Abdennour Abbas

The atomic number Z is so deeply embedded in modern physics and chemistry, and in our understanding of matter, that questioning its foundational status can appear almost sacrilegious. Yet Z was not discovered as a primitive property of nature but introduced historically as an ordering index that later acquired explanatory and ontological authority. Across a series of large-scale empirical studies of atomic spectral data, we show that simple scaling laws, periodic organization, physical-property classification, atomic radii and heavy- element predictions, spectral bifurcation and convergence patterns all emerge from relationships among only two experimentally measured spectral parameters. The two-variable description organizes nearly one hundred elements, ionization sequences, hydrogen spectral families, conductivity classes, magnetic classes, and the heavy-element K-edge series. Remarkably, these structures become visible when atomic spectra are organized by a directly measured, system- specific spectral boundary vmax, rather than by atomic number. The resulting boundary-controlled organization, spanning decades in frequency across different scales, does not emerge naturally from atomic-number representation and, in many cases, is obscured by it. The empirical evidence indicates that atomic number does not generate, constrain, or organize the observed spectral structure. Its spectroscopic utility derives solely from preserving the ordering of atomic K-edge frequencies, which constitute one particular realization of the more general spectral-boundary parameter identified here. This degree of empirical predictive compression warrants a reassessment of atomic number as the primary organizing coordinate of atomic spectroscopy.

Planck's constant h and Boltzmann's constant kB conventionally connect spectral frequency to energy and energy to temperature. Recent analysis of blackbody spectral relations in the frequency domain revealed an unexpected result. Unlike conventional physical constants, which typically appear in the slopes of individual relations as local scaling factors, the ratio h/kB does not appear in any individual relation. Instead, different relations between spectral observables and temperature fail to close, and h/kB emerges as the intercept residual of that non-closure. This suggests that h/kB acts as a global consistency factor required to reconcile the different mappings. This observation raises a deeper question. If h/kB appears neither in the spectral observables nor in temperature itself, but only in the mapping between them, is it describing the empirical structure being observed, or the mapping used to represent it? More generally, are physical constants merely scaling factors between equivalent domains of description, or do they connect fundamentally different domains? We explore this question by distinguishing between an empirical domain of directly measurable spectral quantities such as frequency, phase, and linewidth, and a representational domain of abstract constructs used to represent those observations, including temperature, energy, entropy, and spatial parametrizations. Within this framework, physical constants are interpreted as quantities introduced to achieve internal consistency among different representations of the same empirical structure. We develop this perspective through the concept of Domain Projection and discuss its implications for the interpretation of physical constants, physical laws, and the foundations of scientific description.

Modern science relies on representational models that organize empirical observations through constructs such as electrons, energy levels, spacetime, wavefunction and causal mechanisms. While these models achieved extraordinary predictive success, they also led to an implicit inversion in which these unobservable theoretical constructs are treated as explanatory ground and observable spectral regularities as derivative. Recent large-scale analyses of atomic spectra reveal that diverse physical measurements exhibit unexpectedly low-dimensional and relational organization recoverable directly through frequency-domain relations, without introducing hidden entities or theoretical assumptions. These results show that classification, compression, generalization, and prediction emerge naturally from intrinsic spectral structure prior to representational interpretation. The paper argues that representation reorganizes empirical structure rather than generates it and becomes methodologically secondary once observable relations exhibit predictive closure. The persistent demand for additional mechanistic explanation arises not from deficiencies in empirical structure, but from the expectation that physical understanding must ultimately be translated into narratives of localized objects, linear causality, and particle interactions. Once representational constructs acquire explanatory authority beyond their empirical origin, they tend to stabilize and reify, expanding the theoretical apparatus to accommodate inconsistencies. This motivates the concept of representational debt, whereby additional abstract layers accumulate beyond empirical necessity, increasing descriptive complexity and delaying the discovery of simpler organizing principles. More broadly, the analysis motivates Foundational Science, a discipline concerned not with ontological entities or local causal mechanisms, but with invariant relations, scaling laws, and structural constraints. The Spectral Framework is presented as an empirical realization of this methodological perspective, illustrating how scientific explanation may be re-grounded in simple organizational principles already present in measurements.

Time is commonly treated in physics as a primitive parameter against which observations are indexed. This paper re-examines that assumption from a metrological perspective, arguing that the operational meaning of time arises through calibration hierarchies that ultimately terminate in physically occurring periodic reference. We do not measure relative to time; we construct time from calibrated relations to periodic physical processes. Non-periodic processes may serve as clocks or proxies, but only after calibration against periodic standards. Periodicity is therefore not a contingent feature of measurement technique but a requirement for scale, reproducibility, and quantitative comparability. If periodic reference is required for measurement, it cannot originate within the measurement process itself. The relational and periodic character of measurement is therefore inherited from the physical organization upon which quantitative comparison depends. Analysis of experimental hydrogen atomic spectra supports this view, revealing a relational organization that persists under normalization and changes of scale. These observations suggest that stable physical organization is encoded in invariant relations rather than absolute quantities. Within this framework, phase is the primitive relational descriptor of periodic structure, while frequency provides its operational expression in measurement, rendering the spectral domain implicit in the operational structure of measurement itself.