batch 3

Prompt

RESEARCH QUESTION (plane continua / proximal end pairs; global two-fold planarity beyond ordered tip germs). 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 Research, not audit. Consensus tree T (proximal-pair branch); ledger case PLANAR-RULER / D_g and FR proximal-fold suggestion, 11-05 00:1x and 00:5x. Route tag INDEPENDENT of L2; CHANGE-MAP only if an actual global planarity implication is proved. Technique: representation switch from coalescing end tips to simultaneous fence/fold matchings at one modulus. This task is disjoint from irrational eigenvalue builders, admissible-map extension classification, word-reader absorption and C3 convexity jobs. No lab files are visible. - A continuum is a nonempty compact connected metric space. Nonseparating means its complement in S^2 is connected. Planar means embeddable in R^2. PER asks whether every homeomorphism of every nonseparating plane continuum has a periodic point. PPF means Fix(h^r) is empty for every integer r>=1. - G2 datum: Zhat=product_p Z_p is the full profinite integer group, with diagonal integers Z, and Zhat=Z_2 x Z_odd, Z_odd=product over odd primes. E is compact metrizable zero-dimensional, sigma:E->E a homeomorphism. D:Zhat minus {0}->E is continuous, D(-z)=D(z), D(2z)=sigma D(z). For z not in Z let g(z)=(z,(D(z+n))_(n in Z)); K is its closure in Zhat x E^Z. Set P_0=the cluster set of D at 0 (this notation also identifies that set with its copy in K). Above m in Z, p_m(e) has base m, coordinate e at -m, and D(m+n) at every n!=-m. Above nonintegers the base fibre is a singleton. Write pi:K->Zhat and P_m=p_m(P_0). - The homeomorphisms tau and i0 extend g(z)->g(z-1) and g(z)->g(-z); i1=tau i0. In coordinates tau(z,u)=(z-1,(u_(n-1))_n), i0(z,u)=(-z,(u_(-n))_n). Require tau minimal, sigma(P_0)=P_0, diam(P_m)->0 as |m|->infinity, Fix(i0)=p_0(P_0), Fix(i1)=empty. - Sigma=(R x K)/((t+1,tau x)~(t,x)), nu[t,x]=[-t,i0 x], Y_D=Sigma/nu, Q:Sigma->Y_D. Equivalently Y_D=([0,1/2] x K)/((0,x)~(0,i0 x),(1/2,x)~(1/2,i1 x)). Require a continuum with integral Cech H^1(Y_D)=0; if planar, duality then makes it nonseparating. B maps K homeomorphically onto pi^-1(2Zhat), B g(z)=g(2z), Bp_m(e)=p_(2m)(sigma e). The canonical homeomorphism is h Q[t,x]=Q[2t,Bx]. End j(e)=Q[0,p_0(e)]; ray gamma_e(t)=Q[t,p_0(e)], t>=0. - Two-sided proximal means e!=f and inf_(n in Z) d_E(sigma^n e,sigma^n f)=0. Bi-asymptotic means this distance tends to 0 in both time directions. Distal means no proximal pair. Proximality does not exclude fixed or periodic ends elsewhere. - For q=a/b>0 in lowest terms let C_q be intersection over neighbourhoods U of 0 of closure{(D(z),D(qz)):0!=z in U intersect bZhat}. Lambda_D consists of q for which C_q is the graph of a homeomorphism sigma_q:P_0->P_0. All closures here are in E x E. - Order formulation: on X=K x {0,1} define fence F(x,s)=(x,1-s), fold Phi(x,s)=(i_s x,s). A 2PAGE order means one embedding P:X->S^1 making F-pairs mutually noncrossing and Phi-pairs mutually noncrossing, with both fold colours on the fold page. Crossing means four distinct endpoints alternate a,b,a',b'. F and Phi live on different pages. STRIP means a continuous injection K->R making i0 and i1 separately noninterleaving in that same order; the two fold colours lie on opposite sides of the strip. B. LEMMAS (status; where used) Known (statuses and versions retained; the spec writer has not independently proved the R76 construction). - LAB VERIFIED T1 / T-EXT (cross-family repairs closed, tree 11-05): j(P_0) is precisely the set of points that are not interior points of embedded arcs; every homeomorphism preserves it. Fix(h^r)=j(Fix(sigma^r|P_0)), least periods preserved. The admissible end actions are exactly sigma_q, q in Lambda_D, realised on Y_D; for q!=1 their canonical realisations have periodic points exactly at periodic ends. Each end has one ray arc component, every other arc component is a line. No plane embedding is used in these assertions. - LAB VERIFIED finite-factor/distal boundary (R7/R8 repairs closed, tree 11-05 02:3x): every continuous equivariant map of (P_0,sigma) to a finite permutation system is constant; two-sided distal zero-dimensional systems are inverse limits of finite permutation systems. Consequently |P_0|=1 iff sigma is distal iff all admissible end actions are distal. R76 inherited stale UNAUDITED labels for these inputs; retain the newer limited status. This does NOT say planar implies one end; that stronger statement is OPEN. - LAB DERIVED / UNAUDITED, R76 Astra final answer, exact valuation-staircase datum (use this specimen, not an unspecified E_step): a_n=n/(1+|n|), n in Z; E={-1,1} union {a_n:n in Z}, subspace topology of R. sigma(a_n)=a_(n+1), sigma(-1)=-1, sigma(1)=1. M_0=1; M_r=product_(odd primes p<=r+2) p^r for r>=1. For 0!=w in Z_odd set k(w)=max{r>=0:w in M_r Z_odd}. For (x,w)!=(0,0) set D(x,w)=a_(v_2(x)-k(w)) if x!=0 and w!=0; D(0,w)=1 if w!=0; D(x,0)=-1 if x!=0. Here v_2 is the finite 2-adic valuation of nonzero x. No value at (0,0) is assigned. R76 proves continuity on both axes as well as off them, evenness and covariance. The noninteger witnesses z_r=(2^(r+n),M_r)->0 give D(z_r)=a_n for large r; the axes supply +/-1, so P_0=E. It checks all G2 clauses, including H^1=0 by the two-collar Mayer-Vietoris quotient C(K,Z)/(I_0+I_1), I_s={f:f i_s=f}. The explicit map B(z,u)=(2z,v) has v_(2j)=sigma u_j and v_(2j+1)=D(2z+2j+1). Lambda_D=2^Z, sigma_(2^l)=sigma^l. For the converse use w_r supported at odd prime p with p-coordinate p^r: D(z_r)=a_n but D(qz_r)=a_(n+v_2(q)-v_p(q)); two primes with different v_p(q) defeat graphness whenever q is not a power of two. Named proximal pair: a_0,a_1, with forward iterates converging to +1 and backward iterates to -1. The compatible linear order inherited from R is preserved by every admissible action. For delta<1/2 the compact tip product (e,t)->gamma_e(t), 0<=t<=delta, embeds; its plane model (e,t) gives disjoint tips. Extend sigma increasingly to S:R->R by interpolation, identity outside [-1,1]; (x,t)->(S(x),2t) realises all scale germs. Whole-host planarity remains OPEN. - LAB DERIVED consequence, reconstructed here: this specimen has two fixed ends j(-1),j(1); h is not PPF. Indeed the two accumulation points of E are intrinsically distinguished, so every homeomorphism of Y_D permutes this two-point set and has a periodic point. Embedding this specimen would settle the stronger one-end conjecture negatively, not refute PER. - LAB R76 comparison / Sonnet mining (not an audit): Sol 6.1 gives the analogous one-point compactification Z union {infinity}; Sol 6 gives the two-ended version with factorial odd heights. Kimi's Global Ray-Separation Lemma remains unproved. Argon's saved answer is truncated before its announced datum / CRT-SPLIT; neither is an available theorem. No model gives a whole-host embedding or obstruction. Local ordered tips already pass; do not repeat that construction. C. THE TARGET Can the displayed proximal G2 datum be planar? Prefer an obstruction specifically using proximality and the recurrent global fold structure, or give a whole-host embedding. Alternatively prove a properly stated proximal-pair obstruction for a larger nonempty abstract G2 class and apply it. A YES embedding with a PPF canonical map would refute PER; verify PPF and all infinite coherence clauses before saying so. The displayed fixed-end specimen itself cannot have that consequence. D. ASK 1. Beyond the tip model, extract from an embedding a law about returning portions of both folds. Force alternation at one common modulus, or explain how this datum places those returns consistently. Locate the use of the odd-prime axis; explain CE if the proof also works over Z_2. 2. An obstruction needs one q>=1, finitely many clopen E-codes in finitely many graph coordinates, and cylinder neighbourhoods fixing pi(x) mod q; the four endpoints must be distinct, in one candidate order on one page. Explain why every compatible global order yields a forbidden configuration, with q allowed to depend on that order. Do not require a code partition to be fold-invariant. Prove inheritance from an actual embedding. 3. Builder: give one continuous injection Y_D->R^2, or exact inverse data with a proved continuous injective limit realising all strands, both folds and all accumulation. A replacement aperiodic reader needs every G2 clause checked. Different embeddings at each depth do not suffice. 4. PARTIAL needs a narrower quantified lemma, its proved implication, and a new datum-dependent fact. Separate the specimen, the general proximal class and PER. End KILL: gluing/order law | certificate or failure | hypothesis used. E. CONTROLS (all must hold) (C1) Constant reader E={e}, D=e: planar nonseparating G2 rainbow chart, one end, h fixes it. No proximal pair; an obstruction must allow it. (C2) CE: LAB repaired historical planar control over binary Z_2, outside full-base G2. Infinitely many distal adding-machine ends and a regular canonical fixed point. It defeats cluster-cardinality obstructions with no full-base hypothesis. (C3) A Denjoy Cantor system (split the orbit of 0 under irrational rotation by sqrt(2)-1) preserves circular order and has asymptotic proximal pairs. Its planar Cantor fan has all hairs meeting at a fixed vertex; it is not the G2 disjoint ray/line decomposition. It defeats order-only and local-tip arguments, not a proved global G2 obstruction. (C4) The displayed R76 specimen has a simultaneous ordered planar tip model and fixed ends. A proof must explain what fails only at the recurrent whole-ray completion, or supply that completion. LAB audited nonplanar Y_* has proximal splice ends; it rules out an abstract proximality-implies-planarity inference but is not a premise needed to prove this specimen nonplanar. F. DEAD ENDS (not progress) Finite permutation factors alone; proximal tips forced to intersect merely because they converge; arclength scaling for arbitrary H_q inferred from end transport; one-depth order choices; an unproved GRSL/CRT-SPLIT renamed as the answer; cluster cardinality implies nonplanarity; a fixed-end embedding refutes PER. Preflight prior kills: a compact periodic fence is impossible under length doubling even on the planar constant-reader control, so that absence is no proximal obstruction. Fold-invariant finite quotients may collapse entire exceptional fibres; do not impose them as an exhaustive embedding model. Embeddings of all proper closed pieces do not assemble a global embedding. These are method limits, not conclusions about this datum. Take your time, be extremely meticulous, I believe in you. G. FORMAT Line 1 exactly: VERDICT: NONPLANAR (specified datum or class, obstruction proved) | PLANAR (whole-host embedding, periodic-point status stated) | PARTIAL (new quantified lemma) | OPEN-OBSTRUCTION (exact missing global step). 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: a proximal pair called periodic-point-free; embedding only tip arcs; circular order assumed covariant; different moduli or orders substituted for one common configuration; a finite certificate called an infinite embedding; a restatement called a reduction