TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/clock-source-twist-transport-20261015/PROOF.md
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000078 | of the original fixed charge operator either.
000079 | 
000080 | ## 3. The positive result: one covariant operator family
000081 | 
000082 | Retain the entire covariant tuple, rather than identifying its members:
000083 | 
000084 |     delta_a=J^a delta,   q_a=J^a q,   a in Z4.
000085 | 
000086 | All four delta_a are distinct in either quarter case; delta_2=-delta. For
000087 | the original source offset j define
000088 | 
000089 |     nu_j=((k+j)/8)_k,
000090 |     tau_(alpha,j)=nu_j+alpha q/8,
000091 |     t_(alpha,j)=exp(2 pi i H(tau_(alpha,j))_0).
000092 | 
000093 | This is the full uncompensated torus transition already retained in the
000094 | source contract: `exp(2 pi i alpha Q/8) g0`. The alpha term is not discarded
000095 | or replaced by its square. Put tau_(a,alpha,j)=J^a tau_(alpha,j).
000096 | 
000097 | For the phase-corrected unitary U_J constructed below,
000098 | 
000099 |     K_(delta_(a+1)) U_J = U_J K_(delta_a),
000100 |     Q_(q_(a+1)) U_J = U_J Q_(q_a),
000101 |     t_(a+1,alpha,j) U_J = U_J t_(a,alpha,j),
000102 |     U_J^4=I.
000103 | 
000104 | These equations hold on the full finite-energy core and extend to the
000105 | corresponding diagonal selfadjoint operators and their unitary groups. The
000106 | first equality is the exact quadratic identity
000107 | 
000108 |     |Jp|²/2+(J delta_a).Jp+N = |p|²/2+delta_a.p+N.
000109 | 
000110 | The others follow from equality of the *full* cocharacter pairings, not just
000111 | their mod-1 or adjoint shadows. On the Heisenberg oscillators U_J acts by J,
000112 | which preserves the total level N. On charged root currents it acts as
000113 | 
000114 |     U_J Y(e^p,z) U_J^-1 = eta_J(p) Y(e^(Jp),z).
000115 | 
000116 | The phase eta_J(p) is fixed by the actual source cocycle and the explicitly
000117 | stated basis convention below. Current central terms and adjoints are
000118 | preserved. In particular the quarter-transition character of a charged
000119 | current is transported verbatim. The checker covers both quarter cases,
000120 | four frames, all eight j offsets, and the lattice basis. Basis equality
000121 | proves the character relation for every lattice charge; the 240-root
000122 | comparison is not substituted for an all-charge argument.
000123 | 
000124 | The coefficient q must be transported too. If q is held fixed while J is
000125 | applied, this is not the same tuple and the Q equality fails. Transporting
000126 | q is a real limitation, not an omitted gauge compensation.
000127 | 
000128 | ## 4. The actual ordered CAR/E8 cocycle, not a new sign model
000129 | 
000130 | The checker reads the original functions `eps_car`, `split_even_lattice` and
000131 | `eps_e8` from `seam-joint-ward-state-20261014/operator_transport_check.py`.
000132 | No replacement Hamming/Pauli cocycle is inserted. The original construction
000133 | uses the ordered CAR cocycle on D8 and adjoins X=E_s by
000134 | 
000135 |     X E_d X^-1=(-1)^(s.d) E_d,   X²=-E_q.
000136 | 
000137 | Every p has the unique form d+c s, d in D8 and c in {0,1}. The resulting
000138 | epsilon restricts exactly to the source ordered CAR cocycle on D8. Its
000139 | associativity and all-charge commutator were proved in the cited source;
000140 | here its code and source proof are pinned by SHA-256.
000141 | 
000142 | Let b1,...,b8 be the exact original integral E8 basis from that same file.
000143 | The lifted central group has generators z, t1,...,t8, where z²=1 is central,
000144 | the ti have unrestricted integer powers, and
000145 | 
000146 |     [ti,tj]=z^(bi.bj).
000147 | 
000148 | There are no extra finite-order relations on the ti, because Ls is free
000149 | abelian. The assignment
000150 | 
000151 |     z -> z,   E_bi -> E_(Jbi)
000152 | 
000153 | preserves all these commutators: J preserves the inner product. Since J is
000154 | an integral lattice automorphism, this defines an automorphism of the
000155 | *same* central extension. Negative powers use their actual inverses,
000156 | `E_b^-1=epsilon(b,-b) E_-b`; ignoring that sign would change the lift.
000157 | 
000158 | For n in Z8, multiply the ordered generator powers in the actual source
000159 | cocycle. Write
000160 | 
000161 |     product_i E_bi^ni = c_B(n) E_(Bn),
000162 |     product_i E_(Jbi)^ni = c_JB(n) E_(JBn).
000163 | 
000164 | Then
000165 | 
000166 |     eta_J(Bn)=c_JB(n)/c_B(n)
000167 | 
000168 | is defined for every lattice charge. The induced operator
000169 | `U_J|p>=eta_J(p)|Jp>` is unitary on the charge Hilbert space, with the ordinary
000170 | orthogonal second quantization J on the Heisenberg Fock factor. It fixes the
000171 | vacuum and preserves adjoints. Its all-charge cocycle identity is
000172 | 
000173 |     eta_J(u+v) epsilon(u,v)
000174 |        = eta_J(u) eta_J(v) epsilon(Ju,Jv).
000175 | 
000176 | The identity follows from the generator-extension construction; the 1,024
000177 | generator-pair checks are regressions of its implementation.
000178 | 
000179 | Since J⁴=I, the fourth power of this extension automorphism is a lattice
000180 | character. The exact calculation gives +1 on each of the eight original
000181 | basis generators. It is therefore +1 on **every** lattice vector. The
000182 | oscillator action has fourth power I as well. The vacuum normalization
000183 | removes a global scalar: U_J⁴=I on the complete stipulated Fock space.
000184 | 
000185 | The displayed choice E_bi -> E_(Jbi) is a phase convention, not a physical
000186 | selection rule. Other cocycle-compatible lifts differ by a torus character
000187 | chi. Their fourth-power correction is the norm character
000188 | `product_(a=0)^3 chi(J^a p)`, which is one because
000189 | `I+J+J²+J³=0`. Thus the order-four result is not achieved by secretly tuning
000190 | a new physical coupling or replacing the original signs. It does not
000191 | canonically select the lift's torus character.
000192 | 
000193 | ## 5. State, positivity and what the four frames mean
000194 |