m42

Prompt

RESEARCH QUESTION (plane continua / BFMOT outchannels, KP strips and composants; DP-K (K_s != X) for the deep M_F, M_G doors of a canonical strip of an R-instance with breathing and h^2 continuum-minimal: after Argon 71 DP-K is the single residual node under T1-EX and DOOR-SAME-COMP; which argument that uses the DOOR TYPE and the dynamics of h can force a proper pocket trace?). 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 Argon 71, its Sol audit codex_aud_argon71 and the lab lanes b_opus_ssdyn, b_opus_imptransport, b_opus_l2_incoherent, b_opus_hair_sscof; single-agent lab statements, UNAUDITED unless a status says otherwise). - R-instance: X is a nondegenerate nonseparating plane continuum, U := C \ X, h: X -> X a homeomorphism with no periodic point, 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: C -> C are perfect (continuous, closed, compact fibres) plane extensions of h and h^-1; G is NOT F^-1 off X, and neither is a plane homeomorphism. By BFMOT Theorem 6.2.1 and Lemma 6.1.3 (SOURCE, standing hypothesis checked in the audit) each of F, G has a unique outchannel E_F, E_G (prime ends of U, principal set X); assume E_F != E_G. Consequences (SOURCE/DERIVED, audited): X is indecomposable, int X = empty, every subcontinuum of X is nonseparating and X is hereditarily unicoherent (planar). Comp(x) := union of all proper subcontinua of X containing x (composant); the composants are dense, pairwise disjoint, uncountably many. - A crosscut q is an open arc in U whose closure is an arc with distinct ends q^-, q^+ in X; U \ q has two components, the bounded Sh(q) and the unbounded Out(q). q is E_f (f = F, G) if Sh(q) contains the bounded end of the external ray R_f, otherwise M_f (and then Out(q) contains a tail of R_f). K_q := cl Sh(q) cap X. - Canonical strip of E_F (SIDEC, BFMOT Lemma 6.1.2 and Lemma 4.4.8): an embedded narrow strip S = Phi([r_0, inf) x (-1,1)) in U with core containing a tail of R_F, sections c_e, sides H_1, H_2 made of KP-chords and points of X, nested chain of KP-chords defining E_F. A door is a maximal side chord s = H_i((alpha, beta)) in U with ends s^- = H_i(alpha), s^+ = H_i(beta) in X; depth is the section parameter e. Hypothesis H-M: s is M_F, equivalently the pocket Pk(s) (the component of U \ s missing S) equals Sh(s). A door FAILS when it is M_F, M_G and K_s = X. DP-K: every M_F door with diam cl s <= m_1/8 has K_s != X. The deep form used downstream (SS-COF, DP-K(cofinal)): there is r_* beyond which no door of the canonical strip fails. - Breathing (b_opus_l2_incoherent, PARTIAL, UNAUDITED): for a small chord c across the outchannel, var(h^2, c) = 1 + E(c) with E(c) in {-1, -2} at every scale (h pushes inward at every scale), as opposed to outward-monotone maps, for which L2 holds. Used here only as an extra hypothesis you may call. B. LEMMAS (status; where used) Known (statuses as given). Section names are descriptive; the printed numbers of Argon 71 disagree with its own counter. - H-PK-CONN (Argon 71, Sol audit CWR): for every nondegenerate nonseparating plane continuum X with empty interior and every crosscut s, K_s is a continuum. EB / DOOR-SAME-COMP <=> DP-K (same audit, CWR): K_s is the least subcontinuum of X containing s^- and s^+; so K_s != X iff s^-, s^+ lie in one composant. - Disjoint-pocket ray-end uniqueness (Argon 71, Sol CONFIRMED): the pockets of distinct doors of one strip are pairwise disjoint, so a ray has an essential end in at most one pocket. In particular R_G enters at most one door pocket essentially, and a cofinal sequence of failing doors is NOT caused by R_F or R_G entering the pockets. Bi-density (CWR): a failing deep door has X in the closure of both Sh(s) and Out(s). The claim 'hence a third neutral full-impression prime end in the closed cap of s' is GAP in that report; it is Theorem IMP* below. - Theorem IMP* (b_opus_imptransport, DERIVED from indecomposability only, UNAUDITED; confirmed independently in m24 by Argon and GPT-6 Sol): for any crosscut c with K_c = X, the closed shadow arc I_c of c in the prime-end circle contains a prime end of U with impression X. The shadow arcs I_s of the doors of one strip are pairwise disjoint. So cofinal failing doors give infinitely many full-impression prime ends theta*_n besides theta_F. IMP3 (no full-impression prime end other than E_F, E_G) <=> FT (every small crosscut that is M_F and M_G has K != X); IMP3 is the same question, not a reduction. - T1-EX (Argon 71, CWR as an implication): under DP-K, CP1(G) (uniform variation threshold, audited CWR) and the standing source setting there is no short E_F crosscut with ends exactly h(s^-), h(s^+). DP-K is the only missing input, and it is automatic when diam cl Sh(s) < diam X. - Intrinsic exact law (DERIVED, trivial): h(K(x,y)) = K(h x, h y), where K(x,y) is the least subcontinuum containing x, y (exists by hereditary unicoherence). So 'x, y in one composant' is h-invariant. It is door-blind: pairs of close points in different composants occur in every ball (Lemma GEN of b_opus_hair_sscof). - What failed (details in DEADENDS): codex_sscof (variation of deep doors is zero); m24 (five answers: cm on pocket traces gives X; impression transport blocked by the HOM kill test: a circle model with failing doors at every depth satisfies every transport law); m27 (prime-end rotation, free-disk chains for perfect maps); A196 (composant cocycle: no constraint); m36 (abstract composant data cannot force SS-COF: unanimous); b_opus_hair_sscof (hair separation: Lemma NH, Lemma GEN, no set-valued transport law). C. THE TARGET DP-K for deep doors, in the form SS-COF: on every canonical strip of every R-instance (with E_F != E_G, breathing allowed) there is r_* beyond which every door has K_s != X; equivalently, the two ends of every deep door lie in one composant. A conditional or partial form is acceptable only if the condition is a named lemma that is not a restatement. D. ASK Part 1 (method). Name ONE method that is none of the DEADENDS and say why it sees a full-trace door where the failed ones could not. The method must use at a named line (a) the door type M_F, M_G together with depth along the strip, and (b) at least one global dynamical hypothesis (cm of h^2 along the whole orbit, m_1 > 0, fixed-point-freeness or breathing). Candidates, stated exactly if used (none is claimed to work): (i) POCKET STACKING: two failing doors s, s' of one strip cut U into three domains Sh(s), Sh(s'), W, each with X in its closure; compare with Wada-type and Brouwer-Kuratowski theorems (a plane continuum that is the common boundary of three domains is indecomposable; here the domains are pieces of ONE domain cut by crosscuts, so they do not apply as stated); (ii) the least-continuum map K(.,.) and the orbit of the failing end pair (a_n, b_n) in X x X: the set of pairs with K(x,y) = X is G_delta and h x h invariant; study the closure of the h^2-orbit of the pair and where cm enters (the semicontinuity class of (x,y) -> K(x,y) is UNKNOWN to me, state what you use); (iii) a quantitative prime-end count: the arcs I_{s_n} are disjoint, so their harmonic measures from infinity are summable, while each contains a full-impression prime end; combine with upper semicontinuity of impressions and a harmonic-measure or extremal-length estimate for a small arc s_n bounding a domain of diameter >= diam X; (iv) a depth-compatible Baire argument on the strip parameter: the good depths (ends in one composant) form an F_sigma set, and the composant of a bank point changes only across doors (a path in X lies in one composant). Part 2 (carry it out). Prove DP-K in the SS-COF form, or reduce it to one named lemma L (gate G1-G4: L is not equivalent to DP-K, to IMP3 or to DOOR-SAME-COMP, and is not a level-cut restatement), or prove a no-go for the method in Part 1 with the first hypothesis it cannot reach. If you use breathing, say which statement about var(h^2, .) enters and on which chord. Part 3 (decisive test). Try to build the static planar data of cofinal failing doors: a strip S with doors s_n, ends a_n, b_n in different composants, M_F and M_G, pockets pairwise disjoint with K_{s_n} = X, E_F != E_G, unique outchannels, prime ends theta*_n with impression X, plus the composant labels. Record the first law of the R-instance (a line of LEMMAS or of DEFS) that fails, or the exact list of laws the data satisfy. No claim about an h-dynamic realisation unless every law is checked. 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 before you start): a new method, even a partial one with an honest gap, is worth more than a polished restatement. Take your time: check every step, hypothesis and quantifier. This is a hard open problem. E. CONTROLS (all must hold) (C1) Bucket handle (Knaster, standard embedding): K_s = X occurs for small crosscuts (a crosscut joining two points of different composants through the exterior domain); the method must not exclude them, so it must use the door type, depth and an R-instance hypothesis at a named line. (C2) HOM / circle model: an explicit model with a failing door at every depth satisfies every impression-transport and composant-cocycle law; the circle separates the plane, so a method must use nonseparation (hereditary unicoherence) or perfectness of F at a named line. (C3) Finite depth: excluding failing doors only at depth <= n with a threshold depending on n does not prove SS-COF. (C4) Small pockets: DP-K is automatic when diam cl Sh(s) < diam X; say what your argument does when every deep pocket has trace X. (C5) Abstract composant data with an h-invariant equivalence relation, cm and m_1 > 0 and E_F != E_G (m36) are consistent with cofinal failure; planar geometry must enter. (C6) A196: a solenoid translation is cm and fixes a composant, so 'h^2 moves every composant' does not follow from cm alone. F. DEAD ENDS (not progress) (1) Variation, index and degree counts at deep doors: all zero (codex_sscof), DEG may be +-1 but gives no invariant. (2) cm of h^2 on Hausdorff limits of pocket traces: the limit is X (m24, five answers). (3) Impression transport and prime-end rotation: blocked by the HOM kill test (b_opus_imptransport, m24, m27). (4) Brouwer/Franks free-disk chains for perfect extensions (m27, Astra proper-map counterexample). (5) Composant cocycle bookkeeping alone (A196) and abstract composant labels (m36). (6) Hair separation on an indecomposable host: Lemma NH (the complement of a proper subcontinuum is one piece, input = output), Lemma GEN (intrinsic families are door-blind), plane families have no set-valued transport law (b_opus_hair_sscof). (7) A second G-outchannel in the pocket and channel-membership laws (b_opus_ssdyn, Props SB, 5.1, 5.2); the transplant TR (closed-strip gluing gap). (8) Cofinal essential R_G entries into pockets: excluded by disjoint pockets. (9) Re-proving H-PK-CONN or DOOR-SAME-COMP <=> DP-K, or reporting 'T1-EX reduced to DP-K' as new. 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: DP-K PROVED (deep M_F, M_G doors have K_s != X; method named, the line where each R-instance hypothesis is used) | FAILING DEEP DOORS CONSISTENT (a static planar configuration with failing doors at cofinal depth 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: using a variation, index or degree count (all zero at deep doors); using impression transport or prime-end rotation (the HOM model passes them); using cm of h^2 on Hausdorff limits of pocket traces (the limit is X); a static argument with no R-instance hypothesis (the bucket handle has full-trace small crosscuts); proving only a finite-depth statement for every n; assuming CP1, SEC or E_F != E_G silently; claiming T1-EX, L2 or PER