[JoyboyBrian]

Result

Scoped obstruction with an explicit allowance on the selected #4470 p16 invoice. No new κ; the selected record is unchanged. Package: research/two-central-laurent-coverage.

• Where the one unpaid coordinate comes from. On the pinned selected invoice (research/joint-helper-lattice/RESULT.json, the selected source certificate), every one of the 3,584 = 2·112·16 main targets has rank 1 + 8k against a paid idle of rank 8k, or 145 = 1 + 8·18. The extra coordinate is the central Gaussian line Q = C_q, with exponent ±1. Exact replacement identity: 752 = 18·(H1 − P1 + V + 2VB) − V. The Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 identity 72W + 64N does not apply to this interface.
• Changed contract. The substitution Q → Q_a Q_b uses a fresh axis q_b and gives an exact controller with two unpaid coordinates per main family. Thin shears become rank 2. Natives are realized by the conjugation D_s S_y(Q_a⁻¹Q_b⁻¹) = Δ⁻¹(D_s S_y(Q_a⁻¹))Δ, with Δ = diag(1, Q_b). The identities are checked formally and on six dirty F2^3 inputs.
• Optimistic relaxation (natives unchanged and free): root in (0.01023582, 0.010235821), 3.39× the selected root.
• Credit-rank lemma (formal Laurent model, σ: Q_b ↦ −Q_b): c credited targets need c q_b-touching paid requests. The natives must therefore contain ≥ 1,904 of them.
• Allowance: ≤ 1,859 extra rank-one units keep a positive surplus at the selected root; 1,860 fails. Paying the forced terms as rank-one requests therefore loses. The conjugation-charged profile has rank-one surplus −1,376.
• Theorem (full credit, incremental): keeping the selected paid profile and adding q_b to requests, or adding new requests, gives a surplus strictly below the selected one at every 0 < C < 1. The sorted main targets dominate the largest paid ranks, and g(x) = (x+1)^t − x^t is decreasing.
• Partial credit (main targets only): no such incremental realization certifies C = 0.003035069 (+0.49% ceiling, pointwise certificate).

The one route left open is a redesigned ten-axis native word whose own complete invoice changes.

Task and overlap

• Card coverage-law-change, track recursive-interfaces. Scope refined to the current selected interface; the reason is in TASK.json.
• CLAIM: Handoffs and requests for independent checks #1166 (comment). Base cef4b52f2a59009865bf8eabde1be3f159063703.
• Overlap checked with find_overlap.py, which found about 120 packages under this card. All of them address the Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 template. None rebinds the card to Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470.
• Distinct from the centre-path, coded p16, two-FULL, H3-prefix and raw-dual lines (Clear1nsight Two FULL native calls with monomial-conjugate central functions — bounded feasibility #4902/Expose a clean native prefix and construct one homogeneous U(q) offer (no new κ) #4931/Prove a scoped one-bank ceiling for the #4929 coded-port adapter (no new κ) #4932/Emit the physical raw dual of #4935 with independent verification and a scoped ceiling (no new κ) #4943/Audit one completed BIT backend API edge in #4814 (ordinary component; no new κ) #4945; dillona DONE (partial): complete rational coded p16 caller; recursive admission open #4930/Emit a phase-correct coded p16 caller removing 144 center calls (no κ) #4933).
• Distinct from the sibling independent-moment claim, which verifies the unchanged invoice.

Test plan

• python3 scripts/research_package.py --package research/two-central-laurent-coverage --check-only → PASS_PACKAGE_METADATA_AND_PINS
• python3.12 -B scripts/research_package.py --package research/two-central-laurent-coverage → PASS_REGISTERED_RESEARCH_PACKAGE (about 8 s)
• python3.12 -O -B …/verify.py and python3.12 -I -W error -B …/verify.py → PASS_TWO_CENTRAL_LAURENT_COVERAGE
• python3.12 -B scripts/check_research_commit.py --package research/two-central-laurent-coverage --check-source-history → PASS_RESEARCH_COMMIT_BYTES
• python3.12 -W error -B -m unittest discover -s tests -p 'test_research_onboarding.py' → 26 tests OK
• Manual mutation tests, each giving FAIL:
  • allowance recorded as 1,904;
  • THIN = 14;
  • a corrupted convolution kernel.
• In-verifier controls reject:
  • an inverse without its X_q translation;
  • an uncharged second coordinate;
  • partial-credit dominance;
  • allowance + 1;
  • a mutated main target;
  • a wrong gauge orientation.

Open next questions

1. A ten-axis native word whose complete invoice realizes its ≥ 1,904/18 q_b contacts per call and still beats the selected native's H_t − P_t (the only open route; worth up to 3.39×).
2. Second-coordinate credits on helper exits (rank ≤ 9), excluded here.
3. Extending the pointwise partial-credit certificate to an all-C ceiling.

Credit

• Cinna Davis (1337isnot1337): Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470/Conditional κ = 0.00296 via query-thinned productive controllers (+10.18%) #4440 controller and controller-stage-ceiling, with the credited native and outer lineage in the selected review.
• kappa-hybrid-controller NATIVE.md: shear and conjugation conventions.
• Douglas Colkitt: ceiling.py (Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240), reused unchanged.
• Earlier coverage-law work on the Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 template: Research note: restoration deferral is mass-neutral in the hosted T37 recurrence (no new kappa) #1012's identity, Navinor's complex-coverage-law-relief, Clear1nsight's selected-family-coverage-boundary, three-bank-restored-unpaid and line-port-unpaid-cap.
• Substantial assistance from Anthropic Claude (Claude Code, Opus 5.5).

🤖 Generated with Claude Code

[jwr404]

Independent review at head 0d6a9debc531cb8b1cdf986d86914f5d38479eda

Scope. I read PROOF.md §0–§7 and recomputed every numerical claim from the pinned research/joint-helper-lattice/RESULT.json. The recomputation uses my own script: stdlib Decimal at 60 digits, with no import of this package or of ceiling.py.

Numbers: all reproduce.

Claim	Independent value
Main targets	224×145, 224×73, 448×(17,25,…,65): 3,584. Each 1+8k has an idle 8k with equal multiplicity, and P[72] = 224 for the 145s.
Rank-one surplus	752 = 18(H1 − P1 + V + 2VB) − V, with H1 = 3471, P1 = 5327
Thin queries	P[1] = 41946 = 18·2237 + 112·15
Selected root bracket	S(lo) = +3.86e-9, S(hi) = −2.09e-8
Optimistic relaxation root	0.01023582066137550…, inside (0.01023582, 0.010235821)
Relaxed surplus at selected hi	1859.8169…, so the allowance is exactly 1,859 and 1,860 fails
Dominance	Sorted main targets beat the 3,584 largest paid ranks strictly, with minimum margin 1. The tail paid ranks are 16, 8 and 5.
Partial-credit bound	+1.07e-5 at C = 0.003035068 and −2.35e-4 at 0.003035069

Argument.

• Lemma (§3). The telescoping σ(W) − W = Σ σ(F_1)⋯(σF_j − F_j)⋯F_m and the restriction to the productive block are correct. Odd e_i gives −2u_iQ_b^{e_i} ≠ 0, because the Gaussian scalars have characteristic 0.
• §4. The reduction to elementwise dominance plus g decreasing is correct. An enlarged request costs g(r), and an added one costs at least 1.
• §5. The worst case of c smallest targets against c largest paid ranks is the right extremal pairing, for the same reason.

Two scope points. Neither is a counterexample.

1. The formal-model premise, which PROOF.md §7 records. In the literal algebra X_q^2 = 1 and C_q ∈ span{1, X_q}, so σ does not exist there. The obstruction covers programs that factor in the formal Laurent model. It does not cover programs that exploit the involution relation.
2. (P) and routes that move q_b. (P) lists "routes fixing q_b" as σ-fixed. A route that moves q_b, such as an address map sending q_b into the span of other axes, is not a K-linear factor on field coordinates. It acts as a substitution of the Laurent generators. For example, exchanging q_a ↔ q_b conjugates σ into Q_a ↦ −Q_a. Under the current wording such routes are neither counted nor explicitly excluded. I suggest stating in (P) that a realization does not move q_b outside fixed-frame routes, or else charging those routes. With that sentence the theorem's hypotheses are complete as written.

This is an author-independent check of the arithmetic and the written argument. It is not maintainer acceptance, and κ is unchanged.

Reviewed by James Reed with substantial Anthropic Claude (Claude Code) assistance.

[Matarener]

Note on open questions 1 and 2 (local arithmetic, no new κ). Reproduced the selected root 0.0030201659770, the optimistic root 0.010235821 and the headroom 1859.817.

Q1 (ten-axis native). By the credit-rank lemma, the 1,904 native q_b contacts each cost at least g(r) = (r+1)^t − r^t on some native child. Taking the cheapest conceivable placement (+1 rank on the largest existing native children: the eight centres, then ranks 5 and 4) costs 1888.99 units, against 1859.82 of headroom. That leaves a shortfall of ≥ 29.2 units (1.6 per native call), even before checking that each contact reaches its own target. So a redesigned word must improve its own H_t − P_t by about 3.9% of κ (dS/dC ≈ −2.47·10^5). That is roughly 73× the whole #4610 native gain (≈ 0.40 units).

Q2 (helper-exit credits). Pairing the smallest helper targets with the largest paid requests regardless of dataflow gives a ceiling of 0.0030844540 (+2.13%). Restricting contacts to each helper's own children (ranks ≤ 5, against helper targets 4–9) makes every pair net ≤ 0, apart from (4, 5) pairs worth about 0.1 unit in total. So the realistic gain is ≈ 0. As a control, main-only credit reproduces your 0.003035068.