m41

Prompt

R41_ASTART_prompt.txt 100 % RESEARCH QUESTION (plane continua / Kuratowski-Pelczynski chords and canonical strips of BFMOT; A-START (the bridge from the original canonical strip near the outchannel to a strip with an M_G initial chord, input of DP-K(cofinal) <=> SS-COF, Route I): does a deep E_F, M_G KP-chord g_1 span the original strip (no bulge), so that S(g_1) is a tail of S(g_0)?). 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 Arena A193/A194 and codex_dpk_consolidate; UNAUDITED lab statements unless marked SOURCE). - R-instance: X is a nondegenerate nonseparating plane continuum, U := C \ X, h: X -> X a homeomorphism with h^2 continuum-minimal (cm) and m_1 := min |h(x) - x| > 0; F, G are perfect plane extensions of h and h^-1 (not homeomorphisms) with unique outchannels E_F, E_G (prime ends theta_F, theta_G of U; psi: U -> disk conformal), external rays R_F, R_G. A crosscut q is an open arc in U whose closure is an arc with two distinct ends in X; Sh(q) is its bounded side, I_q its shadow arc in S^1. q is E_f if Sh(q) contains a tail of R_f, else M_f (equivalently theta_f in the open arc I_q, or not). NEST: if q' lies in Sh(q) and cl(q), cl(q') are disjoint, then Sh(q') is inside Sh(q). - KP-chords (SOURCE, BFMOT arXiv 1004.0214v2, Thm 4.1.5 and text after it): U is partitioned by KPP = {U cap convH(B cap X) : B a maximal ball}; each hull is a chord (two points) or a gap (three or more); KP-chords are the chords and the boundary arcs of gaps in the open ball. KP-chords are pairwise disjoint and never cross; limits of KP-chords with distinct limit ends are KP (Lemma 4.1.6). SOURCE (Lemma 4.4.8): for each x in Pr(E_t) and eps > 0 there is a chain of KP-chords in B(x, eps) defining E_t. - E_t is NEGATIVE starting at g in KP (SOURCE, Definition 5.1.1) iff every KP-chord in X cup Sh(g) that crosses E_t essentially has negative variation. Canonical strip (SOURCE, Lemma 6.1.2, second part, papers/lit/bfmot_1004.0214.txt L2992-2996): if E_t is negative starting at the KP-chord g, then S(g) := the union of the hulls convH(B cap X) contained in X cup Sh(g) having a chord that crosses R_t essentially is an infinite narrow strip (image of an embedding Phi of [0,inf) x (-1,1), continuous on the closure, diam Phi({e} x [-1,1]) -> 0), remainder in X, bordered by g and two halflines H_1, H_2 of chords in KP and points of X. Sigma(g) := S(g) with sections c_e = Phi({e} x (-1,1)), sides H_i(e) = Phi(e, +-1); e in Sec iff H_1(e), H_2(e) in X. ST1-ST5 (lab reading): Phi injective on the closed strip, remainder in X, ST2 (c_0 = g is a crosscut with S in Sh(c_0)), the core Phi(., 0) contains a tail of R_F; a door is s = H_i((a,b)) maximal in U, its pocket Pk(s) the component of U \ s missing S; eligible = H-M and diam <= m_1/8 (H-M is the A193 notation; its exact reading is not restated here, UNKNOWN). - Original strip: Sigma_0 := Sigma(g_0), with g_0 an E_F KP-chord and E_F negative starting at g_0. Target forms. (A-START-T) when E_F != E_G there are a KP-chord g_1 in Sh(g_0), E_F and M_G, and e_1 in Sec(Sigma_0) with g_1 = c_{e_1}, such that S(g_1) = Phi_0([e_1, inf) x (-1,1)) and the doors of Sigma(g_1) are exactly the doors of Sigma_0 beyond e_1. (A-START-W) the same without g_1 = c_{e_1}: S(g_1) inside S(g_0), S(g_0) \ S(g_1) relatively compact in U cup the initial arc, doors of Sigma(g_1) = doors of Sigma_0 inside Sh(g_1). B. LEMMAS (status; where used) Known (UNAUDITED unless stated). - A193 (Arena, Sol-audited conditional on ST1-ST5, SEC): Lemma D (every pocket lies in Sh(c_0); every door is H-M), Theorem T (E_G sections form an initial segment; at most one door is G-essential, straddling the E_G -> M_G transition), NE-START (a strip with M_G initial chord has no G-essential door), NE-DEEP (for each strip there is r_G(Sigma) beyond which no door is G-essential), NE-EQ (E_F = E_G: no G-essential door). If E_F != E_G and SEC holds, Theorem C (conditional on PE0-PE2) gives eventually M_G sections of Sigma_0 (not necessarily KP). The transplant TR has an unproved global closed-strip gluing step (new X-vertices may coincide with old wall points); it is not used. - A194 (Arena, PARTIAL, UNAUDITED; notes/web_queue/manual/arena/A194_answer.md): DERIVED: for a KP chord g_1 in Sh(g_0) with cl(g_1) disjoint from cl(g_0), NEST gives Sh(g_1) in Sh(g_0), so negative-starting at g_0 implies negative-starting at g_1 and S(g_1) is contained in S(g_0) (no new hulls); if theta_F != theta_G a deep defining-chain chord g_1 is E_F, M_G, lies in Sh(g_0) with var(F, g_1) = -1, so an A-START candidate always exists (N-START). OPEN: that g_1 is a level section of Sigma_0 (A-START-T) or S(g_1) a tail (A-START-W). CONTROL (toy, NOT a KP system): in a planar half-strip a crosscut joining two points of the same wall, disjoint from g_0, is a 'bulge' that does not span; so noncrossing alone does not force spanning. A level-cut plus tail-hull condition is a RESTATEMENT. A194 left unsettled: shared endpoints of g_0, g_1 (NEST does not apply), downstream door correspondence, a genuine full-KP bulge obstruction. - Ledger lead (root, 2026-11-03 10:5x): the gap is exactly 'deep E_F, M_G KP-chords span the strip (no bulges)'; next method suggested: exclude bulges via the maximal-ball / KPP structure (does a bulge chord bound a region inside Sigma_0 whose hulls are ineligible?). C. THE TARGET A-START-T (or, failing that, A-START-W): a deep E_F, M_G KP-chord g_1 of the defining chain (or another E_F, M_G KP-chord constructed from the strip) is a section of Sigma_0 and S(g_1) is the tail of S(g_0) beyond it; or a full-KP static configuration where every E_F, M_G KP-chord in Sh(g_0) bulges. D. ASK Part 1 (method). A194 used set-inclusion monotonicity under NEST, existence of g_1 from a defining chain (Lemma 4.4.8 and the prime-end form), and a toy planar bulge control. Name ONE method that is none of these and say why it sees spanning where noncrossing alone does not. Candidates, stated exactly if used (none is claimed to work): (i) prime-end disk model: transfer the strip to the unit disk by psi and read sections, bulges and hulls as nested arcs I_c of S^1; a bulge chord has both feet on one wall and its shadow arc is then forced to lie inside a single arc of the wall, which may contradict that a tail of R_F lies in Sh(g_1); (ii) construct g_1 FROM the strip: take a gap hull of S(g_0) straddling a section c_e deep enough to be M_G and use its boundary KP arcs, so that g_1 is by construction a chord of the strip and a section-like cut (then prove the straddling hull exists using that sections need not be KP but the sides are); (iii) separation / duality: apply Alexander duality or a Janiszewski-type argument to the closed set cl(S(g_0)) cup X cup g_1, whose complement components detect whether g_1 separates the initial piece from the tail; (iv) the maximal-ball structure: a bulge chord and the KPP hulls it bounds lie in the strip S(g_0) between its wall and g_1; show a hull there contains a chord crossing R_F essentially (so the bulge hulls are eligible) or not. Part 2 (carry it out). Prove spanning (A-START-T), or A-START-W, or the exact reduction to one named lemma L with gate G1-G4, or build the full-KP bulge configuration (a maximal-ball family and KPP partition, not a toy half-strip) and name the first KP law it violates. If NEST needs disjoint closures, say what replaces it when g_1 shares an endpoint with g_0 or a wall point of Sigma_0. Part 3 (what is used downstream). State which statement about the doors of Sigma_0 follows from the version you prove (DP-K at doors beyond e_1 iff SS at those doors; NE-DEEP for the retained doors), and which universal statement does NOT (T1-EX quantifies over every door, shallow doors are discarded). No claim about T1-EX, RIG-nonXS or DOOR0. 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) E_F = E_G: no E_F chord is M_G (the method must keep E_F != E_G visible and say where it is used). (C2) A chord shallower than the E_G -> M_G transition is E_G, not M_G: g_1 must lie beyond it. (C3) K5: a strip on the wrong side of its first chord breaks Lemma D (ST2 is used). (C4) Control that the failed methods also pass: the toy half-strip with a same-wall bulge and the quoted monotonicity S(g_1) inside S(g_0) are both consistent with every statement A194 proved, so a solution must use the maximal-ball family / KPP partition or the prime-end structure at a named line, and must fail (or be inapplicable) on the toy. (C5) Knaster-type static data (chainable) are not an R-instance; use them only to test a step. (C6) Constructing Sigma(g_1) is not identifying it with a tail; say which each lemma gives. F. DEAD ENDS (not progress) (1) NE-DEEP and the conditional tail observation of A193: per strip, no KP initial chord. (2) The transplant TR (closed-strip gluing gap). (3) Dynamic arguments from h, F, G alone: strip-blind. (4) Re-proving Theorem C or Lemma D. (5) Assuming E_F = E_G or SEC silently. (6) Monotonicity under NEST and the defining-chain existence of g_1 (A194 Part 1): settled, give a candidate but not spanning. (7) The toy bulge picture and the level-cut tail-hull condition: not a KP system, respectively a RESTATEMENT (A194). 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: A-START PROVED (A-START-T or A-START-W, with the method named and the line where the maximal-ball / KPP structure is used) | BULGE CONSISTENT (a full KP static configuration where every E_F, M_G chord in Sh(g_0) is a bulge chord; laws checked) | 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: answering 'a canonical strip exists at g_1' (the target is AGREEMENT with the old strip); assuming SEC, injectivity of Phi on the closed strip, or E_F = E_G silently; a toy half-strip picture without the maximal-ball family and the KPP partition (A194's bulge control is exactly that); calling a level-cut condition a reduction (it is the missing conclusion); using dynamics of h, F, G alone (strip-blind); claiming DP-K, T1-EX, RIG-nonXS, L2 or PER Zobrazuje se R41_ASTART_prompt.txt.