Batch 18: R-I(3) root cases, Baire localisation, common fixed set

Prompt

RESEARCH QUESTION (prover first: classify the root cases of R-I(3)). Self-contained. Label every claim PROVED (you prove it), SOURCE (exact reference), 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. Take your time and be extremely careful; no helper agents. A. DEFINITIONS AND SETTING PRE-DISPATCH CERTIFICATE (batch 18, microeval channel). Shared map: Siege 1 CAN iff P2EX; P2EX Step 7 holds iff G2 and not R-I and not R-II; this batch attacks the residual R-I(3); PER arrow R1 / E_p^-. (C1) R1 / E_p^-. Target: exclude R-I(3) through the root b~. YES: every root case reduces to CUBE-ROOT or to an ABSORB / cm_strong(b~) contradiction, so R-I(3) is excluded. NO: the exact rooted 3-cycle odometer and its first failing clause. (C2) Controls: positive = BR2 (planar chainable calibration, p = 2) must not be contradicted at its scope; negative = ARC-4^- (decomposable: composant and indecomposability steps must fail). (C3) New input: CUBE-ROOT and THREE COPIES (11-09, two GPT lanes). (C4) Duplicate: GPT and Claude lanes ask single pieces; this batch tests five other models on the full case split. Definitions: a continuum is a nonempty compact connected metric space; subcontinua are closed and connected; nondegenerate means more than one point. Planar means embeddable in R^2, hence in the sphere S^2 = R^2 + {inf}; nonseparating means S^2 minus X is connected. Tree-like: for every eps>0 an onto map to a finite tree with all fibres of diameter < eps (tree-like continua are hereditarily unicoherent). Indecomposable: not a union of two proper subcontinua; then every proper subcontinuum is nowhere dense, the union of two proper subcontinua that meet is proper, and the composant Comp(x) (union of all proper subcontinua containing x) partitions X into uncountably many pairwise disjoint dense composants; a proper subcontinuum meeting Comp(x) lies in Comp(x). For a homeomorphism g: cm_strong(g) means no proper nonempty g-invariant subcontinuum, singletons included (so g has no fixed point). Per(g), Fix(g): periodic, fixed points. PFPP: the open plane fixed point problem. Cartwright-Littlewood (SOURCE, via Fokkink-Mayer-Oversteegen-Tymchatyn Thm 10.3): an orientation-preserving homeomorphism of the plane leaving a nonseparating plane continuum invariant has a fixed point in it. Statuses: LAB = earlier lab answer, unaudited unless noted; DERIVED = lab proof, audit owed. Never assign VERIFIED or CLOSED. CLASS A_p+ (LAB). (K1) p is an odd prime. (K2) X is a nondegenerate planar nonseparating tree-like indecomposable continuum, U = S^2 minus X contains inf, X = boundary(U) (X has empty interior). (K3) b: X -> X is a homeomorphism with an anchor cycle A = {a_i : 0 at most i at most p-1}, b(a_i) = a_{i+1} (indices mod p). (K4) cm_strong(b). (K5) R-A: Per(b) = A. (K6) The anchors lie in pairwise distinct composants K_i = Comp(a_i). (K7) F: S^2 -> S^2 is an ORIENTATION-REVERSING homeomorphism with F(inf) = inf and F|X = b^p (so every anchor is F-fixed; F and b^p are NOT cm). (K8) h is the prime-end map of F on U. Write G = F^2 (orientation preserving), H = h^2, E = Fix(H). There is NO assumption that the h-fixed ends are full-principal. Prime ends. phi maps D* = {|z|>1} + {inf} conformally onto U, phi(inf) = inf; F need not commute with phi (it does NOT map radial rays to radial rays). Prime ends = points of the unit circle T. Imp(e) (impression) and Prin(e) (principal set) are subcontinua of X with Prin(e) inside Imp(e). SOURCE (Pommerenke 1992, Thm 2.16): Imp(e) is the unrestricted cluster set of phi at e; Prin(e) is the radial cluster set, equal to the cluster set in any Stolz angle at e and to the INTERSECTION of the cluster sets along all curves in D* ending at e (a single tangential curve can have a larger cluster set, up to Imp(e)). e is accessible iff Prin(e) is a point L(e) (the radial curve lands there). Full-principal: Prin(e) = X. Theta = {e : Imp(e) = X}. INPUTS (LAB/DERIVED, all usable here). I1: h is a homeomorphism of T with Imp(h e) = F(Imp e), Prin(h e) = F(Prin e), L(h e) = F(L e). h reverses orientation of T, so it has exactly two fixed ends, and H preserves orientation. I3 (Moore, SOURCE): collapsing one nonseparating subcontinuum C of X to a point gives a sphere and a map that is a homeomorphism off C. I7: accessible ends are dense on T. Unique access (Brechner 1978 Thm 3.2 via F, LAB): each point is L(e) for at most one accessible end e. STEP 2: every subcontinuum of X is nonseparating. GENERALIZED L-SC (STEP 3): if ends alpha, beta alternate on T with two ends zeta_0, zeta_1 of Prin = X (fixed or not), no proper subcontinuum of X contains Prin(alpha) u Prin(beta) (proof: collapse C, the two radial images form a Jordan curve meeting q(X) only at the collapse point, each side contains a full-principal ray, so q(X) is in the curve). ARC IMPRESSIONS: for a closed arc J of T, Imp_all(J) = intersection over eps of the closures of phi of the eps-neighbourhood of J in D*; it is a subcontinuum equal to the union of Imp(e), e in J. BIS: if a closed arc J misses Theta, Imp_all(J) is a PROPER continuum containing Imp(e) for every e in J. Theta is closed, nonempty and h-invariant. E is closed; H fixes the endpoints of each gap (component of T minus E), is increasing there, and its orbits in a gap converge to the two endpoints. B. THE R-I(3) SETTING (LAB, statuses as given). R-I is the case of a tuple of A_p+ in which Prin(xi_0) = Prin(xi_1) = X at the two h-fixed ends (P7), Theta = E = Fix(H), every component J of T minus E is H-invariant and misses Theta (Type I; it carries one 'lit' composant K(J) with F(K(J)) = K(hJ) != K(J)), and no anchor composant contains an accessible point (DARK). BC-TRANSFER (LAB): the double cover branched at an F-fixed point turns the b-free clauses into an object (Y, f) with f = F~|Y, a deck involution tau commuting with f, Fix(tau) = {0}, and three f-fixed points A = {0, a'_1, a'_2} in three distinct dark composants; R-I(3) needs in addition a ROOT: a homeomorphism b~ of Y with b~^3 = f, A one b~-cycle, cm_strong(b~). IMPORTANT: b~ is a homeomorphism of Y only; only f extends to the plane, so accessibility, impressions and darkness are f-data. C(f) = centraliser of f in Homeo(Y). DINF (audited, repairs open): if some u in C(f) moves 0, then s = tau u tau u^-1 is in C(f), has infinite order, and s 3-cycles A (in the rooted setting such s exist). Write K = Fix(s) (compact, f-invariant, misses A). PC-NO (audited CWR): no component of K is f-periodic. CQ (audited CWR): the component space Q of K is compact zero-dimensional, f_* is a homeomorphism of Q, and a minimal set M of f_* is a Cantor ODOMETER (nested cyclic clopen partitions, periods q_n dividing q_(n+1)). HULL LEMMA: by hereditary unicoherence every set Z of Y lies in a smallest subcontinuum <Z>, and g<Z> = <gZ> for homeomorphisms g. PACKET-SATURATION (DERIVED): every nonempty clopen packet B of components of K that is f_*^n-invariant has <union B> = Y (a proper hull is s- and f^n-invariant, so Cartwright-Littlewood puts exactly one anchor in it, but s 3-cycles the anchors). ABSORB (DERIVED, unaudited): a closed f^m-invariant O inside K with K' u O connected, K' a finite union of proper subcontinua, is impossible. COMP-MEAGRE (DERIVED, unaudited): for every composant L, the members of M lying in L form a meagre subset of M. BLIND (DERIVED, unaudited): a composant with no accessible point meets no proper impression; comeagre-many members of M lie in such dark composants, where every prime-end argument is blind. CUBE-ROOT (DERIVED, unaudited): b~ normalises C(f); b~ K = Fix(s') with s' = b~ s b~^-1, components map homeomorphically and f_* is intertwined; IF s commutes with b~ and a minimal odometer set is b~_*-invariant, contradiction (the centraliser of the 3-adic adding machine is its translations and 3c = 1 has no solution in Z_3). THREE COPIES (DERIVED, unaudited): b~ cyclically conjugates the component spaces of Fix(s), Fix(s'), Fix(s''); an abstract 3-adic datum with Fix(s) = Fix(s') = Fix(s'') exists but is disconnected. MISSING LEMMA (open): PACKET-LOCALIZATION, some periodic packet with a proper hull, which would contradict PACKET-SATURATION and close R-I(3). C. TARGET. (1) State exactly what CUBE-ROOT needs: does it require s to commute with b~, or only that b~ (or a power) maps one minimal set of ONE component space to itself? (2) Split: (a) b~ maps some minimal set of Q to itself after identifying the three copies, (b) b~ 3-cycles three disjoint minimal sets M, b~M, b~^2 M. In case (b) decide whether O = |M| u b~|M| u b~^2|M| (closed, b~-invariant, f-invariant) contradicts ABSORB, COMP-MEAGRE or cm_strong(b~); note <O> is b~-invariant, so <O> = Y. (3) If case (b) survives, give the most explicit p = 3 symbolic model and name its first failing clause. D. ASK Prove or refute. If you prove, give a numbered linear argument citing each input above by name. If you refute, give the model and check every class clause K1-K8 and the R-I clauses on it. Then run each control and say where it exits. E. CONTROLS (all must hold) ARC-4^- (lab control): X = [-1,1] x {0}, f the time-one map of the flow of v(x) = (x^3 - x)/(1 + x^4) (increasing homeomorphism, Fix(f) = {-1,0,1}, no other periodic points), F(x,y) = (f(x), -2y) (orientation-reversing). Prime ends: x^+ and x^- for -1 < x < 1 plus one end at each endpoint; h(x^+) = f(x)^-, h(x^-) = f(x)^+; the endpoint ends are the h-fixed ends (accessible), {0^+, 0^-} is a 2-cycle landing at (0,0); X is decomposable and chainable, Theta is empty. Circle reversal: z -> conj(z) has two fixed points and period-2 pairs, so orientation reversal alone gives nothing. Rotating p-od: p arcs meeting at o, b rotating the arms; the only b-invariant point is o (forbidden only by cm_strong). Pseudo-arc: hereditarily indecomposable chainable, has the fixed point property (never a host), but Oversteegen-Valkenburg Ex. 5.4 embeds it with every impression the whole continuum and many full-principal ends: impressions alone give no contradiction. A disk with a whisker fails empty interior. State each control's first exit. A proof that works verbatim on ARC-4^-, on a chainable host, or without cm_strong(b) is suspect: name the step where indecomposability, tree-likeness, nonseparation or cm_strong(b) is used. F. DEAD ENDS (not progress) Prime-end or crosscut arguments on dark members (BLIND); hulls of single components or of packets (PACKET-SATURATION); fixed-point index counts. G. FORMAT Line 1 exactly: VERDICT: PROVED | COUNTEREXAMPLE-CANDIDATE | PARTIAL | OPEN-OBSTRUCTION | RESTATEMENT. Then SUMMARY (at most 100 words), the argument (at most 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; give exact theorem numbers or say UNSURE. RED FLAGS: using cm(F) or cm(b^p) in place of cm_strong(b); treating F as commuting with phi or mapping radial rays to radial rays; confusing impression with principal set; assuming the h-fixed ends are full-principal (that is the P7 packet assumption, NOT available in A_p+); using KEY-STRONG ('a proper continuum through an accessible point misses its F-image'), which needs full-principal fixed ends; assuming P u F(P) is b-invariant.