Read the result in its research context.

Result

For actual erasure a_a=bar(u_a)(1) and original W_beta=(v.h(-1))e_beta, exactly [a_a,W_beta,1]=bar(c_a)E_beta,1+bar(b_a)W_beta,2 with b_a=u_a.beta,c_a=u_a.v. All20 identities on5 original code states and4 erasures agree exactly. Original witness b2=-i,c3=4-4i has zero a0/a1 terms; a2 gives iW2 and a3 gives(4+4i)E1. No coupling was chosen.

Required assumptions

• Given local E8_1 vacuum representation and explicit cocycle

  Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation.

• Actual native G31 event representation

  Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.

• Symmetric quartic and actual erasure intertwiner

  Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.

• Actual positive Cartan quartic Fock embedding

  Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.

• Original neutral root-pair moment fields

  Use existing E8_1 neutral_source_response reader:64 chirally selected roots, original omission of one SU4 family weight line, B48=12T_E8+R_neut, AR=4P_A2-8P_U1. Family omission is an explicit existing input, not derived here from P1; neither it nor the R port or physical coupling is newly selected.

• Original displayed J-equivariant root isometry

  Use existing S=_SIMPLE and T=_TARGET_SIMPLE from cartan_source._glue_deck_character_certificate: O=T S^-1/sqrt2 maps actual Construction-A roots r/sqrt2 to original Chevalley roots r/2, preserving metric and J. This is the existing concrete map, not a newly fitted embedding; physical P1 selection stays separate.

• Original positive adjoint two-insertion response

  Compute o(W_beta), W_beta=E_beta,0 R_neut, with unchanged original real transverse v_beta. Ordered return is positive Fock Gram o(W)*o(W) between code or original trimer states without intermediate projection. Allow W=sum a_beta W_beta for arbitrary complex input coefficients; distinct lattice charges force nonnegative weights |a_beta|². Nonadjoint returns, other complex transverse fields, longer histories or physical instruments are not thereby selected.

Original evidence

• SOURCE-ERASURE-WARD-AND-CODE-CLOSURE.md lines 24–68

  Source checked at this snapshot

  Original location and fingerprint

  experiments/theory-contracts/joint-native-algebra-selection-20261016/SOURCE-ERASURE-WARD-AND-CODE-CLOSURE.md

  b365eebb2d5419fdf6306d2a41d8ae40bb538b5b195e01ccec91636c1fc8cd38

finding:original-source-erasure-ward-terms-derived