m13

Prompt

RESEARCH QUESTION (plane continua / dynamics; L2_PPF(TL) attacked by the displacement-angle (van Kampen type) route on the deleted product, not by the strip invariants of Route I). 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 (compressed from research/ROUTE_MAP_L2.md section 2a, the lab note 0_B_novel_1031_0238 and the van Kampen notes 0_T5pp_Mtheta_vanKampen_opus, 0_T5ppb_vanKampen_equation_opus; every DERIVED item is a single-agent claim, UNAUDITED; lines marked (INFERRED) are my bridging). - Continuum = compact connected metric space. cm (continuum-minimal): a homeomorphism g of C is cm if no proper nondegenerate subcontinuum K of C has g K = K. PPF: g has no periodic point. - L2_PPF(TL) (the target, ROUTE_MAP_L2 section 2a, R2): for every nondegenerate planar tree-like continuum C and every cm homeomorphism g of C without periodic points, g^2 is not cm. This is the L2 input of the PER edge: PER <=AND= L2_PPF(TL) & NOT-BLOCK*(2) (a Sonnet audit gave CONFIRMED-WITH-REPAIRS; a second audit is pending). L2_PPF(GEN) is the same for arbitrary C. - m_k := min over x in C of d(x, g^k x) > 0 for every k >= 1 (PROVED, compactness, no uniformity in k). Since g has no periodic point, g^k x != x for all x and k >= 1. - X*2 := (C x C) minus the diagonal, with the swap tau(x,y) = (y,x), a free Z2-action. Gamma_k := {(x, g^k x) : x in C}, a closed subset of X*2 homeomorphic to C (a connected set). (INFERRED, direct) tau(Gamma_k) meets Gamma_k iff (x, g^k x) = (g^k y, y) for some x, y, i.e. y = g^k x and x = g^(2k) x. So under PPF, tau(Gamma_k) and Gamma_k are DISJOINT closed sets, at positive distance, for every k >= 1. - Displacement angle. If a: C -> R^2 is a planar embedding and C is tree-like (or Cech H^1(C; Z) = 0), then u_k(x) = (a(g^k x) - a(x))/|a(g^k x) - a(x)| : C -> S^1 lifts to a continuous theta_k: C -> R (circle logarithm; SOURCE r126s_nh7_transline_answer section 6.1, recalled, not re-proved here; for tree-like C it is Lemma A.2 of 0_B_novel_1031_0238, DERIVED). Equivariance: Phi(x,y) = (a(x)-a(y))/|a(x)-a(y)| : X*2 -> S^1 satisfies Phi(tau p) = -Phi(p), so Phi is a Z2-map to the antipodal circle, and u_k is Phi restricted to Gamma_k. - Mechanism A (0_B_novel_1031_0238, Theorem A, DERIVED): for continuous g with a(gx) != a(x) and a lift theta, D(x) = theta(gx) - theta(x) must vanish somewhere (a bounded lift cannot have D of one strict sign), which gives x with a(x), a(gx), a(g^2 x) distinct, collinear and in that order. It uses no cm hypothesis and no invariant measure, and applies to nonplanar tree-like spaces too. B. LEMMAS (status; where used) Known (DERIVED unless marked; all UNAUDITED). - Route I (A162, PARTIAL, Arena, one agent, unaudited). Under PPF each power g^k has its own positive displacement scale m_k; the k-th image sections eventually miss the diagonal access point by m_k/2. LAB-conditional: a stable diagonal integer v_k. OPEN: no restriction on v_k for k >= 3, no cm of powers, no thick band control. A building block only. - Mechanism A fails as a connector (0_B_novel_1031_0238): the collinear triple does not give a fixed or periodic point; an order-3 planar tree map (origin fixed, every other point of exact period three, noncollinear orbits) is consistent with Theorem A, and the connector uniformity needed for a transport bound is false (a conjugated rotation-like example keeps connector diameters near 1 while the displacement tends to 0). - Minc M96 (Theorem 5.32, SOURCE, full text on disk): a tree-like continuum with a periodic point free homeomorphism exists. (INFERRED, not checked here) whether it is planar or has a cm power is not settled in the lab files read for this prompt. On a Denjoy chart host (planar, every power cm, PPF) L2 is the node itself. - van Kampen notes (research/GATE_LEDGER_FJI.md line 94, H-NH8): two PARTIALs. theta = 0 for two-fold chart continua is equivalent to Z-2PAGE (an integer relaxation of 2PAGE); solved for unblown hosts; theta(B~_1) and theta(M_theta) undecided. Parked only as a method; the re-entry condition is a computation of theta(B~_1) != 0. - A164 (PARTIAL, WRAP-MPN): at most floor(1/kappa) qualifying pieces per crosscut, and cm does not give wall invariance. Relevance here is only a warning: cm is not a metric hypothesis. C. THE TARGET Statement VK-PPF. Let C be a planar tree-like continuum and g a cm homeomorphism of C without periodic points. Decide whether the pair (Gamma_1, tau(Gamma_1)) of disjoint closed Z2-related subsets of X*2, together with the Z2-map Phi to the antipodal circle, yields a Z2-equivariant obstruction (a Stiefel-Whitney-type class, an index, or a winding number deg(Phi) along a loop made of a path in Gamma_1, a connector, a path in tau(Gamma_1) and a connector) that is nonzero exactly when no proper subcontinuum K has g^2 K = K, or that otherwise forbids g^2 cm. The honest outcomes are three: an obstruction that carries information; a proof that every such class vanishes for tree-like C (NO CHANGE); or an explicit pair (C, g) with g PPF, g cm and g^2 cm, which would refute L2_PPF(TL) (check chart and Denjoy hosts carefully: g^2 must be cm and the embedding must be planar). Why this is not Route I: it never cuts a strip or uses an outchannel; it uses only the free Z2-space X*2, the connected graph Gamma_1 and the antipodal map Phi. D. ASK Part 1 (the invariant). For C an arc, a triod, a Denjoy chart host and a Knaster bucket handle, compute the restriction to Gamma_1 u tau(Gamma_1) of the Stiefel-Whitney class w_1 of X*2 -> X*2/tau (in H^1(.; Z2), Cech), and say which cases depend only on the pair (Gamma_1, tau Gamma_1) and which need a loop through both, where the connector is exactly the thing Mechanism A could not control. State whether the class is zero for every such pair when C is tree-like (INFERRED: Gamma_1 and tau Gamma_1 are each homeomorphic to C, H^1(C; Z2) = 0, so each is individually trivial; the union is disconnected, so the question is whether a connecting arc in X*2/tau can carry a nonzero class, not the individual classes). Part 2 (where cm enters). Name the single step at which cm of g (or the failure of g^2 to be cm) is used. If no step uses it, say so: the route then cannot decide L2_PPF(TL), and the verdict is NO CHANGE. A proper subcontinuum K with g^2 K = K gives the g-invariant set K u gK and the Z2-invariant closed set (K x gK) u (gK x K) in C x C; say whether it misses the diagonal under PPF (INFERRED: K and gK may intersect, so avoiding the diagonal needs K cap gK empty or an argument) and what its class is. Part 3 (restatement test and gate). If you answer CHANGES, the residual lemma L must pass: (RT1) L has no occurrence of Gamma_k, theta, Phi, "strip" or "RIG": hypotheses on C and g only. (RT2) the steps of "L implies VK-PPF" are listed with the hypothesis each uses; PPF, or some m_k with k >= 2, or cm, must be used at a named step. (RT3) give a configuration (a finite model, a chart germ, a control) that satisfies L but is not excluded by Mechanism A and Route I data alone. Without this the verdict is NO CHANGE or RESTATEMENT. Part 4 (only if Parts 1-3 are done). Is there an analogue of Mechanism A giving an ordered pair of collinear triples with a prescribed orientation at some x, and does that contradict m_2 > 0 on any host where g^2 is cm? Give a proof or a control where it holds trivially. Each claim must be labelled PROVED, SOURCE, LAB, DERIVED, CONJECTURE, OPEN or RESTATEMENT. NO CHANGE is an acceptable verdict if you give a reason (a theorem that all classes vanish, or a control on which the route sees nothing). E. CONTROLS (all must hold) (C1) Y_2-type chart host: planar, H has a fixed point so g is not PPF and Gamma_1 meets the diagonal; say what a nonzero class would mean here. (C2) An irrational rotation of the circle: PPF, minimal, every power cm, but the circle is not tree-like and the lift theta of the displacement angle does NOT exist (winding number 1). The route must say that this is where tree-likeness is used, or show it is not. (C3) Denjoy host with rotation labels: planar, PPF, every power cm; the node is L2 itself, so a claim that the route excludes such a host must be checked against the fact that g^2 is cm there. (C4) An order-3 homeomorphism of a planar finite tree (origin fixed, others of exact period 3, noncollinear orbits): not PPF; off the origin Gamma_1 meets tau(Gamma_2) at period-3 points while tau(Gamma_1) misses Gamma_1. (C5) Dyadic solenoid translation: nonplanar, no periodic point; any step that uses planarity must say where. F. DEAD ENDS (not progress) (a) Mechanism A as a connector: the collinear triple gives no periodic point and the connector uniformity is false (0_B_novel_1031_0238, section A.3 and its Minc control). (b) Route I (D_k): each power has its own displacement scale m_k, but v_k is a bare integer for k >= 3 (A162). (c) Strip-tail continuity plus fold analysis gives RIG: false on Y_2. (d) Length-area and conformal modulus bounds: g is not conformal. (e) Hull mass invariants: they fall only at folds. (f) Dold congruences and the index of F^n on a contour: under PPF the periodic blocks are empty, so the route is off the L2 path. (g) Finite tables of Z2-indices for small trees are Tier 0 evidence and prove nothing about the infinite object. (h) The Argon 55-57 headlines "REDUCED to X plus Lemma L" where L restates the node; Argon 58 and the gate classify them as RESTATEMENT. (i) Scale arguments with cm used as if it were a metric uniformity (A164: cm gives no wall invariance). 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: SETTLES (an obstruction class proves L2_PPF(TL), or an explicit planar PPF cm pair with g^2 cm, inputs listed) | CHANGES (one named new law or named lemma L that passes RT1-RT3 and the gate) | NO CHANGE (the route adds nothing beyond known facts; reason) | PARTIAL (one named sub-claim settled) | 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: declaring REDUCED with a residual lemma that restates L2_PPF or uses no cm hypothesis (Argon 55-57 did this); using an ambient extension of g or prime ends; using that g^2 is cm without saying at which step; counting a periodic g (an order-3 map) as a test; treating finite tables as the infinite object; reproving Mechanism A or Route I (D_k) as if new