batch5

Prompt

RESEARCH QUESTION (planar tree-like continua; periodic-point-free homeomorphisms; the full periodic quotient and the open lemma NAC-PLANAR). 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 (written before any mathematics). (C1) Arrow R1 (TL-PER iff NoTCM and NoTower_full), its NoTower_full branch (NT). Statement: NAC-PLANAR below, all quantifiers fixed. YES (proved): the lab re-decomposition "TL-PER follows from NoTCM and NoPA_per, given NAC" applies to every planar tree-like PPF system, so the box NoTower_full, for which no tool is known, is replaced by NoPA_per, a one-level statement about periodic-atomic maps (of homeomorphism-FPP strength; TL-PER does not imply it). NO: a counterexample that is planar is itself a planar tree-like PPF system, a PER counterexample candidate (Tier 1, refuters needed); a counterexample that is only abstract shows that planarity must enter and leaves NoTower_full as the open arrow. Neither outcome closes PER. (C2) Controls (positive and negative, with first exits) are in section E: Minc 1996 (abstract positive), irrational rotation of the closed disk, circle rotation by 1/q, Y_* and the T-EXT q=3 host, bucket handle and Feigenbaum. Each is run against the statement before any proof. (C3) New input: the lemma with all quantifiers fixed; two reductions of NAC failure in (K5) (constant-period exhaustion; limit collapse of non-null families), derived while writing this spec by one agent, unaudited; the explicit Moore-quotient structure (K3). (C4) Duplicate check: the lemma is the first open lemma of one Opus report and none was found in any other prompt, report or live job (preflight: no hit above threshold). Live neighbours that do not ask it: a Sol audit of the derivations (K1)-(K3) only, and a job on descent of divisible-power towers to q-cycle blocks. Research, not audit. Route CHANGE-MAP (proposed). You see no lab files; statuses are as the lab recorded them. Everything marked LAB below comes from a single Claude Opus report (UNAUDITED; a cross-family audit of K1-K3 is running). - Setting. X is a nondegenerate tree-like continuum in the plane R^2: compact connected metric, and for every eps>0 there is an eps-map (fibres of diameter < eps) onto a finite tree. Subcontinua of tree-like continua are tree-like. A planar tree-like continuum does not separate the plane and is hereditarily unicoherent (HU). h: X->X is a homeomorphism; Per(h) := union over r>=1 of Fix(h^r); PPF := Per(h) is empty. A subcontinuum is a nonempty compact connected subset, proper if it is not X. h is continuum-minimal (cm) if no proper subcontinuum A has h(A)=A (equivalently none has h(A) subset A: intersect the iterates h^n(A)). - (S) := h is PPF and cm. IND := X is indecomposable (not a union of two proper subcontinua). In the planar case (S) implies IND (Bell 1967, through the lab carrier theorem, CLOSED; it uses nonseparation, fixed-point-freeness and cm). - Periodic continua. P(h) := {Z : Z a proper subcontinuum with h^r(Z)=Z for some r>=1}; under PPF every Z in P(h) is nondegenerate. R_0 := {(x,y): x=y, or x and y lie in one Z in P(h)}. R := the smallest CLOSED subset of X x X that contains R_0 and is an equivalence relation. PA (periodic-atomic) := every element of P(h) is a point. NAC := R is not X x X. - NAC-PLANAR (OPEN). For every nondegenerate tree-like continuum X in R^2 and every homeomorphism h of X such that (i) h is PPF, (ii) h is cm, (iii) P(h) is nonempty: R is not X x X. Equivalent form (NAC-EQ, DERIVED): some h-equivariant monotone surjection of X onto a nondegenerate continuum has every element of P(h) inside a fibre. NAC-ABS := the same statement without "in R^2". B. LEMMAS (status; where used) Known, with status. Finite computation is not proof of an infinite statement. (K1) F1 (LAB, DERIVED, UNAUDITED). Under (S) and IND: if Z and Z' in P(h) meet, then Z u Z' is a continuum, proper by IND, invariant under h^(rr'). So R_0 is an equivalence relation, and each R_0-class is an increasing (directed) union of members of P(h). (K2) F2 (LAB, DERIVED, UNAUDITED). Under (S) and IND a maximal element of P(h), if one exists, is unique in its composant (union of proper subcontinua through a point); its translates are maximal, pairwise disjoint, in distinct composants, with least period >=2. (K3) FPQ (LAB, DERIVED modulo Moore's decomposition theorem and standard planar facts, UNAUDITED). Assume (S), IND, NAC, and let mu: X->Xbar be the monotone part of the monotone-light factorisation of X->X/R. Then mu is h-equivariant; hbar := mu h mu^(-1) is a homeomorphism of the nondegenerate continuum Xbar; every Z in P(h) lies in a fibre; hbar is fixed-point free, cm and PA; if (iii) holds, hbar has a periodic point (the image of a fibre that contains a periodic Z); Xbar is indecomposable and HU; if X is planar then Xbar is planar and tree-like (fibres are nonseparating and cellular, so the Moore quotient of the plane is the plane). (K4) Consequence (LAB, DERIVED). If NAC-PLANAR holds, TL-PER follows from NoTCM and NoPA_per, where NoPA_per := no planar tree-like continuum carries a fixed-point-free PA homeomorphism with a periodic point (hbar would be one). The cost is stated plainly: NoPA_per is of HFPP type (homeomorphism fixed-point strength), so TL-PER does not imply it. (K5) Mechanisms of failure of NAC. (LAB, DERIVED) NAC holds if some element of P(h) is maximal and the maximal ones form a null family; NAC fails if some h^r-invariant composant is an increasing union of h^r-invariant proper continua with dense union. Root observations (derived while writing this spec; please re-derive): (a) constant period: if a chain Z_1 subset Z_2 subset ... in P(h) has dense union, put H_i := union of the h^(r_1)-translates of Z_i (r_1 the period of Z_1); all translates contain Z_1, so by IND H_i is a proper subcontinuum, H_i is h^(r_1)-invariant, increasing, with dense union. So exhaustion may be taken with one period. (b) limit collapse: R is closed, so if Z_n in P(h) converge in the Hausdorff metric to a continuum W with more than one point, W lies in one R-class although W need not lie in P(h). Hence NAC needs control of non-null families as well as of exhausting chains. (K6) Planar inputs on the table (LAB list): Bell 1967 (IND); Moore decomposition; BFMOT (for a cm fixed-point-free pair (C,g), C a nondegenerate planar tree-like continuum: a unique outchannel with variation -1, index Ind = Var + 1 = 0); subcontinua of tree-like planar continua are nonseparating. No planar input to NAC is known. NAC-ABS is open. (K7) Minc 1996 (SOURCE: an explicit tree-like continuum with a PPF homeomorphism; nonplanar, LAB VERIFIED). LAB report: a monotone equivariant map takes its periodic-carrier system onto a nonplanar arc-proper periodic-atomic block Bone, so R is not total (step M-ii, conditional on one unaudited lab carrier theorem; step M-i, Bone has no periodic arc, is a sketch). C. THE TARGET Decide NAC-PLANAR: prove it with every planar input named; or build a counter-candidate (abstract first, then say whether planarity can exclude it); or reduce it to one named lemma with the reduction proved. D. ASK 1. Abstract first. Decide NAC-ABS: either prove that for every tree-like X and h with (S) and (iii), R is not X x X; or build an abstract refuter (inverse limit or composant construction) satisfying everything except planarity with R total, for instance an invariant composant exhausted by invariant proper continua, or a non-null family of periodic continua whose Hausdorff limits collapse X. It must differ from Minc. If NAC-ABS holds, say which planar box remains. 2. Planar input. If NAC-ABS fails or is unproved, name the planar fact that could exclude your refuter (Bell, Moore, BFMOT outchannel and variation, nonseparation, accessibility or prime ends) and prove NAC-PLANAR with it; or show the fact is insufficient on the refuter, so that planarity is not the entry point there. 3. Reduction. If neither is reachable, reduce NAC-PLANAR to one named lemma with a complete proof of the reduction. Candidates: NO-EXHAUST (no invariant composant is an increasing union of invariant proper continua) with NULL (the periodic continua form a null family inside each composant). 4. Sub-cases. Treat (K5a) constant-period exhaustion and (K5b) non-null families separately; a full proof of one is a partial answer. 5. Control step. With (X1): does Minc satisfy NAC without the unaudited carrier theorem? Do not re-prove F1, F2, FPQ or NAC-EQ; do not ask for NoPA_per or fixed points of hbar (separate arrow). State residual quantifiers. No audit. No code is needed; if you run any: at most 2 workers, under 2 gigabytes, no background jobs. E. CONTROLS (all must hold) (X1) Minc 1996. Positive for the abstract statement NAC-ABS (the lab report says its quotient is Bone, so NAC holds). Exits at the word planar (nonplanar). Any NAC-ABS proof must be consistent with it; an abstract refuter must differ from it. (X2) Irrational rotation of the closed disk. Every invariant disc is periodic and R_0 is total, so NAC fails; it exits at tree-like (dimension 2), at PPF (fixed centre) and at cm. Each of (i), (ii) and tree-likeness is used. (X3) Rotation of the circle by 1/q, q>=2. Fixed-point free and cm, h^q=id, so every proper subcontinuum is in P(h) and R_0 is total; exits at IND (decomposable), nonseparation and tree-like (circle) and PPF (all points periodic). It says nothing about the planar lemma. (X4) Y_* and the T-EXT q=3 host with its multiplier-3 map. PPF; for ARC-PROPER hosts (every proper nondegenerate subcontinuum an arc; LAB for G2 hosts) the map is totally cm, so P(h) is empty and R_0 is the diagonal. Exit at (iii). Planarity of the T-EXT host is open and is live elsewhere. (X5) Knaster bucket handle shift and the Feigenbaum inverse limit. Exit at (S) (fixed point, periodic points). They show the hypotheses cannot be dropped. F. DEAD ENDS (not progress) Single q-cycle blocks as hosts (mis-posed for towers: Minc's blocks carry a cycle for every admissible prime); abstract coverage and abstract NoTower (killed by Minc: SHRINK); "regular recurrence forces towers" (false: solenoid translation); mapping-torus solenoid obstruction (killed by Feigenbaum); Conley zeta for non-injective extensions (BFMOT index 0 is consistent); a-priori bounds (no planar source). Take your time, be extremely meticulous, I believe in you. G. FORMAT Line 1 exactly: VERDICT: PROVED (NAC-PLANAR, each planar input named) | COUNTEREXAMPLE-CANDIDATE (abstract refuter, planar entry point located) | REDUCED (one named lemma, reduction proved) | OPEN-OBSTRUCTION (exact residual). 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: periodic continua assumed null or maximal elements assumed to exist; the non-closed relation R_0 used where the closed relation R is needed; indecomposability dropped; periodic-point-free confused with fixed-point-free; finite computation as proof; an abstract refuter that is not tree-like or has a proper invariant subcontinuum; the decomposable circle rotation used as evidence about the planar lemma; single q-cycle blocks treated as hosts; Kuratowski-type obstructions sought