TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/W8-ORIGINAL-STRESS-RECONSTRUCTION.txt
Original file SHA-256: e6d33823484092c52b709ff16ddf6aa166e0614c4d599248dc3f2a588933bd90
Inclusive original lines: 96-145
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.

000096 | 4. What the smaller folded stress space misses
000097 | ----------------------------------------------
000098 | At N level1, the closed Lhat-generated Virasoro space has only the ray
000099 | 
000100 |  span{Lhat_-1|d>}=span{d.h_-1|d>}.
000101 | 
000102 | The true W8 first response has
000103 | 
000104 |  v_perp=v_1-d(d.v_1)/d^2,
000105 |  ||v_perp||^2=51311452775625/9345848836096 >0.              (6)
000106 | 
000107 | This excludes v_1 only from that smaller ray and excludes the full smoothed
000108 | response only from the smaller CLOSED Lhat cycle. Equation(5) and the exact
000109 | 22-word CAR reconstruction show that the original Lcal cycle contains it.
000110 | The corrected statement does not impose a new nonstress-source requirement.
000111 | 
000112 | There is a simple intrinsic source formula for the missed direction. For
000113 | 
000114 |  H8(x)=sum_(a in Phi(E8))(a.x)^8-30(x.x)^4,
000115 | 
000116 | the actual coefficient is
000117 | 
000118 |  v_perp=grad H8(d)-d[d.grad H8(d)]/d^2.                  (7)
000119 | 
000120 | All16 quantum contraction monomials Q_ab are O8 invariant. Their level1
000121 | coefficient at the charge background is radial, so they cannot change(7).
000122 | The checker reconstructs the constant and the entire first response from
000123 | all17 old coefficients and the independently enumerated240 standard roots.
000124 | Both match the old sparse PBW answer exactly. There is no new fitted vector.
000125 | 
000126 | The old nonzero eighth stress-jet overlap with the native F8 remains valid.
000127 | A nonzero overlap, the present span statement, and a specified single
000128 | implemented intervention are three different claims.
000129 | 
000130 | 5. Full separated response and source-state reconstruction
000131 | ----------------------------------------------------------
000132 | Equation(5) applies separately to EVERY v_ell, since each is neutral and
000133 | has original grade8ell. Therefore the existing positive-semigroup smoothing
000134 | 
000135 |  Psi_epsilon=sum_ell exp(-epsilon ell)v_ell, epsilon>0,
000136 | 
000137 | belongs to the closed ORIGINAL stress cycle. Its norm is finite by the
000138 | already proved full W8 two-point function. It lies outside the smaller
000139 | folded cycle by(6), and inside the original cycle by(5). These are compatible.
000140 | 
000141 | This is an all-level theorem. The grade8 computation is its smallest direct
000142 | W8 certificate, not a claim that finitely many levels prove all the others.
000143 | The weight8 homogeneous W8 source state is at original grade64; it too lies
000144 | in the original stress span, by(5), without a million-dimensional scan.
000145 | No finite approximation is presented as the whole infinite source.