[petalas]

Result (scoped proof, no new κ)

zxv's #5554 (research/kappa-architecture-frontier) reduces every FULL/QUERY mask of the rank-b productive-helper path to one canonical invoice per FULL count, assuming each QUERY costs v rank-one jobs. Its Open next question 3 asks about a changed query law. This package changes only the query price to w*v and proves, with t=1-C:

• For 1 <= w <= w* = b^t + (2b+1)^t - (3b)^t, the canonical reduction holds at every n, for all pair counts, splits and masks. The minimum is C_n + (w-1) v |n-3|. Convexity, the three-sign exclusion and the price frontier/ratio gate transfer with c_n -> c_n + (w-1)|n-3|.
• For w > w*, merging the two outer 10 pairs into one 11 pair beats the canonical invoice at every n >= 6, by 2v(w-w*). Research: the productive-helper path (#4440 lineage) can't pass κ ≈ .00304 without cheaper helpers #5554's cap must not be transferred.
• For w < 1, the family is unbounded below at every n.

At b=8 and C=.0034, w* = 1.03859829952. A per-query surcharge above about .0386 rank-one units per port breaks the canonical form.

The proof reuses #5554's moves, sorted by their effect on the query count. Moves that delete queries only improve when w >= 1. Moves that keep the count are unchanged. The 11-split adds two queries and costs w - g_k, with g_k increasing. A new move handles the k=0 case: end bit 0 plus pair 11 becomes end bit 1 plus pair 10, which is strictly favorable by subadditivity.

Evidence and reproduction

~/git/integer-mult-bounds/.venv/bin/python3.12 -B research/path-query-price-threshold/verify.py
# PASS_PATH_QUERY_PRICE_THRESHOLD masks_p_le=6 block_ranks=2 exponents=4 checks=456  (~7 s)
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/path-query-price-threshold
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/path-query-price-threshold --check-source-history

All three pass locally on CPython 3.12. The verifier evaluates the literal equation (1) with query price w on every mask with p <= 6 and every split. It uses b in {2,8}, t in {1/2, 9/10, 4983/5000, 999/1000} and 60-digit decimals. It checks that the canonical invoice is minimal below w*, the merge identity and gap above w*, monotonicity of g_k, and the 00-pair gain below w=1. It pins the bytes of #5554's PROOF.md.

Limitations

• These are finite controls on a written proof. There is no all-p search and no proof-kernel or independent human review.
• The result covers Research: the productive-helper path (#4440 lineage) can't pass κ ≈ .00304 without cheaper helpers #5554's relaxed raw invoice family with one changed tariff. It does not cover native words, compiler closure, BIT/outer funding or physical query realization.
• Research: the productive-helper path (#4440 lineage) can't pass κ ≈ .00304 without cheaper helpers #5554's exchange argument, inherited from Prove an unbounded FULL/QUERY controller ceiling (no κ gain) #5405/Independently verify the controller-family ceiling by convexity (no new κ) #5409, is cited as a premise and not re-verified.

Open next questions

1. Which masks are optimal for w > w*, once 11 side pairs survive?
2. Does the reduction survive a rank-q query that also earns a rank-q target?
3. Does a surcharge below w*-1 move Research: the productive-helper path (#4440 lineage) can't pass κ ≈ .00304 without cheaper helpers #5554's optimal FULL count, 16 for the Improve complete conditional κ to 0.00304153 via coupled P104 contacts #5551 bill at C=.0034?

Related open work

• No open PR studies the query tariff of the Research: the productive-helper path (#4440 lineage) can't pass κ ≈ .00304 without cheaper helpers #5554 path family. Joint bit+complex supplier cannot beat the average of the two p16 invoices under the whole-job premise (negative result) #5298 (Matarener) studies a joint BIT/complex invoice under a whole-job premise. It concerns a different object, and no number is shared.
• Prove the exact 406-merge threshold for the frozen joint invoice #5932 (this account) prices cross-side merges in Joint bit+complex supplier cannot beat the average of the two p16 invoices under the whole-job premise (negative result) #5298's frozen joint invoice. It is a different invoice and model, and no number is shared. No other open PR from this account studies the Research: the productive-helper path (#4440 lineage) can't pass κ ≈ .00304 without cheaper helpers #5554 path or its query tariff.

Credit

The path family, equation (1), the canonical reduction and the price frontier come from zxv's #5554, which builds on Ali Romman's #5405/#5409 and the predecessors credited there. The query-price parameter, threshold, k=0 move, sharpness witness and unboundedness result are new here.

CI notes

All checks passed on head 6bc26ee287c074011576f067d19c1e97ef1344ff: plan and replay (research/path-query-price-threshold), run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40524001602.