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Prompt

RESEARCH QUESTION (plane continua / dynamics; node AP-p for general p (h^p gamma = gamma, h has a periodic point) in an L2 strip system; iterated fixed-point indices (Dold congruences) against the shadow formula). 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 notes b_opus_ap_period2 and b_opus_rig_with_cp1; DERIVED-U there, one agent, unaudited here; lines marked (INFERRED) are my own bridging and may be false). - X is a nondegenerate nonseparating plane continuum, h : X -> X a homeomorphism, and h^2 is continuum-minimal (cm): no nonempty proper subcontinuum C of X has h^2(C) contained in C (singletons count). So h^2 has no fixed point, and no point of X has h-period 1 or 2. F : plane -> plane is a BFMOT perfect extension of h (arXiv 1004.0214); T(S) is the filled hull of a Jordan curve S. - var(g, c) is the variation of a plane map g along a chord or crosscut c; ind(g, S) is the fixed point index of g over T(S) when g has no fixed point on S. SOURCE (BFMOT): ind(g, S) = var(g, S) + 1; and if g has no fixed point in T(S) then ind(g, S) = 0 (Theorem 3.1.4). - LAW1 and LAW2 (SOURCE, BFMOT Theorem 6.2.1, under their standing hypothesis): var(F, g) = -1 and var(F^2, g) = -1 for every small chord g crossing the external ray R_t of the unique outchannel E_t essentially. They are available for F and F^2 only, because only h and h^2 are fixed-point free. For F^n, n >= 3, no such law is known. **Setting.** (S), (XS) as in b_opus_rig_with_cp1 sections 1, 3: X nondegenerate nonseparating plane continuum, h homeomorphism, h^2 cm, F perfect extension; strip phi: [r_0, inf) x [-1,1], sides H1, H2 in X, continuous, injective, disjoint images on a tail; sections c_r with widths w_r -> 0; c_r^+ runs from H1(r) to H2(r); core kappa; rho_T the access from kappa(T) along the strip and out. Facts used: Sh(c_e) is the strip beyond e; Pr(E_t) = X; each side is dense (Lemma D there); I(c_s^+, rho_T) = [T > s], which is Lemma 4.4 of v5 with kappa(T) in Sh(c_s) iff T > s. **Definition AP= (answers A132 Tier 1 item 1).** The ordered pair (l1, l2) of rays of X **equals gamma = [H1, H2]** if there are a permutation pi of {1,2} and increasing homeomorphisms f1, f2 of tails with l1 = H_pi(1) o f1, l2 = H_pi(2) o f2, and diam H_pi(1)[f1(r), f2(r)] -> 0. pi = id is "no swap", pi = (12) is "swap". The same limit then holds on the other wall, since each section between joins the walls with width w -> 0. **Composition (DERIVED).** If G = h^p fixes gamma with G H1 = H_pi(1) f1, G H2 = H_pi(2) f2, then k := G^2 satisfies k H1 = H1 o phi1, k H2 = H2 o phi2, no swap, and the pairing diam H1[phi1 r, phi2 r] -> 0 holds. Proof for the swap case: phi1 = f2 f1, phi2 = f1 f2. H1[f2(f1 r), f2(f2 r)] = G(H2[f1 r, f2 r]) is small by uniform continuity, and H1[f2(f2 r), f1(f2 r)] is small by the pairing at f2 r moved across sections. QED - Notation for the target. p0 >= 2 is the least period of gamma under h. M := p0 if h^p0 does not swap the walls, M := 2 p0 if it does. k := h^M fixes both walls up to tail reparametrisation, without swap and paired (Composition). Z := Fix(k), a closed h-invariant subset of X. By the cm hypothesis every point of Z has exact h-period d with d | M and d >= 3 (DERIVED, elementary). B. LEMMAS (status; where used) **Lemma S (exact shadow; P yes / M any continuous extension E of G = h^q fixing gamma / C none; DERIVED).** Let E be a continuous extension of G with G H1 = H_pi(1) f1, G H2 = H_pi(2) f2, paired. Put s = f1(r), epsilon = +1 for no swap and -1 for swap, and Gamma_r := H_pi(1)[f1 r, s] . c_s^(epsilon) . H_pi(2)[s, f2 r] (the first arc is trivial). Then Gamma_r runs from G H1(r) to G H2(r), inside a ball B_r about G H1(r) with radius -> 0, together with E(cl c_r), and: (a) I(Gamma_r, rho_T) = epsilon [T > s] for every T != s. No homotopy is involved. (b) If kappa(T) is off B_r, then I(E o c_r^+, rho_T) = epsilon [T > s]. **Sign dichotomy (DERIVED).** Let k be fixed-point free with k H_i = H_i phi_i, paired. Then phi_i(r) != r, so phi_1 - id and phi_2 - id have one sign each on a tail. They agree: phi1(r) < r < phi2(r) would put H1(r) inside the small arc H1[phi1 r, phi2 r], so |k H1(r) - H1(r)| -> 0 along it, against min |k x - x| > 0. So k is **lowering** (phi_i < id) or **raising** (phi_i > id). **Theorem POS-XS (P yes / M any fixed-point-free homeomorphism k of X with a perfect extension f / C none; DERIVED modulo SOURCE BFMOT, one agent, unaudited).** Let X be a nondegenerate nonseparating plane continuum, indecomposable with empty interior. Let (phi, H1, H2) be a strip with sections forming a chain of a prime end E with Pr(E) = X, H1, H2 injective with disjoint images. Let f be perfect with f|X = k a homeomorphism without fixed points. Then **var(f, c_r) >= 1 for all r >= D is impossible**. **Theorem AP-p (P yes / M h with h^p fixing gamma / C: a fixed point of h^(2p); DERIVED modulo POS-XS, one agent, unaudited).** Let (X, h) carry a strip as in POS-XS (for example (S), (XS)). If h^p gamma = gamma in the sense of AP= then: - with no swap, h^p has a fixed point; - with swap, h^(2p) has a fixed point. **Corollary 5.1 (p = 2 in L2 systems).** Under (S), (XS), h^2 gamma = gamma forces: swap, h^4 has a fixed point z, and z, hz, h^2 z, h^3 z is a 4-cycle (h^2 has no fixed point, as it is cm). Also V2 is then -1 (swap, lowering) or 0 (swap, raising, and then var(G^2, c_r) = -1). C. THE TARGET Residual (OPEN in the lab, node AP-p(per)): (S), (XS), h^p0 gamma = gamma, and k = h^M has a fixed point (so POS-XS does not apply; Theorem AP-p gives only that Z is nonempty). For p0 = 2 with swap this is AP-2(per) below. **AP-2(per) (OPEN; P yes / M h with h^2 cm / C exclude).** (S), (XS), h^2 gamma = gamma with swap, and h has a 4-cycle. Then b := h^2 is cm with a 2-cycle, the strip is b's outchannel strip in the lowering case, and var(b^2, c_r) = +1 on deep sections where defined. That is a flip-outchannel block in the sense of b_opus_planar_step l.187: "Every level is a flip-outchannel BLOCK*(2) whose square has its two fixed points (index bookkeeping Ind(F^2) = Var + 1 = 2 ...)" (heuristic there), with the extra data of a square root and X-sided walls. Missing law, named NO-FLIP-ROOT (lane b_opus_ap_period2): no planar cm map b with periodic points has an X-sided outchannel strip whose walls b swaps, paired, with a square root h. It is not implied by LAW1, LAW2, POS-XS or the lab's other scalar laws. Proposed new route DOLD (INFERRED, mine, unchecked; it may be false; I think it is also too short to be right, so the point of this task is to find where it breaks). For a Jordan curve S with X in the inner domain of S and no fixed point of F^n on S, write i(n) := ind(F^n, S), the sum of the local indices over Fix(F^n) in T(S). (W1) i(1) = i(2) = 0. [Needs: F and F^2 fixed-point free on T(S), also off X. INFERRED from the standing hypothesis.] (W2) Lowering case. Lemma S and ind = var + 1 give i(n) = 2 for every multiple n of p0 when there is no swap; with swap, i(n) = 0 for odd multiples of p0 and i(n) = 2 for even ones. [Needs: var(F^n, S) on the whole loop equals the chord value var(F^n, c_r) from Lemma S, i.e. the rest of S contributes 0 although F^n now has fixed points; BFMOT show this only when F^n is fixed-point free near X (cf. KP-NEUT). UNPROVED for n >= 3. The raising case would be i(n) = 1 for no swap; the lane's Remark 5.3 says some raising rows pass LAW2, so reconcile.] (W3) Dold's congruences for the iterates of F: sum over d | n of mu(n/d) i(d) = 0 mod n, equivalently i(n) = sum over d | n of d a_d with integers a_d. [Dold, Invent. Math. 74 (1983), recalled from memory, NOT checked here. Hypotheses to check: Fix(F^n) compact inside the region; the region closed along F-cycles of its fixed points; no fixed point of F^n on S, for all n | M.] (W4) Arithmetic (INFERRED). By (W1), a_1 = a_2 = 0, and Z has only cycles of length >= 3. No swap: i(p0) = 2 is a sum of d a_d over divisors d >= 3 of p0. If all those divisors share a factor g >= 3 (p0 prime >= 3; p0 = 4, 6, 8, 9, 10, 14, 16, 18, ...), then g divides 2: contradiction. With swap and p0 prime or p0 = 2: i(p0) = 0 forces a_p0 = 0 (p0 = 2 uses i(2) = 0 directly), and then i(2 p0) = 2 = 2 p0 a_{2p0}: contradiction. Undecided by this arithmetic: p0 whose divisors >= 3 have gcd 1 (12, 15, 20, 21, 24, ...). If DOLD works it removes the finiteness hypothesis in the divisibility lemma C1 of the Arena run A137 (which needed Fix(F^4) finite and inside X) and treats all prime p0 at once. D. ASK (A) Check (W1)-(W4) step by step. For each of (W1), (W2), (W3), say: valid, valid with a named extra hypothesis (state it exactly), or false. For (W3) give the exact local form of Dold's theorem that applies to a continuous plane map F and the region T(S), say what you are sure of and what you recall; decide whether fixed points of F^n off X inside T(S) and orbits leaving T(S) are excluded, and by what. (B) If the route survives for some p0, prove AP-p(per) for that p0 completely, with (W1)-(W3) as named inputs and the extra hypotheses written out. If it fails, give the cheapest explicit abstract configuration (finite data: an index sequence i(n), n | M, the cycle structure of Z, and the walls' reparametrisation signs) that satisfies every law listed above and defeats (W4); no planar embedding is required, but say which law you could not check. (C) Otherwise reduce AP-p(per) to ONE named lemma, strictly weaker than the statement itself, and prove the reduction. In the reduction, state whether the undecided p0 of (W4) need an extra idea, and which. E. CONTROLS (all must hold) (C1) p0 = 1, no swap, lowering is excluded in the lab already: the route must reproduce this (i(1) = 2 against W1) without Dold. (C2) p0 = 2 with swap and a 4-cycle: say what the local index of F^4 on each of the four points must be for W3 to hold and what W2 forces. (C3) The Denjoy host and the K = 2 coherent flip are not excluded by the lab's laws, and the germ G4 passes them on one octave of depth; for each, name the hypothesis of DOLD that it violates (G4: h has a fixed point, so cm fails). (C4) Show that cycle-closedness is necessary: give a plane map with a 2-cycle {a, b} and a region U containing a but not b with i(F^2, U) = 1 and i(F, U) = 0, so that W3 fails there. F. DEAD ENDS (not progress) (1) Arena A137 (non-Claude, OPEN-OBSTRUCTION, p = 2 only): its divisibility lemma C1 needs Fix(F^4) finite and contained in X (left OPEN), and its tile-trapping CONJECTURE C2 is not proved (a limit of wall tiles may be all of X). Do not redo A137; the new part here is general p and the removal of finiteness. (2) Localising POS-XS and its N2 step to a wall tail that misses Z: the BFMOT contradiction needs F fixed-point free on the hull, and a fixed point of index <= -1 absorbs it (b_opus_ap_period2, section 4, 'Why fixed-point freeness is load-bearing'). (3) Reducing p0 to 1 by taking a power: cm is not inherited by powers. (4) The A132 point on AP-1 is repaired by Lemma S; do not reopen it. (5) RIG on strips reaching Sigma_0 and the nested-barrier and shadow-repair topics are other tasks. G. FORMAT Line 1 exactly: VERDICT: ROUTE CLOSES AP-p(per) for the stated p0 (proof) | ROUTE FAILS (named gap + configuration) | REDUCED (to one named lemma, proved reduction) | 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^n has no fixed point for n >= 3; computing i(n) on a different curve S for each n; using Dold's congruences without checking the region is closed along F-cycles and that Fix(F^n) is compact inside it; assuming Fix(F^M) finite; treating lowering and raising alike; re-proving AP-1 or POS-XS (assume them)