m31

Prompt

RESEARCH QUESTION (plane continua / Cantor fans in planar tree-like indecomposable continua; NOT-BLOCK*(2) case beta, node NO-FAN, route (ii) not-EXIST-F_pl, FAR-MIX with an ODOMETER blade action AFTER the tight level-presentation class has been eliminated (farmix_odometer_opus_1102)). 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 (single-agent lab lanes, all UNAUDITED; lines marked INFERRED are the spec writer's bridging). - Y is a planar continuum (it embeds in R^2): tree-like, indecomposable (not the union of two proper subcontinua), with p in Y. A Cantor fan Phi in Y with vertex p is a subset homeomorphic to (C x [0,1])/(C x {0}), C a Cantor set; its blades beta_c (c in C) are arcs with beta_c cap beta_c' = {p} for c != c'. f: Y -> Y is a homeomorphism with f(p) = p, f(Phi) = Phi, f(beta_c) = beta_{sigma c}. sigma is the blade action, here an ODOMETER (an inverse limit of cyclic permutations of Z/2^n; minimal, equicontinuous, hence distal). f is intrinsic: NO extension of f to the plane is assumed. Nothing about periodic points of f other than p is assumed. - EXIST-F = this configuration without planarity (K_sigma satisfies it, nonplanar by Corollary K). EXIST-F_pl = with planarity. Not-EXIST-F_pl implies FP-2F implies NO-FAN (fp2f_opus_1102 Prop E), a branch of NOT-BLOCK*(2), which the lab needs for the periodic point theorem PER (research/ROUTE_MAP_L2.md rows 35, 84). - Planar circular order O on C: for four distinct blades, {d, d'} separates {a, b} if the directions interleave after straightening the 4-od (Schoenflies). A pair (a, b) is O-adjacent if one open O-interval between a and b is empty. Vertex transfer T (fixed heights): (a, b) in T iff a != b and there are x_a in beta_a minus p, x_b in beta_b minus p such that for all eta > 0 and blades d, d' not in {a, b} some continuum Z in Y minus (beta_d u beta_d') lies in the eta-neighbourhood of beta_a u beta_b and meets both balls B(x_a, eta), B(x_b, eta). T is f-equivariant. - Level class ODO-T (farmix_odometer): finite trees T_n with points p_n, homeomorphisms f_n fixing p_n, onto piecewise monotone bonds g_n: T_{n+1} -> T_n with g_n(p_{n+1}) = p_n and g_n f_{n+1} = f_n g_n; p_n has valence exactly 2^n, its arms U_{n,i} (i in Z/2^n) are cycled by f_n, each carries a spine arc from p_n, and g_n maps the spine of arm i of T_{n+1} homeomorphically onto the spine of arm (i mod 2^n) of T_n. Y = lim(T_n, g_n). An ODD CROSSING at level n: some arm U_{n+1,j} has a point w with g_n(w) in an arm U_{n,i}, i not congruent to j mod 2. AC-tight: embeddings e_n of T_n in R^2 with sup |e_{n+1} - e_n g_n| < eps_n < theta_n(e_n, g_n). B. LEMMAS (status; where used) Known (all DERIVED, one agent each, UNAUDITED; the kfan, block2 and farmix_odometer lanes are the same model family, so agreement among them is NOT an independent audit). - Theorem 1 / Lemma ST / Theorem M (kfan_planar_opus_1102): if Y is planar, every transfer and every bridge is O-adjacent; the order topology of O equals the topology of C; each blade has at most one O-adjacent partner; so T is a countable partial matching of O-gaps. Theorem BA: every pair in T is bi-asymptotic for sigma. Corollary D: for DISTAL sigma, hence every odometer, T is EMPTY in every planar host. Corollary K / Lemma FL (block2_opus_1102): K_sigma is nonplanar for every aperiodic sigma; planar level-fan constructions are decomposable. - Lemma HU (farmix_odometer): in a tree-like Y (hereditarily unicoherent) every subcontinuum Z with Z cap beta_a and Z cap beta_b nonempty for blades a != b contains p. So mixing of blades away from p exists only as Hausdorff limits of continua missing p. - Proposition DEC / Corollary DEC-1 (farmix_odometer): an indecomposable ODO-T host has odd crossings at infinitely many levels n. Theorem AC (farmix_odometer; Step 5 COMPUTED by check_adj.py with three negative tests): an AC-tight planar realisation of an ODO-T host has no odd crossing at any level n >= 2 (the V-hook of a crossing is spread by the single 2^n-cycle of f_n to 2^n hooks with a common odd difference d; planarity forces d-adjacent pairs in the cyclic order of spokes at radius rho, while sibling spines force 2^{n-1}-adjacency; three distinct neighbours per spoke contradict 2-regularity). Hence no indecomposable ODO-T host has an AC-tight planar realisation. - MICROEVAL m12 (5 answers, UNAUDITED, Tier 0): the STATIC case (no f) is settled: blowing up one point of a planar Knaster bucket handle into a straight Cantor fan gives a planar indecomposable continuum with a Cantor fan whose T is a countable matching (planar separation). It is not tree-like (unverified) and has no odometer or Denjoy equivariance. One answer: a continuum Z in other composants missing Phi (Phi lies in the composant of p) exists near any two blade tips by density, but it may wander far from beta_a u beta_b; this is why far mixing is not a transfer. C. THE TARGET FAR-MIX-ODO (what farmix_odometer left OPEN, STILL OPEN items 1-2): is there a planar tree-like indecomposable Y with f(p) = p and an f-invariant Cantor fan whose blade action is an odometer? The eliminated class is: ODO-T with an AC-tight planar realisation. Two exits remain, and a complete answer covers both: (B1) Y has no ODO-T presentation (extra non-spine branches at the vertex, valence above 2^n, level actions that are not one 2^n-cycle on arms, bonds not spine-homeomorphic, or no equivariant finite-tree presentation); (B2) Y is ODO-T and planar but only through a NON-tight embedding. Statement N: no such Y exists (N for odometers). The lab records that a yes to both "every planar inverse limit of an ODO-T system has an AC-tight realisation, perhaps after telescoping" and "every odometer host has an ODO-T presentation" would prove N, and that each is a separate OPEN statement. D. ASK Part 1 (method). Name ONE method that does not read a level presentation or an epsilon-closeness scheme, and say why it sees what Proposition DEC and Theorem AC could not (they use only the finite-level combinatorics of an inverse limit and uniform closeness of embeddings; they never use tree-likeness or indecomposability of Y itself beyond DEC). Intrinsic candidates you may use or reject (say which; mark any cited classical theorem SOURCE with exact reference, or UNKNOWN if unsure it exists): Hausdorff limits of subcontinua of other composants that miss Phi; the hyperspace action of f on the continua missing p (an odometer acts equicontinuously on C, but f need not on Y); hereditary unicoherence applied to Z u beta_a u beta_b; the complementary sectors of Phi in the plane and the countable set of O-gaps, compared with the minimality of sigma; Moore-type finiteness statements for families of triods or arcs in the plane. A method that is only another finite-level count is on the dead list. Part 2 (carry it out). Prove N, or reduce it to ONE named lemma L STRICTLY weaker than N, or prove that a particular exit (B1 or B2) is impossible. State exactly which of Lemma HU, Theorem 1, M, BA, Corollary D you use and at which line f, indecomposability and tree-likeness each enter. Part 3 (builder or decisive test). Choose a concrete candidate for B1 or B2 (for instance: ODO-T with one extra non-spine arm at every vertex p_n and a non-tight embedding; or a Toeplitz or odometer variant of the m12 bucket-handle blow-up made equivariant) and run the laws in order: planar embedding; tree-like; indecomposable; f(p) = p and f-equivariance; Theorem M; Theorem BA/Corollary D (T empty); Lemma HU; DEC. Report the FIRST law violated, or the exact list satisfied. If you build, state whether the construction avoids ODO-T or only tightness. Each claim labelled PROVED, SOURCE (exact reference), DERIVED, CONJECTURE, OPEN or RESTATEMENT. UNKNOWN if unsure a cited result exists. Restatement gate: if you reduce N to L, give (G1) L and N side by side with the same hypotheses, (G2) a proof that L implies N, (G3) a configuration meeting L but not yet excluded or built, (G4) where L's own hypotheses are used. 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) Abstract Cantor fan with the odometer action f(c, s) = (sigma c, s): tree-like, planar (any Cantor fan embeds), decomposable, an ODO-T presentation with no odd crossing. It passes Lemma HU, Theorem 1, M, BA, Corollary D, DEC and AC, so a solution using only these proves too much; it must use indecomposability at a named line. (C2) Static host of m12 (planar, indecomposable, Cantor fan, T a countable matching, no f): passes Theorem 1, M, and BA vacuously; an argument using only static planar facts plus indecomposability cannot exclude it, so f (or tree-likeness) must enter at a named line. (C3) K_sigma (abstract, tree-like, indecomposable, odometer): a planar argument must fail there at planarity only (T contains all pairs (d, sigma d), uncountable). (C4) The hook dendroid of block2_opus_1102 (Denjoy cone with V-strands on one gap orbit): planar, decomposable, bridges exactly on that orbit; it shows Theorem BA is sharp, and its Denjoy action is not an odometer. F. DEAD ENDS (not progress) (1) Level (inverse limit of finite trees) presentations with tight embeddings: Proposition DEC + Theorem AC (farmix_odometer): indecomposability forces odd crossings, tightness forbids them; this proves nothing about B1 or B2. (2) Showing K_sigma nonplanar, or that any planar fan has T empty for an odometer (Corollary K, Corollary D): known. (3) Lemma FL: planar level-fan constructions are decomposable. (4) Local data at p alone (Proposition LOC: every local datum at p has a planar equivariant hairpin model with no bridges). (5) Prime-end rotation, accessibility, Bell, Cartwright-Littlewood: need an ambient extension of f. (6) The candidate reduction "every planar host would have an ODO-T presentation with a tight realisation": RESTATEMENT (farmix_odometer gate G3 fails). (7) Kelley approximation at one blade toward two others (Sol 5.6, m12): not a strict reduction by its own gate. Related lanes, do not repeat: kfan_planar_opus_1102 (Theorem M, BA), block2_fanbridge_prove_opus_1102, fp2f_opus_1102. 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: ODOMETER FAR-MIX EXCLUDED (no planar tree-like indecomposable Y with an f-invariant Cantor fan with odometer blade action; proof, inputs named) | PLANAR ODOMETER HOST BUILT (explicit Y, f, Phi with every listed law checked) | REDUCED (to one named lemma, gate G1-G4 passed) | PARTIAL (one sub-claim settled: B1 or B2 below) | 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 a level (inverse-limit of finite trees) presentation or an epsilon-tight embedding scheme (these are the eliminated class); using an ambient extension of f to the plane, prime-end rotation at p, Bell or Cartwright-Littlewood (f is intrinsic); proving only that K_sigma is nonplanar (known); letting the blade action be periodic; counting a decomposable host as an example; claiming NO-FAN, NOT-BLOCK*(2) or PER