m38

Prompt

R38_TT2_TORN_prompt.txt 100 % RESEARCH QUESTION (plane continua / prime ends and perfect extensions; channel tearing (Theorem TT2, Route I): every small KP chord across the outchannel of an R-instance is F-KP-torn or F^2-torn; what does tearing at cofinally many chain chords force about the image pairs, their composant classes, and the excursions of F(c) through X?). 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 b_opus_sscof_cl, b_opus_imptransport and b_opus_ssdyn; single agent each, UNAUDITED unless a status says otherwise). - 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) and m_1 := min_x d(x, hx) > 0, m_2 := min_x d(x, h^2 x) > 0 (R-instance). F, G are perfect plane extensions of h, h^-1 (continuous maps of C extending them; NOT homeomorphisms; F(c) may meet X). Unique outchannels E_F != E_G with prime ends theta_F != theta_G; X indecomposable when composants are used (x ~ y iff a proper subcontinuum of X contains both). - Crosscut q: open arc in U whose closure is an arc with two distinct ends in X; Sh(q) bounded component of C \ (X u q); NEST: disjoint-closure crosscuts q' in Sh(q) give Sh(q') in Sh(q). A KP chord is a Kuratowski-Pelczynski chord of BFMOT; c crosses the channel essentially if theta_F lies in the open shadow arc I_c. Feet c^- (alpha), c^+ (beta). U3 (cross-family audit CONFIRMED-WITH-REPAIRS): every small KP chord c has var(F, c), var(F^2, c) in {-1, 0}, and the value -1 if c crosses the channel essentially. By Lemma P, var(F, c) = iota(h c^-, h c^+; w) = s_d * 1_{Sh(d)}(w) when h c^-, h c^+ are the ends of a short crosscut d (the join) with orientation sign s_d. - (T1) h(c^-), h(c^+) are the feet of a small KP chord d_1 (F-KP-untorn). (T2) h^2(c^-), h^2(c^+) are the feet of a small crosscut d_2 (F^2-untorn). UNTORN-2 := (T1) and (T2). A chord is F-KP-torn if (T1) fails, F^2-torn if (T2) fails; torn := F-KP-torn or F^2-torn. The image feet are close (distance at most omega_h(diam c), tending to 0 with diam c), so tearing is a statement about U, not about the metric. - Excursions (Prop EXC): for a crosscut s let T := (F o s)^-1(X); the excursions of F(s) are the closures of components of F(s) \ X, each a path in U with ends in X. (E1) some excursion has its two ends in different composants. (E2) every excursion has both ends in one composant, so the composant of F(s(t)) changes (as t varies over T) only at points of T that are accumulated by other points of T. - Theta_X is the closed set of prime ends with impression X. Theorem IMP*: for ANY crosscut q with K_q = X (or ends in different composants), the closed cap of q contains a Theta_X point. B. LEMMAS (status; where used) Known (statuses as given). - Theorem TT2 (b_opus_sscof_cl section 4.2, DERIVED; inputs Lemma P, U3 for F and F^2, NEST): for a small KP chord c crossing the channel essentially, (T1) and (T2) cannot both hold, since they force var(F^2, c) = +1, which U3 forbids. Proof is four lines of sign bookkeeping: var(F,c) = -1 makes the join d_1 an E_F crosscut over c; var(F, d_1) = -1 makes the h^2-join d_2 an unflipped crosscut over d_1; so c lies in Sh(d_2) with the same orientation and var(F^2, c) = +1. Corollary TT2': every small chord across E_F is F-KP-torn or F^2-torn, for E_F = or != E_G. - UNTORN-2 is exactly where invertibility is used (in HOM, d_1 = F(c) and d_2 = F^2(c)); it fails at every small chain chord of every R-instance, and any law implying it would prove R-instances do not exist (sscof_cl section 4.3). Theorem HU (D391, recorded as audited by Opus; family-correct audit status UNKNOWN): an FPF h in a one-channel embedding must KP-tear every small chord across the channel. TT2 removes the condition E(h) = E(h^-1) of the older statement, at the price of allowing an F^2-tear. - Prop JOIN (b_opus_imptransport, DERIVED): if a failing crosscut s (K_s = X, ends a not~ b) has a short join C' of (ha, hb), then Theta_X meets the closed cap of C'. Prop EXC (DERIVED): if F(s) meets X, the dichotomy E1 or E2 holds. In E1 anchored transport of the Theta_X point holds; E2 is NOT excluded by plane topology (RECALL-level control: the local Cantor-bundle picture of a Knaster-type continuum realises it for a path). The variation data var(F, s) = 0 impose nothing on the composant function in E2. OPEN (E1-LAW): at a failing M_F door, F(s) has a composant-changing excursion; the lab has no law excluding E2. This is the composant form of the tearing statement of Theorem HU. - Door-blindness (sscof_cl section 4.5): TT2 and PAR read only chords of variation -1; a door has variation 0 for F and F^2, and running the steps of TT2 at a door gives only the consistent order (D) (s not in Sh of the join of the image pair). C. THE TARGET For an R-instance with cofinally many small KP chain chords c_e across the channel (the chain defining E_F), all F-KP-torn or F^2-torn (TT2), determine what cofinal tearing forces: (a) about the pairs (h c_e^-, h c_e^+) and (h^2 c_e^-, h^2 c_e^+), (b) about their composant classes, (c) about the excursions of F(c_e) and F^2(c_e), E1 versus E2. Decide whether cofinal tearing alone contradicts a law of an R-instance (cm of h^2, m_1 > 0, unique outchannels, Theorem IMP*), or is realised. D. ASK Part 1 (method). Name ONE method not on the dead-end list and say why it sees tearing where the failed methods saw only variation -1. Failed methods evaluated sign identities on chords (Lemma P, U3, cocycle identities) and needed a join to exist; a torn pair has no join, so those methods are silent exactly when they matter. Candidates, stated exactly if used (none is claimed to work): (i) prime-end geometry of the image pair: when h c^- and h c^+ are accessible with prime ends theta^-, theta^+ (endcuts), tearing means no short crosscut joins them; measure the conformal length of the shorter arc [theta^-, theta^+] of S^1 and its behaviour along the chain, comparing with |I_c| -> 0; (ii) a homotopy invariant: the class of the path F(c) closed up by a short segment of C in the fundamental group of C \ (a finite set of points of X near h c^+), when F(c) meets X; (iii) a Baire argument on the closed set of x in X whose small neighbourhood meets U in a single component near each torn chord; (iv) your own. Part 2 (carry it out). Prove one of: cofinal F-KP-tearing of the chain forces the E1 alternative for F(c_e) (so a Theta_X point lies in the closed cap of an excursion), OR forces h c_e^- and h c_e^+ into different composants, OR forces (T1) to fail only because the join is not KP (so a non-KP short join exists), OR show each of these is consistent. If tearing is F^2-torn only (T1 holds), do the same for h^2. Reduce to one named lemma L with gate G1-G4 if that is all that holds. Part 3 (builder or decisive test). Build the local picture of a torn chord: a chord c across a channel, two close points u = h c^-, v = h c^+ of X with no short crosscut and all of Lemma P, NEST, U3 intact, for two scales of the chain, compatible with the F(c) data (an explicit path through X). Record the first violated law, with the line where the HOM argument would have used d_1 = F(c). A finite number of torn chords is not a cofinal configuration. Each claim labelled PROVED, SOURCE (exact reference), LAB, DERIVED, CONJECTURE, OPEN or RESTATEMENT. UNKNOWN if unsure a cited result exists. What already failed (read it before you start): see DEADENDS. Do not repeat these as progress; a new method, even a partial one with an honest gap, is worth more than a polished restatement. 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) HOM control: if F is a plane homeomorphism then (T1) and (T2) hold with d_1 = F(c), d_2 = F^2(c), and var(F^2, c) = +1 in the reversing case gives a fixed point of h^2; nothing tears and the R-instance is contradictory. Your Part 2 argument must say which hypothesis (F maps a crosscut to a crosscut) fails for a perfect extension and where. (C2) Control that the failed methods also pass: the circle model of b_opus_imptransport (reversing reading) satisfies Lemma P, NEST, U3 and every impression-transport law with door caps untouched, so a proof that only reads variation or impression laws cannot distinguish torn from untorn. (C3) Bucket handle with a shift (m_1 = 0) has a fixed point, so TT2's inputs are void; it is a planarity/hypothesis test only. (C4) E2 is realised for a path by the Knaster local Cantor-bundle picture, so an argument using only plane topology of the path cannot force E1. F. DEAD ENDS (not progress) (1) Sign identities for tearing: TT2 and PAR evaluate var at chords of variation -1 and need a join (sscof_cl section 4.4, 4.5). (2) Replacing UNTORN-2 by a law: impossible, since it fails at every small chain chord of an R-instance (sscof_cl section 4.3). (3) Running TT2 at a door: var = 0, only the consistent order (D) results (ssdyn NX; sscof_cl). (4) Impression or type transport (TT): satisfied by the HOM circle model with failing doors at every depth (imptransport section 6). (5) Plane topology of the path F(c) for E1: the Knaster local control realises E2 (imptransport Prop EXC). (6) The 701 Corollary U.1 route needing E(h) = E(h^-1) and the circle map of 978 Lemma IT (replaced by TT2). 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: TEARING FORCES (a named consequence of cofinal tearing, with proof, e.g. a composant change of the image pair or an excursion of type E1) | TEARING CONSISTENT (explicit data of cofinally torn chords meeting every listed law; first missing global law named) | REDUCED (to one named lemma, 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: using a variation count at a door (zero); assuming F maps crosscuts to crosscuts (that is HOM, where nothing tears); using the untorn hypothesis UNTORN-2 (TT2 shows it fails at every small chain chord); treating torn as merely 'image feet far apart' (they are close: omega_h(diam c) -> 0); confusing F-KP-torn (no small KP chord) with F-torn (no small crosscut); claiming SS-COF, DP-K, L2 or PER