batch 13

Prompt

RESEARCH QUESTION (P2EX siege, Step 7: the TAIL gate T under (R-A)). 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 PRE-DISPATCH CERTIFICATE (batch 13, owner microeval channel). (C1) R1 / E_p^-, arrow CAN <=> P2EX (siege 1), Step 7. Target T: under (R-A), an FP2 pair forces a proper continuum containing a forward or backward G-orbit tail (G = F^2) of some accessible point. YES: with the day's implications this excludes FP2 under (R-A) (Step 7 conditionally closed, audit owed). NO: an explicit relaxed configuration where FP2 holds and every G-orbit tail of an accessible point has closure meeting every proper continuum's complement, with the first clause it breaks. Neither alone is CLOSED. (C2) Controls: ARC-4^- (T holds trivially there: name the step that uses indecomposability); canonical case Fix(h^2) = {xi_0, xi_1} (T must fail or be vacuous there, else the argument proves too much); Oversteegen-Valkenburg Example 5.4; rotating p-od. (C3) New input: the day's missing list collapses toward T (FIX-CARRIER => T, M2 => T, Argon, unaudited). Nobody has attacked T with h^2(e) = e AND AP-UNIVERSAL together. Lab provenance ids (not available to you): Argon 100, S7TAIL. (C4) Duplicate check: S7TAIL attacked T-or-S (shrinking connectors) and found chain gluing fails; batch 12 never targeted T. 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, HU: two subcontinua meet in the empty set or a continuum; every subcontinuum of a planar tree-like continuum is tree-like and nonseparating). 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 of x (union of all proper subcontinua containing x) partitions X into uncountably many pairwise disjoint dense composants. For a homeomorphism g: cm(g) means no proper nonempty g-invariant subcontinuum; cm_strong(g) is the same with singletons counted as subcontinua, so g has no fixed point. Per(g), Fix(g): periodic, fixed points. FPP (fixed point property): every continuous self-map has a fixed point; PPF: periodic-point-free; PFPP: the open plane fixed point problem (does every nonseparating plane continuum have FPP?). Cartwright-Littlewood (SOURCE): an orientation-preserving homeomorphism of the plane leaving a nonseparating plane continuum invariant has a fixed point in it (Bell, Brown extensions). Lab vocabulary: R1, E_p^- = lab route labels (setting below); siege 1 = the lab's current attack on P2EX. Statuses: PARTIAL/OBSTRUCTION = author's verdict; UNAUDITED = no audit yet; CONFIRMED-WITH-REPAIRS = audit passed, repairs written in, not yet closed; one GPT agent = one GPT run; never assign VERIFIED/CLOSED. Take your time, be extremely meticulous, I believe in you. SETTING (route R1, branch E_p^-; LAB; package UNAUDITED). X is a nondegenerate planar nonseparating tree-like indecomposable continuum in R^2, U = S^2 minus X (a simply connected domain containing inf), and X = boundary(U) (clause BD; automatic, as planar tree-like X has empty interior). p is an odd prime. b: X -> X is a homeomorphism with cm_strong(b) and anchor cycle A = {a_0,...,a_{p-1}}, b(a_i) = a_{i+1}; the a_i are called anchors. Sources disagree on the spectrum hypothesis: (R-A) Per(b) = A or (R-B) only Fix(b^p) = A; OPEN, state which reading each step uses. F: S^2 -> S^2 is an ORIENTATION-REVERSING homeomorphism with F(X) = X, F(inf) = inf and F|X = b^p; so F^2 preserves orientation, F^2|X = b^{2p}, and b^p, b^{2p} are NOT cm (anchors are fixed singletons): cm(F) is never a substitute for cm(b). Complement PPF: F has no periodic point in R^2 minus X. Prime ends. Let phi map D* = {|z|>1} + {inf} conformally onto U with phi(inf) = inf (normalisation only; F need not commute with phi, so F does NOT map radial rays to radial rays). Prime ends of U correspond to points zeta of the unit circle T. For a prime end e, the impression Imp(e) and the principal set Prin(e) are subcontinua of X with Prin(e) inside Imp(e). SOURCE (Pommerenke, Boundary Behaviour of Conformal Maps, 1992, Theorem 2.16, checked on the full text): Imp(e) is the unrestricted cluster set of phi at zeta, and Prin(e) equals the cluster set of phi along the radius ending at zeta, which also equals the intersection of the cluster sets along all curves in D* ending at zeta and the cluster set in any Stolz angle at zeta. e is accessible iff Prin(e) is one point (its landing point); e is full-principal if Prin(e) = X. F induces a homeomorphism h = h_F of T with Imp(h e) = F(Imp e), Prin(h e) = F(Prin e). Since F reverses orientation, h has exactly two fixed ends xi_0, xi_1, swaps the two open arcs of T between them, and h^2 preserves orientation; every h-periodic end has least period 1 or 2. A 2-cycle is a pair {e, e'} with h(e) = e' != e; its two ends lie on opposite arcs, so {e, e'} and {xi_0, xi_1} are LINKED (alternate on T). P2EX: no 2-cycle exists, i.e. Fix(h^2) = {xi_0, xi_1}. The lab's canonical prime-end law CAN is equivalent to P2EX (LAB, audited). LBACC (LAB standing package of this branch) includes: the two fixed ends are full-principal, Prin(xi_0) = Prin(xi_1) = X; it gives no prime-end action of b itself, only of F = b^p. Oversteegen-Valkenburg Example 5.4 (SOURCE, not rechecked): a planar pseudo-arc embedding with every impression the whole pseudo-arc and a dense second-category set of full-principal ends, so many full-principal ends carry no static contradiction. B. LEMMAS (status; where used) Step 3 (LAB, unaudited): (i) if Prin(e) = {x} then Prin(h e) = {F(x)}; (ii) at each point of an indecomposable X = boundary(U) at most one prime end has singleton principal set. Hence under (R-A) every 2-cycle {e, e'} has nondegenerate P = Prin(e), P' = Prin(e') with F(P) = P', F(P') = P. L-SC (sector collapse; LAB, 7/7 refuters no defect): if two ends e, e' are linked with two full-principal ends xi_0, xi_1, then NO proper subcontinuum C of X contains P u P'. Proof: collapse C to a point c (Moore; quotient is a 2-sphere); the images of the radial rays of e, e' form a Jordan curve J meeting q(X) only in c; the two fixed-end radial sectors lie in different components V_0, V_1 of S^2 minus J; full-principal rays cluster on all of q(X), so q(X) lies in cl(V_0) cap cl(V_1) = J, so q(X) = {c}, contradiction. DERIVED: a 2-cycle is (FP2) P = P' = X or (SPLIT) P, P' proper, disjoint, in different composants. Under (R-A) SPLIT is excluded (Cartwright-Littlewood for F^2 on P), so P2EX under (R-A) is EQUIVALENT to excluding FP2. Under (R-B), SPLIT: the b-orbit of P has length 2p, K = union b^j(P) is proper and disconnected, the component of K containing P has b-period q in {2, 2p}. Facts (several models, unaudited): a proper F^2-invariant continuum contains at most one anchor; proper b^p-invariant continua have pairwise disjoint b-translates; no proper subcontinuum contains two anchors; a subcontinuum meeting two composants is X. Under (R-A) every proper F^2-invariant subcontinuum contains exactly one anchor and lies in its composant (A6(a): it contains no accessible point). Every composant containing an accessible point is moved by F, so anchor composants are inaccessible. DISTINCT (4 models, 4 families, one third-family audit CONFIRMED): the p anchors lie in p pairwise distinct composants. Crosscut domains, one-sided cluster sets and the four open-arc landing closures at FP2 ends are all X (dead invariants); prime-end rotation of F^2 is 0; any reversing circle map extends PPF to the prime-end disk. Day 11-07 (unaudited unless noted). RIGIDITY (Opus): h is determined by F|X = b^p and the embedding (Step 3(ii) + density of accessible ends), so FP2 is a property of (X in S^2, b^p). (R-B) with T = Fix(b^{2p}) minus A empty implies (R-A) (Bonino 2004 Thm 3.1 + complement PPF; UNSURE of exact numbering); under (R-B), "every proper F^2-invariant subcontinuum lies in an anchor composant" is EQUIVALENT to T empty. NO-GO: continua generated by one landing orbit have closure X; a valid exclusion must use h^2(e) = e, not only Prin(e) = X. MISSING LIST under (R-A), G = F^2, J1 = one open arc between xi_0, xi_1: FIX-CARRIER (FP2 yields a proper G-invariant subcontinuum containing an accessible point); M1 (every FP2 pair is anchored at some anchor, defined via landing orbits of accessible ends tending on T to e, plus the full-principal analogue of "no fixed point of indecomposable X anchors a 2-cycle"); M2 (an accessible x and a proper continuum M containing x, G(x) with diam G^n(M) -> 0 as |n| -> inf); M3 (FP2 yields a point of least F-period 2; immediate contradiction under (R-A)); T (TAIL: a proper continuum containing a forward or backward G-orbit tail of an accessible point). Argon (one model): every accessible end or its h-image has a G-orbit converging on T to a noncanonical fixed end (AP-UNIVERSAL), so FIX-CARRIER => T; Local Component Stabilization: if continua M_n joining x_n, x_{n+1} (consecutive orbit points) eventually lie in a closed ball B(a,r) not covering X, the x_n stabilize in one component of X cap B(a,r), giving T; hence M2 => T; M1 => M3 (conditional on Bonino). So T is the single (R-A) gate apart from M1/M3. Index route dead (every exterior Jordan curve gives F^2-index difference 0). SPLIT-QUOTIENT (2 models PROVED, audit owed): under (R-B) collapsing the components of K gives a host meeting the setting with {e, e'} accessible, so SPLIT reduces to an accessible 2-cycle; NOT checked: that xi_0, xi_1 stay full-principal in the quotient; the quotient is again (R-B). Known obstruction (S7TAIL): orbit points occupy infinitely many components of every small closed neighbourhood of the limit anchor; component limits are inaccessible anchor continua; two null chains can shrink to the anchor while fixed exterior rays block every small bridge. C. THE TARGET (1) State T with quantifiers. (2) Use the FP2 pair: the G-fixed end e in J1 splits J1 into two h^2-invariant arcs; accessible ends in each converge (AP-UNIVERSAL) to e or to xi_0/xi_1. Prove or refute that some accessible G-orbit has its tail inside one proper continuum (e.g. a component of X cap B(a,r) for the limit anchor a). (3) Where does h^2(e) = e enter, and why does the same argument fail in the canonical case? D. ASK Give the first failing step if not PROVED, with the exact quantifier that breaks. Separate (R-A) from (R-B). Test on the controls and name the first exit. No novelty/VERIFIED/CLOSED claims; no helper agents. 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 of [-1,1], Fix(f) = {-1,0,1}, no other periodic points), F(x,y) = (f(x), -2y) (orientation-reversing; R^2 minus X is PPF). Prime ends: x^+ and x^- (upper and lower sides) for -1 < x < 1 plus one end at each endpoint; h(x^+) = f(x)^-, h(x^-) = f(x)^+; the endpoint ends are the fixed ends (singleton principal sets, NOT full-principal) and {0^+, 0^-} is a genuine 2-cycle landing at (0,0) from two sides. X is decomposable and chainable. Interval reflection: b(t) = -t on [-1,1]; Fix = {0}, all other points have period 2; every symmetric [-c,c] is invariant. Circle reversal: z -> conj(z) has two fixed points and period-2 pairs, so reversal alone cannot give P2EX. Rotating p-od: p arcs meeting at a centre o, b rotating the arms; translates of the terminal half of one arm are pairwise disjoint; the only b-invariant point is o (forbidden only by cm_strong). Pseudo-arc: hereditarily indecomposable chainable; it has FPP, so it is never a PPF host, but it is the static calibration for full-principal ends (Example 5.4). Pseudo-circle and SPIRAL(3) separate the plane; a disk with a whisker fails BD. State each added control's first exit. A proof that works verbatim on ARC-4^-, on a disk with whisker, or on a chainable host is suspect; name the step where indecomposability, tree-likeness, nonseparation or cm_strong(b) is used and show it fails on the control. F. DEAD ENDS (not progress) Orbit-generated continua (closure X by the no-go); crosscut domains; index differences; single-gap monotonicity; cm(F) or cm(b^p). 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; UNKNOWN if unsure a cited result exists (give exact theorem numbers or say UNSURE). RED FLAGS: using cm(F) or cm(b^p) in place of cm(b); treating F as commuting with phi or mapping radial rays to radial rays; confusing impression with principal set; assuming Fix(b^{2p}) = A under (R-B); assuming P u P' is b-invariant; dropping the canonical full-principal fixed ends xi_0, xi_1 as a negative test.