batch6

Prompt

RESEARCH QUESTION (monotone coverage of planar tree-like totally continuum-minimal periodic-point-free systems by G2 hosts; the positive-entropy sector that is not hereditarily indecomposable). 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 (lab rule; answered by the dispatcher, not by you). (C1) Arrow R2: NoTCM follows from COV_tcm^M together with T_Lambda (lab Theorem SPLIT, VERIFIED building block); this prompt serves the uncovered half COV_tcm^M on the sector Pos defined below. YES (a proof, or a reduction to named lemmas): every member of Pos is monotonically covered by a planar G2 host with a periodic-point-free homeomorphism, so Pos is empty as soon as the T_Lambda arrow (no planar G2 host carries a PPF homeomorphism) holds. NO, abstract form (a step fails, with the invariant, or abstract nonplanar data): planarity must be used, which is the content of the open arrow. NO, planar form (explicit planar data): an uncovered member is itself a planar tree-like PPF system, i.e. a TL-PER counterexample candidate (Tier 0 as a candidate; an explicit infinite planar object would be Tier 1). (C2) Controls with first exits (details in CONTROLS): positive abstract control Y_* (covered by the identity, positive entropy, not HI, tcm, PPF; first exit: planarity); negative controls Minc 1996 (exits at tcm, by SHRINK), Handel's pseudo-circle map (exits at HI and at tree-likeness), the irrational circle rotation (exits at entropy and at tree-likeness), the pseudo-arc (exits at PPF). (C3) New input: the exact R2 split (NoTCM iff COV_tcm^M and T_Lambda), Lemma B (monotone images of tcm systems are tcm and PPF), the two uncoverability invariants HI and zero entropy (which are removed from the sector here), the Bell indecomposability of tcm carriers, and the decomposition of coverage into the four steps S1-S4 of the TARGET (the dispatcher's decomposition, not a lab theorem). (C4) Duplicates: the HI sector and the zero-entropy sector are handled by other jobs (a web prover and an Opus lane); candidate (ii) in its self-similar substitution form is already dead (K6); this is the first prompt on the remaining sector (positive entropy, not HI); no existing lab report found for this sector. Research, not audit. The reports behind the inputs were written by one agent in sequential roles (not independent checks). No lab files are visible; grades are retained. Setting. A continuum is a compact connected metric space; planar means embeddable in R^2. Tree-like: for every eps>0 there is an eps-map onto a finite tree. Fact (SOURCE, secondary): a planar tree-like continuum is nonseparating (its complement in the plane is connected). A homeomorphism g is PPF (periodic-point-free) if no g^n, n>=1, has a fixed point. g of a continuum C is tcm (totally continuum-minimal) if for every s>=1 and every nonempty subcontinuum A of C, g^s(A)=A implies A=C. For nondegenerate C, tcm implies PPF (a periodic point is a proper invariant singleton); every power of a tcm map is tcm. HI means hereditarily indecomposable. Entropy is topological entropy. G2 host. Data: an infinite compact zero-dimensional metric space K, a minimal homeomorphism tau of K, an involution i_0 with i_0 tau = tau^{-1} i_0. Put S=(R x K)/((t+1,tau x)~(t,x)), flow Phi_u[t,x]=[t+u,x], R[t,x]=[-t,i_0 x], Y=S/R. A G2 host is such a Y where K=K_D is the closure of {(z,(D(z+n))_{n in Z}): z in Zhat not in Z}, Zhat=product of the Z_p over all primes, D: Zhat minus {0} -> E continuous and even into a compact zero-dimensional E with D(2z)=sigma D(z), with tau minimal, singleton fibres of K over Zhat off Z, a dyadic lift B(g)(z)=g(2z) giving the canonical homeomorphism h[t,x]=[2t,Bx], the set P_0 of ends (images of Fix R) nonempty, and Cech H^1(Y)=0. G2 hosts are 1-dimensional. Coverage. COV_tcm^M: for every nondegenerate planar tree-like C and every PPF tcm homeomorphism g of C there are a G2 host Y, a PPF homeomorphism H of Y and a monotone (connected fibres) continuous surjection pi: C -> Y with pi g = H pi. T_Lambda: no G2 host that embeds in the plane carries a PPF homeomorphism. NoTCM: no planar tree-like PPF system has a power-carrier (Z nondegenerate subcontinuum, h^R(Z)=Z) whose return h^R|Z is tcm. The sector: Pos := all (C,g) with C nondegenerate planar tree-like, g a PPF tcm homeomorphism of C, h_top(g)>0, and C not HI. Pos is conjecturally empty (a member refutes TL-PER); you are asked about the coverage statement on Pos, not about existence. B. LEMMAS (status; where used) KNOWN, with status. Finite computation is not proof of an infinite statement. (K1) DERIVED (lab, audit CWR, repairs closed; Sol recheck running; from Bell 1967 Theorem 1: a fixed-point-free map of a nonseparating plane continuum has an invariant indecomposable subcontinuum Q, Q=f(Q)): every member (C,g) of Pos has C indecomposable (Q is nondegenerate since a point would be fixed; tcm gives Q=C). So Pos consists of indecomposable continua that are not HI, i.e. contain a nondegenerate decomposable subcontinuum. (K2) Lemma B, DERIVED (one agent, UNAUDITED; short proof): if pi: C -> Y is monotone, onto, Y nondegenerate, pi g = H pi and g is tcm, then H is tcm and PPF (preimage of an H^s-invariant subcontinuum L is a g^s-invariant continuum, hence C); Y is planar and nonseparating when C is planar tree-like (Moore, cellular fibres). So the PPF clause of COV_tcm^M is automatic and every nondegenerate monotone image of a member is again planar, nonseparating, tcm, PPF. (K3) Uncoverable invariants, DERIVED (UNAUDITED): (a) HI: a monotone image of an HI continuum is HI and G2 hosts contain arcs, so no HI member is covered; (b) zero entropy: ENT-G2 (below) shows a cover forces h_top(g) >= h_top(H) > 0. Both are removed from Pos by definition; no other invariant is known. (K4) ENT-G2, DERIVED relative to a flagged reading of the lab T-EXT theorem (VERIFIED building block): every homeomorphism H of a G2 host has h_top(H) >= |log lambda(H)| for its rational multiplier lambda(H); PPF forces lambda(H) not 1. Hence entropy of a G2 target is positive, and positive entropy of g is necessary for coverage but not obviously sufficient. (K5) Planarity enters only through Moore (images stay planar) and through Bell: SOURCE, secondary (lab ledger): the restriction of a plane homeomorphism to an invariant nonseparating plane continuum has a fixed point. So g never extends to a plane homeomorphism (nor does any power, nor any monotone equivariant image). Hamilton 1951: chainable continua have the fixed point property, so C is not chainable. (K6) Lemma CA and Cutting Lemma, LAB (audit CWR, repairs closed): every proper subcontinuum of a G2 host is an arc or a point (ARC-PROPER). Lemma TCM: a PPF homeomorphism of an ARC-PROPER continuum is tcm. Candidate (ii), DERIVED (UNAUDITED, plus a COMPUTED finite check with verifier): folded suspensions of self-similar palindromic primitive substitution systems have a periodic end in the isotopy class of the identity and of every inflation power, so they die at PPF. Prop E, DERIVED: for a folded suspension over a rank-1 flow-eigenvalue group Gamma containing Z with a multiplier a/b, the base has only the periodic point 0 for every period iff Gamma=Q. (K7) LAB, Astra/Argon, UNAUDITED: there is no continuous phi from a non-universal base group G_b to Zhat with phi(z-1)=phi(z)-1, so a non-odometer base cannot reach a G2 host by a base-respecting map; a monotone cover that does not respect the base is OPEN. (K8) Abstract status: COV_tcm^abs (the same statement with 'planar' dropped from the domain) is OPEN; no abstract tree-like tcm PPF system that is not covered is on disk; every abstract tcm PPF tree-like system on disk (Y_*, the T-EXT hosts) is a G2 host. Minc 1996 is a tree-like PPF homeomorphism with SHRINK (every power-carrier contains power-carriers of arbitrarily small diameter), so by Lemma NF it has no tcm power-carrier and admits no G2 cover; it is not in Pos. C. THE TARGET Decide COV_tcm^M on Pos, in the following proof architecture (the dispatcher's decomposition; the steps are not known to be forced in this order). A cover (Y,H,pi) realizes: S1: Y is ARC-PROPER and nondegenerate; S2: Y is a folded suspension S/R of (K,tau,i_0) as in the definition; S3: (K,tau,i_0) is a reader chart over Zhat with the dyadic lift; S4: H^1(Y)=0, P_0 nonempty, and H (necessarily with a rational multiplier) is PPF. For each member (C,g) of Pos, determine which of S1-S4 follow from the hypotheses (planar tree-like, tcm, PPF, h_top>0, not HI, K1-K8), which fail, and which are open, and either prove all four (a COVERED verdict) or isolate the first step that can fail together with an invariant or abstract data. D. ASK 1. S1. Does C (indecomposable, not HI, positive entropy, planar) have a nondegenerate monotone equivariant ARC-PROPER image? Treat the sub-question 'does C contain an arc?': non-HI alone does not give one (decomposable continua without arcs exist), so say what extra hypothesis is used or give the structure that replaces arcs. 2. S2-S4. For an ARC-PROPER image with a PPF homeomorphism, derive the folded-suspension form (S2) and the reader form (S3) or show S3 can fail (a minimal Cantor base without odometer factor, K7). In particular decide whether a monotone cover that does not respect the base can exist. 3. Entropy. State precisely what h_top(g)>0 adds to S1-S4 beyond K4 (for example: forces an expansive-type structure, excludes equicontinuous quotients, controls the multiplier). Any proved consequence counts. 4. If some step fails: give abstract data (drop planarity, keep tree-like, tcm, PPF, positive entropy, not HI, no G2 cover) and test it against every control; or give the exact property of planarity that would have to be used (not Bell alone: Bell only kills extendable maps, see the sibling question on non-extendability, which you need not answer). 5. Sub-statements: coverage for members whose arc components have a Cantor transversal; for members with a prescribed multiplier; for members whose maximal proper decomposable subcontinua form a closed family. 6. Do not re-prove K1-K3 or rework the HI, zero-entropy or substitution cases; no audit. State residual quantifiers. Code is optional (at most 2 workers, under 2 GB each, one heavy foreground job, small run first). Output: line 1 as specified; SUMMARY (<=100 words); table step | status (follows / fails / open) | key line; STILL OPEN EXACTLY. <=2000 words. E. CONTROLS (all must hold) Control 1 (positive, abstract): Y_*, a G2 host with canonical homeomorphism: tree-like, PPF, tcm (Lemma TCM), not HI, h_top >= log 2 by the canonical bound of K4; covered by the identity, so S1-S4 hold trivially; first exit of any construction aimed at it: planarity (Y_* is nonplanar, lab claim). A coverage proof for Pos that never uses planarity would also cover abstract members (K8). Control 2 (negative, tcm): Minc 1996 is not tcm (SHRINK), hence not in the domain; an argument not using tcm would wrongly cover it. First exit: tcm. Control 3 (negative, HI): Handel's pseudo-circle map is planar, HI, minimal (hence tcm), PPF, circle-like; it is not tree-like (it separates the plane) and not covered (K3a). First exit: tree-likeness, then HI. Control 4 (negative, entropy): the irrational rotation of the circle is planar, tcm, PPF, zero entropy, not tree-like; any cover would need positive entropy (K3b). First exit: tree-likeness, then entropy. Control 5 (negative, PPF): the pseudo-arc is planar, tree-like, HI, chainable, and every homeomorphism has a fixed point (Hamilton). First exit: PPF. F. DEAD ENDS (not progress) Coverage by identity for G2 hosts; arguments valid for abstract tree-like systems presented as planar; positive entropy treated as an arc source; the self-similar substitution class (K6); non-monotone factor maps (no planarity transfer); Bell for extendable maps only; invariants inferred from finite examples; re-deriving Lemma B. G. FORMAT Line 1 exactly: VERDICT: COVERED (coverage proved on the sector, hypotheses named) | STEP FAILS (name the step S1-S4 and the invariant that blocks it) | CANDIDATE (explicit data for a member that is not covered) | 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: using positive entropy as if it produced an arc; treating the self-similar substitution suspensions as the general non-odometer case; proving coverage of G2 hosts themselves; using HI or zero entropy (excluded here); forgetting that planarity must enter because the abstract statement is open; treating Minc 1996 as a member; calling a finite example a proof; confusing monotone with arbitrary factors