m27

Prompt

RESEARCH QUESTION (plane continua / fixed-point-sensitive invariants (Brouwer translation arcs, Conley-Lefschetz index, transverse foliations, prime-end rotation); SS-COF (no failing door at cofinal depth of a canonical KP strip near the outchannel): which method that genuinely uses fixed-point-freeness of h can see a full-trace door?). 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 codex_sscof, b_opus_ssdyn, b_opus_imptransport and microeval m24; all single-agent or model answers, UNAUDITED except where said). - X is a nondegenerate nonseparating plane continuum, indecomposable (so with empty interior), U := C \ X, h: X -> X a homeomorphism with h^2 continuum-minimal (cm: no nonempty proper subcontinuum K of X has h^2(K) inside K) and m_1 := min_x d(x, h x) > 0. In particular h has NO fixed point and no point of period 2 (FPF = this fixed-point-freeness, a GLOBAL property of h on all of X). F, G are perfect plane extensions of h and h^-1 (continuous, proper, F|X = h; G is NOT F^-1 off X; F(s) may meet X). Each has a unique outchannel E_F, E_G (prime ends of U with principal set Pr = X), E_F != E_G, and Theta_X := {prime ends theta of U with impression X}. Theta_X is closed; E_F, E_G are in it. - A crosscut s is an open arc in U whose closure is an arc with distinct ends in X; Sh(s) the bounded component of C \ (X u s); I_s its cap on the prime-end circle; K_s := the smallest subcontinuum of X containing both ends (K_s = X iff the ends lie in different composants, X indecomposable). A door s is M_F (M_G) if Sh(s) misses a tail of the external ray of E_F (E_G). A door FAILS if it is M_F, M_G and K_s = X. - A canonical KP strip near E_F: a nested chain of Kuratowski-Pelczynski chords defining E_F; its doors are its chords, depth n = the n-th. SS-COF: there is n_0 such that no door of depth >= n_0 fails. FT (strong form): every crosscut q with diam q <= m_1/8 that is M_F and M_G has K_q != X. B. LEMMAS (status; where used) Known (statuses as given; everything UNAUDITED unless marked). - Theorem IMP* (b_opus_imptransport; confirmed independently by m24, Argon 2.3 and GPT-6 Sol; DERIVED, uses only indecomposability): for ANY crosscut c with K_c = X (or ends in different composants) the closed cap of c contains a prime end with impression X. Hence SS-COF <=> no Theta_X point in the caps of deep doors; IMP3 (Theta_X = {E_F, E_G}) <=> FT, so IMP3 is not an independent input. - Pocket-trace collapse (m24, all five answers, PROVED): a failing door s has cl Sh(s) cap X = X. - Index blindness (codex_sscof, OPEN-OBSTRUCTION; conditional on the KP laws): beyond a finite depth every canonical door has variation zero for F, G and F^2; for a failing door s exposed outside a fine KP auxiliary Jordan disk Y_d, the pocket-side arc A_d tends to X with zero chord variations and the opposite arc B_d carries -1; so the enclosing curve s u B_d sees no change. A single exposing curve cannot see cofinal doors (codex_sscof sect 5). - HOM kill test (b_opus_imptransport sect 6.3, DERIVED by hand, Tier 0): on an abstract prime-end circle R/Z with E_F = 0, E_G = 1/2, a circle homeomorphism phi (phi(x) = 4x on [0, 1/16], symmetric, Fix phi = {0, 1/2}), door caps I_n = (4^-n, 1.5 * 4^-n) with phi(I_n) = I_{n-1}, Theta_X = {0, 1/2} u closure of one phi-orbit, composant classes shifted by phi, has a failing door at EVERY depth and satisfies every impression-transport law (IMP*, JOIN, EXC, LIM-FT, TT, phi(Theta_X) = Theta_X). Conclusion of that lane: any proof of SS-COF must use an input violated by this model, i.e. it must use FPF globally (in a plane homeomorphism with nonseparating invariant continuum SS-COF is true only via the Cartwright-Littlewood / Bell fixed point theorem, a global result). - Composant cocycle (m24, GPT-6 Sol, PROVED): J_C(s) := 1_C(a) - 1_C(b) for a composant C and a crosscut s from a to b is additive and nonzero for every full-trace crosscut: an additive integer invariant that sees K_s = X exists but is not a KP-local count and by itself proves nothing (it is satisfied by the circle model). - OEXT (D2010, a_opus_oext, UNAUDITED): if some extension F of h has Brouwer degree of one sign on every open set compactly inside N minus X (N a neighbourhood of X), then h has a point of period at most 2, contradicting m_1 > 0 and cm; the one-sign hypothesis is itself equivalent to L2 (OPEN). So a degree-type fixed-point method is already in the lab; a chosen method must say what it adds beyond OEXT. C. THE TARGET SS-COF: no failing door at cofinal depth of a canonical KP strip near E_F, in every R-instance (X, h, F, G as above). D. ASK Part 1 (choose and state the method precisely). The route "equivariant transport" is closed; a Cartwright-Littlewood style index of F on a single curve is being run elsewhere. Choose ONE of the following four methods (or a precisely stated fifth that is neither on the dead list nor an index on one curve) and state it as a theorem you will use, with every hypothesis. (M1) Brouwer translation arcs / free-disk chains (Franks) for the plane homeomorphism case and their replacement for a perfect F: what plays the role of a translation arc through a door and its image, and which disk of the chain would contain a full-trace pocket. (M2) A Conley or Lefschetz index of F on a compact neighbourhood N of X without fixed points, relative to an exit set cut by a door and a ray (a disk through the door-ray pair); say which relative class a failing door changes. (M3) Le Calvez transverse (Brouwer) foliation of the plane minus Fix for an orientation-preserving fixed-point-free plane homeomorphism, and what survives for a perfect map: the sign of the first return of a transverse path to a door. (M4) Prime-end rotation: the circle correspondence of F on the prime-end circle (defined through the proper disk map U -> U), its rotation interval, and the relation of the fixed prime ends E_F, E_G to Theta_X. For the chosen method say (a) which object sees K_s = X although variation, degree and impression counts do not, (b) the exact line where FPF of h on ALL of X (not only on a neighbourhood) is used, and (c) why the HOM circle model, whose phi has fixed points, violates that line. Part 2 (carry it out). Prove, or reduce to ONE named lemma with its gate G1-G4, either (i) SS-COF, or (ii) the implication: cofinal failing doors (equivalently, by IMP*, Theta_X points in the caps of doors of every depth) produce an object of the chosen method that FPF forbids. Say whether the argument needs F to be an orientation-preserving plane homeomorphism; if it does, state the weakest statement the perfect-map case then reduces to. Part 3 (decisive test). Build the strongest model for the chosen method in which the failing doors occur at every depth and the first law of the method is violated; report the first violated law by name (a model that passes every law of the method is a METHOD EXCLUDED result). Label every claim PROVED, SOURCE (exact reference), LAB, DERIVED, CONJECTURE, OPEN or RESTATEMENT; UNKNOWN if unsure a cited theorem exists. 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) HOM circle model above: your method must fail on it and must name the clause that uses FPF; a method that the circle model also satisfies cannot prove SS-COF. (C2) Knaster bucket handle without h: it has full-trace crosscuts, so the argument must say at which line it uses h (m_1 > 0, cm, or the perfect extension); an argument not using h excludes them and is false. (C3) A plane host where h has fixed points (a planar odometer-end host Y_2 or a rotation-type model): the method must not conclude there; say which hypothesis fails. (C4) Finite depth: an exclusion at depth <= n for each n with a threshold depending on n does not prove SS-COF. F. DEAD ENDS (not progress) (1) Variation / index counts at deep doors and KP enclosing curves (codex_sscof sect 3: all zero). (2) Relative-homology invariants: no invariant factoring through H_1 of U minus (R_F u R_G) sees K_s = X (Argon, m24). (3) cm on Hausdorff limits of pocket traces: the trace is X, never a proper continuum (m24, all five). (4) Impression transport, TT, JOIN, EXC, LIM-FT: the circle model of b_opus_imptransport sect 6.3 satisfies them all. (5) IMP3 or FT as input: circular (IMP3 <=> FT). (6) Literal Prop TR at cofinal depth: its band is compact; Prop HOM stops because F(s) may meet X. (7) Channel membership and hyperspace laws are side-blind (Prop SB). 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: SS-COF PROVED (named fixed-point-sensitive method; proof, with the line where fixed-point-freeness enters) | METHOD EXCLUDED (the named method is blind to full-trace doors; HOM-type model satisfying all its laws) | 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: using a variation or index count of KP enclosing curves at deep doors (all zero); using impression transport or TT or JOIN (a HOM circle model satisfies them); using cm on Hausdorff limits of pocket traces (the trace is X); assuming IMP3 or FT as input (circular); assuming F is a plane homeomorphism without saying so; treating a finite depth for every n as one cofinal configuration; repeating a Cartwright-Littlewood style index of F on one curve (another lane does that); claiming DP-K, L2 or PER