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Treating inverse temperature and magnetic field as coordinates on a thermodynamic control manifold, we show that the mixed response field \u03a9\u03b2h = N\u00a0cov(m, e) arises naturally as a curvature-like quantity that measures correlations between magnetic and energetic fluctuations. Monte Carlo simulations reveal a strongly localized mixed-response ridge that emerges from the critical point and extends into the finite-field crossover regime. Analysis of the susceptibility, specific heat, and mixed-response maxima demonstrates distinct scaling behavior in the magnetic, energetic, and mixed fluctuation sectors. When represented in normalized response coordinates, trajectories obtained at different magnetic fields collapse onto a common curve, indicating that the evolution of the mixed response is strongly constrained by the susceptibility. This collapse suggests the emergence of a low-dimensional response manifold and points toward a geometric description of critical crossover based on relations among response functions rather than equilibrium states alone. The framework establishes a direct connection between fluctuation correlations, critical scaling, and geometric thermodynamic response.<\/jats:p>","DOI":"10.1063\/5.0349278","type":"journal-article","created":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T11:12:06Z","timestamp":1790161926000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Mixed response geometry and critical crossover in the Ising model"],"prefix":"10.1063","volume":"165","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0775-9664","authenticated-orcid":false,"given":"Eric R.","family":"Bittner","sequence":"first","affiliation":[{"name":"Department of Physics, University of Houston , Houston, Texas 77204, and , 1375 Avenue Th\u00e9r\u00e8se-Lavoie-Roux, Montr\u00e9al, Qu\u00e9bec H2V 0B3,","place":["USA Canada"]},{"name":"Institut Courtois and D\u00e9partement de Physique, Universit\u00e9 de Montr\u00e9al , Houston, Texas 77204, and , 1375 Avenue Th\u00e9r\u00e8se-Lavoie-Roux, Montr\u00e9al, Qu\u00e9bec H2V 0B3,","place":["USA Canada"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,9,23]]},"reference":[{"key":"2026092307101949800_c1","doi-asserted-by":"publisher","first-page":"2479","DOI":"10.1063\/1.431689","volume":"63","year":"1975","journal-title":"J.\u00a0Chem. 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