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Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_time_compatibility_20261027/AFFINE-SPECTRAL-IDENTITY.txt
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000001 | EXACT SOURCE SPECTRAL IDENTITY AND ITS LOCAL FIELD DISCRIMINATOR
000002 | 27 October 2026
000003 | 
000004 | Verdict: established within the retained four D8 module parent with the
000005 | actual folded source Hamiltonian. This is not an identification of local
000006 | fermion and E8 fields. The separate independent review checks the identity.
000007 | 
000008 | 1. Data and proof on the full charge lattice
000009 | 
000010 | Use the source D8 chart, not an untransported canonical compiler chart.
000011 | For 0<a<1 set
000012 | 
000013 |   delta_j=(j+a)/8-1/2, j=0,...,7,
000014 |   p*=(1,1,1,1,-1,-1,-1,-1)/2,
000015 |   H=8(N+|p|^2/2+delta.p), Q=sum_j p_j.
000016 | 
000017 | N is the nonnegative integer oscillator level. Let A=R^4 exchange channels
000018 | j and j+4. The two exact identities
000019 | 
000020 |   A delta=delta+p*, A p*=-p*
000021 | 
000022 | give an affine involution T(p)=A p+p* with
000023 | 
000024 |   T^2=1, T(p)+delta=A(p+delta), Q(Tp)=Q(p).
000025 | 
000026 | Completing the square proves H(Tp,N)=H(p,N) for every lattice vector.
000027 | Permuting the eight oscillator channels by A preserves their canonical
000028 | commutators, occupation multiplicities, and total oscillator level N.
000029 | Thus this is a bijection of the full orthonormal Fock label basis, not a
000030 | finite root shell coincidence or a numerical low energy approximation.
000031 | 
000032 | The four discriminant sectors are integer even/odd charge 0,v and half
000033 | integer even/odd charge s,c. T exchanges
000034 | 
000035 |   0 <-> s, v <-> c.
000036 | 
000037 | 2. Full thermal traces
000038 | 
000039 | Define Z_b(beta,u)=Tr_H_b exp(-beta H+sum_j u_j a_j,0), beta>0.
000040 | These traces converge for every fixed finite u in C^8. Quadratic charge
000041 | growth dominates the linear term; the eight oscillator products converge.
000042 | 
000043 | The affine change of variables gives the more precise identities
000044 | 
000045 |   Z_s(beta,u)=exp(u.p*) Z_0(beta,A u),
000046 |   Z_c(beta,u)=exp(u.p*) Z_v(beta,A u).
000047 | 
000048 | For u=lambda q, q=(1,...,1), both A q=q and q.p*=0. Therefore
000049 | 
000050 |   Z_s(beta,lambda q)=Z_0(beta,lambda q),
000051 |   Z_c(beta,lambda q)=Z_v(beta,lambda q).
000052 | 
000053 | It follows for every beta>0 and finite total charge chemical parameter:
000054 | 
000055 |   Z_(E8_c)(H,Q)=Z_(0+c)(H,Q)=Z_(0+v)(H,Q)=Z_(NS CAR)(H,Q).
000056 | 
000057 | The other E8 completion has Z_(0+s)(H,Q)=2 Z_0(H,Q). It carries only
000058 | even total Q. A common ground energy or Casimir normalization may be
000059 | included on both sides without changing these identities.
000060 | 
000061 | This statement concerns the actual shifted source H, not unshifted L0
000062 | characters, modular S transformations, or arbitrary Cartan backgrounds.
000063 | 
000064 | 3. The smallest field discriminator
000065 | 
000066 | For a<1/2 the unique lowest vector-sector charge is e0, with H=a and Q=1.
000067 | Its image in c is
000068 | 
000069 |   T(e0)=p*+e4,
000070 | 
000071 | also with H=a and Q=1. The individual Cartan charges differ: the zeroth
000072 | charge is 1 on e0 and 1/2 on T(e0). In particular exp(2*pi*i*a_0,0)
000073 | is +1 on all integer charges and -1 on all half integer charges.
000074 | For a>1/2 use -e7 and T(-e7)=p*-e3. At a=1/2 both pairs contribute.
000075 | 
000076 | Consequently the common H,Q trace does not preserve the full specified
000077 | current readout. The v and c representations of the retained D8 current
000078 | algebra are inequivalent; their unshifted conformal ground weights are
000079 | 1/2 and 1. A map preserving the given D8 currents cannot intertwine them.
000080 | 
000081 | A Hilbert space unitary can be defined by the identity on sector 0 and
000082 | the label bijection T on sector v. It maps NS 0+v to E8_c 0+c and
000083 | preserves Omega, H and Q. Its existence is weaker than a local field map:
000084 | the definition uses global sector projections and does not intertwine the
000085 | retained D8 Cartan currents. No cocycle or local field lift is claimed.
000086 | In addition, NS vector fields are fermionic, while the E8 root spinor
000087 | currents have conformal weight one and bosonic local statistics.
000088 | 
000089 | 4. Consequence for the original research question
000090 | 
000091 | Even an exact equality of all thermal traces resolved by total energy and
000092 | total charge does not solve the requested same-source operator dictionary.
000093 | It is now possible to exhibit the first disagreement on a single charged
000094 | state, with the actual source time kept throughout.
000095 | 
000096 | This is not a universal impossibility result for P1 or a proof that E8_c
000097 | is its output. The original twice-root charge marking selects E8_s in
000098 | the already established transported lattice dictionary. E8_c does not
000099 | preserve that marking: q/2 belongs to E8_s and not to E8_c.
000100 | 
000101 | 5. Source and review
000102 | 
000103 | The original H, both spinor minima, marked charge obstruction and Ward
000104 | state selection occur in
000105 |   experiments/theory-contracts/seam-joint-ward-state-20261014/PROOF.txt,
000106 |   sections 2 to 5, contract UR.SEAM.JOINT_WARD_STATE.01.
000107 | 
000108 | The actual four-sector winding and source time also occur in
000109 |   experiments/theory-contracts/canonical-seam-fermionization-20261025/
000110 |   SAME-SOURCE-AFFINE-DEFECT.txt.
000111 | 
000112 | The current companion ALL-CHARGE-GROUND-AND-SOURCE-SELECTION.txt supplies
000113 | the complete minimum proof and original P1/Perron source audit.
000114 | ALL-CHARGE-GROUND-INDEPENDENT-REVIEW.txt checks the affine identity,
000115 | oscillator transport, convergent traces and the distinct local question.
000116 | 
000117 | The additional Clustering test is recorded separately. Matching H,Q
000118 | spectra neither proves nor repairs its local field requirements.