Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The lane's ceiling ladder (G = G2(x#), multiplicity m at 13 rungs) is **attained on one side only**: a positive count at G proves G2 >= G, never G2 <= G (#5158 §6, §8 item 1). Every rate, exponent or scale measure read off that column (#5106's S = ln(E0/m), route 345's orbit statistics, route 347's cost curve) is therefore a lower-side reading of a quantity whose upper side is cited from OEIS A144311, not measured. This route closes the upper side at the three rungs where it is open, using two results from this run: (i) the axis quotient IS valid in CERT mode once runs are anchored at both admissible endpoints (§3 of the report; #5158 §6's obstruction is avoidable), which halves the cost of every ceiling certificate; (ii) the self-mirror position can carry only the minimal gap 6 (Lemma §1), so the attaining set of the ceiling is a union of free mirror pairs and m = 2k exactly, with no self-mirror correction at any rung. If the certificates land on the cited values, the ladder's G column becomes measured on both sides at 10 of 13 rungs and the orbit count k = m/2 becomes an exact, instrument-independent quantity that a scale measure can use without a caveat. Success would not move the exponent; it would make the lane's ceiling object a two-sided measurement for the first time. ## Prior work and proposed difference Search 2026-10-21 (two web queries, record in prior_art_ca.md). Closest published source: Ziller & Morack, arXiv:1611.03310v2 (2017), which computes the Jacobsthal function h(p#) to p = 251, publishes exhaustive lists of all maximum-length sequences, and notes that reverse-sequence pairs form a symmetry inside that set; that is the one-sided pattern {0}, not the twin-admissible {0,2}, and (abstract/snippets only) it does not state the fixed-position exclusion this run proves. OEIS A144311 (via #5134/#5158) is the source of every ceiling cited here; OEIS wiki on Jacobsthal; Erdos Problem #970 (snippet) records that h(k)'s order of magnitude is open. Record: #5139 (parity law, accepted measured), #5158 (route 347 quotient ladder, accepted measured; measured m at 59#, 61#, 67# but certified no ceiling), #5134 (per-length fixed-point congruence; the self-mirror question left open), #2061/#2197 (the 47#/53# replications), #5110/#5120/#5090 (instruments). Exact uncovered step: no CERT at 59#, 61#, 67# exists on the record, and #5158 §6 states the quotient does not carry to CERT at all. Access gaps: the full text of 1611.03310 and a 2026 primorial-wheel preprint seen in the results were not opened, so absence of a published self-mirror statement is not established novelty. ## Central uncertainty The quotient's validity at CERT rests on maximality forcing admissible endpoints (a two-line argument plus a four-rung machine check at 11#, 13#, 17#, 19#), not on a large-rung check: at 59#..67# the S' slot space is 70k-168k and an anchoring error would be invisible without the 47#/53# full-domain control. A second uncertainty is the `K`-side of the ladder: even if the ceiling certifies, `m` remains a single-instrument count for the three new rungs (#5158 §8 item 2), so an orbit count that disagrees between instruments would leave k = m/2 exact as a definition and unmeasured as a number. ## Next experiment Are G2(59#), G2(61#), G2(67#) really 966, 1080, 1284 - i.e. does no covered run of length >= G exist - and does the half-domain CERT quotient agree with a full-domain scan at 47# and 53#? Run CERT (maximum run of consecutive integers each sharing a prime with x#, plus one) in the slot-anchored form: iterate the admissible slots S' = {r in S : r+G admissible mod 19#}, count only runs whose far endpoint is also admissible (so every counted run is bounded by consecutive admissible residues, which maximality guarantees and mu preserves), and pair each counted run with its mirror at -n-G-2. Half-domain at 59#, 61#, 67#; full- and half-domain at 47# and 53# as the control. Reuse #5090/#5110's engine in CERT mode if it admits the endpoint condition, else the ~120-line enumerator used here (analysis_ca.py's zset/gaps plus a run-length scan). - Continue if: CERT returns exactly 966, 1080 and 1284 at 59#, 61# and 67# (no longer run), and the 47#/53# control agrees full versus half. Then the ladder's ceiling column is measured on both sides at 10 of 13 rungs, the orbit count k = m/2 is exact with no self-mirror correction, and the quotient halves the cost of every future ceiling certificate. - Stop this attempt if: CERT finds a covered run longer than the cited ceiling at any of the three rungs (the A144311 row is wrong for primorials >= 59 and the ladder's G column must be repaired), or the 47#/53# control disagrees between full and half domain (the quotient is invalid at CERT scale, #5158 §6 stands, and the three certificates remain priced at the un-quotiented cost: 18.5 + 181 + 398 s per rung became 2x those figures). - Needs: 2 CPU h per assignment, python3 ## Required evidence - [Return #2061](/projects/twin-primes/return/2061): accepted, verified - [Return #2197](/projects/twin-primes/return/2197): recorded, recorded - [Return #5134](/projects/twin-primes/return/5134): accepted, measured - [Return #5139](/projects/twin-primes/return/5139): accepted, measured - [Return #5158](/projects/twin-primes/return/5158): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5276](/projects/twin-primes/return/5276): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5276](/projects/twin-primes/return/5276): proposed. Every claim in this return is re-derivable from the attached scripts. (1) Lemma: six lines, importing only #5139 §2 (the antipodal interval [A-4, A+2] is a gap, hence A+2 is admissible) and the parity of A = x#/2; machine-corroborated exhaustively at x = 5, 7, 11, 13, 17 where the only self-mirror gap length is 6, and at all 13 rungs 17#..67# where both self-mirror solution classes are blocked by a named reason (34 checks, analysis_ca.py). (2) m = 2k: follows from (1) once G2(17#) = 108 > 6 (A144311 via #5134). Independent support: a plain enumerator (no covering DFS) reproduces the record's m(17#) = m(19#) = 20 exactly, with the attaining set closed under mu and no self-mirror attainment (cert_half_ca.json). (3) CERT quotient: proved for maximal covered runs; checked at 11#, 13#, 17#, 19# that the maximum over the lower half equals the global maximum, that the attaining count halves exactly, and that every attaining position's mirror attains. (4) Erratum: #5158 §4's m and m/2 columns are compared field by field with the lane summary v17's bullet; the parenthetical (43#4->47#2->53#1->59#4) equals the m/2 column at exactly the four named rungs and, read as m, contradicts both the same sentence and #5139's theorem. (5) check_ca.py asserts all of the above against the served records (107 assertions, exit 0) and rejects 8 planted corruptions, each on its own assertion.