[petalas]

Result

Scoped exact finite computation with a finite certificate, in the PR1898 abstract tariff. No κ. In the path-lowered even-subset class of #2677 (research/mutable-source-789q-invoice), every eight-query word needs at least 74 readouts R and at least 4 source XORs X, and one closed walk attains both. The eight-query row of #2677's table was a lower bound (R >= 74, X >= 0, tariff >= 452). It is now exact:

min over eight-query words of  b*Q + a*R + x*X  =  8b + 74a + 4x   for all b, a, x > 0
tariff at (b, a, x) = (38, 2, 2):  460, which is 24 above the six-query endpoint 436
witness (R = 74, X = 4):  closed prefix walk (0, 1, 3, 2, 6, 14, 30, 20, 0)

What was known and what is new. R >= 74 is #2677's census. X >= 4 is the connector consequence recorded in #5935 (this author) with the component inequality of #2455; a two-line proof for the path-local class is included for self-containment. New here:

• One walk attains both floors, so the Pareto frontier at Q = 8 is the single point (74, 4) and the optimum is price-independent. The same holds at Q = 6, (96, 8), and Q = 7, (82, 6), which are existing values this package reproduces.
• Exact census spectra over every set of 5, 6 and 7 connector values. Repeated prefix values force R >= 82, so the exact minimum at eight queries is 74 with repeats allowed, attained only with seven distinct nonzero values. This answers Open next question 2 of Research: 7–9Q mutable-source invoices cannot beat six-query tariff 436 #2677 and the readout half of Open next question 2 of #2686 within the class.
• Sharp XOR floor 4 at eight queries for arbitrary restored words, from Prove connected restored source-XOR words need 2(n-1) calls (no new κ) #5935's lower bound and the attaining word (Open next question 2 of #2643).
• Exhaustive X-floor layers at Q = 6..10 (271,170, 604,800, 730,800, 453,600 and 113,400 walks, equal to the closed-form count) and an attained value 486 at Q = 9 (floor 470).
• Crossovers among Q = 6, 7, 8: f8 < f7 iff b < 8a + 2x, f7 < f6 iff b < 14a + 2x, f8 < f6 iff b < 11a + 2x. At (38, 2, 2) all fail, so the six-query endpoint stays optimal among Q <= 8.

Evidence and reproduction

Python 3.12, standard library only. From the repository root:

~/git/integer-mult-bounds/.venv/bin/python3.12 -B research/eight-query-path-lowered-tariff/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/eight-query-path-lowered-tariff
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/eight-query-path-lowered-tariff --check-source-history

All three passed locally. Timings: verifier 53.6 s, registered replay 49.7 s. Expected marker: PASS_EIGHT_QUERY_PATH_LOWERED_TARIFF Q8=(R74,X4) tariff=460 floors_attained_Q6_Q7_Q8 Q9_in_470_486 no_new_kappa, then PASS_REGISTERED_RESEARCH_PACKAGE, then PASS_RESEARCH_COMMIT_BYTES.

The verifier imports #2677's model.py as a pinned live input after a SHA256 check. It checks the readout census (3,535,767 value sets, 3,097,024 covering), the path-local XOR floor exhaustively on 638,832 covering walks in F2^2..F2^4, every X-floor layer with its closed-form count, literal dirty-row replays of the Q = 8 witnesses, the Q = 7 champion, the six-query word and the Q = 9 witness, the crossover equivalences on a price grid, and 6,000 seeded random walks against the pinned model. The exhaustive search that found the witnesses was a separate unpackaged program; every number quoted is recomputed by verify.py.

Limitations and unrun obligations

• The exact row is for path-lowered even-subset words with Research: 7–9Q mutable-source invoices cannot beat six-query tariff 436 #2677's tariff. Readout and spectrum claims are not checked for arbitrary XOR-interleaved restored words.
• The XOR lower bound for arbitrary restored words is Prove connected restored source-XOR words need 2(n-1) calls (no new κ) #5935's with Prove eight source XORs are necessary at the six-query endpoint #2455's component inequality. It is cited, not re-proved here, and Prove eight source XORs are necessary at the six-query endpoint #2455's inequality is taken as proved for five sources.
• Nine queries are not settled (between 470 and 486). The ten-query layer is a cross-check only: it reproduces the read-only 64-readout repair and the four-XOR requirement at 62 readouts that earlier packages record. No ten-query claim is made.
• The tariff is the abstract complete-call fixture. No physical supplier, hosting or precision claim. Independent human and formal proof-kernel review are unclaimed. No Lean or Docker run.

Open next questions

1. What is the exact tariff at nine queries? Enumerate the X = 6 layer (weight total 12) and compare its least readout count with 70.
2. Does the readout floor 74 hold for arbitrary XOR-interleaved restored words at eight queries? The census uses only the set of connector values, so the first step is to check that a restored word's connector values obey the same even-subset readout constraint.

Related open work

• #2817 (VictorLiwentao): transfer-coupled (5,1,1) seven-query readout obstruction. Same abstract tariff, different interface; no numbers compared.
• #6695 (this author): toggle alphabets in the Research: sharp four/five-source dirty-connector boundary (no new kappa) #943/Research: one all-five toggle has no favorable price (no new kappa) #3321 connector batch. A different interface from these common-B mutable-source words. Its 72 queries and 74 additions are counts in that interface, unrelated to the readout floor 74 here; no numbers compared.

CI notes

At head 2f4add4b548b7eb0ae38abb61fa5dc15d5a2a857, both hosted checks passed, plan and replay (research/eight-query-path-lowered-tariff), in run 41721128970.

This contribution was produced by Nick Petalas.