# Exact source scope Frozen record OWR-2654831-006, record_id 30004563, UnsolvedMath v1.6.0, commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e. Statement SHA-256: 4a47075d8e57cd6470bad47b36aba0cd96386b3bbe898e4e17cbf43881d02346. Question: for alpha-realizations (called immersions in the report) with alpha > 1, is three the maximal number of possible centers of a cube move? Primary source: Paul Melotti's contribution, OWR 35/2020, printed p.1785, following Theorem 4, DOI 10.4171/OWR/2020/35, published 2021. The final journal article by Melotti, Ramassamy and Thevenin (2023), DOI 10.4171/AIHPD/163, Remark 4.8, p.811, states the same expected upper bound for intersections of two construction curves with one shared focus. Definitions 2.4 and 2.5, pp.789-790, allow crossed edges and require six distinct outer vertices and noncollinear triples of each color. Accepted answer: no. At alpha=10 an explicit rational outer hexagon and rational input center admit at least five distinct output centers. The strict sign certificate also persists for alpha in a neighborhood of 10. This is one complete negative answer to one frozen record, not five results. No upper bound of five, exact root count, or statement about proper embeddings or alpha <= 1 is asserted. No inherited upstream research proof was supplied. # Five output centers: complete counterexample Use Euclidean distances in R^2. Let U=(0,0), V=(1,0), W=(3,1/20), P=(1/3,5/6), b=(41/36)^5-(29/36)^5>0, c=(27809/3600)^5-(29/36)^5>0. We seek simultaneous solutions of |X-V|^10-|X-U|^10=b and |X-W|^10-|X-U|^10=c. For s>=0 set B=(s^5+b)^(1/5), C=(s^5+c)^(1/5), x=(1+s-B)/2, y=10(3601/400+s-C-6x), H(s)=x^2+y^2-s. All roots are positive real roots. Squared-distance subtraction proves a bijection between nonnegative zeros of H and the desired points X=(x,y). At a zero, |X-U|^2=s, |X-V|^2=B, |X-W|^2=C. Conversely every solution has this form. Different s give different points because their squared distances from U differ. Exact rational interval arithmetic proves the following enclosures: | s | H(s) lies in | |---|---| | 3/4 | [1.821692,1.821693] | | 9/10 | [-0.724507,-0.724506] | | 6/5 | [1.272250,1.272251] | | 3/2 | [-1.032219,-1.032218] | | 2 | [9.395978,9.395979] | | 4000 | [-396.749376,-396.749375] | Decimal endpoints are rational. The full integer fifth-power certificate is displayed in the manuscript Table 1 and in checks-normal.json. It uses Q=10^12 and integers k with k^5<=Q^5(s^5+b)<(k+1)^5, and analogously for c. Interval subtraction and squaring then give the bounds, with no floating-point rounding. The intermediate value theorem gives a zero in each of the five disjoint open intervals. No numerical root count or simplicity assumption is used. To make this a genuine cube move, set A2=U,A4=V,A6=W and A1=reflection of P in WU, A3=reflection of P in UV, A5=reflection of P in VW. Their coordinates are A1=(3899/10803,-17755/21606), A3=(1/3,-5/6), A5=(1805/4803,-8315/9606). All six boundary vertices are distinct, with orientation determinants det(A3-A1,A5-A1)=71690/51886809 and det(V-U,W-U)=1/20. The circumcenter of A1,A3,A5 is Q=(2550/7169,-30396/35845), distinct from all outer vertices. Reflection makes the two boundary edges in each input quad equal, and Q makes the two remaining edges equal. Thus the three input quads satisfy the alpha-quad identities for every alpha>0. For alpha=10, the output quad UA3VX is exactly the first difference equation, and WA1UX is exactly the second. Subtracting them gives |X-W|^10-|X-V|^10=c-b=|A5-W|^10-|A5-V|^10, exactly the third quad. None of U,V,W solves both equations. None of the three reflected points does either: it shares two distances with P; if it shared the differences too, it would share all three distances to the noncollinear foci and hence equal P. But P is on none of the reflecting lines. Thus all five output centers differ from every boundary vertex. This proves the full negative answer. For exponents near 10, keep the rational boundary fixed and replace b,c by the corresponding alpha-power distance differences of P. Replace each fifth root by (s^(alpha/2)+b_alpha)^(2/alpha). Positivity and all six strict signs persist by continuity. The same scalar equivalence and reflection argument therefore yield at least five output centers on an open interval of alpha values containing 10. No quantitative interval width is claimed. The paper main.tex is the publication rendering of this argument; the source ZIP includes the exact standard-library checker. This is not a formal proof assistant certificate and has not received independent review. # Adversarial self-audit, 27 October 2026 Accepted as COMPLETE_COUNTEREXAMPLE by the originating researcher. There is no outstanding mathematical obligation for the stated negative answer. - The source concerns realizations/immersions, not only proper embeddings. Both same-color triples are noncollinear and all six boundary vertices distinct. We explicitly build the input center, rather than assume an arbitrary pair of curves can arise from a cube move. - Euclidean lengths to power ten are squared distances to power five. The scalar parameter is squared distance s, not the numerical search's tenth-power parameter t. The manuscript and exact checker consistently use s. - Positive fifth roots are unique. The scalar correspondence is reversible; subtracting squared distances gives x and y with coefficient y/10 because W has second coordinate 1/20. The constant |W|^2 is 3601/400. - b and c come from P and are strictly positive. All radicals are defined and continuous on the entire nonnegative parameter axis. - The six endpoint signs are proved with rational enclosures and integer fifth powers. The precision is absolute 10^-12, including the endpoint4000. Interval arithmetic handles dependencies by enclosing them, never treating cancellation as exact unless algebraically justified. Squaring across zero uses a zero lower bound. The margins remain strictly away from zero. - Five disjoint intervals give five distinct roots by continuity. No assumption of a simple root, uniqueness in an interval, or absence of distant roots is made. The large fifth interval is intentional. - Injectivity from s to the solution set uses |X-U|^2=s. Roots cannot represent the same point. All output centers are proved different from the outer vertices. - Reflection formulas and the circumcenter were checked with exact fractions. The input faces satisfy the identities by paired equal distances for every positive exponent. The third output equation is the difference of the first two. - The persistence corollary follows from finitely many strict signs and continuity; it claims existence of an exponent interval, not a numerical width. - The checker has 67 active checks under normal Python and python -O. These are a reproducible finite arithmetic certificate for an existence proof using IVT, not empirical evidence offered as a universal theorem. - No maximum of five is claimed. No result for proper embeddings, alpha<=1, general integrability, or other questions in the source report is counted. - Current bounded primary-source searches found the conjecture in the final 2023 paper and did not locate an earlier five-center counterexample. This is not an exhaustive literature census or an absolute-priority certificate. - No independent reviewer, outside model, or proof assistant has approved it. The user has authorized self-audited complete results without such a gate. # Bounded source and novelty search Cutoff: 2026-10-27. This is not an exhaustive priority certification. 1. Melotti's OWR35/2020 contribution, pp.1783-1785 (published2021), https://doi.org/10.4171/OWR/2020/35. The conjectured bound is on p.1785. 2. Melotti, Ramassamy, Thevenin, Cube moves for s-embeddings and alpha-realizations, AIHPD10(2023),781-817, https://doi.org/10.4171/AIHPD/163. Definitions2.4-2.5 separate realizations from embeddings; Remark4.8 p.811 retains the three-intersection conjecture. 3. https://arxiv.org/abs/2003.08941v2 identifies the final preprint revision dated4October2021 and journal publication. The final journal text is the main reference, avoiding obsolete numbering from preprintv1. 4. https://dimers.science/events/oberwolfach/slides/Melotti_MFO.pdf, contemporaneous author slides, preserved as corroborating source material. 5. https://www.normalesup.org/~ramassamy/publications/alpha_realizations.pdf, author-hosted text also indexed with Remark4.8. No separate new result. Searches used exact title and combinations of alpha-realizations, construction curves, intersections, three, five, counterexample, Melotti/Ramassamy/Thevenin, and 2024/2025. Returned relevant primary materials were the source article, report, arXiv record, slides and author copy. No earlier five-center counterexample was located. Unrelated uses of "power distance", Rubik's cubes, and metallurgy were discarded. The existing authors own the definitions, existence theorem, and three-point examples. The present result is the explicit five-center counterexample with an exact sign certificate and a reflection construction giving an initial realization. Scalar trilateration and interval arithmetic are standard methods, not claimed as new. No adjacent source question is treated as resolved.