Two-sided deficiency budget: Theorem BX for b_Q <= 4 (conditional frame, exact residue, no κ claim)

[DreamingOfClouds]

Adds research/two-sided-deficiency-budget/ only; nothing existing is changed. Pure finite combinatorics on the E8 root lines and the unit P112, in the line of #2816, #2826, #2862, #2832 (kill-shot reduction), the LC / O4 lineage and #2985 (ours). No κ claim, no window, no supplier word.

What is proven or checked (README claims box, each row names its proof or verify.sh step)

• Unified budget: |Lo| = ρ + b_Q + dep_Q, so Λ_Q <= 6t + 6 dep_Q − (Σσ − 6 b_Q); Budget BX is Σσ >= 6 b_Q. The combined cost bound that uses it is stated as conditional on G2-BUDGET P2.
• Dependencies at tight m: every q-dependency is X2^T α (normal-form hypotheses rank Ŷ = 8+t, rank X2 = s stated); the α-merge product identity. The induction down to Decoder floors, case (b) for every number of terms: one-sided P112 decoders cost ≥ 64 without corrections (unique optimum) and with ≤ 2 non-full corrections (no κ claim) #2816 is conditional on the one-sided floor beyond Δ <= 2 (proven there for Δ <= 2 / <= 2 non-full corrections) and on re-normalising D_α.
• Theorem BX (size version): generalised Hamming weights d_1 = 53, d_2 = 78, d_3.. >= 92, 100, 106, 109, 111, 112, the deletion lemma, and the deficit table; BX holds for b_Q <= 2 at every t <= 6, b_Q = 3 for t <= 5, b_Q = 4 for t <= 3.
• Exact supports close (b_Q,t) = (3,6), (4,4), (4,5), (4,6): BX for b_Q <= 4 at every t <= 6. Two independent implementations (the audit scripts checks/audit/chk1-8.py as primary; the cover-oracle scripts in checks/rb/ as second); (4,6) is killed directly over all 112 non-frame roots with no orbit reduction.
• Remark: BX and the dependency analysis do not use Theorem DR; any combination with DR needs the shared-swap-line term or |S_a| <= 2.

Open, exactly: b_Q >= 5; R-dep (a fully-low J_P-circuit at tight m); tight m with s + t <= 7; and the two imports not proven here (G2-BUDGET P2; the one-sided floor beyond Δ <= 2).

Reproduce: bash research/two-sided-deficiency-budget/verify.sh (numpy >= 2; two processes; manifest, root-system facts, constants, the audit scripts, the optimiser, the second implementation). The one cited classical input is the conjugacy of Φ ∩ V to a standard parabolic subsystem (Bourbaki VI §1; Humphreys 1.12).

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.

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