Domain Projection: The Geometric Origin of Physical Constants and Laws

Recovery of the Planck–Boltzmann Ratio (h/kB) from Non-Closure in Blackbody Spectral–Temperature Relations | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Recovery of the Planck–Boltzmann Ratio ( h / k B ) from Non-Closure in Blackbody Spectral–Temperature Relations Abdennour Abbas This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8629054/v2 This work is licensed under a CC BY 4.0 License Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Abstract Planck's constant ( h ) and Boltzmann's constant ( k B ) co-occur in blackbody radiation but are typically introduced through theoretical models applied to spectral data rather than derived directly from spectral relations. Here, we introduce a new method for extracting physical constants as compositional residuals from linear regressions, and apply it to blackbody radiation. We show that h / k B emerges as a single irreducible residual from blackbody spectral relations, without invoking quantization or statistical mechanics. Blackbody radiance spectra are analyzed across 53 temperatures (1–200,000 K), extracting three pairwise linear relations among two spectral measurables (peak frequency and spectral width) and temperature. When these relations are composed, they fail to close, producing an intercept residual. Normalized by the regression slopes, this single residual recovers h / k B = 4.798 × 10⁻ 11 K 2 s (0.02% deviation from CODATA value), the Wien frequency displacement factor 2.8214 (3.5 × 10⁻⁵ % deviation), and the Wien wavenumber constant 13.36 K (0.15% deviation). The recovered ratio appears with non-canonical units (K²·s rather than K·s), indicating that the constants are not recovered as fundamental inputs but as structural invariants of a transformation between representations. No free parameters are adjusted, no physical assumptions beyond the empirically validated blackbody spectrum are required, and the derivation does not invoke the functional form of Planck’s law. Dataset admissibility is established through leave-one-out intercept stability analysis independent of the extracted values. These results show that h / k B and the Wien factors appear as intercept residuals only when empirical spectral relations are mapped into a temperature parametrization, and do not appear in any individual spectral relation, indicating that they are not properties of the empirical spectral structure but arise from the consistency constraints of its temperature-based representation. Mathematical Physics Theoretical Physics Thermodynamics and statistical mechanics Physical constants Blackbody radiation Planck’s constant Boltzmann constant Wien displacement factors Non-commuting mappings Intercept residual Compositional residuals Full Text Additional Declarations The authors declare potential competing interests as follows: Competing Interests: The author is an Associate Professor of Nanotechnology at the University of Minnesota. This work was carried out independently and is not part of or funded by the University. The author is also the founder and shareholder of Spectral Dynamics, Inc., a startup company formed on January 6, 2026. The company has filed three patent applications related to potential downstream applications of this work. The company had no role in the conception, execution, interpretation, or reporting of this work. Cite Share Download PDF Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8629054","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":581313374,"identity":"ac4aec86-6f36-4a13-95a7-96c31bd7e3db","order_by":0,"name":"Abdennour Abbas","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEElEQVRIiWNgGAWjYDACCR4gUQHjFTCDqQNAzINfyxkQC6TagFgtjG1oWvAC/tm9Bz/+nFeXx89+/tiHHwbWidvZTyce+PGHQYYflyV3ziVL8247XCzZk8w8s8cgPXFnT+6Gg71tDDySDTj03MgxkGbcdiBxww1mZgYeg8OJGw7kbjjM2ABkH8CuQ/5GjvHPn3PqEvcDtTD+AWk5/3bDYYY/uLUY3Mgxk+BtYE7cIMHMzAy25QbQFgY23FoM75xLs+Y5drhY4kyyMbOMQbrxhhtvQX6RwOkXudu9h2/+qAGGWPvBx4xvKqxlN5zP3fzhxx8be1whBgMJMIYj1GwJAhqQtNgTVDoKRsEoGAUjDgAAYaBgstjEIrkAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0003-1957-8948","institution":"N/A","correspondingAuthor":true,"prefix":"","firstName":"Abdennour","middleName":"","lastName":"Abbas","suffix":""}],"badges":[],"createdAt":"2026-01-18 03:53:15","currentVersionCode":2,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":true,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8629054/v2","doiUrl":"https://doi.org/10.21203/rs.3.rs-8629054/v2","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107705325,"identity":"75485147-11f5-41c8-a2f5-56b8cb35673b","added_by":"auto","created_at":"2026-04-24 09:11:30","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":482841,"visible":true,"origin":"","legend":"","description":"","filename":"PRXPaperAbbas041326v14.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8629054/v2_covered_4e0251e3-5f4b-4a7e-95d9-f12edd75fe75.pdf"}],"financialInterests":"The authors declare potential competing interests as follows: Competing Interests: The author is an Associate Professor of Nanotechnology at the University of Minnesota. 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Here, we introduce a new method for extracting physical constants as compositional residuals from linear regressions, and apply it to blackbody radiation. We show that \u003cem\u003eh\u003c/em\u003e/\u003cem\u003ek\u003c/em\u003e\u003csub\u003eB \u003c/sub\u003eemerges as a single irreducible residual from blackbody spectral relations, without invoking quantization or statistical mechanics. Blackbody radiance spectra are analyzed across 53 temperatures (1–200,000 K), extracting three pairwise linear relations among two spectral measurables (peak frequency and spectral width) and temperature. When these relations are composed, they fail to close, producing an intercept residual. 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