TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-ENERGY-SOURCE-PROOF.txt
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000001 | ORIGINAL WHOLE GREEN SOURCE, ENERGY TRACE AND MIXED SCALE INSERTION
000002 | 31 October 2026, Sitzung015, R042/R043. Source-reviewed mathematical scope.
000003 | No new original source, normal geometry, physical time, Y/MR or scale chosen.
000004 | 
000005 | 1. Actual source and question
000006 | R040 proves normal profile recovery IF the complete first energy response is
000007 | available. Execute that response from the GIVEN whole normal Green/Poisson
000008 | source, rather than assuming P(0) already supplies it. Preserve the existing
000009 | homogeneous collar, normal N, entire finite Higgs/MR coefficient and labeled
000010 | faces. D0=N(partial_r+B), N*=-N, N^2=-I, {N,B}=0, B*=B. The given quarter
000011 | class has E=|B|>=delta>=1/4. Ell is the original collar width, not a new scale.
000012 | All paired spectral blocks of B are retained; no finite cutoff proves this.
000013 | Normal-Green-to-Calderon review:1-155; APS-DOMAIN-AND-STATE:145-194;
000014 | whole-scale reconstruction:1-130; geometric source001:15-126.
000015 | 
000016 | 2. Energy insertion is forced by the SAME operator
000017 |  (D0-z)u=0 iff (partial_r+B+zN)u=0.
000018 | Set A_z=B+zN, T_z=exp(-ell A_z). It follows that
000019 |  A_z^2=B^2-z^2 I, d_z L_z^-1=-L_z^-1 N L_z^-1.
000020 | There is no new energy vertex. It is N from the original principal symbol.
000021 | For the original Ksigma=fD0f, v=fu gives insertion z exp(sigma(r))N.
000022 | For nonconstant sigma use the ordered interval transfer, not an arbitrarily
000023 | extended sigma on an AP double. No profile is selected by this equation.
000024 | 
000025 | 3. Execute the already given AP normal double, length TWO ell
000026 | For the existing homogeneous double, away from the seam,
000027 |  S_z(t>0)=exp(-t A_z)(I+exp(-2ell A_z))^-1,
000028 |  S_z(t<0)=-exp(-(t+2ell)A_z)(I+exp(-2ell A_z))^-1.
000029 | The jump is I, and S_z(t+2ell)=-S_z(t). This is the inverse of L_z on
000030 | the original AP double wherever invertible, not a new spin/closure choice.
000031 | The Poisson field S_z(r)f0-S_z(r-ell)fell has the interior trace
000032 |  C_z=[[a,T a],[T a,T^2 a]], a=(I+T^2)^-1.                 (1)
000033 | The lower-right entry is -S_z(0-), not -S_z(0+).
000034 | C_z^2=C_z and its range is graph T_z. For unbounded B, interpret the graph
000035 | as the closed full spectral graph and the displayed bounded combined entries,
000036 | not separate exponentially growing factors on the whole Hilbert space.
000037 | 
000038 | 4. Orthogonal Cauchy space reader: real energy and imaginary energy differ
000039 | For REAL z, A_z*=B-zN is not A_z. The raw C_z generally is not selfadjoint.
000040 | The original positive boundary metric instead gives the unique orthogonal
000041 |  Q_z=[I;T_z](I+T_z* T_z)^-1[I,T_z*].                    (2)
000042 | At z=0, T0*=T0 and C0=Q0 exactly, even with reciprocal +/-E eigenvalues.
000043 | At the actual spectral reader points z=+/-i omega, A_z is selfadjoint, so
000044 | C_z=Q_z there too. Q_z is not holomorphic in a complex neighborhood; the
000045 | analytic raw projector C_z can be selfadjoint along the imaginary axis.
000046 | Do not differentiate real-energy Q_z as though it were analytic C_z.
000047 | This is consistent with Booss-Bavnbek/Lesch/Zhu, arXiv:0803.4160, section5.3,
000048 | Definition5.9, Proposition5.12 and Remark5.11: a Poisson trace idempotent
000049 | and the orthogonal projection onto its range need not be the same operator.
000050 | The following exact formulas are derived here, not quoted from that paper.
000051 | 
000052 | 5. Exact whole paired-mode formula and uniform derivative
000053 | On any existing +/-E pair let kappa=sqrt(E^2-z^2), real |z|<delta,
000054 |  c=cosh(ell kappa), s=sinh(ell kappa)/kappa,
000055 |  d=c^2+s^2 z^2=1+s^2 E^2, t=tanh(ell kappa)/kappa.
000056 | Since T=cI-sA, equation(1) becomes
000057 |  C11=(I+tA)/2, C22=(I-tA)/2, C12=C21=I/(2c).
000058 | Orthogonalizing this SAME graph, without a new state choice, gives
000059 |  Q11=I/2+cs B/(2d)+s^2 z NB/(2d),
000060 |  Q22=I/2-cs B/(2d)+s^2 z NB/(2d),
000061 |  Q12=(cI+szN)/(2d), Q21=(cI-szN)/(2d).                 (3)
000062 | One derivation computes T*T, uses det(T*T)=1 in each actual two-dimensional
000063 | pair, then inverts I+T*T. Degenerate E eigenspaces carry the same identity
000064 | on each paired multiplicity; no new representation is assumed.
000065 | 
000066 | At zero set a=tanh(ell E)/E, b=sech(ell E), and J=[[0,1],[-1,0]] on FACES.
000067 |  Cprime0=(a/2) diag(N,-N),
000068 |  Qprime0=[Cprime0,C0]
000069 |         =(a^2/2) I_faces tensor NB+(ab/2) J tensor N.    (4)
000070 | Cprime is antihermitian. Qprime is hermitian and off-diagonal with respect
000071 | to Q0, as required for an orthogonal projection derivative. The two terms
000072 | in(4) anticommute, (NB)^2=E^2I and (J tensor N)^2=I. Therefore
000073 |  (Qprime0)^2=(a^2/4)I,
000074 |  ||Qprime0||=tanh(ell E)/(2E)<=min(ell/2,1/(2delta))<=2.
000075 | The last bound applies to the whole original quarter circle, not a sampled
000076 | cutoff. Equation(3) is a bounded orthogonal direct sum. On compact real
000077 | energy intervals inside (-delta,delta), its derivatives have uniform bounds
000078 | from the displayed hyperbolic coefficients; hence(4) is its operator-norm
000079 | derivative. Tprime itself can grow exponentially and is not globally bounded.
000080 | 
000081 | 6. The actual energy response supplies R040's input
000082 | Differentiating the SAME T_z, either from(1),(2), or the full Green derivative,
000083 |  Tprime0=-int0ell exp(-(ell-r)B)Nexp(-rB)dr
000084 |          =-N int0ell exp((ell-2r)B)dr.
000085 | For normalized B v_n=E_n v_n,
000086 |  M_n=exp(-ell E_n)<v_n,N Tprime0 v_n>
000087 |     =(1-exp(-2ell E_n))/(2E_n).                       (5)
000088 | This is the actual complete flat response used in R040, now supplied by
000089 | the given Green/Poisson producer. Modewise T=Q21 Q11^-1=C21 C11^-1.
000090 | No uniformly bounded inverse Q11 on the full circle is claimed. R040's
000091 | Muntz theorem then returns exactly the existing flat profile1 in its entire
000092 | normal-dependent class. It does not select a physical flat metric from P0.
000093 | For a given nonconstant profile, its full weighted source supplies the
000094 | corresponding moments. Do not scan this already decided profile class again.
000095 | 
000096 | 7. The old FIXED APS inverse supplies the same full Cauchy range
000097 | Preserve P=P_-(h0) from the actual old source, Q=I-P; do not replace it by
000098 | P_-(B_H) under Higgs insertion. On a mode T=exp(-ell A_z), M=PTP|RanP,
000099 |  G_z(r,s)=1_(r>s) exp(-(r-s)A_z)
000100 |           -exp(-rA_z)P M^-1 P exp(-(ell-s)A_z).        (6)
000101 | At zero, M>=I in the actual odd Higgs class; nearby invertibility also follows
000102 | from the existing bounded fixed inverse and the bounded insertion zN.
000103 | The already proven ||L_fixed^-1||<=4 gives a guaranteed local radius1/4.
000104 | The formula holds on the SAME fixed domain; moving-domain corrections are
000105 | not silently dropped. The individual exponential summands are interpreted
000106 | modewise before taking their bounded inverse combination.
000107 | 
000108 | Prescribe incoming a in RanQ at0, b in RanP atell. The actual boundary columns
000109 | of(6) give exactly
000110 |  u(r)=G_z(r,0+)a-G_z(r,ell-)b
000111 |      =exp(-rA_z)[(Q-PM^-1PTQ)a+PM^-1 b].              (7)
000112 | The bracket parametrizes all homogeneous initial vectors. Thus both endpoint
000113 | columns yield graphT; orthogonalizing their range gives exactly(2). This
000114 | does not equate the APS vacuum with the whole collar state's correlations.
000115 | It compares solution ranges using the existing whole source information.
000116 | 
000117 | A decisive exclusion: ONE cross kernel G_z(ell,0) does not suffice. At the
000118 | adapted zero-energy source with P=P_-(B), G0(ell,0)=exp(-ell B)P_+.
000119 | Its first energy derivative is zero: every integrand in -G0 N G0 between
000120 | these endpoints contains P_+ N P_+=0. Yet Tprime0 in(5) is nonzero. Thus an
000121 | endpoint propagator alone loses the very response that the full two-column
000122 | Poisson source retains. This is an actual spectral pair, not a new model.
000123 | 
000124 | 8. Read the entire additional bosonic scale insertion from the same source
000125 | Original K=fDgeo f+Brel and h=delta sigma obey
000126 |  deltaK=-{h,K}/2+W,
000127 |  W=deltaBrel+{h,Brel}/2+f(deltaDgeo)f.                 (8)
000128 | For the ACTUAL SAME fixed-domain conditional resolvent G_z=(K-z)^-1,
000129 |  deltaG_z={h,G_z}/2+z G_z h G_z-G_z W G_z.            (9)
000130 | Proof: deltaG=-G deltaK G; use KG=GK=I+zG in the two anticommutator terms.
000131 | No commutation of h and K is assumed. Consequently the source readout
000132 |  D_z={h,G_z}/2+zG_z hG_z-deltaG_z=G_z W G_z            (10)
000133 | uniquely determines the extra operator as a quadratic form on the actual
000134 | dense domain by (K-z)D_z(K-z)=W. Products with unbounded K require that
000135 | domain/form interpretation, not a bounded inverse-G assertion.
000136 | Both original +/-iomega resolvents may be used with their original source,
000137 | reference, eta, normalization and domain transport. This reads the whole
000138 | operator defect, not only a potentially vanishing residue of its trace.
000139 | 
000140 | 9. Actual Higgs consumer and the normal source: precise remaining identity
000141 | Normal B_Ea=sigma_normal nabla^(a)+Phi_a on normal S2 is first order on a
000142 | different bundle. Its index/kernel is not Brel. R035 already proves this and
000143 | supplies exact kernel metric/vertex transport, not a free overlap exponent.
000144 | The GIVEN Wedge->full96 Higgs vertex is retained. The true inner CHANGE is
000145 | L(H)-DY; the existing seed retains DY and DM. Do not add a second Phi.
000146 | For its actual inner variation I_sigma=f gamma5 L(deltaH)f,
000147 |  delta_sigma I_sigma=-{h,I_sigma}/2,
000148 | so delta_H W=0 when derivatives commute and Dgeo's H convention is fixed.
000149 | A rigid outer I instead gives delta_H W={h,I}/2. These are distinct tested
000150 | source contracts, not permission to choose either as the original Brel.
000151 | 
000152 | To determine W from the normal producer one must evaluate the actual
000153 | derivative of its complete reduction/vertex map, with the existing G/T
000154 | normal metric, connections and eliminated-block source response. Equation
000155 | (10) is an alternative direct readout IF the original common source supplies
000156 | that scale family. A kernel dimension, index, or free scalar c(sigma) is not
000157 | this map. If Brel actually transforms as f Brel0 f, its W is zero; if it is
000158 | rigid and Dgeo fixed, W={h,Brel}/2. The readout distinguishes them.
000159 | In the special COMMUTING scalar case Brel=c(sigma)B0,
000160 |  W=(1+partial_sigma log c)hBrel. Pure congruence requires c=exp(-sigma)
000161 | up to the actual initial value, not exp(-sigma/2). No such c is selected here.
000162 | W=0 is sufficient, not necessary for a vanishing single spectral residue.
000163 | 
000164 | 10. Dynamic Higgs and absolute source: do not invert an averaged propagator
000165 | Equation(9) applies at fixed H to the actual conditional operator. With the
000166 | ORIGINAL complete joint weight rho_sigma(H), Z0=int rho, and ell_h=delta log rho,
000167 |  delta< O >=<delta O>+<O ell_h>-<O><ell_h>,
000168 |  delta log Z0=<ell_h>.                                (11)
000169 | Use the corresponding density derivative where a logarithm is unavailable.
000170 | Rho includes the actual bosonic action/measure, conditional fermionic scalar,
000171 | reference and phases. The covariance is algebraic and presumes no newly
000172 | postulated positive measure. The nonzero Z0 and differentiability conditions
000173 | are the original regulated source conditions; continuum passage needs them.
000174 | Thus (9) under the integral has an additional weight covariance. The inverse
000175 | of <G_H> does not identify a conditional K or W. The whole mixed H/source
000176 | words and the absolute Z0 from R022-R027 provide the proper information to
000177 | transport; no Gaussian replacement of the interacting source is made.
000178 | G_H denotes an operator resolvent. It equals a conditional fermion two-point
000179 | reader only in the actually bilinear charged/skew chart with its correct
000180 | Berezin/Pfaffian convention. Additional original fermionic vertices require
000181 | the full unnormalized source polynomial/differential insertions of R026.
000182 | Equation(9) alone is not their complete interacting two-point equation.
000183 | Its operator inverse does not turn that full source into a Gaussian one.
000184 | 
000185 | 11. Verification, corrections, whole-picture scope
000186 | 31 direct controls pass using the actual historical p=1/4,m=1/3 block and
000187 | the same AP normal Fourier mode. Full-circle conclusions follow analytically
000188 | from(3)-(5), not from the finite checks. No new Lean formalization or ledger
000189 | promotion. No claim Ladm=Ksigma^2, no physical-time/geometry origin supplied.
000190 | Original geometric Tr_rel is a defined expression; its concrete original
000191 | normal reduction/domain/trace realization is still the physical comparison.
000192 | FPzeta(0), logdet, source scalar, and eta remain different original readers.
000193 | R041's local Higgs divergence and the earlier factor-six test remain intact.
000194 | Yukawa/Flavor/Alpha, E8, clocks, state/time/RP and all memory are preserved.