m48

Prompt

RESEARCH QUESTION (plane continua / homotopy tracks and variation (H-EQ, the equality of the outchannels of F and F^2; wall W-TRANSPORT; H-EQ is the standing hypothesis of the Route I node L-MPN-REL and is not itself a ROUTE_MAP_L2 node): the composant covering law UCA that would give H-EQ, and its negation BRC under breathing). 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 b_opus_homavoid and b_opus_heq_kinf; single-agent statements, UNAUDITED unless marked). - (S): X is a nondegenerate nonseparating plane continuum, h: X -> X a homeomorphism with h^2 continuum-minimal (cm: no nonempty proper subcontinuum K has h^2(K) inside K). LAB consequences: h is cm, X is indecomposable (composants are dense, connected, and countable unions of proper subcontinua), m_k := min |h^k(x) - x| > 0 for k = 1, 2, and m := min(m_1, m_2). F is a perfect plane extension of h (NOT a homeomorphism); E_t is the outchannel of F with ray R_t, E_s that of F^2 with ray R_s. H-EQ (the node): E_s = E_t. Breathing: E_s != E_t. For a crosscut c = (a, b) with ends a, b in X write var(h^k, c) := var(F^k, c, X) (BFMOT variation). U3: for small crosscuts g and f in {F, F^2}, var(f, g) = -1 if g crosses E_f essentially and 0 otherwise. So H-EQ holds iff var(h^2, c_n) = -1 on cofinally many E_t-chain crosscuts c_n. - Angle function (Lemma SP; X nonseparating, Borsuk): for v in U there is a continuous Theta_v: X -> R with z - v = |z - v| exp(i Theta_v(z)). For a path beta with ends in X missing v, I_v(beta) := (change of arg(. - v) along beta - [Theta_v(end) - Theta_v(start)]) / 2 pi, an integer equal to I(beta, J) for every closed ray J from v missing X; it is additive, and equal to the winding number about v for loops. Lemma VF: for a small c = (a, b), v in c and k = 1, 2, var(h^k, c) = I_v([h^k(a), h^k(b)]). - Coincidence-free homotopy: H: X x [0,1] -> C continuous, H_0 = h, H_1 = h^2, m_H := min |H(x, s) - x| > 0, modulus omega_H; tracks tau_x := H(x, .). Such H exist (Eilenberg-Bruschlinsky, H^1(X) = 0). delta_H := the supremum of diam c with c small (T1: F, F^2 fixed-point free near cl c) and diam c + omega_H(diam c) < m_H (T2). B_v := {x : v lies on the track tau_x} is closed; on X \ B_v put N_v(x) := I_v(tau_x), continuous and integer valued. B. LEMMAS (status; where used) Known (DERIVED in the reports unless marked; UNAUDITED). - LADDER: if diam c < delta_H and v is in c, then a and b are not in B_v and var(h^2, c) - var(h, c) = N_v(b) - N_v(a). Prop TAUT: for any paths from h(a) to h^2(a) and h(b) to h^2(b) missing a ray J from v, the ladder loop has winding 0 iff var(h^2, c) = var(h, c): the endpoint form is a restatement, and it equals the second-iterate swept-angle coboundary G_v = Theta_v o h - Theta_v (Sol S1 and wrapmpn Lemma SA), since N_v = (A_v - G_v o h) / 2 pi with A_v the swept angle of the tracks. - Theorem CMP: if a and b lie in one quasi-component of X \ B_v (in particular if some composant is disjoint from B_v: it is dense and connected, so X \ B_v is connected), then var(h^2, c) = var(h, c). - UCA (candidate law, neither derived nor refuted): for cofinally many n there are a coincidence-free H_n with diam c_n < delta_{H_n}, a point v_n in c_n and a composant C_n of X with v_n not in H_n(C_n x [0,1]). Cor CMP-EQ: UCA implies H-EQ (CMP gives var(h^2, c_n) = var(h, c_n) = -1, so by U3 c_n crosses E_s essentially cofinally and R_s enters every Sh(c_n)). UCA is not a restatement: it is a covering property of one map on X x [0,1] at the scale of c_n. - BRC (breathing covering law): assume breathing, H any coincidence-free homotopy and c = (a, b) with diam c < delta_H crossing E_t but not E_s essentially. Then for every v in c: B_v separates a from b in X; B_v meets every composant, i.e. c lies in H(C x [0,1]) for every composant C; B_v lies at distance at least m_H - diam c from a and b (a track through v starts at x with |x - v| at least m_H, Lemma OFF); B_v has infinitely many components (Lemma Kinf, sides of a separating neck have infinitely many components); by Baire on the arc c some proper subcontinuum K of each composant has H(K x [0,1]) containing a subarc of c. - AV and TEAR: for closed W with W cap X empty and C \ W connected (a point of U, a compact subarc of R_t, a shallow tail) some coincidence-free H has H(X x [0,1]) disjoint from W. Under breathing, if H misses a point of c then max_s |H(a, s) - H(b, s)| is at least m_H - diam c, so diam c is at least delta_H. Avoidance is free away from X and costs exactly the scale of c: SCALE-AVOID is the whole content. - RAD (explicit homotopy): the radial homotopy (out from h(x) to radius R, around, in to h^2(x)) has B_v given by two angle functions r_1, r_2 and satisfies UCA's hypothesis at c only if both angle jumps vanish, i.e. only if var(h^2, c) = var(h, c) (so one explicit H decides nothing; the quantifier is ALL H). Straight-line, geodesic-in-U, conformal and Alexander-trick homotopies fail: h induces no map of the prime-end circle, tracks pass through X. - pro1103f04 (web, UNAUDITED): a UCA witness forces the integer D_v to be constant on an annulus A(v, r), diam c_n < r < m (necessary only); prime-end rotation gives only tau(g^2) = 2 tau(g); no explicit (S)-host is known, so no finite record can test UCA. C. THE TARGET Decide H-EQ along the covering route: prove UCA (which H_n, which composant C_n, and why the sheet misses a point of c_n), or prove that the covering conclusions of BRC for ALL H contradict (S) (then breathing is impossible and H-EQ holds), or give Tier 0 data in which the BRC conclusions hold at a fixed scale with every law above met, or reduce to one named lemma L (gate G1-G4). D. ASK Part 0 (check). Lemma LADDER step 1 (a, b not in B_v), Theorem CMP (the clopen level sets, the quasi-component, the composant step), BRC (i) to (v) and TEAR, with the quantifier order H, delta_H, c, v, C. Say which parts use cm of h^2, nonseparation and indecomposability. Part 1 (method). Name ONE method not on the dead-end list and say why it sees the covering at the scale of c_n. Candidates, none claimed to work: (i) CECH OBSTRUCTION: N_v is an integer 0-cocycle on X \ B_v; use H^1(X) = 0 and H^1(X x [0,1]) = 0 (X nonseparating) to compute the obstruction class of (H - v)/|H - v| on X x [0,1] \ H^-1(v), and show the jump N_v(b) - N_v(a) = +-1 forces B_v to carry a class that H^1(X) = 0 forbids, or find the class that survives; (ii) SHEET ASSEMBLY: BRC gives a proper continuum K with H(K x [0,1]) containing a subarc of c, and H(K x [0,1]) contains h(K) and h^2(K); use hereditary unicoherence to try to force K cap h^2(K) nonempty or a proper h^2-subinvariant continuum, then UBS (INFERRED, CHECK: can the sheet geometry force the intersection?); (iii) HOMOTOPY OF HOMOTOPIES (INFERRED, CHECK the class count): coincidence-free homotopies fall into homotopy classes indexed by an integer (the winding of tau_x followed by the reverse of tau'_x about x, locally constant, hence constant on the connected X); var(h^2, c) - var(h, c) = N_v(b) - N_v(a) does not depend on H, so follow it along a path of coincidence-free homotopies inside one class from the avoiding H of AV (scale lost) to one with delta_H above diam c, and locate where a track of a or b would cross v (T2 must be re-checked along the path); (iv) your own. Part 2 (carry it out). As in TARGET, with every use of cm, PPF, nonseparation, indecomposability, the perfect extension F and U3 marked. Part 3 (control run). Run your method on the circle control and on the radial homotopy of RAD, and say which hypothesis of (S) the circle lacks. Each claim labelled PROVED, SOURCE (exact reference), LAB, DERIVED, CONJECTURE, OPEN or RESTATEMENT. UNKNOWN if unsure a cited result exists. What already failed (read DEADENDS first): a new method, even partial with an honest gap, beats a polished restatement. Take your time: check every step, hypothesis and quantifier. This is a hard open problem. E. CONTROLS (all must hold) (C1) Circle: X = S^1, h an irrational rotation by alpha below 1/2 (h^2 minimal, no periodic point, but X separates the plane); H(x, s) = x exp(2 pi i alpha (1 + s)) is coincidence-free with tracks in X, so B_v is empty for v outside the disc and CMP gives equality, harmlessly since the outchannel theory does not apply. A proof must use nonseparation. (C2) Six-point record: v = 0, J = [0, infinity), h(a) = 3 - i/10, h(b) = 3 + i/10 (segment crosses J once), h^2(a) = -3 + i, h^2(b) = -3 + 1.1 i (segment misses J): every pair of ladder paths has I_v(tau_b) - I_v(tau_a) = +-1. It encodes var(h) != 0 = var(h^2) and is planar topology of six points; it contains no X, composant or continuity of H, so it cannot test UCA. (C3) If h is induced by a plane homeomorphism then (S) is empty (Cartwright-Littlewood); VF, LADDER, CMP, AV and TEAR are statements about (S)-data and must prove nothing false there. (C4) AV applied to W containing v always gives an H, but then diam c is at least delta_H (TEAR), so there is no contradiction with BRC. (C5) UCA lets H_n depend on n; BRC is for every H: do not swap the quantifiers. F. DEAD ENDS (not progress) (1) Endpoint-track HOM-AVOID (b_opus_homavoid Prop TAUT, ledger l.1199-1204) and the second-iterate swept-angle coboundary (Sol S1, wrapmpn Lemma SA, WALLS row W-TRANSPORT), which LADDER re-derives: both reduce to planar topology of six points. (2) Impression transport (Argon 66, IMP-TRANSPORT-FIN); moving-base-point degree transport HT (b_opus_doublewrap); crossing-index bookkeeping (Sol S2, A199); F^4-cluster index transport (Sol S3); Brown-Bell variation counting (A211); hull-shrink exterior degree bound (Argon 67): each stops at a signed transport law that is not supplied (WALLS.md). (3) Neck and lollipop squeezes (b_opus_pocket_orbit, ledger l.1014-1018, 1061-1063): POCKET holds at every admissible neck pair under breathing (Theorem PA), so the squeeze route is closed; a second location law (Argon 70 PT-INV) needs the open condition SQ_FG. (4) Modifying H near R_t, straight-line, geodesic, conformal or Alexander-trick homotopies (b_opus_homavoid section 5): they gain avoidance only by losing the scale of LADDER (TEAR). (5) Hyperspace trapping, UBS, Lemma U and stationary measures (UD, SH, UG; pro1103f04, ledger l.1361-1367): they convert approximate overlap plus a uniform hole into a proper invariant continuum, but no overlap is extracted from a homotopy. 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: UCA PROVED (a homotopy and composant at the scale of each c_n with the track sheet missing a point of c_n; every law used named) | BRC CONTRADICTION (the covering conclusions of BRC contradict (S); proof) | BRC CONSISTENT (explicit Tier 0 data in which the BRC conclusions hold at one scale and every listed law is met, with the first law that cannot be checked 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: using only the six endpoint points (planar topology of six points proves nothing, TAUT); one explicit homotopy (RAD decides nothing, the quantifier is ALL H); assuming h extends to a plane homeomorphism or acts on prime ends (then (S) is empty or the action is unavailable); avoiding points away from X (free by AV); using the circle control as a model (X separates); claiming H-EQ, L2 or PER as proved