m44

Prompt

RESEARCH QUESTION (plane continua / access transport through the perfect extensions F, G (Route I, ISO-F_i): is h(A_U) inside A_U at the transition witnesses of K_F, what do the collision fibres of F over X look like, and which R-instance law decides it?). 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, the m40 answers, lanes 700 and 841, b_opus_tt2_torn, b_opus_imptransport; 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 (cm of h^2 already forces m_1 > 0). F, G are perfect plane extensions of h and h^-1 (perfect = continuous, closed, compact fibres; not homeomorphisms) with unique outchannels E_F != E_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). - ADMISSIBLE extension (SOURCE-READING of papers/lit/bfmot_1004.0214.txt lines 2617-2625, the standing hypothesis): F is perfect, F|X = h, F has no fixed point on X, and some delta > 0 gives g cap F(g) empty for every KP-chord g (Kulkarni-Pinkall chord of U) of diameter < delta; likewise G with h^-1. BFMOT lines 405-406: every continuum map extends to a perfect map of the plane; lines 2607-2612 (quoting BO09): the extension may be modified on C \ X. The lab states results for one fixed pair F, G; here every conclusion must hold for EVERY admissible pair, or say which pair it needs. - Access. x in X is accessible if some path gamma: [0,1] -> C has gamma(0) = x and gamma((0,1]) inside U (an access arc; may be taken simple). A_U := the set of accessible points. The defect set Sigma := {x in A_U : h(x) not in A_U} depends only on the embedding of X and on h, NOT on F or G. LA-lite (DERIVED): an access arc at x gives one prime end theta_x of U with principal set Pr(theta_x) = {x}; Im is the impression and Theta_X := {theta : Im(theta) = X}. - Collisions: D_F := F^-1(X) \ X and D_G := G^-1(X) \ X (inside U). - The line K_F. Y_F := the hull of X together with the KP+ chords of F (KP-chords of nonnegative variation, BFMOT Def 4.4.10; the sign is lost in the text extraction; so Y_F depends on F); its boundary chords form the line K_F with both ends at the channel theta_F. A boundary chord q FAILS when K_q = X (K_q := the smallest subcontinuum containing the ends of q). Theorem Lambda (DERIVED from a 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 F-chord, or is accessible from outside Y_F with Pr(theta) one point of K_F cap X. A transition toward end i of K_F is a failing boundary chord or a limit transition (a point z of K_F cap X approached monotonically by points not ~ z); the witnesses are the ends of failing chords and the points z; they are accessible. ISO-F_i: only finitely many transitions toward end i (Prop 5.4(c), DERIVED; equivalent to Theta_X missing a one-sided neighbourhood of theta_F). - ACC-TR(W), W a set of accessible points (the object asked for): (i) POINT: h(W) lies in A_U; (ii) SCALED: some omega(eps) -> 0 gives, for each access arc at w in W of diameter <= eps, an access arc at h(w) of diameter <= omega(eps). B. LEMMAS (status; where used) Known (all UNAUDITED). - Trace-access dichotomy (m40 Sol 6.1, PROVED from continuity, F|X = h, G|X = h^-1): for an access arc gamma at x, either a terminal segment misses D_F and F o gamma is an access arc at h(x), or there are z_k in gamma cap D_F, z_k -> x, and y_k := G(F(z_k)) in X has F(y_k) = F(z_k), y_k -> x: arbitrarily small fibres of F meet both X and U. Spec-author remark (check it): r := G o F is a retraction of A := F^-1(X) onto X, and F^-1(h(y)) = r^-1(y) meets X only in y, so the collision fibres are exactly the fibres of r. - ATO (m40 Sol 6, PROVED): if x in A_U and h(x) not in A_U, then EVERY access arc at x meets D_F in (0, eps) for every eps > 0; same for G and h^-1. The hypothesis concerns (X, h) only, the conclusion holds for every admissible F. If D_F and D_G both miss a collar of X then h(A_U) = A_U (landing points only; no prime-end map is induced). - No collar from perfectness plus m_1 (m40 Sol 6 bump construction; m40 Sol 6.1 and Astra: P(s,t) = (s, t^3 - t) is proper, fixes [-1,1] x {0} and sends (s,1) into it): |F(z) - z| > m_1/2 on a collar of X, compatible with arbitrarily small fibres. Neither example is shown admissible. - Homeomorphisms need not preserve access: the Lewis embedding P_s of the pseudo-arc (OV 2011 Example 5.4: distinct accessible points lie in distinct composants) and homogeneity give a homeomorphism g with g(p) = q, p accessible, q not (lane 700, DERIVED from SOURCE and memory of homogeneity). g has fixed points, so m_1 = 0: ACC-TR cannot follow from planarity and homeomorphism alone. - Tearing forces collisions (b_opus_tt2_torn Theorem CT: cofinal tearing at the channel is automatic; m38 Sol 6: an F^2-torn chord has an interior point sent into X by F^2; an F-KP-torn chord has an interior F-hit or a short non-KP join). So collisions of F and F^2 near theta_F exist; open is their fibre structure and location. Prop EXC: if F(s) meets X then E1 (an excursion with ends in different composants) or E2 (all excursions inside one composant); plane topology does not exclude E2. - UBS (m40 Astra, PROVED; Sonnet audit queued): for every r > 0 there is T = T(X,h,r) 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 (else W = closure of the union of the h^j(K) is proper with h(W) in W). A composant contains such a proper K iff sigma(P) = P, so wandering labels carry no bridges. - Labels (m40 Argon, PROVED): classes of K_F cap X are order-convex, consecutive transition classes are pairwise distinct composants, sigma-orbits of labels wander or are periodic. Models with infinitely many transitions, sigma(C_n) = C_{n+1}, cm, m_1 > 0 and every static law exist: the north-south circle model and the solenoid Sigma_2 with a translation (nonplanar). Adjacent, not repeated: IMP-TRANSPORT-FIN (impression transport when F(cl s) meets X in finitely many proper subcontinua; gap at INF-TEAR) and the BRIDGE law of b_opus_noswap_acc. C. THE TARGET Decide ACC-TR for W = the transition witnesses toward an end i of K_F (beyond a threshold) in an R-instance, for every admissible F, G. Equivalent shape: prove Sigma cap W empty, or show that a witness w in Sigma forces a named collision configuration of D_F, D_G along every access arc at w that an R-instance law excludes, or realise it. (Not asked: whether the composant of h(w) is again a label of K_F; ACC-TR alone does not give ISO-F.) D. ASK Part 1 (method). Name ONE method not on the dead-end list and say why it sees accessibility where the failed ones saw only continuity. They used F only through F|X = h and continuity: never the global structure of the collision fibres, nor G and F together, nor the freedom in F. Candidates, none claimed to work: (i) FIBRE-RETRACTION: the nontrivial fibres of F over X are compacta attached to X at one point y (the fibres of r = G o F): semicontinuity, components, shape; what an access defect at x forces of the fibres through the y_k; (ii) EXTENSION-FREEDOM: Sigma does not see F while ATO gives every admissible F the collisions, so look for an admissible F' (F composed with a homeomorphism fixing X pointwise, or F changed off a collar) for which a local variation or index computation on small KP-chords through the collisions contradicts g cap F(g) empty or Lemma P; (iii) CROSS-LINE: use K_F, K_G and G o F = id on X, and ask whether h sends witnesses of K_F into K_G cap X or into pockets of failing chords; (iv) your own. Part 2 (carry it out). Prove ACC-TR(W) (POINT at least), or a dichotomy 'h(w) in A_U, or the configuration C' with C excluded by a stated law or realised, or show by Tier 0 data that every listed law is consistent with infinitely many witnesses in Sigma, or reduce to one named lemma L with gate G1-G4. Say which admissibility hypothesis each line uses. Part 3 (builder or decisive test). Build the local picture of an access defect: accessible w in a planar piece of X, h(w) inaccessible, an admissible-looking F whose collisions lie on every access arc at w (ATO), compatible with the m_1 collar, g cap F(g) empty for small KP-chords, and tearing. Record the first law violated or the exact list met. Finitely many witnesses 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. (C1) Lewis pseudo-arc P_s with a homogeneity homeomorphism g(p) = q inaccessible: the defect set is nonempty, every admissible F has collisions on every access arc at p (ATO), and g has fixed points. Say which R-instance law (m_1 > 0, cm of h^2, E_F != E_G, tearing) your proof uses and why g lacks it. (C2) Control that the failed methods also pass: the north-south circle model with sigma(C_n) = C_{n+1} and the solenoid translation meet cm, m_1 > 0, label injectivity and order-convexity with infinitely many transitions, so an argument using only these fails; the new input must be absent from them (planar F, G). (C3) HOM: if F is a plane homeomorphism, D_F is empty and h(A_U) = A_U, but the instance is contradictory (Cartwright-Littlewood); say on which line you use non-injectivity of F. (C4) P(s,t) = (s, t^3 - t) and the bump perturbation have D_F nonempty with an m_1 collar, so perfect plus m_1 cannot prove ACC-TR. (C5) One access arc, or finitely many witnesses, is not cofinal. F. DEAD ENDS (not progress) (1) Composant-permutation transport and tagged copies h^n(K_F cap X): no order on U, no accesses (m40, all four answers). (2) The trace-access dichotomy and ATO: they locate the obstruction (small fibres meeting X and U) and do not exclude it. (3) UBS and cm on bridges: only inside sigma-invariant composants, never extracted from transitions. (4) Symbolic and dynamical label laws (injectivity, periodic orbits, solenoid shift, north-south shift): consistent with infinitely many transitions (m40). (5) The short-fold law L of Argon: a RESTATEMENT; its Theorem 4 is a triangle inequality (diam >= 3 m_1/4). (6) Impression, moving-access and tearing statements at chords (m24, m28, m38): they give 'interior X-contact or short non-KP join', not E1 against E2. (7) Variation, index and Cartwright-Littlewood counts: zero at failing chords, blind to accessibility. 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: ACCESS TRANSPORT PROVED (a named law ACC-TR for the transition witnesses, proved from the R-instance laws for every admissible F, G) | ACCESS DEFECT CONSISTENT (explicit Tier 0 data with accessible witnesses w, h(w) inaccessible, the forced collisions of F and G, 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, C3); reading 'h permutes composants' as transport of accesses (h acts on X only); using the Lewis pseudo-arc homeomorphism as an R-instance (it has fixed points); applying cm to composants; counting variation or index at a failing chord (zero there); assuming perfect means open, monotone or injective on fibres; proving a statement for one F when the claim is for every admissible F; claiming ISO-F, IMP3, SS-COF, DP-K, L2 or PER