PNAS Peer Review Record
Spectral Framework

The characterization of the work as “pattern recognition and curve fitting” does not address the central empirical claim. The manuscript does not merely fit a function to existing observations; it identifies a fixed relation between measured spectral quantities and uses that relation to generate quantitative predictions beyond the data used to establish it.

Once the empirical spacing law and its coefficients are fixed, the predictions are fixed. There is no element-specific adjustment, residual correction, target-specific scale factor, or free parameter available to force agreement. The predictions therefore place the law at genuine empirical risk: agreement constitutes prospective predictive success, whereas disagreement outside the stated uncertainty falsifies or delimits the law.

The distinction is also broader than extrapolation from a single fitted relation. Across the coordinated package, the same low-dimensional spectral organization is tested across independent observables and phenomena, including spectral structure, periodic organization, atomic and chemical properties, and heavy- and superheavy-element prediction. The scientific question is therefore whether a common empirical structure survives independent tests and generates information beyond the data from which each individual relation was obtained.

Calling this “curve fitting” does not demonstrate that the structure is accidental, that the validation fails, that the predictions are quantitatively unsuccessful, or that parameters were adjusted to obtain the reported agreement. Those would be technical criticisms and can be tested directly.

More fundamentally, the label appears to assume that a structure recovered directly from measurements is merely descriptive unless translated into an established theoretical framework. That is an epistemological assumption, not a technical demonstration that the empirical result fails. Whether the reported structure organizes, generalizes, predicts, and compresses the data must be judged from the evidence itself.

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 reviewer 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, compression 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 and accuracy in the prediction of K-edge frequencies, replacing complex, computationally intensive multielectron relativistic calculations with a compact predictive relation derived directly from two 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 reviewer’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 reviewer 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 quantity with a theoretical interpretation used to describe that quantity. 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, QED corrections, 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 reviewer’s reasoning is circular and unfalsifiable. If relativity is assumed to be embedded in every atomic measurement, then any empirical construction using such a measurement can automatically be described as containing relativistic input, regardless of whether relativistic theory enters the equations or calculation. 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.

The relevant technical question is therefore straightforward: what quantities, equations, and parameters actually enter the prediction? Here, the empirical law is constructed from measured spectral quantities and then frozen before prediction. A technical objection would need to identify where relativistic theoretical information enters that mathematical construction or demonstrate that the resulting predictions fail. The subsequent coordinate-mismatch paper (link) strengthens this distinction by showing that the empirical framework predicts not only measured spectral values but also the values obtained by relativistic and quantum calculations without using those theories, and explains where and why the empirical and theoretical predictions diverge.

This objection is therefore epistemological, not a technical demonstration that the empirical prediction fails. We have addressed the broader issue separately in Beyond Representation. Calling an experimental measurement “relativistic input” because relativity can be used to interpret that measurement does not demonstrate that relativistic theory enters the prediction, nor does it provide scientific evidence against the empirical claim.

This is the same theory-first premise that underlies the reviewer’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.

The reviewer conflates fitted parameters with fixed empirical quantities.

The coefficients a and b are fitted from the empirical spectral relation, while the recursion is initialized from a measured K-edge frequency (a quantity) at the beginning of the heavy-element regime.

That measured starting condition is not a fitting parameter by any accepted statistical or methodological definition.

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

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 reviewer and cannot serve as evidence against the work. The K-edge law is one result within a broader coordinated framework whose structure, generalization, and predictive consequences are examined across the companion papers; it is therefore not presented as the “end outcome.”

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

The reviewer conflates two different methodological situations: a theoretical description that may be modified by introducing additional terms or corrections when discrepancies arise, and the empirical relation tested here, whose coefficients are already fixed by existing measurements and whose prospective predictions contain no adjustable parameters.

This distinction is central because the manuscript gives numerical predictions for elements for which, as the reviewer notes, neither calculation nor experiment is presently available. The predictions are therefore fixed before the outcome is known and are directly exposed to future test. There is no element-specific adjustment, residual correction, target-specific scale factor, or remaining free parameter available to force agreement. Agreement within the stated uncertainty supports the law; disagreement outside it falsifies or delimits its domain of validity.

The reviewer’s comment attempts to discredit both possible outcomes of the prospective test in advance. If future measurements agree, the result is dismissed as something the author will merely “argue” demonstrates predictive power; if they disagree, the reviewer “presumes” that the author will alter the formula. This framing discounts prospective success in advance while replacing possible failure with unsupported speculation about future conduct. That speculation is professionally inappropriate and, more importantly, does not address the scientific status of the prediction actually presented, which is already fixed, prospective, and falsifiable.

This is an instance in which the reviewer’s counterargument inadvertently demonstrates the paper’s central claim. The reviewer’s reformulation conflates fitted parameters, measured physical quantities, and an ordinal counting index. In the empirical law, a and b are coefficients obtained from the measured relation between Δν_max and ν_max; ν_max is the measured physical quantity that defines the law; and n is only an integer counter used to propagate the recurrence.

Δν_max = a + bν_max.

Replacing n by Z − Z₀, k, p, or any other integer counter changes only the label used to count successive steps; it does not replace ν_max as the measured variable that defines the empirical law. For consecutive elements,

Z − Z₀ = n − n₀,

so Z − Z₀ can of course serve as an ordinal counter. But this contains no spectral information. If Z genuinely replaced ν_max as the variable defining the law, the corresponding spacing would simply be

ΔZ = 1,

which contains no analogue of the nontrivial, scale-dependent relation

Δν_max = a + bν_max.

Atomic number therefore supplies ordering, not the measured variation in spectral spacing from which the predictive law is obtained. The law was not, and cannot be, discovered as a function of atomic number because Z cannot substitute for ν_max.

The reviewer’s reformulation works only because it retains the spectral coefficients a and b, the measured spectral anchor, and the already-established spectral recurrence. The role of Z − Z₀ is only to specify how many times that spectral relation is propagated.

The reviewer’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 reviewer’s counterargument demonstrates the opposite of what it appears to suggest. In attempting to replace the spectral formulation by a function of Z, the reviewer can replace only the counting index n with Z − Z₀, while the organizing variable ν_max, the spectral-spacing relation, and all of the empirical information derived from them remain indispensable. That is exactly why Z cannot replace ν_max as the organizing coordinate.

The reviewer 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 reviewer 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.

That is an interesting question, but it is not the question addressed here. This comment extends the same theory-first premise expressed in the preceding objections. The reviewer does not identify a failure of the empirical law, its validation, or its predictions. Instead, he states that the work might have interest for physicists if it were used to illuminate the many-electron Dirac equation or QED corrections.

That requirement reverses the scientific question posed by the manuscript. The work asks whether measured spectral quantities contain sufficient empirical structure to organize and predict heavy- and superheavy-element K-edge behavior without requiring those theoretical constructions as inputs. The reviewer instead assumes that the value of the empirical result depends on its ability to contribute back to those same constructions.

The relevant technical question is therefore not whether the empirical law can be translated into the Dirac equation or QED corrections, but whether the law exists, whether it survives independent validation, and whether its fixed predictions are quantitatively successful and falsifiable. The reviewer does not demonstrate failure on any of those grounds.

The statement that the work would only “conceivably have interest for physicists” if it were connected back to Dirac/QED is therefore not evidence against the empirical result. It is a statement about what kind of result the reviewer regards as physically interesting. If the reviewer is interested to learn about the new insight that this work provides regarding Dirac-Fock and QED corrections, we recommend reading the latest paper that directly addresses that question [link].

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, or demonstrate lack of novelty.

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 reviewer 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 reviewer 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 reviewer 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 reviewer’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 reviewer 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 reviewer 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 reviewer’s preceding criticism. The reviewer 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 reviewer’s earlier assessment did not evaluate the work according to the objective that the reviewer 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 reviewer 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 reviewer 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 reviewer’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 reviewer 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 reviewer 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 1–2 and 13–18 form the principal set of nearly parallel, converging trajectories, whereas transition-metal Groups 3–12 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 reviewer 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 reviewer 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 reviewer 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 reviewer 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 reviewer 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.