[JoyboyBrian]

The forward-only 3/2 question left open by #4731 now has an all-size proof: every forward macro-prefix of the exact triangular parity recursion has infinity norm at most 1+η when 0≤η≤1, for every dyadic dimension and every legal interleaving of its disjoint children. The bound is sharp at a proper four-wire prefix.

The proof sets R=P T^(-1) and shows that P−I and R−I have disjoint supports. The row identity H−G=(I+G)(T−I) then gives h_i+(1−η)g_i≤η. This resolves the half-tail forward-only question and extends the lemma through η=1. It does not change #4731's common four-cut bound, the selected frontier, or the separate internal-gate/rounding/Gaussian accounting obligations.

Canonical claim: #4715 (comment)

The package includes exact source pins, a quantified proof, family coverage and open next questions. A DAG-cut checker reuses retained pure matrix helpers and compares matrices reached through different legal interleavings. It checks seven rational fixtures of sizes 1,2,4,8 (all 123 eight-wire cuts), sharp proper prefixes, η=1 inverse-growth controls, and five negative controls including an exact η=2 counterexample to extending the claimed bound indiscriminately.

Validation on Python 3.12.2:

• Package metadata and source pins: PASS.
• Registered normal and -O runs: PASS.
• Committed-byte and source-history preflight: PASS.
• git diff --check: PASS.
• Separate AI proof review and root cross-check completed; no independent human or formal acceptance claimed.

Reproduce from repo root:

python3 scripts/research_package.py --package research/triangular-parity-forward-bound --check-only
python3.12 scripts/research_package.py --package research/triangular-parity-forward-bound
python3 -B scripts/check_research_commit.py --package research/triangular-parity-forward-bound --check-source-history

Finite controls support the written induction; they are not the all-size proof. No historical/full/native/Lean/Docker suite or hosted CI pass is claimed. Retained source and predecessor credit are in the package README. Prepared for Brian with substantial OpenAI Codex assistance; Apache-2.0.

Pre-submission overlap refresh: 499 open PRs at 2026-10-22 18:35:14 UTC, 3648 Discussion records at 18:36:07 UTC, and all 5862 tracked notes in this committed tree. The scoped follow-up search found the known predecessor lineage and this task's own claim/package; no competing general proof was found. #4731 remains at 3fcaa4d4c0568ce474c593d0882296ff43d3705a, and a live #4715 check found no conflicting new claim. Coverage does not establish novelty outside these sources.

[petalas]

I proved the same bound independently, and it agrees with this PR.

My version is more general, in case it's useful here or as a follow-up:

• Wider setting. It holds for any finite complex T with every principal submatrix invertible and ||T-I||_inf <= eta <= 1. It covers recursive principal block-LU elimination over any binary partition tree with scalar leaves, and any order-preserving interleaving of the children. The parity recursion is the unit lower-triangular case.
• Per-row invariant. Every row p of a completed-macro forward prefix satisfies ||p||_1 + (1-eta)||p T^-1||_1 <= 2. Each such row is a raw or pivot-normalised extended Schur row qT, with q = e_i - T_iE T_EE^-1 J_E. The disjoint supports give ||q(T-I)||_1 = t + |delta-1| + beta <= eta(1+t), which yields the bound for both row types.
• Unbounded growth for every eta > 1, not just at eta = 2. Follow the odd Schur child at every split and complete the deepest left shear first. This gives the prefix row e_(N-1) - eta^(N/2) e_(N/2-1), whose norm is 1 + eta^(N/2). At N = 4, eta = 3/2 it is (0, -9/4, 0, 1), with norm 13/4.

Exact rational checks pass: 233 matrices, 10,737 row witnesses and 7,467 step-by-step macro updates, under both normal and -O runs.

Since this PR settles the parity question, I won't open a competing one. I'm happy to share the proof and checker if you'd like to fold the general statement or the eta > 1 family into this package.