[aliromman]

This PR constructs an odd E8 direct-sum decoder and proves a cost obstruction for one native implementation class. For three factors it gives 1,404,928 rank-one binary ports in dimension 25, fitting rank 20,480, and source/reader pairing tariff 311,296, or 76/343 per port. For every odd factor count the tariff per port tends to zero. The added native result shows why this geometric gain does not suffice: separated use of the frozen complete helper banks fails at exponent 151/50000 = 0.00302 for every odd factor count and stage count at least three, under the stated caller inventory.

There is no complete kappa improvement. The selected checkpoint remains 0.003027288235. A direct joint producer, changed helper word, changed main paths or changed caller inventory remains open. This result gives a precise reason to stop the separated frozen-bank implementation. It does not exclude the geometry or all native producers.

Read the geometry proof, geometry certificate, native-lift proof, exact cost certificate, and source pins.

For three factors, the decoder has 21,504 terms of pairing rank 14 and 512 of pairing rank 20. Their source spans have dimensions 15 and 22. The 2,744 product cubes support a scalar involution between orthogonal 256-port classes. This supplies the local K component of K+H+F=I; it does not supply a native H word. The original incomplete budget gates remain valid cost targets, not admitted exponents.

The native obstruction permits H_k = sum_i H_i tensor D_i, where each D_i is any complex linear operator on the other port factors. It need not be diagonal, symmetric or a scalar polynomial. Each coefficient channel retains one complete frozen helper bank. Coefficient maps, extraction of H and other work outside the stated main paths are granted free. The proof requires at least 10,752 channels for three factors, and at least (2k/t)*56^(k-1) for odd k, with t=(k+1)/2. The checker counts all 408 helper paths in both orientations. The exact cost test covers the declared main paths, productive targets, COPY profile and idle cap. It fails for every odd k,N>=3 at 0.00302 and for every larger exponent below one. Joint or shared helper protocols that change the paid word, address-dependent coefficient operators, different base modules and changed caller inventories are outside this claim.

The scientific base is canonical main 1a3ced60b2f7a46bc70280a04f2eeb8ca027c700. The pre-submission refresh reached 32890bc5de034ab8fc93ac88ead7279a6a416d04; relevant guidance, the selected record and both consumed inputs were unchanged. The Discussion finding records the bounded overlap check. Peer-owned fused-T/Global120 and other native caller scopes are excluded.

Verification uses Python 3.12 and the standard library. The geometry verifier and native-lift verifier are both registered:

python3.12 -I -W error -B research/odd-e8-direct-sum/verify.py
python3.12 -I -W error -B research/odd-e8-direct-sum/native_lift.py
python3.12 -B scripts/research_package.py --package research/odd-e8-direct-sum
python3.12 -B scripts/check_research_commit.py --package research/odd-e8-direct-sum

Expected scientific markers:

PASS_ODD_E8_DIRECT_SUM_COMPONENT ports=1404928 dimension=25 rank=20480 pairing_mass=311296
PASS_ODD_E8_SEPARATED_NATIVE_LIFT_GATE channels=10752 exponent=151/50000 odd_factors=all_at_least_3 odd_stages=all_at_least_3

All four registered normal and optimized checks pass locally. Committed-byte checking passes on all 12 package and input paths. A read-only Git-free export from the exact head passes both scripts in both modes and retains identical file bytes. Geometry checks cover all 12,544 base entries, eight feature supports, 7,168 pair supports, 512 triples, 2,744 product cubes and odd-factor coefficients and spectra through 15. Four data and coefficient mutations reject. Native checks cover exact projection identities, all retained helper width partitions, diagonal and nondiagonal operator-pencil fixtures, and two rejecting controls. Rational log/exp intervals certify the cost signs. Infinite ranges and channel bounds rely on the written proofs, not finite enumeration.

Independent human review, formal proof-kernel verification, native operator replay, the selected supplier, compiler, precision and outer admission checks were not run. Full source-history checking was not run in the shallow checkout. Hosted CI remains separate from these local checks.

Credit: @Dugongue's PR #4929 supplies the rank-eight code and retained native source. Dillon's physical source note and task record supply the literal normalizers and source attribution. The helper profiles come from Cao Yifan's retained native word. Retain Cinna Davis, zxv, Jacob Alan Sussman and all predecessors credited in these sources. Multilinear majority, tensor eigenspaces and the rank arguments use standard algebra; no new general identity is claimed.

Prepared by Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. The change adds one opt-in research package and changes no shared code, predecessor evidence or selected frontier.