m55/56/57

Prompt

RESEARCH QUESTION (plane continua / W-PE, PURE-POCKET in the FIN scope: can a small non-theta_F window crosscut e have an IMPURE pocket for j = +-2, with both ends of its image arc in h^j(K_e) and K_e meeting every finite-channel image arc?). 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 the lab reports api_opus_extindep_nofold, api_opus_pure_pocket, b_opus_m2_quad, b_opus_rigid_kill and the audited PROOF_L2_v2 Lemmas 4.2-4.5, Theorem 6.1; single-agent statements are UNAUDITED unless a status says otherwise; INFERRED marks the spec author's own reading). - (S): X is a nondegenerate nonseparating plane continuum, U := C \ X, h: X -> X a homeomorphism with h^2 continuum-minimal (cm: no nonempty proper subcontinuum K with h^2(K) inside K). Then h^-2 is cm (nested intersection of the sets g^-n(A), DERIVED), h and h^2 have no fixed point, X is indecomposable, and the composants (a composant is the union of all proper subcontinua containing a given point) are dense, pairwise disjoint and permuted by h. The class of (S)-instances may be empty (that is the lab's question PER). - Prime ends: a crosscut q is an open arc of U whose closure is an arc with two distinct ends q^-, q^+ in X; Sh(q) is its bounded complementary side; I_q is the open arc of S^1 under the cap psi(Sh q) (psi the conformal map of U u {infinity} onto D). A prime end is full if its impression is X; Theta_X is the set of full prime ends. Lemma 4.5: disjoint crosscuts are nested or have disjoint shadows. - Extensions (BFMOT Theorem 6.2.1, SOURCE, used as CP1): h has a perfect plane extension F (continuous, F|X = h, compact fibres) with a unique outchannel, a full prime end theta_F, Pr(theta_F) = X. A theta_F-crosscut is a crosscut q with theta_F in I_q. Every crosscut q of small diameter has var(F,q) = -1 if q is a theta_F-crosscut and 0 otherwise (Lemma 4.2); likewise for F^2, G, G^2 (G extends h^-1). Lemma 4.3: var(Phi,q) = I(beta, rho_v) for any short path beta from Phi(q^-) to Phi(q^+) in a small convex ball and access rho_v starting at v in q; Lemma 4.4: I(q^(+-1), rho_w) = +-[w in Sh q] (both VERIFIED under CP1, D2013). S9 (BFMOT Lemma 4.4.8, SOURCE): for every x in X and delta > 0 there is a chain of theta_F-crosscuts of diameter < delta converging to x. - Window and SWAP (hypotheses of the lab's records, not proved): (2G): an open arc J containing theta_F with Theta_X cap cl J = {theta_F}; Gamma_+ and Gamma_- are the gaps on the two sides of theta_F with composants C_+ and C_-; GAP-IMP (Sol CONFIRMED): a crosscut with cl I in Gamma_+ has both ends in C_+. SWAP: h(C_+) = C_- and h(C_-) = C_+. - Pocket data. e is a small crosscut with cl I_e inside J and theta_F not in I_e; then var(Phi_j, e) = 0 for j in {1,2,-1,-2} (Phi_1 = F, Phi_2 = F^2, Phi_-1 = G, Phi_-2 = G^2). P := Sh(e); K := X cap cl P, a continuum (the pocket-trace input, RECALL, inherited), proper by IMP* (Sol CONFIRMED); L := h^j(K). B_1 is a small ball around e, B_2 an open convex ball around h^j(e^-) with Phi_j(B_1) inside B_2 and cl B_1, cl B_2 disjoint. An image arc beta is an arc in B_2 from h^j(e^-) to h^j(e^+); its pieces are the components of beta \ X (crosscuts); a theta_F-piece has theta_F in I_q and a traversal sign eps(q) = +-1. FIN(e,j): some image arc has only finitely many theta_F-pieces; tau(beta) := sum of eps over its theta_F-pieces. - Lasso winding (the winding number around z of the closed lasso beta * delta): for z not in L u beta, D(z) := win(beta * delta, z), delta a return path from h^j(e^+) to h^j(e^-) in a thin neighbourhood of L missing z (L is non-separating, so this stabilises); D is locally constant. Omega_e := the component of C \ (L u beta) containing e. STRONG-PURE(e,j): D = 0 on X \ (L u cl B_2); IMPURE otherwise. PURE-POCKET: every such e is STRONG-PURE for all four j (in the window this is NTJ: every theta_F-test jump piece is a theta_F-piece). B. LEMMAS (status; where used) Known (status as given; the FIN theorems are Sol CONFIRMED in exactly the FIN scope). - Lemma CONST (Sol CONFIRMED, FIN): D = tau(beta) on X \ (L u beta) for a FIN arc beta. Proof: for w in U, D(w) = sum over pieces q of eps(q)[w in Sh q] (finite); at z take an S9 chord d near z whose cap lies inside every theta_F-piece cap (finitely many, here finiteness is used); only theta_F-pieces contain d in their shadow. - Lemma MOUTH (Sol): D = +-var(Phi_j, e) = 0 on Omega_e. Theorem SELF-LINK (Sol CONFIRMED): FIN and IMPURE imply that e^+ and e^- both lie in L = h^j(K), since an end off L would have D = tau != 0 and D = 0. - Theorem TRACE (Sol CONFIRMED, all four j): if some FIN image arc beta misses cl P (an actual arc, not merely endpoints), then e is STRONG-PURE. Proof: otherwise tau != 0, Omega_e meets no point of X, K lies in L u beta, so K inside h^j(K) with K proper, hence h^(-+2)(K) inside K, contradicting cm of h^(+-2). Cor ODD-PURE (Sol CONFIRMED given SWAP): j = +-1 has K in C_s and L in C_-s, so the ends are off L and SELF-LINK gives purity; Cor COMB-ZERO follows (it kills the record CS-1' of comb_shadow in FIN). - Lemma REAL (CONFIRMED) and Prop DECOUPLE (Sol CWR): continuous maps on cl P equal to h^j on K extend to perfect extensions, and no choice of extension together with FOLD-2 (CONFIRMED under CP1) forces purity: the FOLD-2 witnesses lie in a channel cap V_r disjoint from every non-theta_F pocket. The literal NOFOLD is too strong (codex_nofold, GPT family, UNAUDITED: small discretionary folds can be added); NOFOLD = FILL (an (X,h) filling problem) and FILL implies INNER-UC, so it is stronger than purity. - Exact residual (api_opus_extindep_nofold Section 2.3, UNAUDITED as a list, each item a consequence of the CONFIRMED theorems), called R-FIN: an IMPURE FIN window pocket must have (i) j = +-2 (j = +-1 only without SWAP); (ii) e^+ and e^- in h^j(K), so K meets h^j(K) and the closure of the union of the sets h^(2n)(K) is connected, hence equal to X by cm (no contradiction); (iii) EVERY FIN image arc meets K; (iv) tau != 0, and since both ends lie in one composant the finite SIDE-COUNT identity (Sol CONFIRMED, finite decompositions only) gives a net side-changing wall count W = 2 tau != 0 on every FIN arc, so by ACC-FULL (a)-(d) (CONFIRMED) the walls accumulate on full prime ends: theta_F-hugs in the window. The alternative R-INF (no FIN arc exists) is the lane of R56. - Record M2-quad (b_opus_m2_quad, Tier 0): its j = 1 impure pieces are excluded by ODD-PURE; its j = 2 pieces would need (ii)-(iv). Record TP-B* (b_opus_rigid_kill): every jump piece has K = X, so L = X and all FIN theorems are vacuous there. C. THE TARGET In (S) with (2G), SWAP and CP1: decide whether an IMPURE window pocket e with FIN(e,j) can exist for j = 2 or j = -2 (equivalently, satisfy (i)-(iv) of R-FIN). Acceptable outcomes: J2-PURE PROVED (each law used named, planarity marked); J2-IMPURE CONSISTENT (explicit Tier 0 data or abstract configuration meeting every law above, first missing law named; a finite record is not a host); REDUCED to one named lemma L (gate G1-G4); PARTIAL. Not asked: R-INF, PER, L2, PURE-POCKET outside the FIN scope. D. ASK Part 1 (method). Name ONE method not on the dead-end list and say why it sees what CONST, SELF-LINK and TRACE cannot (they stop at 'K meets h^j(K), every FIN arc meets K'). Candidates, none claimed to work: (i) use SELF-LINK at j = 2 and at j = -2 for the SAME pocket, e^+- in h^2(K) cap h^-2(K), together with TRACE applied to sub-crosscuts e' of e inside P (K_e' inside K); (ii) Alexander duality / linking of the two lassos K u e and L u beta: the pair is 'mutually linked' (ends of each in the other) and a planar linking number may be invariant under the h^-2 map on X where cm acts; (iii) a first-hit argument along a FIN arc into P, in the style of NO-STALL (uses m > 0 only); (iv) your own. Part 2 (carry it out). Prove J2-PURE, or realise (i)-(iv), or reduce to one named lemma L with gate G1-G4. Mark each use of Pr(theta_F) = X (S9), Lemma 4.2 for all four maps, cm of h^(+-2), SWAP, GAP-IMP, planarity, and where the hypothesis 'ends in L' or 'K meets every FIN arc' is used. State results for h^2, not h^4. Part 3 (builder or decisive test). Try to realise an R-FIN pocket for j = 2: data (K, e, the image arc beta with its theta_F-pieces and signs, L, the walls and their composant labels, D on X-points), checked against CONST, MOUTH, SELF-LINK, TRACE, SIDE-COUNT and cm, or name the first law every candidate violates. Builder gate: name the TARGET law (B0); secure the background laws first (B1); test (B2); end with 'KILL: law | TARGET or BACKGROUND | mechanism | what a proof must use' (B3). What already failed (read DEADENDS first): a new method, even partial with an honest gap, beats a polished restatement. Take your time, be extremely meticulous, I believe in you. E. CONTROLS (all must hold) (C1) FLIP-S (impure pocket on a locally connected tree with a fixed point): Theta_X is empty, there is no theta_F and no deep chord, so CONST does not apply and impurity is allowed; a proof must use Pr(theta_F) = X. (C2) Ambient hosts (Knaster bucket handle with a Barge-Martin homeomorphism, two-endpoint Knaster): pockets are PURE (AMB-P); Lemma 4.2 is unavailable (fixed points). (C3) Lewis pseudo-arc: chainable, Theta_X = S^1, no window. (C4) TP-B*: every jump piece has K = X, so L = X; passes all FIN theorems vacuously, so those theorems alone cannot separate it from a pure pocket. (C5) Record CS-1' (comb_shadow): killed in FIN by CONST (declared A = -1 against tau = 0); a j = 2 argument must reproduce this kill when its extra hypotheses are dropped. F. DEAD ENDS (not progress) (1) Extension and fold laws (COVER, FOLD-2, CHINJ-NO, EXT-INDEP, DECOUPLE): impurity forces every extension to fold, but no extension choice forces purity; folds are the degree form of the variation law. (2) Literal NOFOLD and FILL: too strong or restatement-strength. (3) ODD-PURE and TRACE need j odd with SWAP, or a FIN arc missing cl P; j = +-2 with K meeting every FIN arc is exactly what they miss. (4) The universal slogan 'every passage needs FLIP or DEEP' (SIDE-COUNT) is REFUTED (codex_aud_triple_kill); only the finite identity survives. (5) h^4 cm is not available and the route is eliminated (b_opus_h4cm, Lemma FAIL-SMALL, UNAUDITED). (6) Orbit-wide pocket iteration (b_opus_pocket_orbit), hyperspace trapping UD/EQ, invariant measures, HT plus semicontinuity, clopen continuum-span: the tried list of the wall W-SPAN-PROPER (no assembly of a proper subcontinuum K with h^(+-2)(K) inside K from local data). (7) Signed transport of the winding datum D from h^j(K) back to K by the homeomorphism h^-j is (INFERRED) an instance of the wall W-TRANSPORT, a signed transport law (impression transport, moving-base-point degree transport, second-iterate coboundary identity, crossing-index bookkeeping, Brown-Bell variation counting all stopped there). G. FORMAT Line 1 exactly: VERDICT: J2-PURE PROVED (no IMPURE FIN window pocket for j = +-2 in (S); every law used named, planarity marked) | J2-IMPURE CONSISTENT (explicit Tier 0 data or abstract configuration meeting every listed law; first missing law named) | REDUCED (to one named lemma L, gate G1-G4 passed) | 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: assuming F or G is a plane homeomorphism or has an induced prime-end action (NO-LOCAL-INJ); using h^4 continuum-minimality; claiming the literal NOFOLD (every extension avoids folds) or an extension-choice argument (DECOUPLE); using ODD-PURE or TRACE outside their hypotheses (j odd with SWAP; a FIN arc actually missing cl Sh(e)); using SIDE-COUNT as a universal FLIP-or-DEEP slogan (REFUTED); variation at a failing chord; claiming PURE-POCKET, COMB-ZERO, L2 or PER as proved outside the FIN scope; presenting a restatement of IMPURE as a reduction (gate G3)