Scoped exclusion
Equal actual three-register marginals exclude exact whole-five ground spaces from at-most-three-body terms
In the question: Common clock sectors, quartic geometry and operational reconstruction
Only exact energetic selection of the ENTIRE original five-code on the same four registers. Arbitrary Hermitian terms and frustration allowed. Does not exclude one selected code state,ancillas,thermodynamic systems,effective four-body interactions,open dynamics or approximate selection. Marginal equality is for the two mixed states,not all pure code inputs. No physical-origin derivation.
Read the exact result, assumptions and evidence
Result
rho_C=Pi_C/5 andrho_S=Pi_sym4/35 have exactly equal reductions on all4+6+4 subsets of sizes1,2,3; each three-register reduction isPi_sym3/20. Their global trace distance is6/7. Every Hermitian at-most3-local Hamiltonian on the same four registers therefore has equal mean energy. If the ENTIRE codeC has minimal energyE0, positivity ofH−E0I forces the full35-dimensional symmetric space to be ground as well. An exact five-dimensional code ground space is impossible. The existing4-local moment parent attains the necessary interaction order.
Required assumptions
- Existing even E8 lattice, Gaussian J and sign lifts
Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts.
- Actual clock/complex-reflection subgroup and native quartic space
R=diag(-1,1,1,1) on Gaussian C4 is the complex reflection of root line(1+i)e0; G0=<J,R>=C4×C2. P5 is the existing five-dimensional symmetric quartic subspace in Sym4(V4), W6=exterior²V4. R is not a real codim1 E8 Weyl reflection; P5 is not the full fourth tensor power.
- 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.
Original evidence
- PROOF.txt lines 1–100
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/collective-native-marginal-origin-20261026/PROOF.txt
72ff4cac1b6e32d71cc1d4f0fb39fbf59754935f72455b7e5a3ba5ffce0b369e
- check_collective_origin.py lines 1–147
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/collective-native-marginal-origin-20261026/check_collective_origin.py
f05fb2f189f6a7d21a1f423042cef3cc605e2e52aa8c1e64068cb4ba6d4b4218
- results.json lines 1–70
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/collective-native-marginal-origin-20261026/results.json
af039e7c41aaf067e583d7c90daf9b88cb62bf129ba5caee55d37186f317d606
finding:native-whole-five-ground-space-needs-four-register-interaction