[unpaidmoon]

Change and mathematical scope

On the fixed PR #5745 source, no three transport-grade-below-8 temporal response spaces sum to a nonzero clean mode. An explicit rational relation among 41 dimension-1 frames has clean mode (0, 0, 0, 0, 14, -14, -14, 14). The clean generation number γ therefore satisfies 4 ≤ γ ≤ 41.

This is a scoped exact certificate. It is not a schedule, a paid-profile improvement, or a change to the selected frontier κ = 781451/250000000. It does not extend the nine-channel 8 H_8 floor to ten channels. That extension needs γ ≥ 11, which remains open. Supports 4 through 10 are not ruled out.

Task and overlap

Base commit: 2485805a6d47f36ea83ddde6327d453ac82d44dc.

Task card: joint-query-controller (priority revision 8). Scope status: refined. The card's first experiment says to audit the #5745 delivery claim or change its channel hypothesis, and not to repeat the free-injection search. This package answers the three-frame question left open by research/nine-channel-monotone-bound: pairwise sums miss the clean mode, and the abstract ten-channel control needs a triple that meets it. No such triple exists on this source. No new caller is constructed.

Nearest work:

• Ali Romman, nine-channel certificate and PR Prove nine-channel delivery floor for monotone tariffs #5845: γ ≥ 3 and 8 H_8 for at most nine channels under every positive nondecreasing canonical transport tariff. This package starts from that pair theorem and decides the next open case.
• Ali Romman, PR Prove the sharp all-rank floor for immutable mode delivery #5823: concave power-tariff bound for every finite channel count. Not replayed, and not the claim made here.
• PR Construct a 22-channel monotone delivery counterexample (no new κ) #5881: a 22-channel monotone counterexample. Different channel count and different claim.
• research/three-frame-shared-map: compatibility of three abstract subspaces under a common invertible map. Not this source and not the clean-mode sum.

The census follows research/nine-channel-monotone-bound/independent.py (Ali Romman, Apache-2.0) and does not import it. Source pins are in SOURCES.json. The original word archive is research/filtered-native-kappa004/inputs/original-packet.tar.gz.

Validation

CPython 3.12 and numpy==1.26.4. From the repository root:

python3 -W error -B research/ten-channel-triple-span/verify.py
python3 -W error -O -B research/ten-channel-triple-span/verify.py

Both rerun the orthogonal census (15,752 contacts, 14,816 distinct frame/columns, 12,240 projected columns, 879 frames), the 41-frame witness, the 3,782 mode-free intersecting-pair reductions, and an exhaustive F_631 screen of every unordered triple with exact rational fallbacks. Proof checks raise SystemExit, including under python -O. Local runs printed:

PASS_TEN_CHANNEL_TRIPLE_OBSTRUCTION
upper_support=41
intersecting_pairs=3782 exact=136832 deferred=3256335 small=102715700 large=10090179 extensions=3198419

Witness frames: 27,31,8,12,39,44,80,81,85,99,46,48,49,54,68,71,73,76,69,95,96,97,61,64,66,330,3,18,107,326,328,4,101,23,5,112,16,88,92,7,331.

A nonzero minor modulo 631 is independence over Q. Full modular row rank is the same. Rank-deficient stacks are reduced over Q with fractions.Fraction and are required not to meet the clean mode. The direct sweep count equals C(879,3), and the extension count equals 3,782 × 877. No Lean replay and no selected-frontier checker were run. Metadata check: python3 scripts/research_package.py --package research/ten-channel-triple-span --check-only.

Credit

The census method, the pair theorem, and the nine-channel tariff argument are Ali Romman's. The audited delivery claim is PR #5745; the word is from PR #5720 (flyingviper). Predecessor source attribution is retained in PROOF.md.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

• Is γ at least 11 on this source? That is the missing inequality for an 8 H_8 floor at ten channels under every positive nondecreasing tariff. First step: screen four-frame sums that contain one of the 3,782 intersecting pairs. The triple certificate does not bound supports 4 through 10.
• Can mixed response dimensions undercut the explicit 41-frame relation? The witness is only an upper bound. First step: decide whether 41 is minimum inside the 14-dimensional dimension-1 kernel before searching mixed supports.
• Can the 41-frame mode be scheduled below 8 H_8 after transport and restoration? An algebraic dependence is not a paid supplier. First step: price one witness under the PR Prove sharp 8H8 active-mode delivery cost with temporal reuse #5745 channel contract. No chronological schedule is constructed here.

Workflow feedback (optional)

The registered replay allows 600 seconds per command. A single-core pass of this screen was over that cap. Scheduling disjoint batches across the allowed cores, and building the second projection only after the first minor vanishes, brought each command to a few minutes without dropping a triple.