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: 201-272
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000201 | 
000202 |  (ii) q_heat=exp(-t/4) with the original EM argument
000203 |       q_EM=delta_top exp(-2 alpha_EM);
000204 | 
000205 |  (iii) the ground-normalized t->infinity limit with an actual operation
000206 |        or limiting prescription in the original electromagnetic source;
000207 | 
000208 |  (iv) the signed B_a circle rotation with the original charged CAR time,
000209 |       Euclidean reflection, full source functional and spacetime locality.
000210 | 
000211 | In particular the older cosmology branch sometimes assumes a lowest-block
000212 | multiplicity48. Its large admissible carrier is not the scalar multiplicity1
000213 | circle used in (1). Both are existing source statements in different
000214 | sectors; their common label does not identify them.
000215 | 
000216 | The heat parameter t, the twist a, the spectral-flow path parameter and
000217 | the electromagnetic coupling alpha_EM are four different variables. A
000218 | positive |B| heat semigroup does not fix the orientation of B, the sign
000219 | of the quarter holonomy, or a physical Fock Hamiltonian.
000220 | 
000221 | 8. The positive spectral half is a different canonical trace
000222 | ------------------------------------------------------------
000223 | The same supplied B_a has the canonical positive spectral projector
000224 | P_+=1_(0,infinity)(B_a). It reduces |B_a|. For every 0<a<1, not only
000225 | the quarters, its compressed heat transfer has nonzero eigenvalues
000226 | 
000227 |   T_t^+=P_+ exp(-t|B_a|)P_+ : exp[-t(n+a)], n>=0.
000228 | 
000229 | With s=exp(-t), R_t^+=exp(ta)T_t^+ has nonzero eigenvalues 1,s,s^2,...,
000230 | and its lowest projection is exactly the Toeplitz defect Pi from section2.
000231 | The zero block on Ran(I-P_+) contributes neither trace nor determinant:
000232 | 
000233 |   ||R_t^+-Pi||_1=s/(1-s).                            (13)
000234 | 
000235 | Thus the original positive spectral half gives a canonical simple line
000236 | at every interior twist. This also explains the three-quarter case:
000237 | its positive-half gap is psi_(0,a), whereas the full |B_a| gap is psi_(-1,a).
000238 | There is no contradiction; the two traces use different spectral spaces.
000239 | 
000240 | At a=1/4, keep q=exp(-t/4). The positive and negative halves give
000241 | 
000242 |   D_+(q)=product_(m>=0)(1-q^(4m+1)),
000243 |   D_-(q)=product_(m>=0)(1-q^(4m+3)),
000244 |   D(q)=D_+(q)D_-(q).                                  (14)
000245 | 
000246 | The first full-circle correction (11) comes specifically from the negative
000247 | half's lowest q^3 factor. If the actual EM admissible sector were P_+,
000248 | its opening would instead be q D_+(q)^(-5/4), and
000249 | 
000250 |   q D_+(q)^(-5/4)-q(1-q)^(-5/4)
000251 |       =(5/4)q^6+O(q^7).                               (15)
000252 | 
000253 | All coefficients for either half follow the same log-derivative argument:
000254 | replace sigma_odd(k) in (12) with the sum of divisors d of k satisfying
000255 | d=1 mod4 or d=3 mod4 respectively. The companion checker controls both
000256 | these recurrences and the factorization (14) independently through degree20.
000257 | 
000258 | These are consequences of spectral projections of the old given B_a,
000259 | not newly fitted sectors. They do NOT establish P_adm=P_+. The q^4 and
000260 | q^6 corrections are conditional diagnostics that distinguish the full
000261 | circle trace from its positive spectral half. Neither is an unqualified
000262 | electromagnetic prediction. The physical trace must be recovered first.
000263 | 
000264 | Outcome
000265 | ---------
000266 | The older papers contained enough concrete data for a genuine conditional
000267 | recovery of the single-mode limit. The full finite heat determinant and
000268 | the rank-one determinant are demonstrably different; their first three
000269 | opening coefficients nevertheless agree. The canonical positive half
000270 | agrees through five opening orders. This sharpens the original
000271 | question to a comparison of actual admissible operators and limiting
000272 | prescriptions. No extra source model or numerical parameter fit is used.