TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/quotient-locality-e8-20261014/PROOF.txt
Original file SHA-256: dfa4b094f660d47bc28cf2729e1c12eb29ae0b79cdfda36e72cb40f9406f5714
Inclusive original lines: 85-197
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000085 | finite particle/hole excitations have a number charge N_k with integer
000086 | occupation differences. Normal-ordering/background constants do not
000087 | change Ad exp(2pi i N_k)=identity on the original Hilbert-space operators.
000088 | 
000089 | A nonzero E8 spinor current with respect to these actual D8 Cartans needs
000090 | the integrated Ward action
000091 | 
000092 |     [N_k,S_p]=p_k S_p, p_k=+/-1/2.
000093 | 
000094 | On an invariant finite-energy domain, exponentiation gives
000095 | 
000096 |     S_p=Ad exp(2pi i N_k)(S_p)=exp(2pi i p_k)S_p=-S_p,
000097 | 
000098 | so S_p=0. A multi-interval intertwiner can exist without having this Ward
000099 | action. Its total integral source charge alone does not identify it with
000100 | the eight half-Cartan-charge spinor fields. The ordinary NS even module
000101 | and spinor modules are distinct; see Boeckenhauer section2 and the prior
000102 | mixed-current-origin-20261014 audit. This obstruction is unchanged by the
000103 | quarter phase and by a unitary CAR relocalization.
000104 | 
000105 | 4. A SECOND, INDEPENDENT TEST USING THE SAME PHYSICAL TIME
000106 | 
000107 | In the fold's occupation convention let
000108 | 
000109 |     mu_k=(k+alpha)/8, delta_k=mu_k-1/2,
000110 |     K=H_fold/8=L0+delta dot Q.
000111 | 
000112 | Normal ordering is relative to the transported source sea, with the same
000113 | vacuum constant. Source modes exp(i(j-a)theta) have exp(-2pi i a), whereas
000114 | the fold endpoint has exp(+2pi i alpha); alpha=-a modulo1 in the same
000115 | orientation. A simultaneous charge/sign reversal changes none of the
000116 | spectral conclusions. The physical excitation normalization is
000117 | 8 min{mu_k,1-mu_k}=1/4 for both actual quarter cases.
000118 | 
000119 | For alpha=1/4, delta=(-15,-11,-7,-3,1,5,9,13)/32.
000120 | For alpha=3/4, delta=(-13,-9,-5,-1,3,7,11,15)/32.
000121 | In either case |delta_k|<1/2 and sum|delta_k|=2.
000122 | 
000123 | With the stipulated common E8 current stress/time, a charge p and
000124 | oscillator contribution N>=0 have relative K-energy
000125 | 
000126 |     E(p,N)=||p||^2/2+delta dot p+N.
000127 | 
000128 | Every nonzero integer coordinate gives n^2/2+delta_k n>0. Therefore on
000129 | D8 the unique zero-energy state is p=0,N=0. In the half-integral coset,
000130 | coordinate convexity gives the unique minimizing charge
000131 | 
000132 |     p_*=(1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2,-1/2).
000133 | 
000134 | It has even chirality, ||p_*||^2=2, total source charge0 and
000135 | delta dot p_*=-1. Its relative energy is therefore0. All other charges
000136 | in this coset and all oscillator excitations have positive energy.
000137 | The full lattice conclusion follows from coordinate minimization, not
000138 | from a finite root census. Thus dim ker K is1 on the original even CAR
000139 | module and2 on the stipulated E8 extension with the same time.
000140 | 
000141 | The previously verified M maps p_* exactly to e1+e4, an ordinary D5
000142 | carrier root, orthogonal to the transported U1 charge2(e3+e6). Also
000143 | mu dot p_*=-1, so the cyclic source clock character on this charge is1,
000144 | just as on the vacuum. The missing partner is neutral and clock-invariant.
000145 | Uniform U1 holonomy compensation cannot raise its energy, because its
000146 | total U1 charge is0. An independently imposed sector-energy offset would
000147 | alter the spinor-current time law; it is not source-preserving transport.
000148 | 
000149 | 5. THE PARTNER IS ABSENT IN THE FULL MICROSCOPIC REPRESENTATIVE
000150 | 
000151 | Use the existing original QWZ cylinder, mass1 and width8, both quarter
000152 | sectors. The transverse Fourier block has the exact original form
000153 | 
000154 |     H(k)=[[-sI,B*],[B,sI]],
000155 |     H(k)^2=diag(s^2 I+B*B,s^2 I+BB*) >= sin(k)^2 I.
000156 | 
000157 | This algebraic identity is already proved in
000158 | source-rg-clock-bridge-20260920; it is reused here, not counted as a new
000159 | source-selection theorem. At k=2pi(j +/-1/4)/N the smallest |sin k| is
000160 | sin(pi/(2N)). Hence the WHOLE one-particle source has no zero eigenvalue,
000161 | and its normal-ordered many-body sea is unique. Its scaled excitation gap
000162 | satisfies
000163 | 
000164 |     g_N >= N sin(pi/(2N))/(2pi) >= 1/(2pi),
000165 |     liminf g_N >=1/4.
000166 | 
000167 | The first bound is uniform; the second uses the limit of the displayed
000168 | lower bound. This applies to both edges and all bulk modes, not only the
000169 | known charged current subnet. In a common-Hilbert, source-state/time
000170 | preserving limit this gap also excludes an extra orthogonal zero-energy
000171 | state from nonlinear microscopic words. A different representation,
000172 | source, scaling or extra sector is outside that statement.
000173 | 
000174 | Direct calls of v1033_charged_disorder.qwz_cylinder, without rebuilding
000175 | the Hamiltonian, give six full-matrix checks (sectors1/3, N8/16/32).
000176 | Scaled gaps are0.24839671278605,0.24959859825891,0.24989961328741.
000177 | Every matrix has zero numerical zero-modes. The analytic block bound,
000178 | not numerical tolerance, proves zero-mode absence for this source class.
000179 | 
000180 | 6. RECONCILIATION AND REQUIRED NEXT SOURCE TEST
000181 | 
000182 | The fold and periodic root conjugacy of v16-operator-pullback-20261014
000183 | remain correct. They were conditional on an E8 extension; neither the
000184 | root map nor the charge-matched lift supplies the missing physical state.
000185 | The multi-interval proposal cannot close this within the unchanged source.
000186 | 
000187 | Existing native-source-reconstruction-20261014 already assumes an E8
000188 | charge lattice and a spatial source. Existing source-charged-register-20260920
000189 | uses a declared Fock/register encoding and state rule. Neither is a P1/P2
000190 | derivation of the required neutral carrier partner. They cannot be imported
000191 | as that proof without an operator, state and time identification.
000192 | 
000193 | The specific necessary test for an independently derived primitive source
000194 | is now: exhibit its actual local D5-carrier field for this root, its
000195 | neutral clock-invariant partner, and its common physical energy/Cartan
000196 | action; or derive a different source dictionary/physical current time
000197 | and recheck those actions. The one-copy gapped QWZ source cannot supply