TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/KONSTANTER-ORIGINALBLOCK-REVIEW.txt
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Inclusive original lines: 1-48
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000001 | CONSTANT ORIGINAL CIRCLE BLOCK: INDEPENDENT ANALYTICAL REVIEW
000002 | 3 November2026. relational_logic. No execution or new model.
000003 | 
000004 | PASS. For the actual constant odd coefficient Phi=Phi^*, gamma^2=I,
000005 | {gamma,Phi}=0 and internally scalar p=n+alpha, the full original block is
000006 | h_p=p gamma-i gamma Phi. It is Hermitian. The cross terms in its square
000007 | cancel and (-i gamma Phi)^2=Phi^2, so h_p^2=p^2 I+Phi^2=:R_p^2. The positive
000008 | R_p=sqrt(h_p^2)=|h_p| commutes with h_p. Splitting the exponential series
000009 | into even and odd powers therefore gives EXACTLY
000010 | 
000011 |  exp(-t h_p)=cosh(t R_p)-h_p f_t(R_p),
000012 |  f_t(r)=sinh(t r)/r for r!=0, f_t(0)=t.
000013 | 
000014 | The removable value at r=0 is fixed by the same entire power series, so
000015 | no inverse, gap assumption or discarded zero mode is involved. This is
000016 | a finite matrix function of the ENTIRE original Phi, including all its
000017 | Yukawa/Majorana entries. It does not require diagonal fitting or selecting
000018 | a16-dimensional compression. For nonconstant Phi(theta), or a connection
000019 | which is not the scalar p and does not commute with Phi, the displayed
000020 | Fourier-block formula is not asserted: the original differential cross
000021 | term -i[A,Phi] must then remain.
000022 | 
000023 | This is a statement for each actual finite Fourier coefficient block.
000024 | When h_p has negative eigenvalues its raw exp(-t h_p) grows in those
000025 | directions. Thus these formulas do NOT define a bounded infinite raw
000026 | Fourier semigroup on the original unfilled charged one-particle space.
000027 | The already given positive excitation construction instead uses
000028 | E_exc,p=|h_p|=R_p on particle/hole modes and exp(-tH_exc) on their Fock
000029 | space. The old charged field remains
000030 | a_old(f)=b(Q_+f)+d^*(overline(Q_-f)), with Q_+/- the spectral projections
000031 | of h, not the gamma grading projections used in E. This preserves signed
000032 | h time and C(h); it is not replaced by a bare excitation annihilator.
000033 | 
000034 | At finite compatible regulator the existing sea/reference bookkeeping is
000035 | unchanged: E_sea=Tr(1_(h<0)h), Q_heat(t)=exp(tE_sea)Gamma(exp(-th)). The
000036 | continuum means the intrinsically defined excitation heat, not a product
000037 | with an infinite absolute sea scalar. The original unnormalized scalar,
000038 | reference and eta/Pfaffian factors remain those already transported in G.
000039 | 
000040 | Original formula/domain source checked in L:
000041 | research-source:251a551b4d69f1b19f606881:10-52,
000042 | tfpt-unified-source-derivation-20261029/graded_boundary_functor/
000043 | FINITE-COUPLED.txt,
000044 | SHA25661d3d524801aff0147d393c518ad554e3ab550271165e2000d9efbf613b90234.
000045 | The original source gives h=A gamma-i gamma Phi and
000046 | h^2=A^2+Phi^2-i[A,Phi]. The present constant-p expression is its direct
000047 | functional-calculus specialization. No canonical source was changed;
000048 | the already registered SEED-FIELD.txt was not edited.