math attempt

Prompt

RESEARCH QUESTION (plane continuum theory / dynamics). You are a research topologist. Work carefully and honestly. Label EVERY claim PROVED (you give the proof), SOURCE (exact reference, theorem number and hypotheses; say if you could not check it), DERIVED-BY-US (taken from the project notes below, unaudited), CONJECTURE or OPEN. Say plainly where your argument stops. If an idea only restates a gap in new words, say RESTATEMENT. Do not pad. 1. SETTING AND DEFINITIONS A continuum is a nonempty compact connected metric space. X is a nonseparating plane continuum: a continuum in R^2 whose complement in R^2 is connected (X need not be locally connected; it may be tree-like, hereditarily indecomposable, etc.). h: X -> X is a homeomorphism of X with NO periodic point. h is NOT assumed to extend to the plane (the classical Cartwright-Littlewood / Bell results need an extension; the intrinsic case is the open problem). A homeomorphism g of X is continuum-minimal (cm) if no proper nonempty subcontinuum K of X has g(K) = K (single points count as subcontinua). The master question PER: does every homeomorphism of a nonseparating plane continuum have a periodic point? 2. THE REDUCTION WE WORK ON (D1997 in our notes) L2: for a periodic-point-free homeomorphism h of a nonseparating plane continuum X, h^2 is never continuum-minimal. Known reduction (DERIVED-BY-US, built on Blokh-Fokkink-Mayer-Oversteegen-Tymchatyn, "Fixed point theorems for plane continua with applications", Memoirs AMS 2013, arXiv 1004.0214, called BFMOT, and on Bell's unique outchannel theorem): L2 together with "there is no planar 2-adic tower BLOCK*(2)" implies PER. The common planar mechanism is Bell's unique outchannel: in a minimal counterexample the prime-end map has a unique outchannel E_t(h); BFMOT Thm 3.2.2 says Index = Variation + 1; the parity of the variation of h^2 along the outchannel is the tool. - Case A (the outchannel is COHERENT): L2 is PROVED modulo BFMOT. - Case B (INCOHERENT outchannel): OPEN. Residual R1. - R2: planar 2-adic towers BLOCK*(2) (the parity mechanism cannot see q = 2 on collapsed levels). OPEN. Controls you must respect: the Knaster bucket handle and the pseudo-arc are chainable, so every self-map has a fixed point (Hamilton); a pseudo-circle is hereditarily indecomposable and planar and carries minimal homeomorphisms (Handel) but separates the plane; solenoids are not planar. Nonseparation and planarity must both be used. 3. WHAT THE PROJECT HAS LEARNED TONIGHT (all one-agent or unaudited unless marked; treat as leads, check them, attack them) (a) Hereditary-unicoherence junction rule: every L2 host is hereditarily unicoherent (DERIVED-BY-US, unaudited); applied at the junctions of the short image arc of the outchannel, together with Janiszewski enclosure and transport of endpoints by h and h^-1, it kills separating deep re-entry for chord-decomposable outchannels with a clean X-sided strip (exact finite tests kill all 3588 + 858 law-passing finite profiles). (b) Theorem W (DERIVED-BY-US): an X-sided L2 counterexample has no h-closed family of product charts (no chart cycle under h; well-ordering of passage depths); with one dense channel h never maps the mid band of a product chart onto the mid band of a product chart. This reduces Cantor-chunk re-entry to the open FOREIGN-BAND conjecture FB (exact chunk tests show that one foreign atom per chart pair lets every single-channel law hold). (c) Case B therefore survives only as (i) Cantor-chunk re-entry (conjecture FB) or (ii) outchannels with SIDE CHORDS at arbitrarily large depth (untouched). (d) R2: parity does see q = 2 on the UNCOLLAPSED tower level (F^2 is fixed-point free there); a level whose outchannel is coherent and one-sidedly adjacent for F^2 is impossible. Theorem K (DERIVED-BY-US): at a tower level an X-sided outchannel strip always has arbitrarily deep sections with var(F^2) <= 0; hence no tower level has a clean strip, and the residual (SC) survives only on strips with side chords at arbitrarily large depth (plus incoherent levels). So SIDE CHORDS are the shared residual of both open cases. (e) Tail recurrence (needed for BFMOT 6.2.1 without minimality of the counterexample): the tail limit set must contain the points a_0, p, z, p_z; this holds exactly when they lie in the projection of the outchannel, and a Janiszewski tail lemma proves it whenever the closed tail through a return point does not separate the plane. The full-density form (N2) is BFMOT verbatim; the single-point form (N1) is not. A Case-A/Case-B proof of L2 does not use it. (f) Structure of any planar counterexample with a trapping block (Escape Lemma, audited by one agent): if h is cm-type but not chain transitive, the attractor A and the repeller A* are DISCONNECTED, X is indecomposable with empty interior, and h does not extend to the plane. A chart-host homeomorphism g_D of the planar-candidate family is topologically mixing, hence chain transitive, so those hosts cannot realise that shape. (g) Dead ends (do NOT repeat them as progress): the Moore-triod count (the relevant hosts contain no triods); rotation-system / BKM coherence tests (they reject even the known planar control host); chain-class and Conley-hull decompositions for hereditarily unicoherent hosts (restatement of chain transitivity); Hagopian's arc-component fixed point theorems (they need EVERY arc component mapped INTO ITSELF; h only permutes arc components, so they do not apply without a further idea); local fixed point index and Cartwright-Littlewood arguments (they need h to extend to the plane); Whitney-map / Baire arguments on reach of folds in class-A continua; "h acts distally on a periodic fence, hence does not extend" (the same fences occur in a chainable planar host, so it does not see planarity). (h) Known facts about Minc's examples: for each j there is a hereditarily indecomposable tree-like continuum with a self-map without a periodic point of period at most j (Minc 2000), but those are not shown planar and not shown to carry homeomorphisms. 4. WHAT WE ASK (do as much as you can; stop where your arguments stop) T1. State the SIDE-CHORD residual exactly (one statement covering Case B (ii) and R2 (SC), or two statements if they are different configurations, and say why). List which of (a)-(f) it escapes. T2. Prove or refute its exclusion for a periodic-point-free homeomorphism of a nonseparating plane continuum, using a method NOT on the dead-end list (g). Candidate tools: Janiszewski-type separation by X-arcs applied to the side chord; hereditary unicoherence of the host at the junctions a side chord creates; transport of the chord by h^{+-1} and its image under the outchannel map; the planar two-dimensional facts that fail in R^3 (an arc has codimension one; Tymchatyn-Walker 1998 Lemma 4: a zero-dimensional separator gives a Jordan curve; alternating endpoints cannot be joined disjointly on one side); a count of disjoint triods / chords (Moore) if you can make it apply. T3. Decide the foreign-band conjecture FB (state it exactly with all quantifiers first), or reduce it to one named lemma. T4. Give a finite or explicit model with a planted deep side chord and say whether the stated rules (HU junction rule, Theorem K) keep or kill it, with a control: a configuration with no side chords where the conclusion is already proved must come out as known. T5. If you believe L2 is FALSE: give the host X and the homeomorphism h with enough data to check that h has no periodic point, h^2 is cm, X is planar and nonseparating; otherwise give the exact shape a counterexample must have. T6. Say where PLANARITY and where NONSEPARATION are each used in your argument, and which step fails in R^3. T7. Final section "STILL OPEN EXACTLY": the smallest remaining statements, each with the single next test that would decide it. 5. FORMAT Plain text or LaTeX, at most about 4,500 words. Start with one headline line: SC-EXCLUDED / SC-KEPT / SC-OPEN ; FB-DECIDED / FB-OPEN ; L2-PROVED / L2-REFUTED / L2-OPEN. Never present a finite computation as a proof of an infinite statement. Never cite a theorem beyond its stated hypotheses. If you are not sure that a cited result exists, write UNKNOWN. Prefer one complete correct lemma to many sketches. APPENDIX A. OUR OWN EARLIER SUMMARY OF THE L2 REDUCTION (project report b_opus_planar_step, one agent, unaudited, excerpt; use only as terminology and as a claim to test): VERDICT: PARTIAL - No lemma weaker than PER can imply (a), (b) and (c) together (proved: any such lemma implies PER). The common planar mechanism is Bell's unique outchannel: tcm systems, blocks and tower carriers are all BFMOT "minimal counterexamples". Through a parity effect it yields the single lemma L2 (h^2 is never continuum-minimal), and L2 + "no planar BLOCK*(2)" => PER. L2 is PROVED (modulo SOURCE BFMOT) when the outchannel is coherent and stays OPEN in the incoherent case. LEMMA: PRF RESIDUAL: (R1) L2 for incoherent outchannels: deep chords of the outchannel E_t(h) whose endpoint images are not joined by a single-crossing chord of E_t. (R2) Planar 2-adic towers, i.e. planar BLOCK*(2) (the parity mechanism cannot see q = 2, by design). # The common planar step behind SL-wand, SL-per and BLOCK*(q) 2026-10-29. Claude Opus 5.5 (effort high), one agent, lane b_opus_planar_step, loop target PRF, case H-NH8. Prover, source-auditor and adversarial roles were done one after another by this one agent, so **these are not independent-agent checks**. Nothing was computed: no Python ran apart from `tools/litsearch.py` and `tools/theoremsearch.py`, there was no heavy run, and no other agent was used. Statuses: SOURCE means quoted from a full text on disk, DERIVED means proved here (UNAUDITED), and OPEN and CONJECTURE are used as in AGENTS.md. Sources used (all on disk, text extraction): - **[BFMOT]** Blokh, Fokkink, Mayer, Oversteegen, Tymchatyn, *Fixed point theorems for plane continua with applications*, Memoirs AMS 2013, arXiv:1004.0214v2, `papers/lit_pc3b/bfmot2012.txt`: - Def 2.2.2 and the following remark, var = win(alpha,S,v), lines 909-949; - Thm 3.2.2 (Index = Variation + 1), line 1201; - standing hypothesis, lines 2617-2625; - Lemma 6.1.2, 6.1.3, Thm 6.1.4, 6.2.1, 6.2.2, lines 2987-3364; - Cor 7.1.2, Thm 7.1.3, Cor 7.1.4, lines 3392-3425. - **[BO09]** Blokh-Oversteegen, Fund. Math. 206 (2009), `papers/lit/ms_dl/blokh_oversteegen2009_fm206.txt`: - abstract, lines 13-20; - Thm 4.3 (= FMOT Thms 6.5, 9.1: the outchannel map "changes orientation (so that g flips essential g-crosscuts)"), lines 791-803; - degree -2 prime-end picture, lines 1702-1720. - **[H07]** Hagopian, Open Problems in Topology II §31, `papers/lit_pc3b/hagopian2007.txt` lines 194-206: Bell/Sieklucki/Iliadis Theorem 2, and "exactly one outchannel and its variation must be -1". - theoremsearch pointers, not quoted as load-bearing: - FMOT 2008 Cor 10.4 (0805.1184, oriented maps have a point of period <= 2; the same as BFMOT Cor 7.1.4 on disk); - Boronski 2015 Thm 1.2 (1510.06663: orientation-reversing plane homeomorphism, acyclic continuum with a k-cycle, k > 1, contains a point of least period 2). Project inputs used: - D1954 (PC3b-NEC CLOSED): carriers are indecomposable, planar tree-like continua are nonseparating, Lemma 3.1. - PROOF_PRF_REDUCTIONS_v1 (D1988/D1992). - D1968 Prop 3 as repaired in D1975 (tower level quotients and fibres are BLOCKs). - D1970: thinning. - D1972: BLOCK(2) = BLOCK*(2). ## 0. Notation - X is a nondegenerate planar tree-like continuum, hence nonseparating (D1954 L3.1), so T(X) = X. - A homeomorphism g of X is **continuum-minimal (cm)** if X has no proper nonempty subcontinuum K with g(K) ⊂ K. Singletons count. For a homeomorphism, forward invariance is equivalent to invariance: take the intersection of the g^n(K). - g is **tcm** if every g^k, k >= 1, is cm. - (S), U-per, U-wand and BLOCK(q)/BLOCK*(q) are as in PROOF_PRF_REDUCTIONS_v1 §1, §6 and D1968. - **NoTCM**: no planar tree-like X carries a tcm homeomorphism. - **NoTower**: no planar tree-like PPF system has an infinite carrier tower in the sense of D1954 Thm 7.2. ## 1. Exact logic: what any common lemma must contain **Theorem 1 (DERIVED).** PER <=> NoTCM ∧ NoTower. *Proof.* - (=>) A tcm homeomorphism is PPF, because a periodic point gives an invariant singleton. A tower lives in a PPF system. - (<=) Let h be PPF and let C_0 be a minimal h-invariant subcontinuum (Zorn). C_0 is nondegenerate (PPF) and indecomposable (D1954 5.1). - Put g = h^{p} restricted to C, with C cm for g. If g is tcm, NoTCM is contradicted. - Otherwise let m >= 2 be the least integer such that g^m has a proper invariant subcontinuum. Let K be cm for g^m; K is nondegenerate by PPF. - **Claim:** K, gK, ..., g^{m-1}K are pairwise disjoint. - Suppose K ∩ g^jK ≠ ∅ for some 0 < j < m. Put d = gcd(j,m). - The sets g^{ij}K (i mod m/d) form a chain of consecutively intersecting proper subcontinua of the indecomposable C. Their union L is therefore a proper continuum: two intersecting proper subcontinua have a proper union, then induct. - L is g^d-invariant with d < m. This contradicts either the minimality of m (if d >= 2) or the cm property of g (if d = 1). - So (K, g^m) is the next tower level, with q = m >= 2. Pairwise disjointness of all translates follows by the usual induction. - Iterating, either some level is tcm (NoTCM fails) or an infinite tower exists (NoTower fails). ∎ **Corollary 1.1 (DERIVED).** For PER, branches (a) SL-wand and (b) SL-per are needed only on **tcm** systems. - In a tcm system every minimal set is U, since F needs a proper f^q-invariant hull (PRF-RED §3). - Inside tcm, every periodic composant carries no invariant continuum, so it is a "translation composant" (D1965). - Hence (a) = PCOMP + no-translation-composant, and (b) = no-translation-composant. - The U-minimal sets of non-tcm (S) systems never have to be excluded. PRF over-proves in this respect.