[Matarener]

Requested by Mark

Change and mathematical scope

Negative result (budget screen plus scoped closures); no new κ. The selected κ is unchanged.

zxv's p3 H branch (#4359, H = I + BC, CB = −2I) has a high raw ceiling. It is blocked by three Q K Q interval endpoints whose direct correction costs 672 (complex) and 666 (BIT) rank-one requests. #4431 closed completion-only fresh inverses. This package adds the following:

• Exact budget at κ = 0.00302015 (Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470). At most 322 complex and 397 BIT rank-one calls are affordable; budget + 1 is certified negative. Saving roots: free boundary 0.00638 / 0.00793; one shared head conjugation (224 / 222) 0.00404 / 0.00518; two 0.00172 / 0.00245; the direct three fail even at s = 0. A corrected H branch therefore beats the frontier only if all three endpoints cost at most about 1.44 head conjugations. These are raw rank ceilings, without outer rows, tails, compiler gates or decoder scalar/movement costs.

• Whole-bank Q events cannot fit. At most two fit (224 ≤ 322 < 336), and Conditional κ = 0.0023 (+5.690e-4, +32.87% over #4181): productive helper jobs compiled into a uniform supplier #4359's two-event screen already excludes two. Three would give a complex root near 0.0029.

• Port subsets cannot be skipped. The native H112 has rank(H − I) = 8, but all 112 rows and columns of H − I are nonzero.

• Gather decoders through the rank-8 bottleneck cost ≥ 672. By Q² = I, QKQ − K = QMQ − M with M = K − I of low rank. However, Q on port p is one rank-one request along that port's own line, and requests on different lines cannot be merged. All 112 complex query lines q and all 111 BIT port lines are distinct. With independent inputs for the three roles, any linear gather decoder needs one request per port on each side for each role, which equals the direct bill.

• No controller in this family has fewer Q-conjugated roles. A census of every free noncommuting controller of the retained H probe for p = 1..4 and every idle mask finds 2, 3, 8 and 20 gauge-compatible ones. Each has exactly p Q K Q roles with nonempty K, because each local call stamps its stage letter on both streams of its pair. With H surplus 224p − 8(2p+1) and 224 per repaired role, the direct bill leaves −8(2p+1) < 0 for every p (−56 at p = 3, as Conditional κ = 0.0023 (+5.690e-4, +32.87% over #4181): productive helper jobs compiled into a uniform supplier #4359 reports). Each pair's endpoint repair costs exactly the rank credit the pair brings.

• Shared query lines do not escape (README §6). The local call is I = K + BA with pair reads K only on orthogonal F2 labels. On H112 all labels are odd and J vanishes on all 6,608 non-orthogonal label pairs, so ports sharing a line have BA = I on their class and are served as independent copies. The tail gather is 112 + 112 for P112 and 224 + 224 for a doubled unit (224 ports, 112 lines), which is 2 per port in both cases. For any unit with V ports and rank h the H branch nets 2Vp − h(2p+1) − 2p(V − h) = −h < 0, whatever the line layout.

Remaining scope: the section 4 bound covers linear gather decoders at fixed H112, owners and p3 controller. It is not a theorem about all algorithms. Section 6 extends it to every line layout of a local call of the form I = K + BA. A local identity of another form, or a controller not built from Q-gauged pairs, is outside it.

Task and overlap

• Card. complex-new-helper-inventory, priority revision 7. Scope changed: it screens the existing H interface and closes decoder routes rather than constructing a new interface (reason in TASK.json).
• Base. f626db5c3a3d781d8f7695038776c621651f75e1. All 16 inputs are pinned in SOURCES.json and declared in VERIFY.json.
• Overlap. Conditional κ = 0.0023 (+5.690e-4, +32.87% over #4181): productive helper jobs compiled into a uniform supplier #4359 (AGGREGATE-H-PROBE, PR589-H-GLOBAL-PROBE, JOINT-BOUNDARY-PROBE) gives the direct bills, moments at 0.004 and the two-event screen. Prove a completion-independent native H obstruction from one ordinary edge #4431 gives the completion-independent fresh-sector obstruction. Neither prices the boundary against the current frontier or treats port-subset and gather decoders. A search of PRs This section's polynomial-inverse error is strictly below 2^{-475} (no new κ) #4398–Independently check #4554's native banded inverse: exact elimination bounds, Gaussian tail, exact periodic Woodbury solves (no new κ) #4556 found no other H-branch work after Prove a completion-independent native H obstruction from one ordinary edge #4431. Compile native dyadic blocks: 766 → 698 scalar allowance (no new κ) #4494/Exact native gauges: 698 → 674 scalar allowance, with a zero-scratch option #4499/Borrowed rank-one adapters: 674 → 666 scalar allowance with no extra fields #4502/Levi-adaptive gauges: 666 → 616 scalar allowance; lower suffix error #4518 (scalar allowance 766 → 616) concern cleanup ADDs, not H endpoints.

Validation

python3.12 -B scripts/research_package.py --package research/h-boundary-budget --check-only
python3.12 -B scripts/research_package.py --package research/h-boundary-budget
python3.12 -B scripts/check_research_commit.py --package research/h-boundary-budget --check-source-history

All three pass on CPython 3.12 (Windows). The verifier takes about 4.5 minutes and uses only the standard library. Moments use the retained exact interval h_interval of the source modules; the rank and support of H − I and H² = I are exact over Q. Controls: budget + 1 is rejected, a perturbed H − I fails rank 8, a duplicated query line is detected, p3 keep-43 reproduces the probe parity [0,1,0,1,1,0], an even label is rejected, and one merged port class needs only rank 8. The gather lower bound in README §4 is a written argument, not machine-checked. No Linux run.

The research/kappa-rebuild modules import research/kappa-next-regime from the repository root, which is absent on main. The verifier adds the vendored copy under research/kappa-0023-productive-jobs/dependencies/.

Credit

The H construction, endpoint factorisation, retained profiles, receipts and the two-event screen are zxv's (#4359, #4431) together with their cited predecessors. Mark (Matarener), with substantial Claude Code assistance. Apache-2.0. No independent human review.

Open next questions

See README "Open next questions" and TASK.json:

1. A local call outside I = K + BA, with pair reads not tied to label orthogonality. README §6 closes shared query lines within that form.
2. A controller not built from Q-gauged pairs. README §5 closes controllers with fewer Q-conjugated roles for p ≤ 4.

🤖 Generated with Claude Code