# AIM-GEOMETRY-0322: exact source and interpretation Frozen identity: ulamai/UnsolvedMath v1.6.0, commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e, record 20001984. The immutable source_record.json retains the original wording. The source is F. Bihan's Question A.2, page 3 of the AIM workshop list *Amoebas and tropical geometry*, January 2004. The official HTML entry is https://aimath.org/WWN/amoebas/articles/html/28a/ . The January 2009 compiled list repeats it as item 0.2; it is not a different mathematical problem. The source asks two questions for a compact connected component K of an amoeba intersection: whether its normalized mixed Monge-Ampere mass equals the volume of a corresponding mixed cell, and whether the solutions above K admit a real-preserving bijection with a corresponding binomial system. It states no full-dimensional-support or patchworking hypothesis. Accepted coverage: a complete counterexample to the unrestricted single-binomial real-preserving correspondence, valid for any lattice subdivision of the displayed Newton polytopes. The single-cell volume claim is refuted for the canonical Ronkin-spine subdivision, with an arbitrarily large discrepancy. The word "corresponding" does not specify a subdivision in the source. Therefore we conservatively count ONE exact explicit source-question closure, and ZERO whole-record closures. Not covered: every possible coarsening's complex mass, a union of start systems, full-dimensional-support modifications, non-Archimedean lifting or valid patchworking statements. Coarsening can match the complex mass; it cannot remove the real-root obstruction to one binomial system here. The inherited phase-stability caveat is incorrect for the source's compact component setting. Passare--Rullgard's June 2000 preprint explicitly gives the unrotated local root count in Corollary 1 after Theorem 5. This is credited prior knowledge, not a new theorem. The frozen report remains unchanged and this dated dossier supersedes that caveat. # A compact amoeba intersection with four real roots and four unit mixed cells 27 October 2026. Complete mathematical derivation; publication and novelty assessment are recorded separately. Let Log(z,w)=(log|z|,log|w|), and let A(h)=Log(h^{-1}(0)) in the complex torus. Write S(h) for the **canonical Ronkin spine**, the corner locus of the maximum of the affine extensions of the Ronkin function from all amoeba-complement components. Volumes of fully mixed cells are determinants of their edge vectors. The normalized mixed Monge-Ampere measure is mu=2!*Mtilde(N_f,N_g), in the notation of Passare--Rullgard. ## Main theorem Put P(z)=(z-10)(z-12)(z+11)(z+13) =z^4+2z^3-265z^2-266z+17160, f(z,w)=w-1, g(z,w)=P(z)-525w. Then: 1. A(f) intersect A(g) is one compact connected interval K in y=0. 2. The system f=g=0 has four distinct real torus solutions, all above K, and mu(K)=4. 3. S(f) and S(g) intersect transversely at (log r,0), r=10,11,12,13. Each dual fully mixed cell has volume one. Thus no one of these cells has volume mu(K), or a binomial system with as many complex roots. 4. More strongly, no binomial system formed from one nondegenerate fully mixed cell for these Newton polytopes, in any lattice subdivision and with arbitrary nonzero coefficients, has more than two real torus solutions. A bijection sending the four real original solutions to real solutions of a single such system is therefore impossible. ### The entire intersection is connected For u=z+1/2 and m=533/4, direct multiplication yields P(z)=(u^2-441/4)(u^2-625/4)=(u^2-m)^2-529. For |t|<=525, the four roots of P(z)=t are the analytic branches z_{s,e}(t)=-1/2+s sqrt(m+e sqrt(529+t)), s,e in {+1,-1}, where both square roots use their branch with positive real part. These branches are defined on a neighborhood of the closed disk: 529+t has positive real part, while |sqrt(529+t)|<=sqrt(1054)525. Each curve C_{s,e}={z_{s,e}(525 exp(i theta)):0<=theta<=2pi} is connected and compact in C*. Their union is exactly |P(z)|=525. The radial image of each curve is therefore a compact positive interval. Define alpha=sqrt(m-sqrt(1054)), beta=sqrt(525)/2, gamma=sqrt(541)/2, delta=sqrt(m+sqrt(1054)). The real boundary values at t=+525 and t=-525 show that the four radial images contain, respectively, the intervals [alpha-1/2,beta-1/2], [alpha+1/2,beta+1/2], [gamma-1/2,delta-1/2], [gamma+1/2,delta+1/2]. These four intervals overlap in the displayed order. Indeed, exact squaring gives alpha<41/4, beta>45/4, gamma<47/4, delta>51/4. Consequently beta-alpha>1, gamma-beta<1/2<1, and delta-gamma>1. All intervals have positive endpoints. Since four connected radial images contain an overlapping chain, their union is one compact interval [r-,r+] with r->0. As A(f)={y=0}, the intersection is exactly K=[log r-,log r+] x {0}. No additional components are omitted: the four explicit branches exhaust the entire lemniscate. ### Roots and mass Setting w=1 gives P(z)=525, with roots -1/2 +/- alpha, -1/2 +/- delta. They are four distinct nonzero real numbers. Their logarithms lie in K by the preceding construction. Passare--Rullgard, Theorem 5 and Corollary 1 in their June 2000 preprint, give mu(K)=4. In this example the phase-average identity can also be checked without a constancy theorem: independent coordinate rotations reduce the equations to P(e^{i a}z)=525 e^{i b}. The four explicit branches remain distinct for all such phases, and every root has its logarithm in K. The normalized average root count is four. ### Spines and cells More generally, let Q be monic of degree d with distinct positive root moduli rho_1<...=log|Q(0)|>log L for every x, the top term is inactive on y=0. Each of the d distinct bottom breakpoints gives a transverse intersection of S(h) with S(w-1), and the two dual edge vectors are (1,0) and (0,1). In the quartic example the breakpoints are log 10 through log 13, and the four cells all have volume one. Each binomial system for one such cell has exactly one torus solution. ### The real obstruction survives coarsening The first Newton polytope is the primitive vertical segment. Its binomial equation, up to an invertible monomial factor, has the form aw+b=0 with ab!=0. It fixes one w in C*. A second binomial equation corresponding to an edge of the other Newton polytope becomes z^k=c after substitution, where k is a nonzero integer and c!=0. Nonzero k is exactly the full-dimensional mixed-cell condition. For real solutions, w must first be real; z^k=c then has at most two nonzero real roots, for any complex c. Hence every such binomial system has at most two real solutions. This argument covers any lattice subdivision or coarsening and does not assume inherited coefficient signs. It disproves the requested real-preserving correspondence with a **single** binomial system. It says nothing against a disjoint union of several start systems. ## Unbounded complex discrepancy For every d>=2 there is a monic degree-d polynomial Q with distinct root moduli such that A(w-1) intersect A(Q-w/2) is a single compact component of mass d, whereas each of its d canonical-spine mixed cells has volume one. Let zeta_j=exp(2pi*i*j/d) and Q_e(z)=product_j(z-(1+j*e)zeta_j), e>0. At e=0, Q_0=z^d-1 and all its critical values equal -1. Continuity of the roots of Q_e' and evaluation of Q_e imply that, for sufficiently small e>0, no critical value lies in |t|<=1/2. Consequently the proper polynomial map above a slightly larger disk is an unramified cover of degree d. A cover of a simply connected disk is trivial, so there are d holomorphic inverse branches on a neighborhood of the closed half-disk. At e=0 these branches are zeta_j(1+t)^{1/d}. At the two boundary points t=-1/2,+1/2 their moduli are 2^{-1/d}<1 and (3/2)^{1/d}>1. The implicit-function theorem and compactness permit the inverse branches to be labeled continuously for small e, so these strict inequalities persist. Every branch image of |t|=1/2 therefore has a radial interval containing 1. Their union is one compact positive interval; zero is excluded by |Q_e(0)|=product_j(1+j*e)>=1>1/2. The intersection is thus one compact K. Each phase has d simple roots there, so the normalized mass is d. The root moduli 1+j*e are distinct; the preceding spine lemma gives d transverse intersections and d unit cells. This proves the unbounded discrepancy. The construction is existential for sufficiently small e; finite numerical samples are not used to prove it. ## Source interpretation and limits These statements give a negative answer to the unconditional single-cell picture when its cell is taken from the canonical Ronkin-spine subdivision, and a subdivision-independent negative answer to the single-binomial real-preserving correspondence. The AIM wording does not specify a subdivision. We do not assert that no coarsening can match the complex mass: in the quartic example a coarse length-four bottom edge can have volume four, but its binomial still has at most two real roots. The first polynomial has a one-dimensional Newton polytope. No claim is made about a modified problem imposing full-dimensional supports or patchworking hypotheses. We do not contradict tropical Bernstein or tropical lifting, whose statements count the contributions of all relevant cells under their stated hypotheses. General real homotopy limitations are known; the contribution being assessed is this explicit compact-component construction and its unbounded canonical-spine amplification. # Detailed self-audit, 27 October 2026 Acceptance: COMPLETE_COUNTEREXAMPLE within the explicit scope in canonical_statement.md. No known gap remains in the stated theorem. This is the originating researcher's audit, not independent review or proof-assistant verification. 1. **Is the amoeba intersection only sampled?** No. The four nested radical branches exhaust the entire level set |P|=525. Their exact radial intervals form an overlapping chain. No plot or sample establishes topology. 2. **Are square-root branches globally defined?** Yes. For |t|<=525, Re(529+t)>=4. The modulus of its square root is at most sqrt(1054)<533/4, so both outer radicands have positive real part. Strict margins give a neighborhood of the closed disk. The standard square-root branch works. 3. **Could roots coincide?** No: the inner radical is nonzero, the two squared outer values differ, and neither outer radical is zero. 4. **Could the torus exclude a root?** P(0)=17160>525; w=1. Both coordinates are nonzero. All radial images are compact and bounded away from zero. 5. **Are the radial intervals ordered correctly?** The positive inner, negative inner, positive outer, negative outer branches give exactly the four intervals displayed. Bounds alpha<41/4<45/4