TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/FIXED-CAP-STATE.txt Original file SHA-256: 072ad02ddf3a1298079647e939f7d74c175421d3dd0c622ee097ce39e0c0e451 Inclusive original lines: 1-120 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 | THE FINITE FIXED CAP IS AN EXACT HEAT-PREPARED COUPLED SOURCE STATE 000002 | 29 October 2026. Independent all-mode analytic review of the existing source. 000003 | No new model, physical state selection, cutoff extrapolation, original edit, 000004 | checker run or Atlas mutation. 000005 | 000006 | Result 000007 | The previously computed det_F C is exactly the return amplitude of the old 000008 | sea vector under the positive COUPLED excitation Hamiltonian. Its normalized 000009 | half-time vector has precisely the symmetric middle-cap projector. At finite 000010 | positive collar width it is a genuine coupled pure state, generically not 000011 | stationary under the same coupled source flow. This does not conflict with 000012 | positivity of the excitation Hamiltonian or with the existing full graph. 000013 | 000014 | 1. Actual data and the one-pair identity 000015 | Keep the constant quarter source of FIXED-CAP-DETERMINANT.txt: 000016 | h0=p gamma, h=h0+V, V=-i gamma Phi, {h0,V}=0, 000017 | p=n+alpha, alpha=1/4 or3/4, E=(p^2+Phi^2)^(1/2), 000018 | P0=P_-(h0), Ph=P_-(h). 000019 | For each actual odd singular block choose its harmless internal phase so 000020 | h=p sigma_z+m sigma_y, p>0, m>=0, E=sqrt(p^2+m^2). 000021 | For p<0 the old occupied grading is exchanged and all formulas use |p|. 000022 | These are singular blocks of the same coefficient, not additional species. 000023 | 000024 | In the sea Fock representation of Ph, the old occupied vector has form 000025 | Omega0,p= sqrt(a) Omega_h,p + exp(i theta) sqrt(b) Pair_h,p, 000026 | a=(1+|p|/E)/2, b=1-a. 000027 | Pair_h,p is one particle AND one hole, neutral and even relative to the 000028 | coupled sea. Its excitation energy is 2E; the sea energy is set to0 in 000029 | H_exc. Thus 000030 | 000031 | =a+b exp(-2ell E)=C_p. 000032 | This counts one odd pair per occupied singular slot, not two separate 000033 | copies from the selfdual conjugate test space. 000034 | 000035 | 2. Infinite source, exact scalar and energy domains 000036 | The existing HS sea result is reused, not rederived as a new source choice: 000037 | finding:whole-constant-coupled-source-is-implementable-despite-isolated-insertion-domain, 000038 | primitive_insertions_review/UV-INSERTION.txt:55-78. 000039 | It places the whole old and coupled pure seas in the SAME Fock representation. 000040 | Indeed sum_(p,j)b_pj0. Internal zero singular 000042 | values contribute a=1,b=0. There are48 occupied slots per Fourier p for the 000043 | actual96 component coefficient; no additional conjugate-slot multiplicity. 000044 | 000045 | The positive coupled excitation Hamiltonian is 000046 | H_exc=sum_(p,j) E_pj(n_particle,pj+n_hole,pj), 000047 | with its usual selfadjoint Fock domain and coupled sea energy0. Its commuting 000048 | pair factors give the exact whole-space identity 000049 | =product_(p,j)C_pj=det_F C>0. 000050 | Equivalently this is the Laplace transform of the actual nonnegative sum of 000051 | independent pair excitation energies in the old vector. Since sum b_pj is 000052 | finite, only finitely many pairs are occupied almost surely. Monotone 000053 | limits of these commuting positive factors establish the whole expectation; 000054 | the trace-class cap determinant already proves the same positive product. 000055 | No uncontrolled cutoff extrapolation is used. 000056 | 000057 | Consequently the exact normalized middle vector is 000058 | Omega_mid=exp(-ell H_exc/2)Omega0 / sqrt(det_F C). 000059 | Its occupation probability for a coupled particle-hole pair is 000060 | q_pj=b_pj exp(-2ell E_pj)/C_pj, 000061 | so its mean coupled excitation energy is sum_(p,j)2E_pj q_pj0 the vector is in Dom(H_exc^k) for every finite k, including 000063 | the energy form domain. This follows directly from boundedness of 000064 | H_exc^k exp(-ell H_exc/2), or from the exponential suppression of q_pj. 000065 | At ell=0 the existing old vector is still well defined, although its mean 000066 | coupled energy diverges logarithmically for constant nonzero Phi. That old 000067 | domain obstruction and this positive-time smoothing are fully compatible. 000068 | 000069 | 3. Exactly the symmetric exterior middle projector 000070 | At one pair set t=ell/2. The exterior-transferred old occupied vector is 000071 | v=exp(-t h)e_old 000072 | =(i(m/E)sinh(tE), cosh(tE)+(p/E)sinh(tE)) 000073 | for the displayed p>0 block. Its squared norm is 000074 | M=cosh(ell E)+(p/E)sinh(ell E). 000075 | The positive excitation shift supplies precisely exp(-t E) on this 000076 | one-occupied-particle exterior sector. Therefore 000077 | exp(-2t E)||v||^2=exp(-ell E)M=C_p, 000078 | and its normalized ray is the same Omega_mid,p just constructed in Fock space. 000079 | 000080 | On the whole circle define this subspace modewise through the isometries 000081 | F_t(p)=exp(-t h(p))P0(p) 000082 | [P0(p)exp(-2t h(p))P0(p)|RanP0]^-1/2. 000083 | Then P_mid(p)=F_t(p)F_t(p)*. The direct sum is a bounded orthogonal projector. 000084 | This formulation supplies the correct normalized graph-domain meaning of 000085 | exp(-t h)RanP0; the individual exponential is not asserted bounded for 000086 | unbounded signed h. It is the covariance of the same whole pure heat-cap 000087 | state. Relative to Ph its projector difference has 000088 | ||P_mid(p)-Ph(p)||_HS^2=2 sum_j q_pj, 000089 | which is summable and exponentially suppressed in the UV at ell>0. 000090 | 000091 | 4. Exact stationarity test and time scope 000092 | For the displayed single pair, 000093 | P_mid=|v>=-E sinh(ell E)-p cosh(ell E), h^2=E^2I; 000097 | the variance numerator is E^2M^2-[E sinh(ell E)+p cosh(ell E)]^2=m^2. 000098 | Thus for m!=0 and finite ell the cap state is NOT invariant under the 000099 | already implemented coupled source automorphism exp(is h). Equivalently 000100 | its coupled Fock-energy variance is positive. As ell tends to infinity it 000101 | converges to the coupled sea, since the latter has nonzero old-sea overlap. 000102 | 000103 | A positive Hamiltonian does not require every heat-prepared vector to be 000104 | stationary. The old boundary vector, the finite middle-cap vector and the 000105 | coupled ground vector are different explicitly determined states of the 000106 | same given representation. They are not competing new free source inputs. 000107 | Likewise the pure middle projector is not the one-face thermal marginal 000108 | of the already established full finite-collar Cauchy graph. 000109 | 000110 | The existing graded field/time dictionary identifies its stated source 000111 | unitary flow; the heat in this note is its H_exc imaginary-time semigroup. 000112 | Identification of that mathematical semigroup parameter with ORIGINAL 000113 | physical radial OS time remains a source-intertwiner question. This note 000114 | does not assign the original operational Seed's omega or tau again. 000115 | 000116 | Finally H_exc has the declared coupled-sea zero energy. The finite return 000117 | amplitude det_F C and its normalized vector do not fix an absolute bulk 000118 | energy shift, determinant-line phase, original Z_adm or spectral-action 000119 | measure. The logarithmic bulk term and the original source comparison in 000120 | FIXED-CAP-DETERMINANT.txt remain separately typed.