[petalas]

Result and claim type

On the A_3 line system of the Gram-class price model in
research/p104-duality-ceiling-flag-note, there is an exact three-term design

u_1 = (0, 0, 1, -1)      w_1 = (-1/2,  1/2,  1/2, -1/2)
u_2 = (0, 1, -1, 0)      w_2 = ( 1/2,  1/2, -1/2, -1/2)
u_3 = (1, -1, 1, -1)     w_3 = ( 3/4, -1/4, -1/4, -1/4)

with sum_k u_k w_k^T = Pi_4 = I - J/4 identically over the rationals and term
prices 3, 3, 2, total 8. The natural dual-basis design of the same shape
(u_k = e_k - e_4, w_k = e_k - (1/4) 1) is exact and costs 3 + 3 + 3 = 9,
so the written design is strictly cheaper in the same model. The declared finite
family of three-term designs over the 14 canonical equality-pattern
representatives (1812 full-rank triples) has minimum 8, attained by the written
design.

Claim type: exact finite certificate (written rational design, exact identity
and exact prices) plus a bounded exhaustive enumeration of a declared finite
family. This is not a lower bound over all designs, not a kappa change, and not a
supplier admission.

The predecessor's note states A_4 and every n >= 3 as outside its scope. The A_2
value 4 is reproduced here only as a control and is not claimed as new.

Evidence and reproduction

From the repository root, using the repository's CPython 3.12:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/gram-class-a3-three-term-design/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/gram-class-a3-three-term-design/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/gram-class-a3-three-term-design
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/gram-class-a3-three-term-design --check-source-history

Local results on CPython 3.12:

• verify.py prints
  {"a2_control": {"prices": [3, 1], "total": 4}, "a3_design": {"prices": [3, 3, 2], "total": 8}, "controls": ["perturbed_entry", "zero_carrier", "u_full_rank"], "dual_basis": {"prices": [3, 3, 3], "total": 9}, "family": {"designs": 1812, "minimum": 8, "patterns": 14}}
  with status PASS_GRAM_CLASS_A3_THREE_TERM_DESIGN.
• scripts/research_package.py reports
  PASS_REGISTERED_RESEARCH_PACKAGE research/gram-class-a3-three-term-design outcome=partial.
• scripts/check_research_commit.py --check-source-history reports
  PASS_RESEARCH_COMMIT_BYTES at commit c5de3d6b23c229d5106ef7e12c81ba65fd7cdfe8
  with source-history commit e54523517d0f03e58352e647c403a5af3e525a57.

The verifier checks the rational identity, the three prices, the dual-basis
comparison, the A_2 control, every enumerated design and three rejection
controls, and prints a receipt bound to the package file hashes.

Limitations and unrun obligations

• The enumeration keeps one canonical representative per equality pattern, not
  every rational carrier. Designs with the same pattern but different carriers
  can have different prices, so 8 is only an upper bound on the A_3 Gram
  cost. The exact minimum is open.
• No lower bound on the cost beyond "at least three terms" (rank) is proved.
• The model is the scalar Gram price model with the F2 mask-rank rule. Nothing
  here concerns a physical word, caller, compiler, native implementation,
  precision, restoration or the selected record.
• No new kappa and no supplier admission.
• Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. What is the exact Gram-class minimum on A_3? A lower bound must handle
   arbitrary rational carriers; equality patterns do not determine prices.
   First step: price-multiset case analysis for totals below 8, beginning with
   four-term designs, where the W factor has one free direction
   W = W_0 + q r^T.
2. Is the A_n Gram cost n^2 for every n, or does A_3 start an n^2 - 1
   family? First step: generalise the written pattern to n >= 4 and price it.
   Blocked by: the general pattern is not constructed.
3. Does the A_2 gap (Gram 4 against identity decoder 3) persist for some
   n >= 3? First step: compare the identity decoder cost n+1 with the best
   explicit Gram design at n = 3 and n = 4. Blocked by: question 1.

Related open work

No open PR studies the A_n Gram-class cost for n >= 3. research/dual-basis-gram
concerns a different object (a rank-(9,9) dual basis) and does not compute a
Gram-class price. The predecessor research/p104-duality-ceiling-flag-note is
cited for the model and the A_2 value; it is not audited here.

Open PR #7322 (this account, created after this PR) proves an exact optimum of
the P104 weighted star program on orthogonal 8-sets, a different object and
quantity from this package; no number is compared.

CI notes

Head c5de3d6b23c229d5106ef7e12c81ba65fd7cdfe8. Run
https://github.com/CrocSwap/integer-mult-bounds/actions/runs/43105433200
(job 130807408852): plan failed, replay skipped, so CI certified
nothing for this package.

The plan failure is repository-level and unrelated to this package. The
"Onboarding helper tests" step ran 760 tests with one failure and 24 skipped:

FAIL: test_current_ci_section_matches_selected_record
tests/test_research_current_status.py line 58
AssertionError: '`make frontier7000-verify`' not found in the CI verification doc

On canonical main e54523517, certificates/selected-result.json names
make frontier7000-verify while docs/ci-verification.md still describes
#6030, so the assertion fails on main's own files. I read the job log and
confirmed exactly this cause. The same single test fails, with the same
1 failure, 24 skipped count, in runs 43029932349 and 43030221243 (other
contributors' PRs), which I also read.

Local checks pass on CPython 3.12: the package verifier under -B and -O,
scripts/research_package.py --package … (outcome=partial) and
scripts/check_research_commit.py --check-source-history. No package-level
hosted replay has run; the package has no CI certification.