original representative the ground-mode limit and its simplicity are 000098 | proved, rather than inserted as a free rank-one vector. 000099 | 000100 | The normalization exp(t/4) matters: T_t itself tends to zero in trace norm. 000101 | It does not tend to P_gap. Subtracting the known ground energy is a 000102 | mathematical ground-energy normalization. Its use in a physical EM source 000103 | must be part of the original source prescription, not hidden in notation. 000104 | 000105 | 4. Fredholm determinant and controlled rank-one limit 000106 | ------------------------------------------------------ 000107 | Let Q be a separate determinant argument with |Q|<1. By (2), 000108 | 000109 | det(I-Q R_t)=product_(m>=0)(1-Q r^m). (4) 000110 | 000111 | The product converges since R_t is trace class. Equation (3), or the 000112 | direct convergent product, gives 000113 | 000114 | det(I-Q R_t) -> 1-Q. (5) 000115 | 000116 | For 0<=Q<1 the log error is nonnegative after reversing its sign and obeys 000117 | 000118 | |log det(I-Q R_t)-log(1-Q)| 000119 | =sum_(m>=1)-log(1-Q r^m) 000120 | <=sum_(m>=1) Q r^m/(1-Q r) 000121 | =Q r/[(1-r)(1-Q r)]. (6) 000122 | 000123 | Here -log(1-x)<=x/(1-x) and Qr^m<=Qr justify every inequality. This is 000124 | an infinite-dimensional bound, not an extrapolation from a finite cutoff. 000125 | 000126 | For a supplied susceptibility exponent b=5/4, raising (4) to the power 000127 | -b gives the full transfer factor, whose limit is (1-Q)^(-5/4). Neither 000128 | b nor the determinant prescription is derived from the heat spectrum; 000129 | they are retained original electromagnetic inputs. 000130 | 000131 | 5. The unnormalized full heat determinant 000132 | ------------------------------------------ 000133 | At finite t put q=exp(-t/4). Then the actual unnormalized heat transfer is 000134 | 000135 | T_t=diag(q,q^3,q^5,q^7,...) 000136 | 000137 | in the energy basis, and its complete Fredholm determinant is 000138 | 000139 | D(q)=det(I-T_t)=product_(m>=0)(1-q^(2m+1)), 0=1)(1-q^n). Those are 000147 | different domains, not two expressions for the same transfer. 000148 | 000149 | If the electromagnetic opening uses the COMPLETE transfer (7), its 000150 | mathematical continuation in this operator class would be 000151 | 000152 | Phi_full(q)-phi_base=q D(q)^(-5/4). (8) 000153 | 000154 | The later rank-one prescription gives 000155 | 000156 | Phi_1(q)-phi_base=q(1-q)^(-5/4). (9) 000157 | 000158 | Since D(q)=(1-q)(1-q^3)(1-q^5)..., 000159 | 000160 | Phi_full(q)-phi_base 000161 | =q+(5/4)q^2+(45/32)q^3+(355/128)q^4+O(q^5), 000162 | 000163 | Phi_1(q)-phi_base 000164 | =q+(5/4)q^2+(45/32)q^3+(195/128)q^4+O(q^5). (10) 000165 | 000166 | Thus the first difference is 000167 | 000168 | Phi_full(q)-Phi_1(q)=(5/4)q^4+O(q^5). (11) 000169 | 000170 | The first three opening coefficients agree. Agreement of those coefficients 000171 | cannot distinguish the two original spectral prescriptions. The difference 000172 | in (11) is conditional on using the whole quarter-circle transfer and 000173 | identifying its q with the electromagnetic argument; it is not a new 000174 | physical prediction or a replacement for the original closure by default. 000175 | 000176 | 6. All orders without fitting new coefficients 000177 | ------------------------------------------------ 000178 | Write F(q)=D(q)^(-b)=sum_(n>=0)t_n q^n, t_0=1. Taking a logarithmic 000179 | derivative of the absolutely convergent product gives 000180 | 000181 | log F(q)=b sum_(n>=1) sigma_odd(n) q^n/n, 000182 | sigma_odd(n)=sum_{d divides n, d odd} d. 000183 | 000184 | Consequently every coefficient is fixed by 000185 | 000186 | n t_n=b sum_(k=1)^n sigma_odd(k)t_(n-k). (12) 000187 | 000188 | This is the full series within the displayed operator/determinant class. 000189 | The companion checker independently expands the individual odd factors 000190 | and compares them with (12), exactly through the declared degree. The 000191 | infinite product and recurrence themselves follow analytically here. 000192 | 000193 | 7. What remains an original-source obligation 000194 | ----------------------------------------------- 000195 | The positive results remove a free psi only on the old given circle 000196 | representative, and show how rank one arises in a controlled normalized 000197 | limit. They do not silently replace any of the following identifications: 000198 | 000199 | (i) H_circle into the actual admissible EM spectral sector, preserving 000200 | |B|, its trace, its simple gap line and the original restrictions;