m43

Prompt

RESEARCH QUESTION (plane continua / global assembly of a proper h^2-sub-invariant subcontinuum (W-SPAN-PROPER): from cofinal failing doors of a canonical KP strip one wants a closed connected proper M with h^2(M) in M; the missing input named by b_opus_hair_sscof is a SET-VALUED planar transport law h(A) in A' for strip-defined subsets A of X. Which such law is true, strong enough, and not a restatement?). 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 b_opus_hair_sscof, b_opus_gbridge2, b_opus_imptransport, codex_torn_coupling and Argon 71 with its Sol audit; 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 are perfect plane extensions of h and h^-1 (not plane homeomorphisms, G != F^-1 off X) with unique outchannels E_F != E_G (BFMOT Theorem 6.2.1, SOURCE). Consequences (SOURCE/DERIVED, audited): X is indecomposable, int X = empty, hereditarily unicoherent; Comp(x) := union of the proper subcontinua containing x (composant); composants are dense, pairwise disjoint, uncountably many; every proper subcontinuum is nowhere dense and lies in exactly one composant. - Crosscut q: open arc in U with distinct ends q^-, q^+ in X; Sh(q) bounded side, Out(q) unbounded side; K_q := cl Sh(q) cap X, a continuum (H-PK-CONN, CWR), the least subcontinuum containing the ends, so K_q != X iff the ends lie in one composant. A crosscut is E_f / M_f according to whether Sh(q) contains the bounded end of the external ray R_f or not (f = F, G). - Canonical strip of E_F (SIDEC): Phi: [r_0, inf) x [-1,1] -> C, open strip in U, sections c_e, sides H_1, H_2 of KP-chords and points of X, a nested KP chain defining E_F. A door s = H_i((alpha, beta)) is a maximal side chord in U; its pocket is Sh(s) (hypothesis H-M: s is M_F); it FAILS if it is M_F, M_G and K_s = X. Not-SS-COF: failing doors s_n at cofinal depth (ends a_n, b_n in different composants, diam s_n -> 0, pockets pairwise disjoint). Strip-defined subsets of X (the class STRIPSETS): bank sets B_i := {x in X : x = H_i(e) for some e}, windows W[e,e'] := Phi([e,e'] x [-1,1]) cap X, door traces K_s, the points a_n, b_n, arc components and composants of such sets, finite Boolean combinations, closures, Hausdorff limits as e -> inf. - SETLAW (the object asked for). A pair (T, L): T assigns to each strip-defined A in STRIPSETS a set T(A) in STRIPSETS (A' = T(A)) by an explicit clause in the plane data of the strip (not by h alone); L is the statement h(A) in T(A), or h(cl A) in T(A), for all A in a stated sub-family, in every R-instance. ASSEMBLY: from cofinal failing doors, iterating T along A_n, h^2 A_n, ... yields a nonempty closed connected M != X with h^2(M) in M or h^-2(M) in M. cm then refutes not-SS-COF. B. LEMMAS (status; where used) Known (statuses as given). - W-SPAN-PROPER (research wall): the missing step in W-INF-TEAR, W-INT-T0, W-RIG, H-EQ. Tried and failed: orbit-wide pocket iteration (b_opus_pocket_orbit), hyperspace trapping gap UD/EQ (S4), invariant measures and hyperspace operators (S9), HT + semicontinuity (Argon 68), clopen continuum-span (Argon 69). - Lemma NH (b_opus_hair_sscof, DERIVED from classical facts): if M != X is a nonempty subcontinuum of an indecomposable X, X \ M is connected, dense and contains every composant except that of M. So a family of 'hairs' with complement M is ONE piece, and its connectedness hypothesis is the conclusion 'M is connected'. Corollary NH-cm: for open O with h^2(O) = O, no component of X \ O is h^2-fixed. Lemma GEN: every ball contains two points in different composants (so intrinsic families are door-blind). Lemma C-OPEN: an open union of composants is empty or X. Lemma INV: cm of h^k <=> cm of h^-k. - Exact intrinsic law (DERIVED, trivial): h(K(x,y)) = K(hx, hy) for the least subcontinuum K(x,y); composant relation preserved (codex_torn_coupling: transport of composant relation and of min-continuum diameter). Door-blind. - Step INV (b_opus_hair_sscof): h acts on X only; no map on U, crosscuts, prime ends or accessible points. F(s) may meet X, F(Sh(s)) need not be a pocket, and the audit codex_aud_kill_hom withdrew any 'geometric bank permutation'. Prop JOIN (b_opus_imptransport, DERIVED): an anchored transport holds when F(s) meets X only at its ends or a short join exists; otherwise the E1/E2 dichotomy for F(s) (E2 not excluded by plane topology). - Pocket traces: the trace of a failing pocket is X, so cm on traces gives X (m24, five answers). Non-failing pockets have K_s proper but are the good case. - Positive instance of a transport pattern: b_opus_gbridge2 (Theorem HT0, chart hosts with h^2 cm): free-ended hairs form a topologically invariant family whose complement Y_{C'} is a proper subcontinuum invariant under every homeomorphism, contradicting cm. By Lemma NH that works only where the host is decomposable; on an indecomposable X it has no content. - Counterexamples to plausible laws: A201 (a solenoid translation is cm and fixes a composant: not planar); m36 (abstract composant labels with cm, m_1 > 0, E_F != E_G are consistent with cofinal failure); Astra (a perfect degree-1 fixed-point-free plane map with a free disk returning in three steps); the HOM / circle model (failing door at every depth, rotation 0, satisfies every impression-transport law, but the circle separates the plane). C. THE TARGET A pair (T, L) as in DEFS, with ASSEMBLY, or a precise obstruction to every pair of the stated class. Equivalent aim: W-SPAN-PROPER at the case SS-COF. D. ASK Part 1 (necessary conditions). Derive the properties any successful pair must have, and say which are new. Required: (N1) the produced M lies in ONE composant and is nowhere dense, so the sets A_n cannot contain a nonempty open set of X (use the nowhere-density and composant facts above); (N2) T must distinguish failing from non-failing doors (the intrinsic law does not), so T(A) must involve a plane datum of the door: a crosscut, the pocket Sh(s) or the shadow arc I_s of s; say which; (N3) in the circle / HOM model L holds with T = F (h extends to a plane homeomorphism) and failing doors occur at every depth, yet the rotation is cm and no proper h^2-invariant continuum exists, so ASSEMBLY must use nonseparation or the perfectness of F at a named line (a genuine R-instance has no plane-homeomorphic extension: Sol 6, via Cartwright-Littlewood); (N4) L must be proved from the axioms of an R-instance, not read off a model (the class of R-instances may be empty). Part 2 (name one law, carry it out). Name ONE candidate L that is not on the dead-end list, with exact quantifiers, and prove it from the R-instance axioms, or refute it with static Tier 0 data (planar strip, perfect F with F(s) meeting X, composant labels, all laws of LEMMAS), or reduce it to one named lemma (gate G1-G4). Candidates, none claimed to work: (i) TEAR LAW: for a chord q with F(cl q) cap X = T_q (the torn set, a compact set containing h(q^-), h(q^+)), the law h(K_q) in the K-trace of the component of U \ F(q) that contains F(Sh q) near h(q^-), when F(q) is a crosscut after discarding its X-contacts (the E1 side of the dichotomy); say what happens on E2; (ii) COMPONENT LAW: for the arc component P(x) of x in K_s (or composant of x) the set h(P(x)) lies in the arc component of h(x) in some K_{s'}; find the crosscut s' from the plane geometry of h(s^-), h(s^+) (hint: by DP-K / T1-EX, h(s^-), h(s^+) are never ends of a short E_F chord when the door does not fail; what replaces this when it fails?); (iii) LIMIT LAW: transport of Hausdorff limits L = lim cl A_n of bank-window sets A_n along e -> inf, using upper semicontinuity of F-images (F perfect: F(cl A) = cl F(A)) and that Pr(E_F) = X; (iv) a G-side law: use G (G|X = h^-1) to define T through G(F(A)) cap X, and test whether G o F = id fails off X in a way that breaks every candidate. Part 3 (assembly or builder). If you prove a law, show the ASSEMBLY: which sets A_n are used, why the union and closure are connected, why M != X (avoid Lemma NH: M must not be the complement of a single piece; say which composant contains M) and why h^2(M) in M. If you refute a law or class, give the static data and the first axiom it would violate in an actual R-instance, or state that all axioms hold. 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 law, even partial with an honest gap, is worth more than a restatement of W-SPAN-PROPER. Take your time: check every step, hypothesis and quantifier. This is a hard open problem. E. CONTROLS (all must hold) (C1) Positive control: on b_opus_gbridge2's decomposable chart hosts (thick fibres, h not cm) the pattern 'invariant open family of hairs, complement a proper continuum' works; your law must say why the analogue on an indecomposable X is not Lemma NH. (C2) HOM / circle model: any law proved from continuity and plane topology alone holds there; the assembly must break there at a named line. (C3) Abstract composant models (m36, A201): a law that holds in them cannot prove ASSEMBLY. (C4) Finite depth: a law transporting A_n to A'_m for each n with a threshold depending on n does not give one object. (C5) A law of the form h(A) in A' with A' = X is true and empty; T(A) must be a proper subset of X. F. DEAD ENDS (not progress) (1) Hair separation with a complement-continuum (Lemma NH) and intrinsic families (Lemma GEN, Lemma C-OPEN): invariant for free, door-blind. (2) Pocket-trace cm, hyperspace trapping UD/EQ, invariant measures, HT + semicontinuity, clopen span (WALLS row W-SPAN-PROPER). (3) Plane families with no transport (Step INV); 'geometric bank permutation' withdrawn (codex_aud_kill_hom). (4) Abstract composant laws and cocycles (A196, m36), impression transport (m24, HOM kill test). (5) The exact law h(K(x,y)) = K(hx, hy) alone (door-blind). (6) A law stated only for the pair (a_n, b_n), not for a set, or only at finite depth. 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: SETLAW PROVED (a named law, its proof from the R-instance axioms, and the assembly of M from cofinal failing doors) | NO SETLAW OF THE STATED CLASS (a no-go with static Tier 0 data satisfying every listed law) | 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 the intrinsic law h(K(x,y)) = K(hx, hy) as if it saw doors (it is door-blind); using a hair family whose complement is the candidate M (Lemma NH: input = output); a law true only for plane homeomorphic extensions (the HOM / circle model); a law stated for the pair (a,b) only, not a set; claiming SS-COF, DP-K, L2 or PER