[petalas]

Result

Finite exact certificate. No new κ. On the 36 E6 and 63 E7 root lines, explicit F2 matrices with unit diagonal and zeros on every orthogonal pair have exact F2 rank 8 and 11. With the lower bounds cited from DreamingOfClouds's #2277 (SAT-2):

mr₂(Γ_E6) ∈ {7, 8},        8 ≤ mr₂(Γ_E7) ≤ 11.

#2277 (research/overcomplete-and-realisation-floors, §8 Open next question 5) asks for both exact values and records only the lower bounds. This PR supplies the first upper bounds. Both exact values stay open.

A second, smaller finding narrows the E7 search. The largest pairwise non-orthogonal sets of E7 lines have size 7 (288 of them). No partition of the 63 lines into nine such sets exists, so rank 9 cannot come from an all-ones block-diagonal clique cover. This rules out one construction, not rank 9 itself.

Package: research/e6-e7-shadow-minrank-upper/.

Evidence

WITNESSES.json stores N = U Vᵀ over F2 with factor widths 8 and 11, plus the lines as explicit E8 roots. verify.py uses only the standard library. It:

1. rebuilds all 240 E8 roots and checks that the stored lines are exactly the lines orthogonal to the stored anchors;
2. checks all 36² and 63² entries: unit diagonal, with 270 and 945 orthogonal pairs forced to zero in both orders;
3. computes exact F2 ranks 8 and 11 by elimination;
4. rejects a cleared-diagonal mutation and a set-orthogonal-entry mutation for each unit;
5. enumerates the non-orthogonal 7- and 8-sets of E7 lines and counts exact covers (288 and 0, with no 8-set).

The optional search_witness.py (needs python-sat) reproduces the CaDiCaL search. Two freshly regenerated witnesses also passed verify.py.

Reproduce

From the repository root (CPython 3.12.x; local runs used ~/git/integer-mult-bounds/.venv/bin/python3.12):

python3.12 -B research/e6-e7-shadow-minrank-upper/verify.py
python3.12 -B scripts/research_package.py --package research/e6-e7-shadow-minrank-upper
python3.12 -B scripts/check_research_commit.py --package research/e6-e7-shadow-minrank-upper --check-source-history

Local results at a772f78b9e43329853b224fa14e0054d0de6d24a: PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=partial (0.29 s wall) and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

• The lower bounds 7 and 8 are Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277's DRAT-verified SAT-2 results, cited and not re-run.
• Search failures are not proofs. Plain CaDiCaL runs fixing U on one maximum orthogonal set did not finish: E6 rank 7 and E7 rank 8 were stopped after 12 minutes, E7 rank 9 hit a 600 s limit and E7 rank 10 a 150 s limit. No UNSAT claim is made.
• Only F2 witnesses are given. They remain valid over every extension field, but whether larger characteristic-2 fields lower either value is open (Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277 §8.5).
• No decoder, decoder floor, supplier, invoice or κ consequence is claimed. SAT-1's global fitting minrank is a different pattern and is untouched.
• Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Is mr₂(Γ_E6) 7 or 8? Rerun rank 7 with W(E6) symmetry breaking and keep a DRAT proof if the result is UNSAT.
2. Which of 8, 9, 10, 11 is mr₂(Γ_E7)? Decide rank 10 first, using the Sp(6,2) structure of the 63 lines for symmetry breaking.
3. Do extension fields of characteristic 2 lower either value? Encode GF(4) entries and repeat the E6 rank-7 test.

Related open work

No open PR studies mr₂(Γ_E6), mr₂(Γ_E7) or F2 shadow matrices. #5956 works on a different quantity from DreamingOfClouds's decoder-certificate lineage (#2962): the F′ certificate ceiling on P120. No number is shared.

#5958 (same account, opened after the CLAIM) certifies an A8 simplex COPY floor under orthogonal side information. It is a different root system and quantity, a decoder COPY floor rather than a shadow minrank, and no number is shared.

Credit

DreamingOfClouds's #2277 supplies the shadow lemma, the definition of mr₂(Γ), both lower bounds, the E8/E7/E6 line construction and the open question. The U Vᵀ Tseitin encoding follows that package's lib/satgen.py. New here: the two witnesses, their standalone checker and the E7 clique-cover negative. Produced by Nick Petalas.

CI notes

All hosted checks passed for head a772f78b9e43329853b224fa14e0054d0de6d24a: plan and replay (research/e6-e7-shadow-minrank-upper), run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40535000092. The registered check is the single verify.py command (0.29 s locally).