m34

Prompt

RESEARCH QUESTION (plane continua / dynamics of homeomorphisms of nonseparating plane continua; L2 (h^2 is never continuum-minimal), FRESH STRATEGY after Route I has been blocked at four nodes by the same pattern: local laws are blind to the defect and only a global law can help). 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 and lab lanes; every status is UNAUDITED unless stated). - L2: for every homeomorphism h of a nondegenerate nonseparating plane continuum X there is a proper nonempty subcontinuum K with h^2(K) = K. Working form: an R-instance is (X, h) with h^2 continuum-minimal (cm: no nonempty proper subcontinuum K with h^2(K) inside K); L2 is equivalent (v5 Thm 11.1 plus definition) to: no R-instance exists. L2 may assume h has no periodic point (ROUTE_MAP_L2 section 2a). L2 is intrinsic to X; PER (every homeomorphism of a nonseparating plane continuum has a periodic point) also needs NOT-BLOCK*(2). - Setup in an R-instance: U := C minus X (connected, unbounded), m_1 := min over x of |h x - x| > 0. F, G are perfect plane extensions of h, h^-1: continuous proper maps of the plane with F|X = h, G|X = h^-1. They are NOT homeomorphisms in general, G is NOT F^-1 off X, and F(U) may meet X. A crosscut q of U is an open arc in U whose closure is an arc with distinct ends in X; Sh(q) is the bounded component of C minus (X u q). Prime ends of U are classes of null chains of crosscuts; the impression of a prime end is the intersection of the closed shadows. BFMOT (arXiv 1004.0214) gives, for a fixed-point-free perfect extension F with h^2 cm, a unique outchannel E_F: a prime end with principal set X, dense, and index = variation + 1 (BFMOT 3.2.2 on disk). - Variation var(F, q) in Z of a small crosscut q is the winding-type count of F along q relative to the shadow. Lab laws (CP1, LAWS; DERIVED, no valid audit): (V1) var(F, c) = -1 for every small crosscut c of the outchannel E_F; (V2) var(F^2, c) is in {0, -1}. Route I reaches a contradiction from RIG_S: the image F(c) of a small E_F-crosscut c meets every deeper E_F-crosscut q with intersection number -1, at c and at a constructed shallower crosscut C, which gives V_2(c) = +1. The map of the program says: L2 is EQUIVALENT to RIG_S failing cofinally in an R-instance. So Route I turned L2 into a local geometric statement, not into an easier one. B. LEMMAS (status; where used) What failed on Route I (each line: node, method, reason, source; all DERIVED or COMPUTED by single agents unless noted). 1. RIG_S(c) and RIG_S(C): equivalent to L2 on the R-class (ROUTE_MAP_L2 5A, 5B R1). Restatement. 2. SS-COF (side selection of DP-K, no failing door at cofinal depth): index and variation counts all vanish at deep doors (codex_sscof); cm on Hausdorff limits of pocket traces: a full-trace pocket has trace X, so the limit is X and cm gives nothing (MICROEVAL m24, 5/5); impression transport and TT: a circle model with failing doors at every depth satisfies every law including TT (b_opus_imptransport 6.3); Witness W is equivalent to DP-K and Image I is false for perfect extensions (A176). 3. BESIDE-OUT (RIG-THICK chart branch): every local law, Fact D and L1-SEP are blind to transported two-level adjacency; germs A177, A182, A186 satisfy them all. 4. W2 / RIG-THICK interleaving: an explicit reversal with a rail chord inside a hull has Delta Q = 0 and satisfies CUT, Q, LL (MICROEVAL m23). 5. L-MPN-REL (WRAP-MPN): cm on shrinking carriers fails (carriers collapse to moved singletons), compactness keeps +1 at the access endpoint, and a coherent two-piece laminar tower realises residual +2 with all finite instances (MICROEVAL m25, A188). 6. L-Foreign-Excursion V_2 = +1: the two second-generation branches cancel, V_2 = 0 (MICROEVAL m22). 7. BI-DENSE-EXCL by prime ends: a fixed crosscut has every defining chain eventually on one side, so no second deep family; REDUCED, no exclusion (A171). Dold congruences need a compact cycle-closed block on one region (A155). 8. RIG-XS-Sigma0 / thick band: L-FDC is the Y_2 diagonal mechanism restated (Argon 56); Theorem B (Sigma_0 contact) is necessary only; the planar odometer-end host Y_2 has a thick band re-entering its hole, failing only V1 and m_1 > 0 (it has a fixed point). 9. T2 NEC-EXP: h expands some pair at every small scale; the S3 form of T2 is a restatement (t2_diammono_opus_1102). 10. Route II: the last leaf PW is equivalent to L2 given (R8): restatement. Route III: no exhaustion theorem of host classes (EXH-H missing). Recurring pattern (INFERRED, root reading of lines 2-8): a local law is a statement about one crosscut, one scale, one chart or one pocket. Every one has a finite or bookkeeping model that satisfies ALL of them at every finite scale (circle model, germs, towers, laminar module) while an infinite coherent object is never produced or excluded. cm and the sign of the displacement m_1 > 0 are global laws, but every attempt to use them applies them to a LIMIT of local objects, and the limit degenerates (X, a moved singleton, a boundary point). What is missing is a global statement about the pair (X, h) or (F, X) that is not computed one crosscut at a time. C. THE TARGET Find ONE fresh strategy for L2 (or for the R-instance contradiction), genuinely different from Route I, Route II and Route III, and carry it out on a decisive sub-case that YOU name. D. ASK Part 1 (strategy). Name the method and say in a paragraph (a) which global object it uses (a compact invariant region, a hyperspace or Kelley-continuum dynamical statement for h^2 on the continua of X, a shape or Cech cohomology invariant of the pair (X, h), an index or Lefschetz-Nielsen-Conley count over a region of the plane that avoids prime ends, a Brouwer or Franks-type free-disk argument that survives non-invertible F, an entropy or measure argument, ...), (b) why it can see a defect that crosscut-wise laws cannot, (c) exactly where cm, m_1 > 0, the uniqueness of E_F and the non-invertibility of F enter, and (d) which of the ten failed lines above it avoids and why (one clause each for the lines that look nearest). You may cite classical theorems only with hypotheses; mark SOURCE (exact reference) or UNKNOWN. Part 2 (decisive sub-case). Name a sub-case of the R-instance class on which the method decides (examples, not mandatory: X tree-like; X indecomposable; F a homeomorphism near X but not globally; h with a prime-end action that is compatible; X with finitely many accessible points). Prove the result there, or reduce L2 on that sub-case to ONE named lemma STRICTLY weaker than L2 (restatement gate below). Part 3 (decisive test). Give a concrete configuration (a PL or circle model with all data explicit) that passes the Route I laws V1, V2 and RIG_S at every finite scale you can state, and compute the invariant of your method on it. Report the FIRST violated law, or report that the invariant does not see it (then the strategy is REJECTED, and say what that teaches). Each claim labelled PROVED, SOURCE (exact reference), DERIVED, CONJECTURE, OPEN or RESTATEMENT. UNKNOWN if unsure a cited result exists. Restatement gate: if you reduce, give (G1) the statement of L and of the sub-case target side by side with the same hypotheses, (G2) a proof that L implies it, (G3) a configuration meeting L but not yet excluded, (G4) where L's 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) The homeomorphism case: if F is a plane homeomorphism with F(X) = X, L2 holds by the classical Cartwright-Littlewood and Bell theorems (RECALL, not re-proved here). Your method must either reduce to those or say which hypothesis it uses that they do not. (C2) The circle model of b_opus_imptransport 6.3 (failing doors at every depth, all transport laws satisfied): your global object must be one that this model cannot define or violates. (C3) The disk with an irrational rotation: h^2 has a fixed point and is not cm; it fixes sign conventions and must not be a counterexample to your invariant. (C4) The planar odometer-end host Y_2 (a global homeomorphism with a thick band that re-enters its hole): your method must say which R-instance law it breaks. (C5) A finite-scale tower is not a host: an argument that works for each depth n with a threshold depending on n does not prove an infinite statement. F. DEAD ENDS (not progress) (1) Variation, index or intersection-number counts of crosscuts at any scale or depth (lines 1-6). (2) cm applied to Hausdorff limits of traces, pockets or carriers (lines 2, 5). (3) Prime-end impression, composant or channel transport under F (lines 2, 7): F is not a homeomorphism, so it does not act on prime ends of a perfect extension. (4) Finite germs, towers, level presentations, bookkeeping circle models (lines 3, 5). (5) Route II word classes (PW is a restatement) and Route III host classes (no exhaustion theorem). (6) Nielsen or coincidence numbers of contractible spaces (FAILED_APPROACHES F012: homotopy invariants vanish). 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: FRESH STRATEGY CARRIES (one named method, a decisive sub-case settled by proof, and a test that the Route I methods fail on it) | FRESH STRATEGY REDUCED (to one named lemma, gate G1-G4 passed, with the decisive test) | STRATEGY REJECTED (the proposed method is blind in the same way; first violated law named) | 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: repeating a Route I method (variation, index or intersection counts of crosscuts; cm applied to Hausdorff limits of pockets, traces or carriers; prime-end impression or composant transport; finite-scale germs, towers or level presentations); assuming F is a plane homeomorphism (that is the classical case); using finite depth for every n as one infinite object; naming a classical theorem you cannot state with its hypotheses (write UNKNOWN); claiming PER or L2 unless the proof is complete