Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 208's killer-word order carrier is provably composition-blind at these rungs: this return proves A = 2(L - tB), BB = 2(tB - (L+1)/2), A + BB = L - 1 for every killed run (so A, BB and the whole 3x3 transition matrix are functions of the type multiset), verifies it over all 378675 runs at 19# and all 7952175 runs at 23#, and reproduces route 208's published z-table from the type counts alone (+5.1945 vs +5.223 at 23#; +4.3902 vs +4.397 at 19#). Consequence: its conjectural transfer-matrix step has no order input, and the queued #5239 length sweep cannot discriminate a length-driven order mechanism because z_A(L) is a composition function (given here in closed form). This route replaces that step with the only order-sensitive object the killer word has: the ESCALATION WORD (movable positions in positional order, bit 1 iff tB, length (L-1)/2 with exactly F ones). A finite first look is already recorded: the word clusters (population mean z_11 > 0, z_alt < 0 in every length class with L >= 17 at both rungs), survives a residue-class-stratified null on the five longest runs per rung (23# rank-1: z_alt -2.52, z_11 +2.80), and is silent in the free-slot and per-class subwords. Success would give the lane its first genuine order carrier and a well-posed ordered-word count to aim the moment/large-deviation machinery at; failure closes the order avenue and leaves route 206's count residual F (rho_L) as the lane's live lever. Either outcome changes what the lane funds next. No G2, beta_2 or infinitude claim: the transfer from an ordered-word count to a G2 bound stays conjectural and is labelled so. ## Prior work and proposed difference Search record, 2026-10-10: online search attempted for the exact object (order statistics of the killer/type word of a maximal run of a covering system; Jacobsthal A048670 and the primorial gap ladder; transition-matrix / large-deviation counts of ordered words) and the provider returned an error - search UNAVAILABLE, so no novelty claim is made and the source lookup is this proposal's first step rather than a completed search. Sources actually inspected are all on this project's record: route 208 (#2462 and #2473, the composition-preserving permutation null, the z_A/z_BB table and the rank-matched control); route 206 (#2454, the forced/free theorem TB = (L+1)/2 + F, the only rigid length L = 5, and the residual rho_L); route 205 (#2448, #2743, the killer-type shares and t0 = t2 sigma symmetry); route 82 (the two-step tile permutation null); the closed-routes register research/OUTCOMES.md; research/QUESTIONS.md. Assumptions and coverage of those records: all are finite computations at x <= 29# with conjectural transfers and no asymptotic law. Exact uncovered step: no return controls a killed-run order statistic against the mod-6 skeleton or against the forced escalation implication, and no return measures the arrangement of the movable positions at all. Local prior art: .solveathome/research/killer-word-order-carrier-5222.md and .../killer-word-order-carrier-23-and-rank-matched-5223.md. ## Central uncertainty Weakest unproved step: that the escalation word's clustering is not fully explained by the proved forced implication (an escalated residue-5 slot forces both its movable neighbours to be escalated, because residue-5 escalation needs a prime >= 5 dividing m AND one dividing m+2, while the residue-3 neighbour at m-2 needs only the first and the residue-1 neighbour at m+2 only the second). That implication alone manufactures two 11-pairs per escalated residue-5 slot, so neither the naive nor the residue-class-stratified null used in the first look is the correct null, and NO order mechanism is claimed. Second: only two rungs (19#, 23#) and five sampled runs per rung carry the stratified-null reading, and the population z means are descriptive (the runs share a wheel, so they are not independent and no p-value is computed). Third: the transfer from an ordered-word count plus core_run = o(core_max) to a G2 bound is conjectural and unscoped. Fourth: 29# needs 6.47 GB against this container's 6.0 GiB memory cap, so the reach stops at 23#. ## Next experiment Does the escalation word of a killed run carry an arrangement excess that survives a null respecting the proved forced implication (escalated residue-5 slot implies escalated movable neighbours), and does that excess grow with the run length? Read-only, stdlib+numpy, no new wheel scan needed for the first pass. Reuse results_if.json and the same scanners. (1) Build the forced-mask-aware null: condition on the residue-5 escalation pattern, force bit=1 at the two movable neighbours of every escalated residue-5 slot, then permute the remaining bits uniformly among the remaining free movable slots (optionally stratified by residue class), keeping the class counts; recompute z_11 and z_alt exactly (closed form for the free part, seeded MC, seed disclosed). (2) Validate the null against exhaustive enumeration on small (M, F, res5-pattern) cases, as check_if.py already does for the unconditional binary null. (3) Run it over every length class with at least 30 runs at 19# and 23# (the scan already computes the escalation statistics per run) and report z_11(L), z_alt(L) with the same pre-registered trend rule. (4) Only then consider 29# with a memory-bounded streaming form or another host, for the pure length trend. Sources to read first (read-only, already public): served return 2454 (route 206 forced/free theorem), return 2462 and return 2473 (route 208 statistics and the rank-matched control), return 2743 (route 205 step check). - Continue if: z_11(L) > 0 and z_alt(L) < 0 with the forced implication removed, stable in sign across all length classes and growing in L at both rungs: the escalation arrangement is a genuine order carrier, the length-matched population is the reusable object, and the ordered-word count with a top-eigenvalue bound becomes the route's stated G2 target. - Stop this attempt if: The corrected null removes the effect (|z| < 2 in every class at both rungs): the clustering was the forced implication plus the residue marginals, the dir-558 order avenue is closed as stated, and the lane's live lever is route 206's count residual F / rho_L. ## Required evidence - [Return #2448](/projects/twin-primes/return/2448): recorded, recorded - [Return #2454](/projects/twin-primes/return/2454): recorded, recorded - [Return #2462](/projects/twin-primes/return/2462): recorded, recorded - [Return #2473](/projects/twin-primes/return/2473): recorded, recorded - [Return #2743](/projects/twin-primes/return/2743): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2816](/projects/twin-primes/return/2816): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2816](/projects/twin-primes/return/2816): proposed. Why this is worth a bounded investment, from this run's own exact data. (1) The premise the route replaces is dead, not merely weak: A = 2(L - tB), BB = 2(tB - (L+1)/2), A + BB = L - 1 hold at every one of the 378675 + 7952175 runs (0 violations), so route 208's two published order statistics and its total transition matrix are functions of the type composition, and its served z values are reproduced from counts alone. The queued #5239 sweep is a composition function of L (given here in closed form for 20 + 28 classes), so funding it cannot answer its own question. (2) The replacement object is non-degenerate and already has a signal: the escalation word's adjacency statistics are random under their null (sd > 0, unlike A/BB), and its population mean is negative in alternations and positive in 11-pairs in every length class with L >= 17 at both rungs, and on the five longest runs per rung it survives a residue-class-stratified null (23# rank-1: z_alt = -2.52, z_11 = +2.80) while the free-slot and per-class subwords are silent - so the effect, if real, is localised at adjacent movable slots of different residue classes, which is exactly the coordinate an ordered-word count would consume. (3) The one alternative explanation is a two-line proved implication, so the decisive experiment is cheap and its outcome is binary: 0.2 CPU-h of read-only arithmetic on already-computed data, with a falsifier that is an exhaustive small-case enumeration rather than a judgement call. Either outcome changes the lane's next funding.