TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-source-clock-field-closure-20261102/CANONICAL-SECTOR-RECONSTRUCTION-AND-ACTUAL-FIXEDPOINT.txt
Original file SHA-256: 78aa483f60cbd27c4850cd17ce9dffa2a378bea7b4620c2fda0783ca1c9b7a86
Inclusive original lines: 120-227
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000120 | This does not prove positivity, a unique vacuum, clustering or a physical
000121 | Theta lift for the whole enlarged field algebra. Existing actual zero-energy
000122 | charged-root obstruction is retained: beta=e1+e4 has grade1 and
000123 | 8(1+d.beta)=0 for BOTH retained quarters. The supplied time cannot be
000124 | changed on that charged sector while keeping this reconstruction fixed.
000125 | 
000126 | 5. ACTUAL CURRENT TEST AND CONSTRUCTIVE ALL128 MODULE FACTORIZATION
000127 | 
000128 | Executable: check_actual_fold_fixedpoint.py in this directory.
000129 | Data: actual_fold_fixedpoint_results.json. Python3 -B -OO: PASS.
000130 | It reuses all192 signed mixed OPE rows from the original affine engine,
000131 | not an independently phased or fitted current model. It reads the literal M
000132 | from v16-operator-pullback-20261014/fold-results.json and the original source
000133 | constructor tfpt_explorer/current_block_geometry.py::_actual_current_source.
000134 | For a target root in doubled coordinates r, the actual source grade is the
000135 | common coordinate parity of M^T r.
000136 | 
000137 | All12 family target roots:4 even,8 odd. All192 ordered mixed products:
000138 |  grades(left,dagger,right)  (0,0,0):32 (1,1,0):32
000139 |                           (0,1,1):64 (1,0,1):64.
000140 | The actual coefficient-1 example has doubled target (0,0,0,0,0,0,2,2)
000141 | and source grade1. Thus common-carrier cancellation does not automatically
000142 | make the family current a member of the original even source algebra.
000143 | The full fixed target U has16 even and24 odd D5 roots,4 even and8 odd A3
000144 | roots. The20 actual X currents split10 even and10 odd. The Cartan quartic,
000145 | erasure and stress-descendant polynomials are all fixed by gamma; the actual
000146 | H_exc=8(L0+d.h0) and sigma_d commute with gamma on every charge and grade.
000147 | 
000148 | CONSTRUCTIVE CONSEQUENCE: transport the ONE marked source spinor by M:
000149 |  s_target=M(1/2,...,1/2)=e3+e6.
000150 | For each of the128 actual odd roots beta, alpha=beta-s_target belongs to
000151 | the ORIGINAL even source lattice M D8. In the same lattice vertex formula,
000152 |  Y(e_alpha,z)|s> = eps(alpha,s) z^(alpha.s)
000153 |                   exp(sum_(n>0)alpha(-n)z^n/n)|alpha+s>.
000154 | Hence the actual mode gives exactly
000155 |  (E_alpha)_(-alpha.s-1)|s> = eps(alpha,s)|beta>.
000156 | No oscillator, charge or endpoint projection is discarded in this leading
000157 | coefficient. The test checks all128 charges, integral modes, output weight1
000158 | and the ORIGINAL Chevalley B cocycle. For root coefficients it verifies
000159 | literal equality to the original eps implementation. The coefficient field
000160 | weights0,1,2,3,4 occur with multiplicities1,28,70,28,1; signs64plus/64minus.
000161 | Thus the128 odd currents require one marked charged module and actual even
000162 | lattice coefficients, not128 independently supplied couplings.
000163 | 
000164 | Bounded algebra version: every odd b is aF. A gauge-even word reduces to A
000165 | using F a=rho(a)F and F^2=u. An actual neutral CP response can therefore
000166 | retain charged intermediate branches through rho, precisely as prior
000167 | NEUTRAL-DHR-BILOCAL-SOURCE-TEST equations(1a),(5a) already establish.
000168 | For unbounded point currents, their analytic smearing/domain limits must
000169 | still be transported; the bounded reconstruction alone is not that proof.
000170 | 
000171 | 6. PHYSICAL SCOPE AND NEXT DECISIVE ORIGINAL CONDITION
000172 | 
000173 | The8 odd family currents are not by themselves physical standalone reads.
000174 | Their grading disproves only preservation of the full field dictionary by
000175 | the pointwise projection E(E_beta)=0 on odd currents. It does not disprove
000176 | the DHR reconstruction of neutral histories. No gauge-even scalar output
000177 | has been found that differs between B and its exact DHR representation.
000178 | 
000179 | There is a separate genuine symmetry condition: a compiler transformation
000180 | U_G descends to the SAME observable net A=B^gamma only if it normalizes the
000181 | fixedpoint structure, U_G A U_G*=A. For this Z2 grading, a sufficient exact
000182 | criterion is U_G gamma U_G*=gamma; a purported symmetry mapping even roots
000183 | to odd ones fails. Target U is not automatically a normalizer of source D8.
000184 | This must be tested for the actual physical P2/clock operation, not inferred
000185 | from a coordinate or single-current mismatch. The parallel relational owner
000186 | checked the literal native sigma, and it admits a further positive closure
000187 | statement, independently reviewed here in the original current engine:
000188 | 
000189 | Original verification/v634_st31_structure.py:122-123 gives
000190 |  SIGMA_IDX=[4,5,0,1,2,3,6,7], sigma(r)=r[SIGMA_IDX].
000191 | This exact coordinate permutation is orthogonal, order3, and preserves the
000192 | chosen240 E8 roots including their half-integral chirality condition. It
000193 | therefore preserves the conformal vector. Its vertex lift uses the existing
000194 | actual cocycle cochain, rather than assigning independent root phases; see
000195 | FOUR-STARTING-POINTS-STATE-RELATIVE-EQUIVALENCE.txt:225-243.
000196 | 
000197 | For the ACTUAL source grading sigma sends64 even roots to odd roots,64 odd
000198 | to even, and preserves48 even and64 odd roots. In particular,
000199 |  r=(-2,0,0,0,0,-2,0,0) is even,
000200 |  sigma(r)=(0,-2,-2,0,0,0,0,0) is odd.
000201 | The exact source root set of A plus sigma(A) has176 roots. Taking their
000202 | actual original Chevalley brackets generates all240 roots in ONE closure
000203 | round, with the actual unit cocycles; no new phases were fitted. The same
000204 | script and results JSON record this independent check.
000205 | 
000206 | Hence, IF sigma is independently realized by the primitive source and its
000207 | admissible local algebra is closed under that original operation, the
000208 | minimal connected local same-conformal algebra generated by A and sigma(A)
000209 | is the E8 extension. One odd spinor current forces the full irreducible128
000210 | under D8. This can SELECT completion instead of merely showing that the
000211 | marked sector permits it. Its sufficiency does not need pre-inclusion of
000212 | the full target U. The first remaining premise is the actual P1 source map
000213 | for sigma; a sigma defined only on the already supplied E8 target cannot
000214 | serve as its own source-origin proof. Native sigma is different from the
000215 | charge-diagonal source fold twist sigma_d: the latter does commute with
000216 | the fixedpoint involution and does not alone force this completion.
000217 | 
000218 | After canonical reconstruction the remaining actual source input is the
000219 | P1 to (A,rho_s) local dictionary preserving its eight-arc support, original
000220 | state/reference, domain, reflection, source time and physical readout action.
000221 | For neutral predictions no separate choice of omega_B is required. For a
000222 | charged primitive OS endpoint, its domain and Theta are still required.
000223 | Neither the formal spectral determinant/eta nor a given collar kernel alone
000224 | supplies that endpoint or its reflected norm. The parallel original-domain
000225 | audit locates no spin-changing endpoint in the literal admissible family.
000226 | 
000227 | PRIMARY RECONSTRUCTION SOURCES READ