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R66_EQFACT_prompt.txt 100 % RESEARCH QUESTION (plane continua / dynamics; does the end action (P_0, sigma) of a G2 host (planar or not) admit a nontrivial equicontinuous factor? beyond THM-A (finite cyclic factors) to connected-group rotations). 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 lab notes; the spec author's own bridging is marked INFERRED; everything is single-agent lab work unless an audit is named). - Class G2. Zhat = profinite integers (product of the Z_p), Z inside it diagonally, Zhat = Z_2 x Z_odd (Z_odd = product over odd primes). E: compact zero-dimensional metric space with a homeomorphism sigma. D: Zhat \ {0} -> E continuous, even (D(-z) = D(z)) and D(2z) = sigma D(z). For z not in Z put g(z) = (z, (D(z+n))_{n in Z}); K := closure of these points in Zhat x E^Z. The maps tau g(z) = g(z-1), i0 g(z) = g(-z), i1 := tau i0 extend to homeomorphisms of K (i0, i1 involutions). pi: K -> Zhat is the base projection; its fibres over non-integers are single points; for m in Z the fibre P_m = {p_m(e) : e in P_0} is a copy of the cluster set P_0 := E(D) (all limits of D(z_i), z_i -> 0, z_i != 0), where p_m(e) has coordinate e at index -m; diam P_m -> 0 as |m| -> infinity; tau is minimal; i0 fixes P_0 pointwise; sigma(P_0) = P_0; |P_0| = 1 is allowed. - Host. Sigma = (R x K)/((t+1, tau x) ~ (t, x)), nu[t,x] = [-t, i0 x], Y_D = Sigma/nu, a continuum with Cech H^1(Y_D; Z) = 0 (part of the class). Strip form: ([0,1/2] x K)/((0,x) ~ (0,i0 x), (1/2,x) ~ (1/2,i1 x)). Write q: Sigma -> Y_D and j(e) = q[0, p_0(e)]. Canonical map: B g(z) = g(2z) (a homeomorphism of K onto pi^-1(2 Zhat), B p_m(e) = p_2m(sigma e)) and h q[t,x] = q[2t, Bx], a homeomorphism of Y_D. 'Planar' means Y_D embeds in R^2 (then it is nonseparating, as H^1 = 0). - End set: A(W) := points of W that are not an interior point of an embedded arc [-1,1] -> W. Every homeomorphism of W permutes A(W). - (T1) DERIVED (GPT-6; Claude audit with editorial repairs only): j is a homeomorphism of P_0 onto A(Y_D), and Fix(h^r) = j(Fix(sigma^r | P_0)) for every r >= 1, least periods kept. So the canonical h has a periodic point iff the end action (P_0, sigma) has one; no planarity is used. - (FOLD-A) DERIVED (GPT-6 Pro; Claude audit CONFIRMED, building-block grade): for W = ([0,1] x C)/((0,z) ~ (0,t0 z), (1,z) ~ (1,t1 z)), C compact zero-dimensional, t0, t1 involutions, A(W) = q({0} x Fix t0) u q({1} x Fix t1). For Y_D: Fix i0 = P_0 and Fix i1 is empty. Hence EVERY homeomorphism f of Y_D permutes j(P_0) and induces a homeomorphism g_f of P_0 (f need not be a power of h or commute with sigma). Tree-likeness and planarity are not used. B. LEMMAS (status; where used) Known (status as given). T := R/Z. A continuous f: P_0 -> T with f(sigma e) = f(e) + alpha is an 'eigenfunction with rotation alpha'. - (THM-A) DERIVED (GPT-6 Pro T3, unaudited; checkable arithmetic of Zhat): every finite permutation factor of (P_0, sigma) has a fixed atom (proof: capture D(M Zhat \ {0}) in a clopen V where the factor is exactly equivariant; F := c o D satisfies F(2z) = s F(z); take a prime l > |F|, a prime p | 2^l - 1, a = M e_p, so 2^{l p^j} a -> a and the cycle length of F(a) divides l p^j, hence is 1). The nonplanar aperiodic control Y_* has the same property. - (FA, DF) DERIVED (cloud R4, Claude Opus, unaudited): FA: for N = (x,w) in Z_2 x Z_odd, w != 0, sigma^n D(N) and sigma^n D(0,w) are asymptotic as n -> +infinity. DF: let S := closure of D(Zhat \ {0}) (so P_0 inside S inside E). Any continuous equivariant map phi from (S, sigma) into a compact metric system with no two distinct forward-asymptotic points sends P_0 to one fixed point (D' := phi o D is continuous with D'(2z) = sigma' D'(z); by FA D'(x,w) = D'(0,w) for all x; letting w -> 0 gives one limit c' with sigma' c' = c'). So (S, sigma) has no eigenfunction with rotation alpha != 0, and no odometer or rotation factor. DF concerns S, NOT (P_0, sigma): the Sturmian core X_theta of D_low inside P_0 has a rotation factor that does not extend to S (R4). - (MIN) INFERRED (spec author; verify). If (P_0, sigma) is minimal and phi: P_0 -> (Q, rho) is an equicontinuous factor, Q = G is a compact monothetic group and rho = translation by a topological generator g. Any finite quotient G/H = Z/n with n > 1 would be a finite permutation factor of P_0 without a fixed atom (g generates the quotient), against THM-A; so G has no proper finite-index closed subgroup, i.e. G is connected, and a nontrivial character chi gives an eigenfunction f = chi o phi with rotation alpha = chi(g) of infinite order. So for minimal P_0 'nontrivial equicontinuous factor' means 'eigenfunction with irrational rotation' (EIG). For non-minimal P_0 an equicontinuous factor may be trivial-dynamics (an invariant function) and finite quotients of a minimal subgroup do not extend to P_0; state your own definition of 'nontrivial' and defend it. - (EIG-ERR) INFERRED (spec author; verify the lift and Haar step). Let f: P_0 -> T be an eigenfunction with rotation alpha. E is zero-dimensional, so f extends to a continuous f~: E -> T; put delta := f~ o sigma - f~ - alpha (continuous on E, zero on P_0). For w in Z_odd \ {0} let Gam = closure{2^n} in Z_odd^x (an isomorphic copy of (Zhat, +1), 2 acting as the generator) and G_w(gamma) := D(0, gamma w), continuous on Gam with G_w(2 gamma) = sigma G_w(gamma) (R4 Lemma 1.3). A continuous T-valued function on a profinite group lifts to R; with F := f~ o G_w (lifted) and delta_w := delta o G_w (lifted, small when w is in M Z_odd, M large), n_w := F(2 gamma) - F(gamma) - alpha - delta_w is integer-valued and locally constant. Integrating against Haar measure (which is 2-invariant, and gives rational mass to clopen sets): alpha + integral of delta_w d(gamma) is a rational number, for every w != 0. In particular if G_w(Gam) lies inside P_0 for some w then delta_w = 0 and alpha is rational; so for irrational alpha every odd-axis orbit closure leaves P_0, with mean error = -alpha mod Q, while |delta_w| -> 0 as w -> 0. This is a necessary condition, not a contradiction. - (Cores) D_low (planarity open; s448 audit: candidate confirmed): P_0 contains a Sturmian core X_theta, theta = sqrt2 - 1 (an almost 1-1 extension of the circle rotation by theta) and an uncountable family of splice ends (a Sturmian or mechanical left half joined to a right half), unaudited details. Y_*: Thue-Morse core (odometer factor Z_2 on the core) plus splices. Suppose a splice end z is backward asymptotic to a core point x' and forward asymptotic to a core point y' (sigma^n z and sigma^n y' close for n large, sigma^-n z and sigma^-n x' close for n large). If an extension of the core's rotation eigenfunction f(x) = k * (intercept) to P_0 were continuous, then f(z) = f(y') and f(z) = f(x'); so it can exist only if k * (intercept of x') - k * (intercept of y') is an integer for every splice that occurs. - (ORD) DERIVED (cloud R4 lemma ASY, Claude Opus, and lab Theorem BA for another host class, one agent each, unaudited): if sigma|P_0 is aperiodic and preserves or reverses a compatible circular order of P_0, then there is one minimal set M, M is an almost 1-1 extension of an irrational rotation, and every point off M is bi-asymptotic to one of the countably many gap endpoints of M. INFERRED (spec author): the two sides of a gap of M carry the same rotation coordinate, so under this order hypothesis the rotation eigenfunction of M does extend to P_0 (a Denjoy-type system), and (EIG) becomes exactly the question 'can a Denjoy-type system be the cluster system P_0 of a G2 reader'; R4 leaves this Denjoy escape OPEN (R8). C. THE TARGET Decide (EIG): in G2, is there a continuous f: P_0 -> T with f sigma = f + alpha for some irrational alpha (equivalently, for minimal P_0, a nontrivial equicontinuous factor)? Separate (a) what the arithmetic of Zhat alone decides (no planarity), (b) what a plane embedding adds. Not asked: finite factors (THM-A), equicontinuity of sigma itself (THM-B), the claim 'planar implies equicontinuous' (running job T-ACC), PER. D. ASK Part 1 (G2 only, no planarity). Using (MIN), (EIG-ERR), FA and axis recurrence, prove (EIG) is impossible, or isolate the single obstacle. Explain exactly why THM-A's proof (exact equivariance on a clopen V, so the error is zero) fails for circle-valued maps (the error delta is small but accumulates along 2^k a, k = l p^j -> infinity), and test whether the rational-mean identity of (EIG-ERR), applied along the whole family w -> 0, yields a contradiction through the denominators: a/b = alpha + mean error is determined by the modulus of the integer function n_w, which must deepen as w -> 0 if alpha is irrational. If you find that (EIG) is consistent with all G2 clauses, give an explicit abstract construction (E, sigma, D) with an irrational eigenfunction on P_0, checking continuity of D at every point of Zhat \ {0}; a Liouville-type alpha with ||l p^j alpha|| -> 0 for all primes p is the natural attempt, but test it against the mean identity. Part 2 (planarity). Assume Part 1 leaves a G2 datum with a connected-group factor. Does a plane embedding exclude it? Candidate mechanisms (none claimed to work): the circular order that 2PAGE induces on P_0 (a Sturmian or Denjoy end action preserves one, an odometer-type does not; the lab asks Q-SPL: does aperiodic sigma always violate it in G2), accessibility of the arc germs at the ends, the fold structure of i0 and i1. State clearly whether you use planarity and where, and whether the planar constant reader satisfies your argument (it must). Part 3 (decisive test). On D_low: decide whether the rotation eigenfunction of the Sturmian core extends to a continuous eigenfunction on all of P_0 using the splice ends (the forward/backward asymptotic argument of (Cores)), stating exactly which splice pairs you need and from which statement about E(D_low) they come (flag it UNAUDITED); and on Y_*: decide whether the odometer eigenvalues of the Thue-Morse core extend. Builder gate: name the TARGET law, secure background laws first, test, and end with 'KILL: clause | what a proof must use'. 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) CE (binary base, outside G2): (Z_2, +1) has every 2^k-th root of unity as eigenvalue and P_0 is not collapsed; your argument must name the Zhat clause it uses and fail over Z_2. (C2) Denjoy/Sturmian circle systems have irrational eigenfunctions and are aperiodic: an argument that does not use the G2 arithmetic proves too much. (C3) The Sturmian core X_theta inside P_0 has the rotation factor: an argument must distinguish the core from all of P_0, i.e. must use the splices or the clopen capture of the whole P_0. (C4) The constant reader trivially satisfies every conclusion (one fixed end). F. DEAD ENDS (not progress) (1) DF for P_0: DF is about S; the extension of f from P_0 to S is exactly what is not available. (2) Exact eigenfunction on all of E: impossible by DF (even on S); the question is the approximate one. (3) INV1 (sigma|P_0 preserves or reverses a compatible circular order of P_0) as the route to PER: it holds on the Sturmian core and fails on D_low and Y_* only through splices; Denjoy-type escape inside G2 is OPEN (R4 R8). (4) Finite cyclic factors: THM-A. (5) Assuming planarity gives equicontinuity or a bounded atlas: running job T-ACC. (6) Restrictions of arbitrary homeomorphisms to P_0: running job T-EXT. (7) Lab statements that the maximal equicontinuous factor of a periodic-point-free map is never a nontrivial odometer concern the host map in the planar setting (S), not the end action of a G2 host; their exact scope is UNKNOWN here, do not import them. G. FORMAT Line 1 exactly: VERDICT: EIG EXCLUDED (no continuous f: P_0 -> T with f sigma = f + alpha, alpha irrational; hypotheses marked G2 only or planar) | EIG REALIZED (explicit G2 datum; planarity of it decided or marked open) | REDUCED (to ONE named lemma on the error cocycle, gate G1-G4) | PARTIAL (one sub-case 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: applying DF to P_0 (DF is about the reader system S); forgetting that an extension of f from P_0 to a neighbourhood is only approximately equivariant; treating a factor of a minimal subset of P_0 as a factor of P_0; using order arguments that CE (planar) refutes; assuming planarity gives equicontinuity (running job T-ACC); claiming PER Zobrazuje se R66_EQFACT_prompt.txt.