PNAS Peer Review Record
Spectral Framework

Modern atomic datasets make it possible to test several observables and scaling relations simultaneously across the periodic system. This changes the scope of empirical analysis: instead of identifying isolated spectral laws, one can now ask whether relations in the frequency domain form a closed, explanatory, and predictive structure without requiring a prior model of the atom. That is a fundamentally different scientific question from whether a curve can be fitted to a dataset. Again, this distinction would likely have been more apparent had the referee been able to evaluate the present manuscript in the context of the other papers submitted as part of the coordinated package.

A curve fit and a generative predictive structure are not the same thing. A generative predictive structure is required to do something substantially more demanding than numerical extrapolation. A fitted function may generate unseen values beyond its construction data, but a generative structure must remain fixed, survive independent tests, and generate new empirical structure that was not supplied to it. More importantly, it must generalize beyond a single fitted relation: the same low-dimensional organization must reappear across independent domains, constrain additional quantities, and organize or predict multiple observables, regimes, or phenomena. The scientific question is therefore not simply whether parameters were estimated from measurements, but whether the resulting structure generates and generalizes empirical information beyond the data from which it was obtained. That is the stronger form of generativity tested across this coordinated package. A fitted relation does not, merely by virtue of being fitted, acquire the ability to recover new empirical organization or to generalize successfully across independent phenomena.

“Curve fitting” is rhetorically loaded because it bundles several distinct criticisms together as though they were equivalent. It can mean simply that parameters were estimated from empirical data, which is routine scientific practice across both empirical description and theoretical modeling; in that sense, obtaining the spacing-law coefficients from measured K edge frequencies and spacings does not establish that the relation is merely descriptive. It can also mean that a sufficiently flexible function was used to reproduce the observations from which it was constructed. That is not what occurs here: the quantities entering the construction are fixed by measurement, and the coefficients are fixed by the empirical relations between those quantities. There is no flexible parameter that can be varied to manufacture the resulting progression. Third, curve fitting can mean that free parameters were introduced specifically to absorb the discrepancy under examination or force agreement with the target being predicted. That does not occur here either: once the empirical relation and its coefficients are fixed, the subsequent predictions are fixed as well, with no element-specific adjustment, residual correction, target-specific scale factor, or free parameter available to force agreement. The predictions therefore place the empirical law at risk: agreement constitutes prospective predictive success, while disagreement outside the stated accuracy delimits or falsifies the law.

More fundamentally, the referee’s characterization assumes, without evidence, a particular epistemological hierarchy between theory and measurement: theoretical interpretation is granted explanatory priority, while a structure recovered directly from measurements is dismissed as “curve fitting.” A construction built from measured quantities is not descriptive by definition, any more than a construction expressed in theoretical quantities is explanatory by definition. That distinction cannot be inferred from the provenance of the variables; it must be judged from the evidence and from what the competing constructions actually organize, generalize, predict, and explain.

This is closely related to another epistemological argument raised in this and related reviews: that theoretical content is already “contained” in the measurement. In both cases, theoretical interpretation is granted epistemic priority before the empirical alternative is technically evaluated. We have anticipated and addressed these broader epistemological arguments separately in a series of non-technical papers (Link). They do not constitute a technical scientific evaluation of the claims and results presented here. The present manuscript concerns empirical evidence, reproducibility, predictive structure, and whether a closed organization can be recovered directly in measurement space.

The scientific result is the degree of empirical predictive compression achieved by this common structure: a remarkably small set of measured spectral quantities carries quantitative information across phenomena ordinarily treated as separate atomic problems.

Dismissing this as “pattern recognition and curve fitting” does not demonstrate that the structure is accidental, that its predictions fail, that its parameters were adjusted to the target quantities, or that its cross-phenomenon predictive compression disappears under independent testing. Those are the technical questions, and they require technical evaluation of the evidence.

Across the coordinated package, the result is not merely a fitted relation or an isolated prediction. It is a striking compression of atomic information: the same low-dimensional spectral structure organizes fine spectral behavior, periodic organization, multiple atomic and chemical properties, and quantitative predictions in the heavy and superheavy regime. That degree of empirical compression is itself a physical result.

We therefore expected a technical evaluation of whether the reported structures are real, whether the quantitative relations hold, whether the predictions survive independent tests, and whether the claimed compression is actually present in the data. Instead, the referee substitutes an epistemological classification of what is allowed to count as “physics” or as an “explanation” for a technical evaluation of the claims and results.

The statement that the work has “little physics content” is therefore not evidence against the manuscript. It reflects an interpretive assumption about the form that physical understanding must take. The relevant scientific question is whether a closed predictive structure exists directly in the measurements and whether that structure is sufficient to organize, generalize, and predict the observed phenomena. That question has to be answered by examining the evidence, not by defining the result outside physics before evaluating it. We have anticipated and addressed these epistemological arguments separately in a series of non-technical papers (Link). The present paper is about empirical evidence, reproducibility, and predictive structure. Replacing evaluation of those claims with a prior debate over what is permitted to count as physics does not constitute a technical assessment of the work.

This is a legitimate and important scientific question that we have precisely addressed in a subsequent paper [link], but it is not the question addressed in the present paper. The purpose of this paper is to demonstrate that the spectral coordinate developed in the companion work permits a very high degree of empirical compression in the prediction of K-edge frequencies, replacing complex, computationally intensive multielectron relativistic calculations with a compact predictive relation derived directly from measured spectral quantities.

Understanding how this compressed empirical law bears on the Dirac–Fock description is nevertheless an important next question, because it concerns what the existence of this empirical compression means for our understanding of the theoretical representation of the same atomic system.

The referee’s comment therefore identifies an important next-level question, but it does not invalidate the empirical result reported here. It also highlights the importance of reviewing this body of work as a coordinated package: the individual papers address different but connected levels of the same framework, and evaluating one paper in isolation can make an intentional division of scientific questions appear to be an omission.

The referee assumes that a prediction beginning from an experimentally measured K-edge constitutes “relativistic input” because the measurement is presumed to contain relativistic effects. This conflates a measured physical phenomenon with a theoretical interpretation—relativity—used to describe that phenomenon. The predictive construction uses measured spectral quantities and empirical relations between them; it does not use the Dirac equation, relativistic corrections, relativistic many-electron calculations, or theoretical relativistic quantities as inputs. Interpreting a measured K-edge through relativistic theory does not make that theory an input to an empirical relation constructed from the measurement itself.

The referee’s reasoning is circular and unfalsifiable. If relativity is assumed to be embedded in every atomic measurement, then any empirical model using such measurements can automatically be described as containing relativistic input, regardless of whether relativistic theory enters the model at all. No empirical result could then demonstrate independence from the theoretical machinery, because that independence has been excluded by definition. This grants theoretical interpretation epistemic priority over the measurement itself: the theoretical explanation is assumed to be intrinsic to the observation before the empirical alternative is tested. The argument therefore assumes the very point the manuscript is testing: whether the observed structure and its predictions can be recovered from measurements alone, without relativistic theory as an input.

This distinction is consequential. An observation and a theoretical account of that observation are not the same epistemic object. Treating them as inseparable makes it impossible, in principle, to ask whether the measurements possess an independently sufficient organization of their own. The empirical question posed here is whether a closed predictive structure exists directly in the measurements and whether that structure is sufficient to organize, generalize, and predict the observed phenomena.

The subsequent paper on coordinate mismatch (LINK) shows how treating measurement as inseparable from a specific theoretical interpretation can lead to mistaking a consequence of the theoretical representation for a physical effect.

This objection is therefore epistemological, not a technical demonstration that the empirical prediction fails. We anticipated and addressed this broader issue separately in the non-technical paper Beyond Representation:

“Modern physics often treats empirical observations and the theoretical frameworks historically used to explain them as inseparable. Once a theory succeeds, the data themselves are frequently interpreted as already containing the theory’s assumptions, making it difficult to distinguish between what is directly observed and what is introduced through representation. This tendency can obscure simpler organizational principles present in the empirical relations themselves. … The existence of a successful theoretical framework is thus conflated with the necessity of its variables for organizing the empirical structure that preceded it. Empirical spectral observables preexisted, and can be analyzed independently of these theories, just as planetary trajectories can be analyzed independently of any particular coordinate system.”

The appropriate technical evaluation is to identify the quantities and equations actually supplied to the prediction, freeze the empirical law, and test its predictions. Calling an experimental measurement “relativistic input” because relativity can be used to interpret that measurement does not constitute such a test and does not provide scientific evidence against the empirical claim.

This is the same theory-first premise that underlies the referee’s subsequent claim that an experimentally measured K edge already constitutes “relativistic input.” In the first comment, the existence of the theory is treated as sufficient reason to subordinate the empirical question to that theory; in the second, the theory is treated as intrinsic to the measurement itself. The latter goes further by conflating the measured phenomenon with its theoretical interpretation.

This comment substitutes the referee’s preferred form of explanation for the scientific question actually addressed in the paper. This is therefore another epistemological objection about what is allowed to count as physical explanation, not a scientific demonstration that the empirical structure, its compression, or its predictions fail. It nonetheless warrants clarification.

The manuscript does pursue mathematical understanding: it identifies a scale-invariant relation between measured spectral quantities, determines its structure, tests its stability, and uses the resulting fixed law to generate quantitative predictions across the heavy- and superheavy-element regime. What it does not do is require that this empirical structure first be translated into the conventional microscopic theoretical description before it can count as scientific understanding.

The fitting formula is not the scientific endpoint; the empirical predictive compression it reveals is the result. A remarkably small structure derived from measured spectral quantities reproduces and predicts K-edge behavior ordinarily approached through complex, computationally intensive multielectron calculations. Across the coordinated package, the same low-dimensional spectral framework is tested for its ability to organize, generalize, and predict phenomena ordinarily treated as separate atomic problems. That degree of empirical predictive compression is itself a scientific result that requires explanation.

The statement that the author argues that the formula is “all one needs to know or care about” is not a claim made in the manuscript. It is an attribution introduced by the referee and cannot serve as evidence against the work.

This characterization also illustrates the consequence of evaluating the manuscript outside the coordinated package in which it was submitted. The K-edge law is one result within a broader framework whose structure, generalization, and predictive consequences are examined across the companion papers. Had the referee been able to evaluate the manuscripts as the coordinated body of work submitted to PNAS, the characterization of the fitting formula as the “end outcome” would have been difficult to sustain.

The relevant technical evaluation is whether the claimed empirical structure exists, whether it survives independent tests, and whether its predictive compression is real. The referee does not demonstrate that any of these claims fail.

This is the most troubling referee comment I have read in my entire scientific career.

The referee says, essentially: if future measurements agree, the author will call that predictive success; if they disagree, presumably the author will fit another formula.

The most serious issue is that the referee makes an unsupported attribution of hypothetical future scientific malpractice. By stating that the author would “presumably” fit a new formula or introduce additional parameters if a prediction failed, the referee assumes, without any evidence, that the author would respond to falsification by moving the goalposts rather than accepting the empirical result. A referee should evaluate the methodology, predictions, and scientific conduct actually presented in the manuscript, not invoke an imagined future violation of that methodology as evidence against the work. This is not scientific evaluation; it is prejudgment.

The scientific logic of the comment is equally problematic because it amounts to a non-falsifiable criticism of a falsifiable prediction. The manuscript provides fixed numerical predictions for elements for which, as the referee himself notes, neither calculation nor experiment is available. Those predictions are therefore genuinely exposed to future evidence. Agreement within the stated uncertainty constitutes prospective predictive success; disagreement outside it falsifies or delimits the law. The referee cannot neutralize that falsifiability in advance by imagining that the author would later alter the formula. That creates an almost unfalsifiable standard of criticism: unmeasured predictions do not count because they are untested; future successful tests apparently would not count either because the author would merely “argue” that they demonstrate prediction. Under that framing, no possible experimental outcome can count strongly in favor of the empirical law. Agreement is discounted in advance, while disagreement is converted in advance into hypothetical goalpost-moving.

The comment may also reflect a confusion between a theoretical model that can be refined in response to new measurements and an empirical law whose coefficients and predictions are fixed before the target measurements exist. Here, the quantities entering the recurrence are fixed from existing measurements, the predictions are generated prospectively, and there is no free parameter left to adjust when the future measurements become available. If those predictions fail, the law fails or its domain of validity is reached; changing the formula afterward would constitute a different model, not a rescue of the prediction being tested.

This reverses the logic of prospective prediction. The scientific significance of predicting values before either measurements or calculations exist is precisely that the predictions are frozen before the outcome is known and are therefore genuinely at risk. Future agreement supports the law; future disagreement falsifies or delimits it.

The referee therefore replaces evaluation of a falsifiable scientific prediction with an unsupported and itself unfalsifiable speculation about the author’s future conduct. That is not a scientifically or professionally defensible basis for rejecting the manuscript’s predictive claim.

category error in the referee’s proposed -based reformulation Z − Z₀,

This is an instance in which the referee’s counterargument inadvertently demonstrates the paper’s central claim. The referee is correct that the recurrence can be written in closed form. We deliberately did not present it as f(Z), because doing so obscures the central mathematical and empirical point of the paper: the law was not, and cannot be, discovered as a function of atomic number. The referee’s reformulation commits a category error by confusing the physical variable that defines the empirical law with the ordinal index used only to count successive elements. Remarkably, in attempting to rewrite the law in terms of Z, the referee is able to replace only the counting index n, not the organizing variable ν_max. This demonstrates exactly the point made in the paper: Zcannot replace ν_maxas the organizing coordinate.

The empirical law is constructed from the relationship between two measured spectral quantities,

Δν_max = a + bν_max,

where ν_maxis the measured K-edge frequency and Δν_maxis the measured spacing between successive K-edge frequencies. The index nhas an entirely different mathematical role: it merely labels successive observations in the ordered sequence.

Thus, the manuscript distinguishes two fundamentally different objects:

Physical variable defining the empirical law: ν_maxOrdinal counting index: n.

The referee replaces the second with Z, not the first. That distinction is the entire issue.

For consecutive elements,

Z − Z₀ = n − n₀,

so Z − Z₀can of course be used to count how many successive elements separate the predicted value from the starting point. But this is only an ordinal identity. The same quantity could be denoted n − n₀, k, p or any other integer counter. Changing the name of the counter does not change the empirical variable from which the law was discovered, nor does it alter the source of the information contained in the recurrence.

Indeed, the referee’s reformulation itself demonstrates why Z cannot replace ν_max as the organizing coordinate. If atomic number genuinely replaced ν_maxas the variable defining the empirical structure, then for consecutive elements the corresponding spacing would simply be

ΔZ = 1.

That contains no analogue of the nontrivial, scale-dependent spectral relation

Δν_max = a + bν_max.

Atomic number supplies an ordering and a trivial unit increment. It does not supply the measured variation in spectral spacing from which the predictive law is obtained. The law therefore cannot be discovered from Zalone.

The referee’s proposed expression avoids this loss of information precisely because it does not derive the law from Z. It retains the spectral coefficients aand b, obtained from the measured spectral-spacing relation; it retains a measured spectral anchor; and the quantity being propagated remains the K-edge frequency itself. The only role assigned to Z − Z₀is to specify how many times the already-discovered spectral relation is propagated.

This is why writing the result immediately as f(Z)would be conceptually misleading. It would visually suggest that the prediction is a law obtained from atomic number, when in fact all of the nontrivial information in that expression has already been extracted from spectral space before Zappears at all. The recursive representation was therefore deliberate: it keeps visible the distinction between the empirical variable carrying the structure and the index used to step through the ordered sequence.

The referee’s reformulation is therefore not an independent Z-based alternative to the spectral law. A genuine Z-based alternative would have to recover the nontrivial spacing relation and its predictive content from atomic number itself, without importing the spectral coefficients, spectral anchor, and spectral recurrence. The proposed expression does not do that.

Thus, the referee’s counterargument demonstrates the opposite of what it appears to suggest. In attempting to replace the spectral formulation by a function of Z, the referee can replace only the counting index nwith Z − Z₀; the organizing variable ν_max, the spectral-spacing relation, and all of the empirical information derived from them remain indispensable. That is exactly why Zcannot replace ν_maxas the organizing coordinate.

The referee again replaces the scientific question posed by the manuscript with a different one. The paper does not ask for a microscopic account of the “innermost workings” of K-shell electrons. It asks whether the measured K-edge progression contains a low-dimensional empirical structure capable of organizing and quantitatively predicting heavy- and superheavy-element behavior.

Calling that question non-fundamental simply because it is not formulated in microscopic theoretical variables is an epistemological judgment about what is permitted to count as a fundamental physical question, not evidence that the empirical result fails. Whether a remarkably compressed predictive structure exists directly in the measurements is itself a fundamental scientific question, particularly when that structure reproduces behavior ordinarily obtained through complex, computationally intensive multielectron calculations.

Across the coordinated package, the question is broader still: whether the same low-dimensional spectral organization can generalize across fine spectral structure, periodic organization, and multiple atomic and chemical phenomena. That degree of empirical predictive compression is precisely what requires explanation; it cannot be dismissed by redefining “fundamental” in advance to mean only microscopic theoretical mechanism.

The referee therefore does not demonstrate that the empirical structure is accidental, that its predictions fail, or that its compression disappears under independent testing. The comment states a preferred conception of fundamental explanation; it does not constitute a technical evaluation of the scientific claim made in the manuscript.

The referee further states that we argue that the empirical law is “all one needs to know or care about.” I do not recognize this as a claim made in the manuscript. If the referee believes that I make this argument, we respectfully ask that the specific passage be identified. Otherwise, this statement attributes a position to the author and then criticizes that attributed position rather than the claims actually made in the manuscript.
The referee states that “a physicist normally seeks understanding” and criticizes the work because it does not connect the empirical law to the many-electron Dirac equation or QED corrections. This is an ontological requirement, not a technical criticism of the work. The purpose of peer review is to assess whether the evidence supports the claims made in the manuscript, not to prescribe what form a valid physical explanation must take. The referee does not show that the empirical structure fails to organize or predict the atomic system. Instead, he requires that it be translated back into the theoretical models with which conventional atomic physics is familiar. This is precisely the epistemic problem discussed in Beyond Representation: an independently sufficient empirical structure is treated as incomplete until it is given a familiar microscopic narrative. The additional statement that the author believes the formula is “all one needs to know or care about” is not a technical criticism at all; it is an unsupported attribution of a philosophical position to the author. Neither statement provides evidence against the empirical results or their predictions.

Neither statement belongs in the technical assessment of an empirical scientific paper. The relevant question is whether the empirical evidence supports the claims, whether the analysis is sound, and whether the predictions are falsifiable and quantitatively successful. A reviewer’s view of what a physicist should seek, or what form a satisfactory explanation should take, is not evidence for or against the reported empirical result. Here, those preferences are being used as grounds for rejection without demonstrating a failure of the empirical framework itself.

The referee does not test whether that empirical structure works or whether its predictions are correct. Instead, he criticizes the work by stating: “If the author would use the fitted function to learn something about the solution of the many-electron Dirac equation, or various QED corrections, it might conceivably have interest for physicists.” He therefore faults the work precisely because it does not return to those same theoretical constructs that the evidence shows are not needed for organization and prediction. That misses the central point of the work. The work asks whether those constructs are needed to organize and predict the atomic system. The reviewer assumes they are needed from the start and asks the work to build on them.

This is one of the epistemic problems I anticipated and explained in Beyond Representation: empirical work can be judged not on its own content, predictive power, and predictive compression, but on its relationship to an established theoretical representation. The paper specifically discusses the tendency to demand that empirical structure be translated into particles, forces, fields, energies, or microscopic causal mechanisms before it is accepted as physical explanation. The referee’s statement that the work contains “little physics,” followed by his demand that it be connected to the many-electron Dirac equation or QED corrections, is a direct example of this epistemic problem.

The review recommends rejection under “Conclusions Justified?: No.”

Yet the report does not identify a quantitative failure that would justify that judgment. It does not challenge the reported residuals, uncertainty analysis, cross-validation, parameter stability, held-out predictions, or comparisons with available experimental and theoretical values. Nor does it identify a statistical error, failed prediction, or reproducibility problem.

The manuscripts provide the complete experimental datasets, equations, fitting procedures, calculations, and uncertainty analyses required to reproduce the work in full. With current computational and AI-assisted tools, the reported results can be independently checked in seconds by simply uploading the manuscript. Yet the referee does not identify any quantitative result that is incorrect.

Instead, the negative assessment is directed primarily toward epistemological objections: whether an empirical structure without a Dirac/QED interpretation counts as physics, whether established theoretical representation should be granted explanatory priority, whether theory hides inside empirical measurements, and what form of understanding a physicist is expected to seek.

Those objections do not demonstrate that the conclusions are unjustified. A judgment of “Conclusions Justified?: No” requires identifying where the empirical, mathematical, or quantitative evidentiary chain fails. The referee does not do so. The report does not show that the data are wrong, the calculations fail, the uncertainties are mishandled, the validation is defective, or the predictions are quantitatively unsuccessful. This distinction is consequential. Peer review cannot substitute disagreement over what should count as physical explanation for a technical evaluation of whether the evidence supports the claims actually made. Here, the numerical evidence is fully exposed and directly reproducible, while the stated grounds for rejection are directed predominantly toward the epistemological status of the empirical framework rather than a demonstrated failure of its scientific results.

The reviewer reduces the paper’s result to the generic observation that spectroscopy contains periodic information. That is not the novelty claimed or demonstrated. The paper identifies a specific two-coordinate spectral structure in which periods form bounded bands, groups form scaling trajectories, and spectral lineage can be followed across successive atoms as continuous modes bifurcate into persistent descendant branches. These splitting events coincide with major periodic boundaries, directly linking the evolution of spectral structure to periodic differentiation. The framework further reveals a quantitative contraction of periodic bandwidth and supports prediction of atomic properties.

If these results are already known, the reviewer should identify prior work demonstrating this spectral lineage, its recursive mode splitting at periodic boundaries, the two-coordinate scaling geometry, quantitative bandwidth convergence, and the resulting predictive relations. No such literature is cited. General references showing that spectroscopy contains periodic information do not establish prior discovery of these results.

This comment contains two distinct inaccuracies: a historical inaccuracy in the proposed S → P → A sequence, and a conceptual mischaracterization of what the present work demonstrates.

First, the historical sequence is inaccurate. The periodic system was established primarily from recurring chemical properties and atomic weights, not from spectroscopy. The manuscript explicitly distinguishes these stages: periodic organization was identified first from elemental properties, while spectroscopic regularities were recognized subsequently and later interpreted within atomic theory. Combining “spectroscopic and chemical data” into a single historical category S therefore conflates distinct stages in the development of the periodic system.

Second, describing the present work simply as S → P removes almost everything the paper actually demonstrates. Using only two directly measured spectral frequencies, the framework recovers the chemical grouping of the periodic system and reveals additional groupings associated with physical properties such as conductivity and magnetism, with each group following its own quantitative scaling law that can be used predictively. It also reveals spectral lineage and recursive mode splitting associated with periodic differentiation, quantifies the contraction of periodic structure across the table, and generates predictions of atomic and material properties.

Calling all of this simply “S → P” is therefore not a substantive description of the work. Recovering a known classification from data is one thing; showing that two measured frequencies define a quantitative system that organizes chemical and physical behavior, reveals the structure and evolution of periodic differentiation, and generates predictions is another.

If the reviewer considers these specific results—including the empirical scaling laws and prospective predictions—to have been “repeatedly shown,” then the relevant prior literature demonstrating them should be identified.

The referee appears to have misread the manuscript: most of the historical claims presented here as objections actually agree with the paper, while the concluding criticism attributes to the manuscript a claim that it does not make.

First, the historical criticism is directed at a statement that is not in the paper. The manuscript states that the periodic system was established from empirical trends in atomic weight and chemical behavior and was later interpreted through models of atomic structure. The referee’s statements that the periodic system emerged from chemical behavior and that chemical frameworks were important to its formulation therefore do not contradict the manuscript; they largely restate it.

The referee then concludes that the manuscript claims the periodic system was “devised without chemical or theoretical knowledge.” That is not what the manuscript says. The phrase “without prior chemical or theoretical knowledge” refers to the objective of the present work: whether periodic organization can be derived directly from measured atomic observables without using chemical classification or theoretical structure as input. It is not a historical claim about how the periodic table was devised. The criticism therefore changes the meaning of the manuscript’s statement and then rejects that altered version.

Second, reconstructing the periodic table from chemical data is not equivalent to what is done here. A data-driven reconstruction from chemical properties still begins with chemical information. The present work instead asks whether periodic organization emerges from a coordinate system defined by only two directly measured spectral frequencies, and whether that same empirical structure yields quantitative scaling laws, physical-property groupings, periodic differentiation, and predictions. The manuscript shows that the resulting spectral space recovers periods and chemical and physical groups, reveals recursive spectral branching and periodic contraction, and provides a quantitative basis for prediction.

The historical observations cited by the referee therefore do not refute the manuscript, and reconstruction from chemical data addresses a different question. The central claim is whether periodic organization can be derived, quantified, and predicted directly from atomic observables without supplying the chemical or theoretical structure in advance.

This comment is difficult to reconcile with the referee’s preceding criticism. The referee previously characterized the work as merely reproducing the known relation S → P and therefore lacking conceptual novelty, but here explicitly recognizes the attempt to identify the minimal empirical principles underlying periodicity as valuable and relevant to questions of minimal dimensionality and data-driven discovery. That is precisely the broader objective of the work that the preceding criticism overlooks.

The requested clarification is also already present in the manuscript. The Introduction explicitly states: “Reexamining periodicity in the spectral domain is an attempt to identify the minimal empirical principles from which periodic organization of the elements can be explained, quantified and predicted.” This statement appears before the analysis and directly frames what follows.

The issue is therefore not that the objective was absent, but that the referee’s earlier assessment did not evaluate the work according to the objective that the referee here correctly identifies.

This criticism is based on a clear factual misreading of the Methods. Groups with three members were not excluded from the analysis of periodic structure. The structural analysis spans the periodic groups across the table, including Groups 1 and 2, transition-metal Groups 3–12, and Groups 13–18. The manuscript explicitly analyzes the transition metals and shows that Groups 3–12 form distinct scaling trajectories in the same spectral coordinate system.

The n=3 restriction cited by the referee applies only to the leave-one-out cross-validation of predictive relations. In that procedure, withholding one element from a three-member group leaves only two points for fitting. Those groups were therefore excluded from that specific predictive cross-validation, not from the structural analysis of periodicity. The Supplementary Methods explicitly state this restriction in the section describing the cross-validation procedure.

The referee has therefore attributed to the main analysis an exclusion that does not exist. The subsequent claims that the analysis favors main-group elements, introduces selection bias, and thereby undermines the generality of the periodic derivation all follow from this incorrect premise.

Moreover, the referee’s criticism effectively asks that three-member groups be included in a leave-one-out validation where withholding one element leaves only two observations for fitting. Such a procedure would be statistically invalid: a regression fitted to two remaining points cannot provide a meaningful test of predictive performance. The n=3 restriction is therefore not a source of selection bias but a necessary safeguard against an invalid validation procedure. It has no bearing on the inclusion of these groups in the structural analysis.

This criticism conflates two different types of prediction and incorrectly states that the manuscript lacks statistical validation.

The principal quantitative predictions are explicitly cross-validated, and the results are reported in the Abstract itself. Leave-one-out validation yields median prediction errors of 1.1% for first-ionization frequencies and 2.1% for covalent radii, with errors below 0.3% for the lanthanide-series predictions. The statement that the prediction claims “are not supported by statistical validation” is therefore directly contradicted by the manuscript.

The Cr and Ag superconductivity statement is a separate qualitative prospective prediction, not the basis of the quantitative validation. The referee selects this secondary example and uses it to characterize the entire prediction analysis while ignoring the explicit cross-validation.

The demand for comparison with quantum chemistry also changes the question being tested. This work asks what can be predicted directly from measured spectral observables without supplying a theoretical model as input. Predictive validity is therefore tested against withheld or independently measured data, not by requiring agreement with a theoretical framework.

The referee thus overlooks the statistical validation reported in the Abstract itself, substitutes a secondary qualitative prediction for the quantitative analysis, and imposes a theoretical benchmark that is not required to test the empirical claim.

The distinction is understood, but it does not create an ambiguity in the analysis. “Periodic system” appears only once in the manuscript’s substantive text, in the opening sentence of the Abstract; the remaining occurrences are in cited reference titles. Throughout the manuscript, “periodic table” refers to the empirical organization and classification of the elements, not merely to its tabular graphical representation.

For terminological precision, “periodic system” can be used when referring to the conceptual organization of the elements and “periodic table” when referring specifically to its tabular representation. This terminology does not affect the analysis, results, or conclusions.

This criticism conflates periodic recurrence with the claim that every elemental property follows a perfectly regular monotonic trend. The manuscript makes no such claim.

A central result of the paper is precisely that periodic organization is not described by a single uniform trend, but by distinct spectral scaling regimes. Groups 1–3 and 13–18 form the principal set of nearly parallel, converging trajectories, whereas transition-metal Groups 4–12 define a second family with weaker slopes and local expansion within the d-block. The manuscript further resolves deviations from nominal group trajectories and piecewise regimes within the lanthanides and actinides. Identifying this structured coexistence of convergence, local expansion, and group-specific departures in the same two-coordinate spectral space is part of the novel empirical result.

The behavior of heavier elements is likewise not ignored. The manuscript quantitatively shows that periodic spectral bandwidth contracts with increasing ν_max, causing group trajectories to become progressively compressed and less distinguishable in the lanthanide and actinide regions. Thus, the weakening and blurring of conventional periodic trends in heavy elements is not a counterexample to the framework; it is itself a measured feature of the spectral structure that the paper identifies and quantifies.

More fundamentally, recurrence does not require every local trend to be smooth or exception-free. A subsequent empirical analysis addresses recurrence directly and shows that it follows a quantitative spectral scaling law within the same two-coordinate space [LINK].

The literature cited by the referee may challenge simplistic or universally monotonic formulations of periodic trends, but that is not the formulation used in this manuscript. The present work instead reveals a more structured form of periodicity in which recurrence coexists with distinct scaling families, local deviations, and progressive loss of differentiation—and those features are themselves among the paper’s principal empirical findings.

This criticism is factually incorrect. The statistical analysis and correlation metrics that the referee states are absent are explicitly provided in the Supplementary Information and directly cited in the corresponding main text passage.

Immediately after stating that electrical conductivity and magnetic behavior align along distinct spectral branches, the manuscript states that these branches resolve the different physical property classes “with high statistical significance (Table S3).” Table S3 reports the regression slopes, Pearson correlation coefficients, R², adjusted R², residual sums of squares, degrees of freedom, number of observations, root MSE, and normalized residuals for the conductivity and magnetic branches. The reported correlations are approximately |r| = 0.998 to 1.000, with R² values of approximately 0.996 to 1.000.

The superconducting branch is also tested separately in Table S4 using three different elemental compositions, yielding R² = 0.9971, 0.9971, and 0.9982.

The referee therefore labels the result “qualitative” and states that “no statistical analysis or correlation metrics are provided” despite the main text explicitly directing the reader to the statistical analysis and the Supplementary Information reporting precisely those metrics. The request that these claims be “substantiated with data” asks for analysis that is already present in the submitted manuscript.

The referee correctly notes that the first row deviations are a recognized feature of the periodic system and suggests the “singularity principle” as a related conceptual description.

The deviation of these elements is not treated as an inconvenient anomaly to be discarded from the spectral framework. It is itself part of the empirical structure revealed by the analysis.

The anomalous behavior of B, C, and N emerges directly from the same two measured spectral coordinates, without supplying prior chemical knowledge.

The referee suggests invoking the “singularity principle” as an existing conceptual description of such first row anomalies. That literature may provide historical or chemical context. However, invoking an established name for the anomalous behavior does not explain the empirical spectral result reported here.

The relevant distinction is between recognizing that first row elements are anomalous, which is already known, and showing that this anomaly emerges independently and quantitatively within the newly identified spectral structure. The latter is the result of this work.