m37

Prompt

R37_PAR_LIMIT_prompt.txt 100 % RESEARCH QUESTION (plane continua / composants of accessible points and prime ends; ISO-F and ONE-CLASS (the side-selection node of SS-COF, Route I): Theorem PAR forbids ONE-CLASS under joins, so a strip without failing doors needs cofinal LIMIT transitions of composant class on both banks of the outchannel; do limit transitions on both banks contradict PAR or any law of an R-instance?). Self-contained. Label every claim PROVED (you prove it), SOURCE (exact reference; unchecked?), LAB (from this prompt, status as given), CONJECTURE, OPEN, or RESTATEMENT (a gap reworded). If a definition is ambiguous, state your reading in one line and continue. Say where your argument stops. A. DEFINITIONS Setting (compressed from the lab lanes b_opus_sscof_cl, b_opus_thetax_bound and b_opus_sscof_trace; all single-agent, UNAUDITED unless a status says otherwise). - X is a nondegenerate nonseparating plane continuum, U := C \ X, h: X -> X a homeomorphism with h^2 continuum-minimal (cm: no nonempty proper subcontinuum K with h^2(K) inside K) and m_1 := min_x d(x, hx) > 0 (an R-instance). F, G are perfect plane extensions of h and h^-1 (NOT homeomorphisms; G is not F^-1). Each has a unique outchannel, prime ends theta_F != theta_G of U (psi: U -> disk conformal, S^1 the circle of prime ends). Composants are used only for indecomposable X (the setting of Lemma CL and IMP*); a composant is the union of the proper subcontinua through a point. Write x ~ y if some proper subcontinuum of X contains both (same composant). - A crosscut q is an open arc in U with closure an arc whose two distinct ends lie in X; Sh(q) is the bounded component of C \ (X u q); I_q is its shadow arc in S^1. KP-chords are the Kuratowski-Pelczynski chords of BFMOT. A canonical KP strip near theta_F is a nested chain of KP-chords (sections) c_e, e >= r_0, with I_{c_e} shrinking to theta_F; the feet of c_e are c_e^1 (alpha-foot, bank 1) and c_e^2 (beta-foot, bank 2). var(F, c) in {-1, 0} is the BFMOT variation count; by Lemma P it equals iota(h c^1, h c^2; w), the signed intersection of a short path from h c^1 to h c^2 with a ray from w in c. Case A means var(F, c_e) = var(F^2, c_e) = -1 on deep sections (the sections cross the channel essentially); then each deep section is E_F and M_G (EB2: c_e^1 and c_e^2 are NOT in one proper subcontinuum). - Untorn: for every rho > 0 there are arbitrarily deep sections c with BOTH joins short: h c^1, h c^2 are joined by a crosscut d_1 of diameter < rho and h^2 c^1, h^2 c^2 by a crosscut d_2 of diameter < rho. - Theta_X := the set of prime ends with impression X (closed). Composant labels (Lemma CL): c(theta) := the composant containing Im(theta) for theta outside Theta_X; c is constant on each component of S^1 \ Theta_X, so there are countably many labels. Along the line K_F of boundary chords seen from the F-auxiliary disk, composant classes are order-convex. A transition toward end i is either a chord transition (a failing chord K_q = X: the two feet lie in different classes; a prime end of Theta_X sits behind it) or a LIMIT transition (a point z approached monotonically by points x_k not~ z; a 4th-kind accessible prime end Xi(t_z) of Theta_X with point principal set and impression X). ISO-F_i (side i) means Theta_X misses a neighbourhood of theta_F on side i; ISO-F_i is equivalent to finitely many transitions toward end i (thetax Prop 5.4(c)), and to ONE-CLASS on bank i. Cor Lambda2: every full-impression prime end except theta_F is behind a boundary chord or is a limit transition. - Composant cocycle: J_P(x, y) := 1_P(x) - 1_P(y) for a composant P. It is additive, and J_{hP}(hx, hy) = J_P(x, y). B. LEMMAS (status; where used) Known (statuses as given; all UNAUDITED unless stated). - Theorem PAR (b_opus_sscof_cl, DERIVED; inputs Lemma P, IMP*, EB2, Lemma D, Lemma SR; weakest input: Lemma SR depends on the unaudited IMP*). Assume theta_F != theta_G, case A and ISO-F_2 (ONE-CLASS on bank 2, all deep feet c_e^2 in one composant P_2) . Then no deep section has both joins of diameter < rho. Cocycle form: J_{P_2}(h^k c^1, h^k c^2) = -var(F^k, c) for k = 0, 1, 2 (var(F^0, c) := +1); with k = 0, 1, 2 one gets h^-1 P_2 = h^-2 P_2, so P_2 = h^-1 P_2 and c^1 in P_2, against EB2. Consequence: in an untorn case-A strip, ISO-F fails on BOTH banks (mirror with ISO-F_1). - Theorem TT2 (same lane, DERIVED): every small KP chord across the channel is F-KP-torn or F^2-torn. So an R-instance cannot be untorn at every small chord; PAR concerns only the untorn deep sections that may still exist. - Lemma SR / Lemma CL (DERIVED): if no Theta_X point lies on the closed cap arc between the sections c_t and c_e on bank 2, then c_t^2 and c_e^2 lie in one composant. Labels change only across Theta_X. - SS-COF (no failing door of a canonical strip beyond some depth) kills chord transitions cofinally, so in an untorn case-A strip SS-COF can only hold through cofinal LIMIT transitions on BOTH banks (the strictness gap of sscof_trace section 5): infinitely many points z_n -> theta_F on each bank, each an accessible 4th-kind full-impression prime end. - b_opus_thetax_bound section 7.1 (DERIVED): none of cm of h^2 via bank composants, uniqueness of the outchannel, positive displacement, Baire category of Theta_X excludes cofinal Theta_X points theta*_n -> theta_F; a Tier-0 north-south circle model (circle homeomorphism phi with fixed points theta_F repelling, theta_G attracting; Theta_X = orbit of one point plus theta_F, theta_G; labels h^n(C) for a wandering composant C on the arcs between consecutive orbit points; one label on the other bank) meets every static law and has no h on X. C. THE TARGET Decide whether an untorn case-A R-instance can have cofinal limit transitions (4th-kind full-impression accessible prime ends accumulating at theta_F) on both banks, no failing chord transitions, all laws above holding. Equivalently: can PAR's escape through limit-only transitions on both banks be closed by a law that uses h? D. ASK Part 1 (method). Name ONE method that is not on the dead-end list and say why it can see what the failed ones could not. Failed methods saw either the variation line K_F (index, TT2, PAR at sections of variation -1) or proper continua (cm on h^2), and neither reads the composant of a bank point BETWEEN sections. Candidates, to be stated exactly if used (none is claimed to work): (i) composant-label combinatorics: treat the labels on each bank as a sequence indexed by the transition points z_n and ask what the identity J_P(hx, hy) = J_{h^-1 P}(x, y) iterated along the orbit (not only k <= 2) says about the class sequence, using that F^k has a variation law only for k <= 2 so far; (ii) Cantor-Bendixson induction on Theta_X near theta_F: the lowest-rank accumulation point of limit transitions and the image of its neighbourhood under h; (iii) prime-end arc length: measure each section by the conformal length of I_c and compare the lengths of I_{c} and of the shadow arcs of the joins d_1, d_2 (a tear is a loss of length, an untorn pair keeps it); (iv) your own. Part 2 (carry it out). Prove limit-only transitions on both banks impossible for an untorn case-A strip, OR reduce the claim to one named lemma L with gate G1-G4, OR prove that PAR adds nothing beyond ISO-F failing on both banks (say what is then the first law violated by the data of Part 3). Part 3 (builder or decisive test). Build the data: a circle-level model of the two banks with limit transitions z_n^(1), z_n^(2) -> theta_F, the label sequences, the h-action on labels forced by PAR (h maps the class of a bank-1 foot to the class of the beta-foot of the join), and check the cocycle identity for k = 0, 1, 2. Record the FIRST law violated (cm of h^2, EB2, uniqueness of the outchannel, m_1 > 0, Theorem Lambda, Lemma CL, PAR) or report consistency with the exact list of laws checked. A finite number of transitions is not a cofinal configuration. Each claim labelled PROVED, SOURCE (exact reference), LAB, DERIVED, CONJECTURE, OPEN or RESTATEMENT. UNKNOWN if unsure a cited result exists. What already failed (read it before you start): see DEADENDS. Do not repeat these as progress; a new method, even a partial one with an honest gap, is worth more than a polished restatement. Take your time and be extremely meticulous: check every step, every hypothesis and every quantifier before you write it down. This is a hard open problem, and I believe in you. E. CONTROLS (all must hold) (C1) North-south control: the circle model above (thetax 7.2) satisfies the static laws that failed (cm via bank composants, uniqueness, displacement, Baire) and has cofinal limit-type points on both banks only if one adds a second orbit on bank 2; it has no h, so any proof must name the line where h or F enters (J_P(hx, hy) = J_{h^-1 P}(x, y) or var(F^k)). (C2) HOM control: if F were a plane homeomorphism the R-instance is contradictory (fixed point of h^2 by the index count), so PAR and TT2 are consistent but empty; your argument must say which of its steps fails for a perfect extension that is not a homeomorphism. (C3) Bucket handle (Knaster): Theta_X is one 4th-kind prime end, so ISO-F holds with one transition; the argument must not claim zero transitions there. (C4) Solenoid translation (nonplanar): composants either invariant or wandering with cm holding in both cases; a law that only uses cm of composants cannot decide. F. DEAD ENDS (not progress) (1) Index budget and variation counts at doors or deep sections: all zero at doors (codex_sscof; sscof_cl section 4.5). (2) CL/Bell index transfer: the fixed-point-sensitive steps live on chords of variation -1; method ELIMINATED for SS-COF (FAILED_APPROACHES 2026-11-03). (3) Impression transport and type transport TT: a circle model with failing doors at every depth satisfies all of them (b_opus_imptransport section 6). (4) cm on Hausdorff limits of pocket traces: cl Sh(s) cap X = X, so the limit is X (MICROEVAL m24, 5 of 5). (5) Static laws against cofinal Theta_X points: cm via bank composants, uniqueness of the outchannel, positive displacement, Baire dichotomy (scattered, condensation) are each blind (thetax section 7.1); ACC-COUNT is unsourced. (6) PAR run with ONE-CLASS assumed on a bank: the hypothesis is exactly what limit transitions deny. RESTATEMENT GATE (mandatory; answered BEFORE the headline, in its own section). If your verdict reduces the node N to a residual lemma L (with or without other proved pieces), you must give ALL of: (G1) the exact statement of L and of N side by side, with the same standing hypotheses; (G2) a proof that L implies N (the reduction); (G3) a reason L is STRICTLY weaker than N: either an explicit configuration (or consistent abstract data meeting every standing law) where L holds and the conclusion of N is not yet available, or the named hypothesis of L that the negation of N does NOT supply; (G4) the line of your argument where L's own hypothesis (the defect, the witness, the scale) is actually used; if it is never used, L is a restatement. If G3 or G4 fails, the headline word is RESTATEMENT, not REDUCED, and the answer reports the proved building blocks only. A headline REDUCED without G1-G4 counts as a failed answer. G. FORMAT Line 1 exactly: VERDICT: LIMIT TRANSITIONS EXCLUDED (cofinal limit transitions on both banks contradict a named law of an untorn case-A R-instance; proof) | LIMIT-ONLY CONSISTENT (explicit data meeting every listed law; first missing global law named) | REDUCED (to one named lemma, gate G1-G4 passed) | PARTIAL (one named sub-claim settled) | OPEN. Then SUMMARY (<= 100 words), the argument (<= 2,000 words), STILL OPEN EXACTLY. Short of effort: line 1 + summary; an honest OPEN is valid. No finite computation as proof; no theorem beyond its hypotheses; UNKNOWN if unsure a cited result exists. RED FLAGS: treating the static north-south circle model as an R-instance (it has no h and no F); using a variation or index count at a door (all zero); assuming F is a plane homeomorphism (then PAR and TT2 are vacuous because the R-instance is contradictory); reading composants of inaccessible points; treating finite depth for every n as one cofinal configuration; claiming SS-COF, DP-K, L2 or PER