m20

Prompt

RESEARCH QUESTION (plane continuum theory / BFMOT variation and the Kuratowski-Pelczynski chords; node KP-REPL of Route I (L2): the input R-KP of Lemma W, namely whether the threshold of BFMOT Theorem 6.2.1 is uniform over ALL KP-chords of the eps-neighbourhood). 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 files PROOF_L2_v5 (parts 1 and 4), the lanes b_opus_rig_with_cp1 and b_opus_ndrepair, and the source papers/lit/bfmot_1004.0214.txt; project statements are DERIVED and UNAUDITED; lines marked (INFERRED) are my bridging). - X is a nondegenerate nonseparating plane continuum, h: X -> X a homeomorphism, F a plane extension (BFMOT sense), T(X) the filled hull, U := C \ X. For f = F or F^2 write var(f, g^+, X) for the BFMOT variation of a crosscut g (Def 2.2.2), defined when f(cl g) and cl g are disjoint. Thresholds (lab, DERIVED): if diam g is small then F(cl g) is in a small ball about h(a), disjoint from cl g (m := min(m_1, m_2) > 0). - (S1) SOURCE, quoted from papers/lit/bfmot_1004.0214.txt (the text extraction loses the symbol epsilon; where I restore it the restoration is INFERRED), l.2616-2623, "standing hypothesis": f perfect, no fixed point in T(X), X minimal with f(X) in T(X); fix eps > 0 such that for each KP-chord g in T(X)_eps, g cap f(g) = empty and f is fixed point free on T(X)_eps; then eps "defines variation near X" and (f, X, eps) satisfies the standing hypothesis. KP_eps denotes the KP-chords in T(X)_eps (INFERRED notation). - (S2) Definition 5.1.1 (l.2625-2640): an outchannel is a prime end E_t of U_eps with a chain {g_i} of crosscuts, var(f, g_i, T(X)) = 0 for all i. It is a geometric outchannel iff "for sufficiently small eps, every chord in KP_eps which crosses E_t essentially has nonzero variation". It is negative (resp. positive) starting at g in KP iff every KP-chord h in T(X) cap Sh(g) crossing E_t essentially has negative (resp. positive) variation. - (S3) Theorem 6.2.1 (l.3184-3194): "If (X, f, eps) satisfy the standing hypothesis then there exists a unique geometric outchannel E_t for X, which is dense in X = T(X). Moreover, for any sufficiently small chord g in any chain defining E_t, var(f, g, X) = -1, and for any sufficiently small chord g not crossing R_t essentially, var(f, g, X) = 0." - (S4) End of its proof (l.3340-3350): the case of a positive outchannel is excluded by a Lollipop-lemma argument; "By Theorems 6.1.4 and 3.2.2, X has exactly one negative outchannel and its variation is -1." Earlier (l.3098-3100): "it will follow later from Theorem 6.2.1 that there are no chords of positive variation." - (S5) Lemma 6.1.2 (l.2987-2995): if there is a chord g of T(X) of negative (resp. positive) variation with no fixed point in T(T(X) cup g), then there is a negative (resp. positive) geometric outchannel E_t starting at g; the set S of hulls conv(B cap T(X)) containing a chord that crosses R_t essentially is an infinite narrow strip bordered by a KP-chord and two halflines H1, H2. - (S6) Theorem 6.2.2 (l.3354-3366): the sides H1, H2 of the channel "consist entirely of chords of zero variation and points in X". - Crossing essentially (BFMOT Def 3.6.4, lab reading): a crosscut g crosses R_t essentially if a bounded-side end of R_t lies in Sh(g). B. LEMMAS (status; where used) Known (all UNAUDITED). - Lemma W (lane b_opus_ndrepair, DERIVED, assumes R-KP). A section c_r is tau-walled if there is an arc L_r with the ends of c_r, disjoint from c_r, the deep strip inside S_r := L_r cup cl c_r, and every component of L_r \ X a KP-chord of diameter < tau (W1-W3). If r is tau_2-walled and the width w_r < tau_1, then var(F, c_r^+, X) = -1 and var(F^2, c_r^+, X) is in {0, -1}. Proof: ind(F^j, S_r) = 0 splits into the section term and finitely many KP-chord terms, each evaluated by R-KP. The quantifier order of the lemma is: there are tau_1, tau_2 > 0 (independent of r) such that for all r and all components of L_r \ X, so the chord terms are evaluated at chords that are not part of any one fixed chain. - R-KP (lab, INFERRED reading, the object of this task). For small delta a KP-chord g has var(F, g^+, X) = -1 if g crosses R_t essentially and 0 otherwise; the same for F^2 and R_s; standing hypothesis S2 (cl g and F(cl g) disjoint). The lab flags that the uniform-threshold reading is itself INFERRED from (S3). - LAB (ledger, cp1_source_check, Sonnet, one agent): BFMOT 6.2.1 gives var = -1 / 0 only for chords in KP, and the lab notes that KP_eps is defined through the family of balls B of small diameter, i.e. eps acts as a diameter bound; whether a small crosscut near X is already a KP-chord is UNPROVED. Check this against (S1)-(S3): does 'sufficiently small' in (S3) mean diam B < eps (then (U3), (U4) may hold with delta = eps) or something else? - Lab (K2, A166): the zero clause for crosscuts that are NOT KP-chords is the content of Theorem CP1 and is not available; this task concerns KP-chords only. C. THE TARGET Four statements, in order of strength (eps fixed by the standing hypothesis; "g" ranges over KP_eps; the quantifier order matters). (U1) There is eps_1 <= eps such that every g in KP_eps1 crossing E_t essentially has var(F, g^+, X) != 0. (U2) There is eps_1 such that every g in KP_eps1 has var(F, g^+, X) <= 0. (U3) There is delta > 0 such that every g in KP_eps with diam g < delta that crosses E_t essentially has var(F, g^+, X) = -1. (U4) There is delta > 0 such that every g in KP_eps with diam g < delta that does not cross E_t essentially has var(F, g^+, X) = 0. R-KP-UNIF := (U3) and (U4) (and the same for F^2, E_s). Lemma W needs R-KP-UNIF with delta independent of the chord and of the section r. D. ASK Part 1 (quantifiers). For each of (S2)-(S6) write its quantifier structure (over eps, over chains, over chords, over the order "sufficiently small") and decide which of U1-U4 follows from the quoted text alone, which from the quoted text plus a named standard fact (additivity of variation, the junction formula, Lemma 4.3, Corollary 3.3.2), and which is not implied. For U1 and U2 point to the lines of the source. State what "any sufficiently small chord in any chain" gives: is delta allowed to depend on the chain, and on which chain? Part 2 (the two clauses). Prove U3 and U4, or show where the quoted statements stop. Suggested reductions, not claims: (a) U2 from (S4) and (S5) (no positive chord); (b) a lower bound var >= -1 for a small chord (a chord with var <= -2 and small diameter; use that a negative chord starts the unique channel (S5) and the strip S of Lemma 6.1.2); (c) U4: a small chord outside the strip S has zero variation, with a threshold from the width of S and from (S6). Say for each whether it needs the standing hypothesis S2 uniformly in g. If a step is not available, give abstract data (a sequence of KP-chords g_k, diam g_k -> 0, with their variations and their positions relative to R_t) consistent with every quoted statement (S1)-(S6) and violating U3 or U4, or say why none exists. Part 3 (what Lemma W really needs). Decide whether Lemma W (the quantifier order above) needs U3 and U4 in full uniformity or only a weaker statement (for instance: delta depends on the section but not on the chord within it; or only finitely many chords per section; or only chords with both ends on one side of the strip). Name the weakest statement R-KP-W that makes Lemma W go through, prove R-KP-W implies Lemma W's use, and state whether R-KP-W follows from the source. Do not argue from Theorem CP1. Each claim must be labelled PROVED, SOURCE (exact line of the quoted text), LAB, CONJECTURE, OPEN or RESTATEMENT. If unsure that a cited BFMOT result exists, say UNKNOWN. E. CONTROLS (all must hold) (C1) Quantifier control: a statement of the form "for every chord there is delta" must not be used where "there is delta for every chord" is needed; mark each use. (C2) Planted negative: a sequence of KP-chords g_k with diam g_k -> 0 that does NOT cross R_t essentially but accumulates at an end of the channel; your proof of U4 must say why var(F, g_k) cannot be -1 for them (the strip S and (S6)). (C3) eps is fixed by (S1): no step may shrink eps to make a chord small unless U1 is used for that smaller eps and the chords of KP_eps are all accounted for. (C4) Do not use Theorem CP1, Lemma 4.2 outside KP-chords, or the zero clause of CP1 for non-KP crosscuts. (C5) A KP-chord has both ends in X; a chord of a chain defining E_t is not a generic KP-chord (the chain is selected); say which of your statements concern which. F. DEAD ENDS (not progress) (a) The three sub-questions of K2_KPREPL (an X-connector for a hidden end, a weaker law on sums over components, a choice of short paths with exposed ends): other tasks. (b) Proving Theorem CP1 by the conjugation/lens route. (c) Re-proving Lemmas 4.3-4.6. (d) Counting signed crossings with transversality. (e) Using sums of var(F, d) and I(F(d), sigma_d) separately. (f) Quoting 6.2.1 for a crosscut that is not a KP-chord. 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: R-KP-UNIF PROVED (from the quoted statements, steps listed) | R-KP-UNIF CONDITIONAL (one named extra statement, proof of the rest) | R-KP-UNIF NOT IMPLIED (what is only chain-dependent, abstract data meeting every quoted statement) | 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: exchanging "there is delta for all chords" with "for each chord there is delta"; treating a chord of a chain as a generic KP-chord; using the zero clause of CP1 or Theorem CP1 (this task is the KP-chord case only); quoting 6.2.1 for a crosscut that is not a KP-chord; assuming var is bounded below by -1 for small chords without a proof (the winding of f(g) closed up can in principle be large); using that eps is arbitrary (eps is fixed by the standing hypothesis); claiming L2 or PER