m45

Prompt

RESEARCH QUESTION (plane continua / composant labels along the line K_F (Route I, ISO-F_i): the K_F labels are an infinite wandering sequence of composants if ISO-F fails, h permutes composants (sigma); is the set of K_F labels closed under sigma and sigma^-1 cofinally toward an end, and if not, where does the image composant hide?). 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 b_opus_thetax_bound, api_opus_inftear and the m40 answers; single-agent lab statements, UNAUDITED unless stated). - R-instance: X is a nondegenerate nonseparating indecomposable 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 |h(x) - x| > 0. F, G are perfect plane extensions of h and h^-1 (NOT homeomorphisms) with unique outchannels E_F != E_G; every statement is for every admissible pair (BFMOT standing hypothesis: F perfect, F|X = h, no fixed point on X, g cap F(g) empty for small KP-chords g; likewise G). x ~ y iff a proper subcontinuum contains both; the classes are the composants; a proper subcontinuum lies in exactly one composant. h permutes composants (PROVED, trivial); sigma(P) := h(P). - Access: x is accessible if a path in U cup {x} lands at x; A_U := the accessible points. LA-lite (DERIVED): an access arc at x gives a prime end theta_x of U with principal set Pr(theta_x) = {x}; Im is the impression, Theta_X := {theta : Im(theta) = X}, closed. - The line K_F. Y_F := the hull of X with the KP+ chords of F (variation >= 0; so Y_F depends on F); its boundary chords form the line K_F with both ends at the channel theta_F. For a boundary chord q with ends in X, K_q := cl Sh(q) cap X is a continuum (H-PK-CONN) and q FAILS when K_q = X. For theta in the open cap of q, Im(theta) lies in K_q (trivial direction). Theorem Lambda (DERIVED, SOURCE-READING of BFMOT 6.1.5-6.2.2): each prime end theta != theta_F lies in the closed cap of exactly one boundary chord, or is accessible from outside Y_F with Pr(theta) one point of K_F cap X. Transitions toward an end i of K_F: failing boundary chords and limit points (z in K_F cap X approached monotonically by points not ~ z). ISO-F_i: finitely many transitions toward end i (Prop 5.4(c), DERIVED). - Labels. Lab_F := the set of composants P with P cap K_F cap X nonempty (K_F labels); J_P := P cap K_F cap X is order-convex in K_F (Lemma CL, EB2). If ISO-F_i fails, the tail labels C_1, C_2, ... toward end i are pairwise distinct (m40 Argon, PROVED). LC_i (the object asked for): there is t_0 such that for every K_F label C whose J_C lies beyond t_0 toward end i, both sigma(C) and sigma^-1(C) are K_F labels. LC_i^+ asks for sigma(C) only; LC-INF asks for it on infinitely many tail labels. - HA: h(A_U) = A_U (accessibility is preserved). Whether HA holds is a separate access-transport question; it is granted here to isolate the label question, and a proof of LC_i without HA is welcome (say where HA enters). B. LEMMAS (status; where used) Known (all UNAUDITED). - Lemma FM (spec author, DERIVED from Theorem Lambda, LA-lite and Im(theta) in K_q; CHECK IT FIRST). Assume HA. Let x be in K_F cap X with composant C and x' in {h(x), h^-1(x)}; x' is accessible. If x' has an access prime end theta' that is accessible from outside Y_F, or an endpoint of a cap, then x' is in K_F cap X; if theta' is in the open cap of a boundary chord q that does not fail, then x' is in K_q, a proper continuum lying in the composant of the ends of q, which are in K_F cap X. In all these cases sigma^+-1(C) is a K_F label. So under HA, LC fails at C ONLY by POCKET-HIDING: every access prime end of x' lies in the open cap of a FAILING boundary chord, and the composant of x' misses K_F. Without HA it also fails when x' is inaccessible. - Counting remark (spec author, CHECK): if LC_i^+ fails on a tail under HA, the images sigma(C_n) are pairwise distinct (sigma is a bijection), each hidden under failing caps. So either infinitely many distinct failing chords hide a label, or one failing cap contains infinitely many accessible points in pairwise distinct composants; by TWO-ARC (api_opus_inftear, DERIVED: two accessible points in different composants put full-impression prime ends on both circle arcs between their accesses) Theta_X then has infinitely many points in one closed cap (condensation inside a door pocket, Prop 6.1 case P2 of b_opus_thetax_bound). - Pockets: a failing pocket has trace X (m24), so cm on pocket traces gives nothing. The CL labels (composants of Im(theta), theta outside Theta_X) are countable (Cor 5.2, DERIVED); the K_F labels depend on F through Y_F. - UBS (m40 Astra, PROVED; Sonnet audit queued): for every r > 0 there is T such that every subcontinuum K with K cap h(K) nonempty has diam K >= m_1 and the union of h^j(K), 0 <= j <= T, is r-dense in X. A composant contains such a proper K iff sigma(P) = P, so wandering labels carry no bridges. Also (m40 Argon): sigma-orbits of labels wander or are periodic, an orbit of size p carries at most p labels. - Access (m40 Sol 6, 6.1, PROVED; the ACC-TR question): if x is accessible and h(x) is not, every access arc at x meets D_F := F^-1(X) \ X arbitrarily near x, so small fibres of F meet X and U; perfect plus m_1 > 0 does not exclude this. - Models: the north-south circle model with sigma(C_n) = C_{n+1} has infinitely many transitions and satisfies cm, m_1 > 0, label injectivity, order-convexity and Lemma CL, but no planar X, F, G; the solenoid Sigma_2 with a translation has wandering composants, cm and m_1 > 0 (nonplanar). In the shift model every sigma(C_n) is again a label: LC_i holds there. C. THE TARGET Decide LC_i in an R-instance for every admissible F, G: prove it (state where HA is used), or give Tier 0 data in which cofinally many tail labels have an image composant missing K_F (hidden under failing caps, or inaccessible) with every listed law met, or reduce it to one named lemma. Also decide the consequence: do LC_i, label injectivity and UBS imply ISO-F_i? If not, name the label-level law that is missing (the shift model has LC_i and infinitely many transitions). Part 1 (method). Name ONE method not on the dead-end list and say why it sees WHERE sigma(C) sits relative to K_F, which composant identity, cm, UBS and label injectivity do not. Candidates, none claimed to work: (i) CHORD-DEPTH: the boundary chords of Y_F and their shadows form a nested forest (disjoint or nested); define a depth for accessible points, show how F or F^2 on K_F cap X changes it, and bound the jump into a failing pocket by m_1, the unique channel and tearing (TT2); pocket hiding is a depth jump to a failing cap; (ii) DUAL-LINE: use K_G and G o F = id on X, and test sigma(Lab_F) against Lab_G and the pockets of G-chords instead of Lab_F; (iii) COUNTABLE-SATURATION: the sigma-saturation of the countably many CL labels is a countable h-invariant family of composants with dense meagre union; relate it to the two lines and to Theta_X near theta_F; (iv) your own. Part 2 (carry it out). Prove LC_i (or LC_i^+) with the exact use of each R-instance law, or prove a dichotomy 'LC_i, or the configuration C' with C excluded by a stated law or realised, or show by Tier 0 data that LC_i fails consistently (give the hidden labels and the failing chords carrying them), or reduce to one named lemma L with gate G1-G4. Check Lemma FM and the counting remark before building on them. Say what LC_i would give for ISO-F_i. Part 3 (builder or decisive test). Extend the north-south model by pocket structure: failing chords q_n with caps between consecutive K_F classes, hidden labels P_n inside the caps, and an honest bijection sigma with sigma(C_n) = P_n and sigma(P_n) = C_{n+1}. Check it against sigma bijective, countable labels, order-convexity along K_F, Lemma CL, Theorem Lambda caps, TWO-ARC separation, Prop 6.1, and record the first law violated or the exact list met. Test your claim also on the shift model (where LC_i holds). Finitely many tail labels are not cofinal. Each claim labelled PROVED, SOURCE (exact reference), LAB, DERIVED, CONJECTURE, OPEN or RESTATEMENT. UNKNOWN if unsure a cited result exists. What already failed (read DEADENDS first): a new method, even partial with an honest gap, beats a polished restatement. Take your time: check every step, hypothesis and quantifier. This is a hard open problem. E. CONTROLS (all must hold) (C1) Shift model sigma(C_n) = C_{n+1}: LC_i holds and transitions are infinite, so 'LC_i implies ISO-F_i' needs a planar input absent from it. (C2) Control that the failed methods also pass: the hidden-label extension of Part 3 and the solenoid translation meet sigma bijective, injectivity, order-convexity and cm with LC_i failing (resp. holding), so label bookkeeping, composant identity, cm on continua and UBS cannot decide LC_i either way; the new input is the position of sigma(C) relative to K_F in the plane. (C3) HOM: if F is a plane homeomorphism, accesses and K_F are transported and LC_i is easy, but the instance is contradictory (Cartwright-Littlewood); say on which line you use that F is only perfect. (C4) Knaster bucket handle: one limit transition and no h; LC_i is about cofinal labels, so it is vacuous there. (C5) A closure for finitely many labels, or with a threshold depending on the label, is not LC_i. F. DEAD ENDS (not progress) (1) Composant permutation, tagged copies of K_F cap X, label injectivity and orbit bounds: no information about where sigma(C) sits (m40). (2) UBS and cm on bridges: only inside sigma-invariant composants. (3) Access transport at a point (ATO, trace-access dichotomy): about HA, not about pocket hiding. (4) Pocket-trace cm: the trace of a failing pocket is X (m24). (5) Variation, index and Cartwright-Littlewood counts: zero at failing chords. (6) Baire and Cantor-Bendixson on Theta_X: both branches consistent with every static law (b_opus_thetax_bound). (7) Symbolic label laws (north-south and solenoid shifts, cocycles PAR, m36, m37): consistent with infinitely many transitions. 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: LABEL CLOSURE PROVED (cofinal closure of the K_F labels under sigma^+-1 toward an end, proved from the R-instance laws, with the hypothesis HA marked where used) | LABEL CLOSURE FAILS CONSISTENTLY (explicit Tier 0 data: cofinally many tail labels whose image composant misses K_F, meeting every listed law; first missing law named) | REDUCED (to one named lemma L, 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: assuming F or G is a plane homeomorphism (contradictory instance); treating 'h permutes composants' as 'h permutes the K_F labels' (that is the question); applying cm to composants (cm acts on continua only); a label model with no pocket structure (it cannot see where the image composant sits); counting variation or index at a failing chord (zero there); claiming ISO-F, IMP3, SS-COF, DP-K, L2 or PER