TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/native-arc-charge-source-interface-20261026/ELLIPTIC-CURRENT-SOURCE.txt
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000001 | UR.NATIVE.ARCCHARGE.ELLIPTIC-CURRENT.01
000002 | 26 October2026. Existing same-E Gaussian kernel to full current source.
000003 | 
000004 | 1. Exact question and retained premise
000005 | Does the already derived free elliptic Green/Calderon source determine an
000006 | actual current functional and the neutral native composites, rather than
000007 | only an abstract possible current embedding? Reuse that given source.
000008 | Success requires the original quarter/current conventions, full charge
000009 | sectors and Weyl central phase. A mismatch in the first neutral current norm
000010 | aborts equality with the old Hardy state, not the source construction itself.
000011 | No new geometry,temperature,coupling or primitive measure is selected here.
000012 | The free Gaussian quantization is an EXPLICIT existing premise. It has not
000013 | been derived from the primitive P1 relative spectral action.
000014 | 
000015 | 2. Same source and exact generator scales
000016 | The actual square curve and labelled right annulus face have
000017 |  C_A=(1+exp(2pi B_cov))^-1, B_cov=2(B_1/4 direct_sum B_3/4).
000018 | Quarter mode r therefore has covariance
000019 |  c_r=(1+exp(b(r+alpha)))^-1, alpha=1/4 or3/4, b=4pi.       (1)
000020 | This is a direct specialization of the existing NS cover mode n=2r+epsilon,
000021 | not a fit. The two quarter covariances are independent blocks of this given
000022 | quasifree state; they are not assigned independent new vacua.
000023 | Write the original sea-relative Fock energy as
000024 |  H_alpha=N+F_alpha(Q), F_alpha(q)=q²/2+(alpha-1/2)q.       (2)
000025 | Filling q>0 particles or -q holes proves the charge term; the original
000026 | Maya/current correspondence gives the oscillator degree N.
000027 | The density operator is rho_alpha=Z_alpha^-1 exp(-4pi H_alpha).
000028 | For the cover generator2H_alpha the SAME inverse normalization is2pi.
000029 | For the previous native K_alpha=H_alpha/8 it is32pi.
000030 | These describe one density operator. None is a physical temperature or an
000031 | absolute time calibration.
000032 | 
000033 | 3. Constructive current representation, not a vacuum replacement
000034 | The fermionic partition product converges, since excitation energies grow
000035 | linearly and both positive gaps alpha,1-alpha are nonzero.
000036 | Consequently rho_alpha is a faithful traceclass density on the original
000037 | sea Fock space, with exactly the occupation covariance(1).
000038 | Define the bounded source representation on its thermal cyclic Hilbert space
000039 | by pi_alpha(A)X=A X, cyclic vector rho_alpha^(1/2), Hilbert-Schmidt norm.
000040 | Then <rho^(1/2),pi(A)rho^(1/2)>=Tr(rho A). This is an explicit unital
000041 | normal star representation. This is the full Hilbert-Schmidt representation
000042 | for the full CAR source algebra. When only currents are used,restrict to
000043 | their cyclic subspace closure{A rho^(1/2):A in current algebra};currents
000044 | do not connect different charge blocks. In particular pi(exp(iJ(f))) supplies the actual
000045 | current Weyls. Its source regional algebras are the images of the original
000046 | connected current algebras; it is not an assertion that primitive P1 has
000047 | already supplied this representation or its physical admissible subspace.
000048 | The full annulus graph and this labelled marginal remain different objects.
000049 | The same-E reflection/word positivity is reused from ELLIPTIC-GREEN-RP.txt.
000050 | 
000051 | 4. Full charge and oscillator source functional
000052 | Boson/fermion correspondence or Jacobi triple product gives
000053 |  Z_alpha = Theta_alpha(0) product_(m>0)(1-exp(-bm))^-1,
000054 |  Theta_alpha(t)=sum_(q in Z) exp(-bF_alpha(q)) exp(i t q).  (3)
000055 | This retains the full original charge carry. In particular the original
000056 | sea-relative mean charges are plus/minus0.0413432895814817...,not zero in
000057 | each quarter. The two values cancel for their combined charge.
000058 | For a smooth real profile f, use the ORIGINAL convention
000059 |  J(f)=f0 Q+sum_(m>0)(f_m J_-m+conj(f_m)J_m).
000060 | Independent thermal oscillators with [J_m,J_-m]=m now give EXACTLY
000061 |  omega_alpha(W(f))=Theta_alpha(f0)/Theta_alpha(0)
000062 |    times exp[-1/2 sum_(m>0)m coth(bm/2)|f_m|²].           (4)
000063 | Every Fourier tail of a smooth profile is retained. Properly supported
000064 | profiles are not replaced by finite Fourier polynomials.
000065 | For any finite ordered list use the ORIGINAL central commutator
000066 |  c(f,g)=sum_(m>0)m[conj(f_m)g_m-f_m conj(g_m)],
000067 |  omega(W(f1)...W(fk))=exp[-1/2 sum_(i<j)c(fi,fj)]
000068 |                       times omega(W(sum_i fi)).         (5)
000069 | The covariance,normalization,charge factor and phase in(3)-(5) follow from
000070 | the existing geometry and free source. They are not a physical intervention
000071 | schedule or an identification with exp(-S_prim) for arbitrary operator
000072 | variations. Coherent state interventions and gauge orbits of D remain distinct.
000073 | 
000074 | 5. Independent derivation of the neutral current norms
000075 | Direct fermionic Wick contraction with(1) yields,for every integer m>0,
000076 |  omega(J_m J_-m)=sum_r c_r(1-c_(r+m)),
000077 |  omega(J_-m J_m)=sum_r c_(r+m)(1-c_r).
000078 | Put y=exp(bm). Each term obeys
000079 |  c_r(1-c_(r+m))=(c_r-c_(r+m))/(1-y^-1),
000080 |  c_(r+m)(1-c_r)=(c_r-c_(r+m))/(y-1).