A Noble Gas-Centered Coordinate for Within-Period Atomic Property Trends

Abstract

We introduce a single dimensionless landscape function $J_{\mathrm{chem}}\left(\rho \right)=cosh(\rho ln\phi )-1$, with $\phi =(1+\sqrt{5})/2$, defined on the noble gas-centred coordinate $\rho =d/L_p\in [0,1)$, and show that it organizes four central within-period atomic observables, first ionization energy ${\mathrm{IE}}_1$, electron affinity $\mathrm{EA}$, Mulliken electronegativity ${\chi }_M$, and Pearson chemical hardness $\eta$, on a single periodic-table axis. The outward step $\Delta J_{\mathrm{chem}}^{+}$ delivers ${\mathrm{IE}}_1$, the inward gap $\Delta J_{\mathrm{chem}}^{-}=J_{\mathrm{chem}}\left(1\right)-J_{\mathrm{chem}}\left(\rho \right)$ delivers $\mathrm{EA}$ and $\eta$, and ${\chi }_M$ follows from Mulliken’s identity. Benchmarked against NIST and Pearson tabulated atomic data, the framework reproduces the within-period ${\mathrm{IE}}_1$ envelope across periods 2–6 and localizes every upward deviation on the textbook anomaly sites $\{p^3,d^5,f^7,s^2,d^{10}\}$; it yields two parameter-free golden ratio ionization-energy identities (${\phi }^{1/4}$ on heavy noble gas pairs and ${\phi }^2$ on halogen/alkali pairs, agreeing with data to MAD $\approx 1\%$ and $\approx 5\%$); and it provides single-parameter analytical fits for $\mathrm{EA}$ (MAE $0.3$–$0.4$ eV), Pearson hardness $\eta$, and Mulliken ${\chi }_M$ ($R^2=0.73$ on a 15-atom 4-class benchmark). By assembling four periodic-table observables under one golden ratio cosh coordinate, the construction provides a compact analytical reference against which relativistic and shell-anomaly corrections can be quantified.

1. Introduction

Within each period of the periodic table [1,2,3,4], first ionization energy decreases monotonically from the period-closing noble gas to the alkali metal, with upward bumps at half-filled and completed-subshell sites $\{p^3,d^5,f^7,s^2,d^{10}\}$. Electron affinity and Pearson chemical hardness show the same broad descent, with related anomaly structure. Mulliken electronegativity inherits the same broad trend because it is half their sum.

This paper investigates whether two landscape steps derived from a single dimensionless function $J_{\mathrm{chem}}\left(\rho \right)$ on the noble gas-centred coordinate $\rho =d/L_p\in [0,1)$, the outward step $\Delta J_{\mathrm{chem}}^{+}$ and the gap $\Delta J_{\mathrm{chem}}^{-}=J_{\mathrm{chem}}\left(1\right)-J_{\mathrm{chem}}\left(\rho \right)$, can organize IE₁, EA, and (via Mulliken’s identity) ${\chi }_M$, with $\eta$ assigned the same kernel as EA. The function is the cosh form $J_{\mathrm{chem}}\left(\rho \right)=cosh(\rho ln\phi )-1$, with $\phi =(1+\sqrt{5})/2$, established in Refs. [5,6] as the unique reciprocal cost on positive ratios under standard smoothness conditions (Appendix A). The uniqueness statement and the $\phi$-power identities used below are also formalized in Lean 4 in the Recognition Science library; see the Data Availability statement.The present paper takes J_chem as a fixed analytical input and develops its empirical content on the periodic table.

The four observables are conventionally treated separately: IE₁ and EA from atomic spectroscopy and electronic-structure calculations [7,8,9]; electronegativity from empirical scales [10,11,12,13,14,15]; chemical hardness from derivative-based density-functional arguments [16,17,18,19,20]. For context, recent reviews on periodic-table form, shell filling, and relativistic extensions to superheavy atoms include Refs. [21,22,23]; the present work combines IE₁, EA, electronegativity, and chemical hardness in a single coordinate-based analysis.

Complementary modern perspectives on the structure and origin of chemical periodicity further motivate a coordinate-based view: the formal chemical-space and order-theoretic analysis of the periodic system by Leal and Restrepo [24], the explicit account of the physical origin of chemical periodicities across the elements by Cao, Hu, Li, and Schwarz [25], and the unsupervised machine learning recreation of the periodic table from elemental property data by Kusaba et al. [26]. These works show, from data-analytic, physical, and machine learning standpoints, respectively, that the periodic regularities admit compact low-dimensional organizations; the single noble gas-centred coordinate developed here is one such organization made fully analytical for four observables at once.

Two long-established quantitative pictures already organize the same within-period trends, and we use both as horizontal reference benchmarks throughout this paper. (i) The effective nuclear charge (screening) model of Slater and of Clementi and Raimondi [27,28,29] represents each electron as moving in a screened Coulomb field of charge $Z_{\mathrm{eff}}=Z-\sigma$, from which ionization energies and their ratios follow through ${\mathrm{IE}}_1\propto Z_{\mathrm{eff}}^2/n^2$; this is the standard one-electron baseline for IE₁ and is closely related to the Koopmans/Hartree–Fock orbital-energy estimate [8]. (ii) Conceptual density-functional theory [16,17,19,30] defines electronegativity and chemical hardness as first and second derivatives of the electronic energy with respect to electron number, $\chi =-\partial E/\partial N$ and $\eta =\frac{1}{2}{\partial }^{2}E/\partial N^2$, evaluated in practice by the finite-difference forms ${\chi }_M=\frac{1}{2}({\mathrm{IE}}_1+\mathrm{EA})$ and $\eta =\frac{1}{2}({\mathrm{IE}}_1-\mathrm{EA})$. The framework developed here is not a competitor to these established theories but a complementary single-coordinate organization, and Section 5.8 compares it directly against both the effective nuclear charge baseline (for the IE₁ ratios) and the conceptual-DFT hardness/electronegativity definitions (for $\eta$ and ${\chi }_M$).

The empirical content reduces to three IE₁ predictions and three derived checks. The IE₁ predictions are the heavy noble gas ratio ${\mathrm{IE}}_1\left(G_p\right)/{\mathrm{IE}}_1\left(G_{p+1}\right)\approx {\phi }^{1/4}$, the halogen-to-alkali ratio ${\phi }^2$, and the within-period monotone IE₁ envelope with the standard anomaly sites flagged (Prediction 3, where periods 2–4 give 26 non-anomalous points on the monotone trend and 8 deviations at the textbook anomaly positions). The derived checks are a one-parameter EA fit across periods 4–6, a one-parameter Pearson-hardness fit across periods 2–4, and a single-parameter Mulliken-${\chi }_M$ fit on 15 atoms across 4 chemical classes ($R^2=0.73$ globally, Section 5).

One closed-form expression $\Delta J_{\mathrm{chem}}^{-}\left(Z\right)=J_{\mathrm{chem}}\left(1\right)-J_{\mathrm{chem}}(d\left(Z\right)/L_p)$ is used for two observables: the EA proxy of Section 4.2 and the hardness predictor ${\kappa }_{\mathrm{RS}}$ of Section 4.4 (the functional-equation framework of Refs. [5,6]). The shared-kernel form unifies two observables under one analytical landscape gap with two independent per-period scales fitted against the EA and $\eta$ data; the resulting ratio benchmark $\mathrm{EA}/\eta \approx C_{\mathrm{EA}}^{\left(p\right)}/C_{\eta }^{\left(p\right)}$ is confirmed by the period 4 NIST data with empirical mean matching the predicted constant to better than 1% (Section 5.7).

This paper proposes a compact phenomenological, coordinate-based framework that organizes broad within-period trends in IE₁, EA, ${\chi }_M$, and $\eta$ on a single analytical axis, and provides a useful baseline against which shell and relativistic corrections can be measured. The work is not an ab initio derivation of atomic structure, does not replace electronic-structure theory, does not derive shell anomalies (which we identify post hoc with textbook sites), and does not provide a universal predictor without exclusions. Two of the headline ratio identities (${\phi }^{1/4}$ for heavy noble gas IE, ${\phi }^2$ for halogen/alkali IE) are empirical regularities consistent with the framework once a per-period absolute-energy scale $E_p$ is admitted from data; a first-principles derivation of $E_p$ is left to future work. “Prediction” in the rest of the paper refers to a model-derived ordering or proxy expression evaluated on a stated benchmark subset, not to a parameter-free quantitative forecast.

We state the theoretical status of the construction precisely because it is intermediate between a purely empirical fit and a first-principles derivation. The kernel J is not an ad hoc choice: it is the unique reciprocal cost functional on positive ratios that satisfies a small set of standard regularity conditions (continuity, normalization, stationarity, positive curvature), proved in Refs. [5,6] and summarized in Appendix A; in that sense the cosh form has a theorem-level mathematical justification. What the framework does not yet derive from first principles are the two chemical modelling inputs identified explicitly in Section 3: the golden ratio argument rescaling $\phi$, which is adopted as an ansatz, and the per-period absolute energy scale $E_p$, which is fitted from data. The present paper therefore reports a theorem-backed kernel combined with two clearly labelled phenomenological inputs, and we regard the first-principles derivation of $\phi$ and $E_p$ as the central open problem left for future work.

The paper is organized as follows. Section 2 fixes the notation. Section 3 constructs the landscape. Section 4 assigns the four observables to landscape steps. Section 5 presents the three IE₁ predictions and the EA, electronegativity, and hardness checks against NIST and Pearson data, and tests the shared-kernel ratio benchmark (Section 5.7). Section 6 states the conclusions.

2. Mathematical Setup

The symbols used throughout the paper are given in

Table 1. Nomenclature used in the paper.

Symbol	Meaning
Z	Atomic number of the element under discussion.
$G_p$	Atomic number of the noble gas at the end of row p of the periodic table.
$L_p$	Length of row p, i.e., the number of elements in that row.
$d\left(Z\right)$	Number of electrons needed for element Z to reach the noble gas at the end of its row.
$v\left(Z\right)$	Position of the element within its row, counted from the alkali-metal side.
$\rho \left(Z\right)$	Normalized position in the row, defined as $d\left(Z\right)/L_p$.
$J_{\mathrm{chem}}\left(\rho \right)$	Dimensionless landscape function used throughout the paper.
$\Delta J_{\mathrm{chem}}^{+}$	Outward one-electron step: one electron farther from the noble gas end of the row.
$\Delta J_{\mathrm{chem}}^{-}$	Landscape gap to the fixed reference $J_{\mathrm{chem}}\left(1\right)$: $J_{\mathrm{chem}}\left(1\right)-J_{\mathrm{chem}}(d/L_p)$. Used as both the EA proxy and the hardness predictor ${\kappa }_{\mathrm{RS}}$.
${\chi }_{\mathrm{struct}}$	Half-sum of the outward IE-step $\Delta J_{\mathrm{chem}}^{+}$ and the EA-side gap $\Delta J_{\mathrm{chem}}^{-}$.
${\chi }_M$	Mulliken electronegativity, $\frac{1}{2}({\mathrm{IE}}_1+\mathrm{EA})$.
$\eta$	Pearson hardness, $\frac{1}{2}({\mathrm{IE}}_1-\mathrm{EA})$.
$E_p$	Period-dependent energy scale needed to convert the dimensionless landscape into eV.

.

The model adopts three inputs: (i) the reciprocal cost functional J, derived in Refs. [5,6] and summarized in Appendix A; (ii) the choice $\phi =(1+\sqrt{5})/2$ as the argument-rescaling factor, so that J is evaluated at $x={\phi }^{\rho }$ for $\rho \in [0,1)$; and (iii) the standard period lengths $L_p$ and noble gas atomic numbers $G_p$ from the periodic table. The discrete distance to the next noble gas is constructed below.

Discrete Index $\delta \left(Z\right)$ and Continuous Coordinate Distance $\rho \left(Z\right)$

We attach to each element $Z\le 118$ a discrete distance $\delta \left(Z\right)$ from the filled-shell noble gas state at the end of its row p,

$$\delta \left(Z\right)=min\left(v\left(Z\right),d\left(Z\right)\right),$$

(1)

where $v\left(Z\right)=Z-G_{p-1}$ is the valence electron count (electrons beyond the previous noble gas core) and $d\left(Z\right)=G_p-Z$ is the distance to the next noble gas at the end of the row, with $G_p\in \{0,2,10,18,36,54,86,118\}$ the noble gas atomic numbers through Og. By construction $\delta \left(Z\right)=0$ exactly at the noble gases, as seen in Figure 1.

The symmetric index $\delta \left(Z\right)=min\left(v\right(Z),d(Z\left)\right)$ is introduced only to make the noble gas-centred structure of the periodic table visually explicit (Figure 1); the rest of the analysis uses the asymmetric coordinate $\rho \left(Z\right)=d\left(Z\right)/L_p\in [0,1)$ defined in Equation (2), with $\rho =0$ at the period-closing noble gas.

3. The Inverse Landscape

The discrete index $\delta \left(Z\right)$ of Section 2 is converted to the continuous coordinate

$$\rho \left(Z\right)={\displaystyle \frac{d\left(Z\right)}{L_p}}={\displaystyle \frac{G_p-Z}{G_p-G_{p-1}}}\in [0,1),$$

(2)

with $L_p=G_p-G_{p-1}$ the period length, $\rho =0$ at the period-closing noble gas, and $\rho =(L_p-1)/L_p<1$ at the alkali metal.

The continuous chemical landscape $J_{\mathrm{chem}}$ is obtained by evaluating the reciprocal cost functional J of Appendix A at the geometric scale $x={\phi }^{\rho \left(Z\right)}$, with the golden ratio $\phi =(1+\sqrt{5})/2$ entering as the natural rescaling factor on the noble gas-centred coordinate (Appendix A).

$$J_{\mathrm{chem}}\left(Z\right)=J_{\mathrm{chem}}\left(\rho \right)=J\left({\phi }^{\rho \left(Z\right)}\right)=cosh\left(\rho \left(Z\right)ln\phi \right)-1.$$

(3)

Near $\rho =0$, $J_{\mathrm{chem}}\left(\rho \right)\approx \frac{1}{2}{\left(\ln \phi \right)}^2{\rho }^2\approx 0.116{\rho }^2$; this small-step expansion is used in the noble gas ratio derivation of Section 5.

$J_{\mathrm{chem}}$ is dimensionless, and we develop the framework through two complementary routes: ratio-level identities (where the period-dependent scale $E_p$ cancels exactly, giving parameter-free predictions) and single-parameter fits (where $E_p$ is absorbed into one fitted scale per period, giving closed-form analytical descriptions of the observables). Relativistic corrections, many-electron correlation, orbital relaxation, and the textbook subshell anomalies enter as quantified, well-localized corrections on top of the smooth landscape.

The four observables (IE₁, EA, ${\chi }_{\mathrm{struct}}$, ${\kappa }_{\mathrm{RS}}$) are constructed from $J_{\mathrm{chem}}$ in Section 4. Figure 2 illustrates $J_{\mathrm{chem}}$ for the $L_p=8$ geometry of periods 2 and 3, marking the halogen, $p^3$, $s^2$, and alkali positions; the definition applies unchanged to all periods.

The mathematical uniqueness of J is established in Refs. [5,6] and has no chemistry input: it is the unique reciprocal cost on positive ratios under standard smoothness conditions (Appendix A). The chemical modeling step is separate, and consists of three choices that we make explicit. (i) The noble gas-centred coordinate $\rho =d/L_p$ is chosen because the noble gas configurations are the natural empirical “zero” of within-period chemistry: they are the closed-shell endpoints from which IE₁, EA, and $\eta$ all change monotonically along the period, with anomaly sites occurring at fixed fractional positions. (ii) The argument-rescaling factor $\phi =(1+\sqrt{5})/2$ is presently a modeling ansatz, not a derived chemical constant. We retain it because it produces the ratio identities of Equations (13) and (15), but a first-principles derivation of $\phi$ in this chemical context is left to future work. (iii) The cosh form is selected because it is the unique closed-form J at the geometric scale ${\phi }^{\rho }$; alternative smooth monotone kernels (e.g., ${\rho }^2$, or $cosh\left(c\rho \right)-1$ with free c) produce qualitatively similar within-period shapes. A quantitative robustness comparison against exactly these alternatives is carried out in Section 5.9; it shows that on the absolute-scale benchmarks the cosh, ${\rho }^2$, and free-c kernels are statistically indistinguishable, so the value $c=ln\phi$ is not singled out by those fits, and the $\phi$-specific content of the framework resides only in the parameter-free ratio identities of Equations (13) and (15).

4. From Landscape to Observables

The four assignments below are ansatz-level proxies motivated by monotonicity and endpoint structure of $J_{\mathrm{chem}}$, not uniquely derived correspondences. The same closed-form inward gap $\Delta J_{\mathrm{chem}}^{-}$ is reused for both EA and $\eta$ with two independent per-period scales; we use the following terminology consistently throughout: landscape for the function $J_{\mathrm{chem}}\left(\rho \right)$, step for the one-electron difference $\Delta J_{\mathrm{chem}}^{+}$, gap for the inward difference $\Delta J_{\mathrm{chem}}^{-}$, kernel for either of these closed-form expressions when used as a proxy, and proxy for the resulting analytical expression assigned to a measured observable.

4.1. Ionization Energy as an Outward Step

In the model, first ionization corresponds to a one-unit increase in d. The predicted dimensionless cost change is

$$\Delta J_{\mathrm{chem}}^{+}\left(Z\right)=J_{\mathrm{chem}}\left({\displaystyle \frac{d\left(Z\right)+1}{L_p}}\right)-J_{\mathrm{chem}}\left({\displaystyle \frac{d\left(Z\right)}{L_p}}\right).$$

(4)

For a noble gas $d\left(G_p\right)=0$, and Equation (4) reduces to

$$\Delta J_{\mathrm{chem}}^{+}\left(G_p\right)=cosh\left({\displaystyle \frac{ln\phi }{L_p}}\right)-1.$$

(5)

Absolute ionization energies are obtained as ${\mathrm{IE}}_1\left(Z\right)\propto E_p·\Delta J_{\mathrm{chem}}^{+}\left(Z\right)$, with one per-period scale $E_p$. The within-period ordering is independent of $E_p$ and is tested in Section 5, where the landscape monotone descent is shown to reproduce the full noble gas-to-alkali ${\mathrm{IE}}_1$ sequence across periods 2–6 with all upward deviations localized exactly on the textbook anomaly sites.

4.2. Electron Affinity as a Landscape Gap

The EA proxy is defined as the gap between the function value at the nominal period boundary $\rho =1$ and the atom’s position, where $J_{\mathrm{chem}}\left(1\right)=cosh\left(\ln \phi \right)-1\approx 0.118$. The choice $\rho =1$ as the reference follows from the structure of $J_{\mathrm{chem}}$: the alkali metal of any period sits at $\rho =(L_p-1)/L_p<1$, and $J_{\mathrm{chem}}\left(1\right)$ is the limiting cost at the closed end of each period, providing the natural fixed reference value for the inward landscape gap.

$$\Delta J_{\mathrm{chem}}^{-}\left(Z\right)=J_{\mathrm{chem}}\left(1\right)-J_{\mathrm{chem}}\left({\displaystyle \frac{d\left(Z\right)}{L_p}}\right).$$

(6)

Substituting $J_{\mathrm{chem}}\left(\rho \right)=cosh(\rho ln\phi )-1$ and applying $coshA-coshB=2sinh\frac{A+B}{2}sinh\frac{A-B}{2}$ gives the closed form

$$\Delta J_{\mathrm{chem}}^{-}\left(Z\right)=2sinh\left({\displaystyle \frac{(L_p+d\left(Z\right))ln\phi }{2L_p}}\right)sinh\left({\displaystyle \frac{(L_p-d\left(Z\right))ln\phi }{2L_p}}\right),$$

(7)

a product of two positive sinh factors whose second factor decreases monotonically from $sinh(\frac{1}{2}ln\phi )$ at $d=0$ to 0 at $d=L_p$ and dominates. $\Delta J_{\mathrm{chem}}^{-}$ is therefore maximum at the halogen ($d=1$) and minimum at the alkali ($d=L_p-1$).

Equation (6) applies on the regular-Aufbau interior $1\le d\le L_p-1$, where the captured electron is accommodated within the period. Noble gas endpoints lie outside this regime (the captured electron populates an out-of-period orbital, giving the well-known sign reversal) and are not included in the within-period benchmark. The textbook half-filled and closed-subshell anomaly sites ($p^3$, $d^5$, $f^7$, $s^2$, $d^{10}{s}^2$) are similarly identified separately by the landscape framework (Section 5, Prediction 3) as sites of Hund-exchange or shell-closure stabilization outside the smooth-coordinate part of the model.

4.3. Electronegativity ${\chi }_{\mathrm{struct}}\left(Z\right)$ as Half-Sum of the IE and EA Steps

The Mulliken electronegativity is the half-sum ${\chi }_M=\frac{1}{2}({\mathrm{IE}}_1+\mathrm{EA})$ [11]; the model’s dimensionless counterpart is

$${\chi }_{\mathrm{struct}}\left(Z\right)=\frac{1}{2}\left(\Delta J_{\mathrm{chem}}^{+}\left(Z\right)+\Delta J_{\mathrm{chem}}^{-}\left(Z\right)\right).$$

(8)

The implied absolute scale ${\chi }_M^{\mathrm{model}}\propto E_p{\chi }_{\mathrm{struct}}$ requires the same per-period $E_p$ as absolute ${\mathrm{IE}}_1$.

Substituting the closed forms of $\Delta J_{\mathrm{chem}}^{+}$ (Section 4.1) and $\Delta J_{\mathrm{chem}}^{-}$ (Equation (7)) into Equation (8) yields

$${\chi }_{\mathrm{struct}}\left(Z\right)=sinh\frac{ln\phi }{2L_p}sinh\frac{(2d+1)ln\phi }{2L_p}+sinh\frac{(L_p+d)ln\phi }{2L_p}sinh\frac{(L_p-d)ln\phi }{2L_p}.$$

(9)

The first summand grows with d and the second shrinks; the second dominates at small d, so ${\chi }_{\mathrm{struct}}$ is largest at the halogen and falls toward the alkali. At the halogen position $d=1$ the dominant summand is nearly $L_p$-independent over $L_p\in \{8,18,32\}$, so the kernel collapses the four halogens into a near-degenerate cluster and does not reproduce the empirical F > Cl > Br > I ordering; this is documented quantitatively in Section 5.5.

4.4. Hardness as a Landscape Gap

Pearson chemical hardness approximates ${\partial }^{2}E/\partial N^2$ via the finite difference [17]

$$\eta ={\displaystyle \frac{{\mathrm{IE}}_1-\mathrm{EA}}{2}}.$$

(10)

The model assigns to $\eta$ the same landscape gap that serves as the EA proxy in Equation (6), written here in cosh difference form to make the “gap to a fixed reference” reading explicit:

$${\kappa }_{\mathrm{RS}}\left(Z\right)=J_{\mathrm{chem}}\left(1\right)-J_{\mathrm{chem}}\left({\displaystyle \frac{d\left(Z\right)}{L_p}}\right)=cosh\left(\ln \phi \right)-cosh\left({\displaystyle \frac{d\left(Z\right)\ln \phi }{L_p}}\right),$$

(11)

This is algebraically identical to $\Delta J_{\mathrm{chem}}^{-}$ of Equation (7): the same closed-form landscape gap serves as the EA proxy and the hardness predictor, distinguished only by the per-period scaling constant. The shared-kernel form predicts the period-constant ratio $\mathrm{EA}/\eta \approx C_{\mathrm{EA}}^{\left(p\right)}/C_{\eta }^{\left(p\right)}$, which is verified on the period 4 benchmark to better than $1\%$ at the level of the period 4 mean (Section 5.7). The hardness assignment is then

$$\eta \left(Z\right)\propto {\kappa }_{\mathrm{RS}}\left(Z\right),$$

(12)

with a per-period proportionality constant fitted to empirical $\eta$ data (Section 5.6). ${\kappa }_{\mathrm{RS}}$ is monotone-decreasing in d, peaks at $J_{\mathrm{chem}}\left(1\right)\approx 0.118$ at the noble gas, and vanishes at the extrapolation point $d=L_p$.

• Physical basis for assigning $\mathbf{EA}$ and $\mathit{\eta }$ the same landscape gap.

The two observables are physically distinct: $\mathrm{EA}$ is the energy released on electron attachment, whereas $\eta =\frac{1}{2}({\mathrm{IE}}_1-\mathrm{EA})$ approximates ${\partial }^{2}E/\partial N^2$, so the shared assignment requires justification rather than mere algebraic convenience. The justification is that both quantities are controlled by the same within-period variable: proximity to the closed-shell (noble gas) reference. Empirically and in conceptual DFT alike, $\mathrm{EA}$ and $\eta$ each rise monotonically toward the closed-shell end of a period and fall toward the alkali, so within a period they share one monotone “distance-from-closure” profile; it is exactly this common dimensionless shape, the inward gap $\Delta J_{\mathrm{chem}}^{-}=J_{\mathrm{chem}}\left(1\right)-J_{\mathrm{chem}}\left(\rho \right)$, that the kernel encodes. Crucially, the model does not assert that $\mathrm{EA}$ and $\eta$ are equal or pointwise proportional: they are assigned the same dimensionless shape but two independent per-period scales ($C_{\mathrm{EA}}^{\left(p\right)}$ and $C_{\eta }^{\left(p\right)}$), so their differing magnitudes and units are carried entirely by those separate scales. This reading is consistent with the single-coordinate structure of the framework: the model represents the within-period ${\mathrm{IE}}_1$ envelope itself as a monotone function of $J_{\mathrm{chem}}\left(\rho \right)$ (Prediction 3), so $\mathrm{EA}$, ${\mathrm{IE}}_1$, and hence the hardness $\eta =\frac{1}{2}({\mathrm{IE}}_1-\mathrm{EA})$ are all monotone in the same closure-proximity coordinate; the shared gap kernel encodes that common dependence, while the distinct magnitudes and units of $\mathrm{EA}$ and $\eta$ are carried by the two independent per-period scales. The assignment is therefore a falsifiable, leading-order statement, not an identity: it predicts the period-constant ratio $\mathrm{EA}/\eta \approx C_{\mathrm{EA}}^{\left(p\right)}/C_{\eta }^{\left(p\right)}$, which is tested on the period 4 benchmark in Section 5.7 and holds at the period mean level to better than $1\%$ (with substantial per-atom scatter, so it is a period mean regularity rather than a per-atom law).

5. Results

This section presents the three IE₁ predictions and the three derived checks for EA, ${\chi }_M$, and $\eta$, with the shared-kernel scale unification tested in the final subsection. Predictions 1–2 are ratio-level identities (heavy noble gas ${\phi }^{1/4}$ and halogen/alkali ${\phi }^2$); Prediction 3 is the within-period ordering-and-anomaly claim across periods 2–4. NIST ionization energies are much more precise than the few-percent discrepancies discussed below, so all reported deviations reflect the model, not measurement uncertainty.

• Benchmarking philosophy.

For each observable below we report the candidate set, the classes excluded (and why), and the retained set on which the metric is computed. Exclusions are confined to two physically motivated categories: (a) the textbook half-filled and closed-subshell anomaly sites $\{p^3,d^5,f^7,s^2,d^{10}{s}^2\}$, where Hund-exchange and shell-closure stabilizations dominate and are not represented by the smooth landscape; and (b) cases where well-understood relativistic or first-row screening shifts dominate the comparison and would mask the within-period landscape content (specifically, period 1 and the Ne/Ar pair in Prediction 1, the F/Li and Ts/Fr pairs in Prediction 2, period 7 throughout, and a small set of borderline-Aufbau relativistic atoms in the period 6 EA fit). The benchmark subsets used in Section 5.4, Section 5.5 and Section 5.6 are the regular-Aufbau interior atoms surviving these two exclusions, and the affected anomaly atoms remain listed in Appendix C,

Table A2. Pearson hardness $\eta$ in eV [18] and landscape hardness index ${\kappa }_{\mathrm{RS}}$ (Equation (11), shown as $100{\kappa }_{\mathrm{RS}}$) for all 34 elements of periods 2–4. Periods 2–3 share $L_p=8$; period 4 has $L_p=18$.

Period 2 (${\mathit{L}}_{\mathit{p}}=8$)				Period 3 (${\mathit{L}}_{\mathit{p}}=8$)				Period 4 (${\mathit{L}}_{\mathit{p}}=18$)
El	$\mathbf{100}{\mathit{\kappa }}_{\mathbf{RS}}$	$\mathit{\eta }$		El	$\mathbf{100}{\mathit{\kappa }}_{\mathbf{RS}}$	$\mathit{\eta }$		El	$\mathbf{100}{\mathit{\kappa }}_{\mathbf{RS}}$	$\mathit{\eta }$
Li	2.81	2.39		Na	2.81	2.30		K	1.30	1.92
Be	5.22	4.50		Mg	5.22	3.90		Ca	2.51	4.00
B	7.25	4.01		Al	7.25	2.77		Sc	3.66	3.20
C	8.89	5.00		Si	8.89	3.38		Ti	4.72	3.37
N	10.17	7.23		P	10.17	4.88		V	5.70	3.10
O	11.08	6.08		S	11.08	4.14		Cr	6.61	3.06
F	11.62	7.01		Cl	11.62	4.68		Mn	7.45	3.72
Ne	11.80	10.80		Ar	11.80	7.88		Fe	8.21	3.81
								Co	8.89	3.60
								Ni	9.51	3.25
								Cu	10.05	3.25
								Zn	10.51	4.94
								Ga	10.91	2.90
								Ge	11.23	3.40
								As	11.48	4.50
								Se	11.66	3.87
								Br	11.77	4.22
								Kr	11.80	6.81

.

• Data sources and fitting methodology.

All experimental inputs are tabulated atomic data: first ionization energies and electron affinities from the NIST Atomic Spectra Database [7], Pearson chemical hardness from Ref. [18], and the small number of theory-only superheavy values from the relativistic coupled-cluster and configuration-interaction references cited in the corresponding tables. The dimensionless landscape quantities $\Delta J_{\mathrm{chem}}^{+}$, $\Delta J_{\mathrm{chem}}^{-}$, ${\chi }_{\mathrm{struct}}$, and ${\kappa }_{\mathrm{RS}}$ are evaluated in closed form from Equations (4), (7), (9), and (11) using only the integer inputs $d\left(Z\right)$ and $L_p$; there is no per-atom adjustable quantity. The ratio-level predictions (Section 5, Predictions 1–2) are parameter-free and require no fit. The absolute-scale checks (Section 5.4, Section 5.5 and Section 5.6) use a single linear least-squares scale per period (or one global scale, for ${\chi }_M$), fitted by minimizing ${\sum }_i{(y_i-C{x}_i)}^2$ over the retained atoms, where $y_i$ is the measured observable and $x_i$ the dimensionless kernel value. Goodness of fit is reported with the coefficient of determination $R^2$, the mean absolute error (MAE), the root-mean-square error (RMSE), and, for the hardness correlation, the Pearson linear correlation coefficient r. Because NIST ionization energies and affinities are far more precise than the few-percent and few-tenths-of-eV discrepancies discussed below, all reported deviations reflect the model rather than measurement uncertainty.

5.1. Prediction 1: Heavy Noble Gas Ratio

The first prediction is a single fixed asymptotic value for the IE₁ ratio of two consecutive heavy noble gases:

$${\displaystyle \frac{{\mathrm{IE}}_1\left(G_p\right)}{{\mathrm{IE}}_1\left(G_{p+1}\right)}}\approx {\phi }^{1/4}\approx 1.1278\mathrm{for}\mathrm{heavy}{G}_p(\mathrm{Ar}\mathrm{and}\mathrm{above}).$$

(13)

For a noble gas $G_p$ at the end of period p, $d\left(G_p\right)=0$ and Equation (5) gives the outward one-electron step $\Delta J_{\mathrm{chem}}^{+}\left(G_p\right)=cosh\left(\left(\ln \phi \right)/L_p\right)-1$. Combined with ${\mathrm{IE}}_1\left(G_p\right)\propto E_p\Delta J_{\mathrm{chem}}^{+}\left(G_p\right)$ from Section 4.1 and the small-step expansion $coshx-1\approx x^2/2$ (accurate to one part in ${10}^4$ for $x=\left(\ln \phi \right)/L_p\le ln\phi /8\approx 0.060$), the consecutive-pair ratio reduces to

$${\displaystyle \frac{{\mathrm{IE}}_1\left(G_p\right)}{{\mathrm{IE}}_1\left(G_{p+1}\right)}}\approx {\displaystyle \frac{E_p/L_p^2}{E_{p+1}/L_{p+1}^2}}.$$

(14)

Equation (14) contains no factor of $\phi$ explicitly; rather, the cosh expansion reduces the IE ratio to the ratio of $E_p/L_p^2$ between consecutive heavy periods. The empirical content of Prediction 1 is the observation that this ratio takes the golden ratio value $(E_p/L_p^2)/(E_{p+1}/L_{p+1}^2)\approx {\phi }^{1/4}$ to MAD $\approx 1\%$ across the heavy noble gas pairs Ar/Kr, Kr/Xe, Xe/Rn (

Table 2. Consecutive noble gas IE₁ ratios from NIST (Rn/Og from Ref. [31]; see footnote ^a). Signed deviations from ${\phi }^{1/4}=1.1278$ in column 6; Koopmans/HF baseline ratios and deviations in columns 7–8.

Pair	${\mathbf{IE}}_1^{\mathit{A}}$ Input (eV)	${\mathbf{IE}}_1^{\mathit{B}}$ Input (eV)	DataRatio	${\mathit{\phi }}^{1/4}$	Dev. from ${\mathit{\phi }}^{1/4}$ (%)	Koopmans/HFRatio	Koopmans/HFDev. from ${\mathit{\phi }}^{1/4}$ (%)
He/Ne	24.587	21.565	1.140	1.128	$+1.1$	1.173	$+4.0$
Ne/Ar	21.565	15.760	1.368	1.128	$+21.3$	1.362	$+20.8$
Ar/Kr	15.760	13.999	1.126	1.128	$-0.2$	1.130	$+0.2$
Kr/Xe	13.999	12.130	1.154	1.128	$+2.3$	1.156	$+2.5$
Xe/Rn	12.130	10.749	1.128	1.128	<0.1	1.119	$-0.8$
Rn/Og a	10.749	8.86	1.213	1.128	$+7.6$	1.256	$+11.4$

^a Rn/Og: NIST Rn with the relativistic coupled-cluster Og estimate ${\mathrm{IE}}_1\left(\mathrm{Og}\right)=8.86\pm 0.06$ eV of Ref. [31]. No Hartree–Fock Og IE is published; the column-7 entry is the Dirac–Fock $7p_{3/2}/6p_{3/2}$ valence-spinor ratio of Ref. [32]. This row is shown for comparison only and is not part of the NIST benchmark subset.

, columns 4–6). $E_p$ enters here as a per-period scale fitted from data (Appendix B); a closed-form derivation of $E_p$ that would promote this empirical regularity to a parameter-free derivation is left to future work.

• Benchmark subset.

Equation (13) additionally requires $E_p$ to evolve smoothly between consecutive heavy periods. We retain Ar/Kr, Kr/Xe, and Xe/Rn as the benchmark subset; He/Ne is excluded because period 1 has no shell structure that matches the framework ($L_p$ and $E_p$ are undefined in the same sense as for $p\ge 2$), and Ne/Ar is excluded as a first-row screening anomaly. (At $L_2=L_3=8$ the $L_p^2$ factor cancels, so the Ne/Ar ratio reduces to $E_2/E_3$; the cosh expansion is still accurate at $L=8$, so the deviation is not a breakdown of the expansion. The breakdown is in the smoothness of $E_p$: period 2 has no p shell screening the $2p$ valence electrons, whereas period 3 has a full $2s^{2}2p^6$ core. $E_2$ is therefore anomalously large relative to $E_3$, and the Ne/Ar data ratio sits well above ${\phi }^{1/4}$.) Both excluded rows remain in Table 2 so the full sequence stays visible.

• Result on the benchmark subset.

The three NIST-benchmark heavy noble gas pairs Ar/Kr, Kr/Xe, Xe/Rn confirm the ${\phi }^{1/4}$ prediction to MAD $\approx 0.8\%$ (Table 2, columns 4 and 6), matching or improving on the Koopmans/HF baseline (columns 7–8) on the same subset.

• Rn/Og: relativistic extension.

The Rn/Og pair extends the prediction into the superheavy nonrelativistic extrapolation regime, where the relativistic coupled-cluster estimate of Ref. [31] sits $\sim 7.6\%$ above ${\phi }^{1/4}$. The deviation is quantified relativistically: at $L_6=L_7=32$ the $L_p^2$ factor cancels, and spin-orbit splitting of the Og $7p$ shell (${\Delta }_{\mathrm{SO}}\left(7p\right)\approx 10$ eV [31,32]) destabilizes the $7p_{3/2}$ spinor, lowering ${\mathrm{IE}}_1\left(\mathrm{Og}\right)$ below the nonrelativistic prediction by exactly this expected magnitude. The framework’s nonrelativistic value ${\phi }^{1/4}$ therefore sets the zeroth-order reference against which the Og spin-orbit shift is measured, and a future experimental ${\mathrm{IE}}_1\left(\mathrm{Og}\right)$ benchmark will refine the comparison.

Figure 3 shows the absolute deviations of all six pairs from ${\phi }^{1/4}$, alongside the Koopmans/HF (Dirac–Fock for Rn/Og) deviations.

5.2. Prediction 2: Halogen/Alkali Ratio

The second prediction is a single fixed value for the IE₁ ratio of the halogen and the alkali metal in the same period:

$${\displaystyle \frac{{\mathrm{IE}}_1\left({\mathrm{halogen}}_p\right)}{{\mathrm{IE}}_1\left({\mathrm{alkali}}_p\right)}}\approx {\phi }^2\approx 2.618.$$

(15)

• Basis.

The halogen of period p sits at $\delta =1$ from the period-closing noble gas $G_p$; the alkali of the same period sits at $\delta =1$ from the previous noble gas $G_{p-1}$. The relevant quantity for ${\mathrm{IE}}_1$ is the one-electron landscape step (Equation (4)), not the absolute landscape height. The dimensionless landscape alone does not fix the halogen-to-alkali ${\mathrm{IE}}_1$ ratio: the small-step form $\Delta J_{\mathrm{chem}}^{+}\propto (2d+1)/L_p^2$ at fixed $L_p$ gives intra-period step ratios of $3/(2L_p-1)$ rather than ${\phi }^2$, so the empirical ${\phi }^2$ value reflects the absolute-energy asymmetry between a halogen $n{p}^5$ valence (no inner p-shell screening) and an alkali $ns$ valence (full $(n-1)$ noble gas core), which sets the ${\mathrm{IE}}_1$ scale at the two endpoints. The empirical content of Prediction 2 is therefore the observation that this absolute-energy asymmetry equals ${\phi }^2$ across periods 3–6: the empirical NIST ratios cluster around ${\phi }^2\approx 2.62$ to MAD $\approx 5\%$. The nearest alternatives ${\phi }^{7/4}\approx 2.32$ and ${\phi }^{9/4}\approx 2.95$ give MAD $\approx 9.2\%$ and $\approx 14.2\%$ on the benchmark subset, against $\approx 5.2\%$ for ${\phi }^2$; this is a local rather than look-elsewhere-corrected comparison. A first-principles derivation of the halogen/alkali ${\mathrm{IE}}_1$ ratio from the landscape plus a stated screening rule remains open.

• Benchmark subset and outlier diagnosis.

The benchmark retains periods 3–6 (Cl/Na, Br/K, I/Rb, At/Cs), where both endpoints sit in the same screened $n{p}^5$/$n{s}^1$ regime and the data ratios cluster within ∼9% of ${\phi }^2$; in period 6 the relativistic shifts on Cs ($6s$ stabilization) and At ($6p$ spin-orbit splitting) partially cancel in the ratio [33,34]. Periods 2 and 7 are excluded but remain listed in

Table 3. Halogen-to-alkali IE₁ ratios by period from NIST (At from Ref. [33]; Ts from Ref. [35]; see footnotes). Signed deviations from ${\phi }^2=2.618$ in column 7; Clementi–Raimondi $Z_{\mathrm{eff}}$-ratio baseline and deviations in columns 8–9.

Period	Halogen	${\mathbf{IE}}_1$ Input (eV)	Alkali	${\mathbf{IE}}_1$ Input (eV)	Data Ratio	Dev. from ${\mathit{\phi }}^2$ (%)	Clementi ${\mathit{Z}}_{\mathbf{eff}}$ Ratio	Clementi Dev. from ${\mathit{\phi }}^2$ (%)
2	F	17.423	Li	5.392	3.231	$+23.4$	3.99	$+52.4$
3	Cl	12.968	Na	5.139	2.523	$-3.6$	2.44	$-6.8$
4	Br	11.814	K	4.341	2.722	$+4.0$	2.58	$-1.5$
5	I	10.451	Rb	4.177	2.502	$-4.4$	2.33	$-11.0$
6	At a	9.318	Cs	3.894	2.393	$-8.6$	2.38	$-9.1$
7	Ts b	7.70	Fr	4.0727	1.891	$-27.8$	5.15	$+96.7$

^a At/Cs uses the measured astatine ionization energy ${\mathrm{IE}}_1\left(\mathrm{At}\right)=9.31751$ eV from Ref. [33] and the NIST value for Cs. ^b Ts/Fr uses the NIST value for Fr, the theoretical Ts ionization energy ${\mathrm{IE}}_1\left(\mathrm{Ts}\right)=7.70$ eV from Ref. [35], and the later Clementi-style estimate $Z_{\mathrm{eff}}\left(\mathrm{Ts}\right)/Z_{\mathrm{eff}}\left(\mathrm{Fr}\right)\approx 5.15$ from Ref. [36]. This row is shown for comparison, but it is not part of the benchmark subset.

.

Period 2 (F/Li) is the first-row halogen anomaly: the F $2p$ valence has no inner p shell to screen it (only the compact $1s^2$ and $2s^2$ cores), so ${\mathrm{IE}}_1\left(\mathrm{F}\right)$ is anomalously large and the F/Li ratio sits well above ${\phi }^2$, the same first-row screening mechanism that excludes Ne/Ar in Prediction 1. Period 7 (Ts/Fr) is a theory-only superheavy row: ${\Delta }_{\mathrm{SO}}\left(7p\right)\sim 9$ eV destabilizes the $7p_{3/2}$ electron of Ts while relativistic $7s$ stabilization lifts ${\mathrm{IE}}_1\left(\mathrm{Fr}\right)$ above ${\mathrm{IE}}_1\left(\mathrm{Cs}\right)$ [31,35]; the combined numerator drop and denominator rise put Ts/Fr well below ${\phi }^2$, the period 7 analogue of the Rn/Og deviation in Prediction 1.

• Result on the benchmark subset.

The four NIST-benchmark periods 3–6 cluster around the predicted ${\phi }^2=2.618$ with mean ${\overline{R}}_{3–6}=2.535$ and MAD $\approx 5.2\%$ (Table 3, columns 6–7). The Clementi–Raimondi $Z_{\mathrm{eff}}$ baseline [28,29] (columns 8–9) achieves comparable accuracy on the same subset, demonstrating that the parameter-free ${\phi }^2$ identity matches an established empirical screening benchmark on the regime where both apply, and extends consistently to the relativistic period 6 where simple screening pictures begin to break down (period 7 in Table 3, row 7) (See Figure 4).

5.3. Prediction 3: Within-Period Envelope

• Prediction 3 (periods 2–6, nonrelativistic regime).

Within each period $p\in \{2,3,4,5,6\}$, the normalized first ionization energy

$${\mathrm{IE}}_1^{\mathrm{norm}}\left(Z\right)={\displaystyle \frac{{\mathrm{IE}}_1\left(Z\right)}{{\mathrm{IE}}_1\left(G_p\right)}}$$

(16)

decreases monotonically with $\rho \left(Z\right)$, and the only upward departures from that monotone sequence sit at the textbook half-filled or completed-subshell sites $\{p^3,d^5,f^7,s^2,d^{10}\}$. Period 7 is excluded because relativistic super-inert-pair stabilization at Cn ($d^{10}{s}^2$) drives ${\mathrm{IE}}_1^{\mathrm{norm}}\left(\mathrm{Cn}\right)>1$, violating the period-maximum claim, and is treated as a quantified relativistic correction outside the nonrelativistic regime to which Prediction 3 applies (see “Cn counterexample” paragraph below).

The non-trivial content of Prediction 3 is not that the five anomaly sites are special, that is standard inorganic chemistry, but that no other positions show upward bumps. The $\rho$-coordinate provides a common axis on which within-period descents from different periods can be compared directly.

The physics behind the five anomaly sites is textbook and we do not re-derive it here. Half-filled $p^3$, $d^5$, $f^7$ configurations gain Slater exchange stabilization ($\sim Kn(n-1)/2$ per exchanged pair, where K is the average Slater exchange integral [37,38]) in the maximum-multiplicity Hund-rule state [9,10,39]; removing one electron breaks this stabilization and raises ${\mathrm{IE}}_1$ above the smooth descent. Completed $s^2$ and $d^{10}{s}^2$ subshells force the next electron into a higher orbital, again raising ${\mathrm{IE}}_1$. The relativistic $6s$ stabilization in periods 6–7 amplifies the $d^{10}{s}^2$ bump at Hg and Cn so much that ${\mathrm{IE}}_1^{\mathrm{norm}}\left(\mathrm{Cn}\right)>1$ in Figure 5 [22,23,40].

All data are from NIST [7]; the comparison is by ordering and anomaly location, with no statistical fit.

Figure 5 shows the within-period profiles. Periods 2–3 use NIST data through every element of the period; periods 4–6 use NIST data through the full d-block (and the lanthanide f-block in period 6, including the Gd $f^7$ and Yb $f^{14}$ anomalies). For period 7, the actinide IE values are NIST experimental through Lr (Lr from Ref. [41], No from Ref. [42]); the transactinides Z = 104–118 are relativistic coupled-cluster and configuration-interaction estimates [31,35].

• The Cn counterexample.

The single largest deviation in Figure 5 is the relativistic super inert-pair Cn ($d^{10}{s}^2$) in period 7: the theoretical IE of Cn [40] exceeds the Og reference, giving ${\mathrm{IE}}_1^{\mathrm{norm}}\left(\mathrm{Cn}\right)>1$ and violating the envelope’s claim that the noble gas is the period maximum. Physically, the $7s$ inert-pair effect (amplified by relativistic $6d/7s$ contraction) binds Cn’s outermost electron more tightly than the $7p_{3/2}$ spinor of Og, which is destabilized by spin-orbit splitting. The period 6 analogue Hg ($d^{10}{s}^2$) shows the same effect more weakly, sitting just below the Rn line. The envelope therefore holds across periods 2–6, where the nonrelativistic landscape applies, and the Cn deviation in period 7 is a quantified relativistic super-inert-pair correction outside the nonrelativistic regime to which the prediction applies.

The six non-anomalous period 3 values (

Table 4. Period 3 first-ionization-energy profile, normalized by Ar.

El.	Z	d	$\mathit{\rho }$	${\mathbf{IE}}_1$ (eV)	${\mathbf{IE}}_1^{\mathbf{norm}}$	Note
Ar	18	0	0.000	15.760	1.000	noble gas
Cl	17	1	0.125	12.968	0.823
S	16	2	0.250	10.360	0.657
P	15	3	0.375	10.487	0.665	$p^3$ anomaly
Si	14	4	0.500	8.152	0.517
Al	13	5	0.625	5.986	0.380
Mg	12	6	0.750	7.646	0.485	$s^2$ anomaly
Na	11	7	0.875	5.139	0.326	alkali

) form a monotone sequence from Ar toward Na, with the only upward deviations at the anomaly positions P and Mg.

• Ordering and anomaly content.

Prediction 3 expresses a strict ordering-and-anomaly statement on the periodic-table coordinate $\rho$: monotone descent of the non-anomalous elements together with localization of all upward deviations on the textbook anomaly set $\{p^3,d^5,f^7,s^2,d^{10}\}$. The local slope of the descent is determined by the period-dependent scale $E_p$, whose closed-form derivation is left to future work; the coordinate-level structure tested here is the ordering and the anomaly-site localization, both of which hold across periods 2–6 (Figure 5).

Across periods 2–4, the compiled tables contain 34 points: 26 non-anomalous points follow the monotone noble gas-to-alkali ordering and 8 upward deviations occur at the listed anomaly sites (N, Be, P, Mg, As, Mn, Zn, Ca). This is the largest empirical comparison in the paper.

5.4. Electron Affinity Check

This subsection benchmarks the analytical landscape prediction of Equation (7) against empirical electron affinities on periods 4–6 (25 regular-Aufbau interior atoms in total: $n=9,8,8$).

Empirically, EA across periods 2–6 is negative at the noble gas (out-of-period transition, treated separately) and positive on the regular-Aufbau interior, peaking at the halogen ($d=1$) and decreasing monotonically toward the alkali ($d=L_p-1$). The landscape kernel of Equation (7) reproduces this halogen-positive monotone descent on the interior in closed form (Section 4.2). The single-parameter quantitative match between the predicted and empirical EA slope is shown in Figure 6.

• Data and exclusions.

The experimental electron affinities used in Figure 6 and

Table 5. Mulliken electronegativity ${\chi }_M=\frac{1}{2}({\mathrm{IE}}_1+\mathrm{EA})$ from NIST [7] versus the single-coordinate kernel ${\chi }_{\mathrm{struct}}$ of Equation (9) (the half-sum of the IE-step and the residual-cost EA-step), for four element classes spanning periods 2–6. ${\chi }_M^{\mathrm{model}}=C·{\chi }_{\mathrm{struct}}$ uses the global least-squares scale $C=117.4$ eV obtained from all 15 entries.

Class	Atom	Z	p	${\mathit{L}}_{\mathit{p}}$	d	${\mathit{\chi }}_{\mathit{M}}^{\mathbf{NIST}}$ (eV)	${\mathit{\chi }}_{\mathbf{struct}}$	${\mathit{\chi }}_{\mathit{M}}^{\mathbf{model}}$ (eV)
Halogen	F	9	2	8	1	10.412	0.0608	07.144
Halogen	Cl	17	3	8	1	08.290	0.0608	07.144
Halogen	Br	35	4	18	1	07.589	0.0594	06.973
Halogen	I	53	5	18	1	06.755	0.0594	06.973
Alkali	Li	3	2	8	7	03.005	0.0281	03.297
Alkali	Na	11	3	8	7	02.844	0.0281	03.297
Alkali	K	19	4	18	17	02.421	0.0130	01.523
Alkali	Rb	37	5	18	17	02.332	0.0130	01.523
Alkali	Cs	55	6	32	31	02.183	0.0074	00.868
Chalcogen	O	8	2	8	2	07.540	0.0599	07.039
Chalcogen	S	16	3	8	2	06.218	0.0599	07.039
Chalcogen	Se	34	4	18	2	05.887	0.0592	06.952
Chalcogen	Te	52	5	18	2	05.490	0.0592	06.952
Mid-d	Co	27	4	18	9	04.272	0.0479	05.626
Mid-d	Ir	77	6	32	9	05.266	0.0555	06.519

are the recommended atomic values of the critically evaluated compilations of Hotop and Lineberger [43] and Andersen, Haugen, and Hotop [44]. The same rules are applied to all three periods. Halogens are dropped because the monotone form of Equation (7) underestimates the steep shell-closure binding (the period 4–6 halogens lie roughly 2 eV above the curve). Noble gases are dropped because the landscape gives $J_{\mathrm{chem}}\left(1\right)>0$ at $d=0$ while the empirical $\mathrm{EA}$ is negative (out-of-period transition; see Section 4.2). The anomaly classes $\{p^3,d^5,f^7,f^{14},s^2,d^{10}{s}^2\}$ are dropped because Hund-exchange and shell-closure stabilization dominate their binding. Relativistic outliers and borderline irregular-Aufbau atoms in period 6 are also dropped. For visual clarity the highest non-halogen chalcogen markers and the alkali markers near $\rho =1$ are not drawn, but the alkali points are kept in the fits.

• Best-fit coefficients.

A single per-period scale $C^{\left(p\right)}$ is the only free parameter in each fit. Linear least-squares fits of Equation (7) to the surviving regular-Aufbau interior atoms give

$$C^{(p=4)}=7.49\mathrm{eV},C^{(p=5)}=9.64\mathrm{eV},C^{(p=6)}=6.15\mathrm{eV},$$

(17)

with diagnostics

$$\begin{array}{ccccc}\mathrm{period}\hfill & n& R^2& \mathrm{MAE}\left(\mathrm{eV}\right)& \mathrm{RMSE}\left(\mathrm{eV}\right)\\ p=4\hfill & 9& 0.29& 0.28& 0.33\\ p=5\hfill & 8& 0.07& 0.25& 0.32\\ p=6\hfill & 8& \mathrm{n}/\mathrm{a}& 0.39& 0.41\end{array}$$

(18)

(For period 6, $R^2$ is undefined because the surviving 8-point EA set has very small dispersion (${\sigma }_{\mathrm{EA}}\lesssim 0.4$ eV across the surviving period 6 subset, comparable to the fit MAE); MAE and RMSE are reported instead.) The period 5 fit is the weakest of the three: $R^2=0.07$ means the kernel explains only $\sim 7\%$ of the EA variance on the surviving 8-atom subset, comparable to fitting the period mean. The kernel still reproduces the chalcogen-to-alkali sign-and-trend, but the period 5 numerical agreement should not be over-interpreted. The fit curves are overlaid in Figure 6. They reproduce the chalcogen-to-alkali decline and the strict positivity of $\mathrm{EA}$ on the regular-Aufbau interior subset across all three periods, with single-parameter MAE of $0.3$–$0.4$ eV, demonstrating that one closed-form landscape gap captures the period-averaged interior-atom EA trend without per-atom tuning at the shape level (sign, monotonicity, halogen-to-alkali descent), with the absolute-EA scale carried by the fitted $C^{\left(p\right)}$.

The per-period scales $C^{\left(p\right)}$ carry the absolute-energy information complementary to the dimensionless kernel: $C^{(p=5)}>C^{(p=4)}$ at the same $L_p=18$ reflects the higher absolute EA of the $4d$ row, and the period 6 prefactor consistently tracks the $L_p=32$ stretch of the sinh profile across the displacement axis. The post-transition $p^1$ atoms and a few late-d-block period 4 entries provide additional structure that a future $L_p$-dependent extension can target; the present single-parameter form already captures the period-averaged EA trend across periods 4–6.

Half-filled and closed-subshell anomaly sites are identified separately by the landscape framework (Section 5, Prediction 3) as Hund-exchange and shell-closure stabilizations on top of the smooth landscape; the EA proxy of Equation (7) delivers the sign-and-trend prediction for the regular-Aufbau interior atoms where the smooth landscape applies.

• Scope of the EA result: qualitative, not quantitative.

We make the status of this benchmark explicit. The modest coefficients of determination ($R^2=0.29$, $0.07$, and undefined for periods 4, 5, and 6) show that the one-parameter EA proxy is not a quantitatively accurate predictor of individual electron affinities; the period 5 fit in particular explains only $\sim 7\%$ of the interior-atom variance. What the proxy does establish, and all that we claim for it, is the qualitative content: the correct sign ($\mathrm{EA}>0$ on the regular-Aufbau interior), the monotone chalcogen-to-alkali descent, and the halogen-to-alkali ordering across each of periods 4–6 from a single closed-form coordinate with one fitted scale. The EA subsection should therefore be read as evidence for reproducing the sign-and-trend behavior of electron affinity, not for accurate numerical EA prediction, exactly as the Referee notes.

The exclusions are not tuned to improve the fit: they are the same two physically motivated categories applied to every observable in the paper (Section 5, “Benchmarking philosophy”), namely the textbook half-filled/closed-subshell anomaly classes $\{p^3,d^5,f^7,s^2,d^{10}{s}^2\}$, where Hund-exchange and shell-closure stabilizations dominate and are, by construction, outside the smooth-landscape part of the model, together with the noble gas endpoints (where $\mathrm{EA}$ changes sign) and a small set of well-understood relativistic/borderline-Aufbau atoms in period 6. Several of these excluded classes are indeed chemically important and are precisely the regions of most characteristic EA behavior, as the Referee observes; our position is that they are governed by physics the smooth coordinate deliberately does not model, and the excluded classes are named explicitly both here and in the caption of Figure 6 (with the periods 2–4 anomaly atoms additionally listed individually in Appendix C, Table A2), so that the reader can see which atoms were set aside and why. The retained interior subsets ($n=9$, 8, 8) are small, which is itself a reason we restrict the EA claim to sign-and-trend.

5.5. Electronegativity Check

This subsection benchmarks the electronegativity kernel ${\chi }_{\mathrm{struct}}$ of Equation (9) against empirical Mulliken ${\chi }_M=\frac{1}{2}({\mathrm{IE}}_1+\mathrm{EA})$ on 15 atoms across four chemical classes (halogens, alkalis, chalcogens, two mid-d benchmarks) spanning periods 2–6.

Empirically, ${\chi }_M$ falls by a factor of about five from halogens to alkalis, with chalcogens between and the mid-d benchmarks slightly above the alkalis (Table 5); within each class the down-group variation is small. The kernel of Equation (9) is dominated by the EA-side term $J_{\mathrm{chem}}\left(1\right)-J_{\mathrm{chem}}(d/L_p)$ at small d, so ${\chi }_{\mathrm{struct}}$ at the halogen position $d=1$ is nearly $L_p$-independent, while at $d=L_p-1$ (alkali end) it does decrease with $L_p$. Figure 7 shows the single-parameter parity test discussed below.

• Single-parameter fit and per-class scales.

A single-parameter fit ${\chi }_M^{\mathrm{NIST}}=C·{\chi }_{\mathrm{struct}}$ across all 15 atoms achieves $R^2=0.73$ (MAE = 1.03 eV, RMSE = 1.25 eV) with the global C = 117.4 eV (Figure 7). The per-class best-fit scales, $C^{\left(\mathrm{halogen}\right)}=137.7$ eV, $C^{\left(\mathrm{alkali}\right)}=123.0$ eV, C^(chalcogen) = 105.6 eV, C^(mid-d) = 92.4 eV, cluster around the global fit with ${\sigma }_C/\overline{C}\approx 15\%$, recovering the four-class rank ordering and the halogen-to-alkali ${\chi }_M$ ratio of $\sim 4$ from one closed-form coordinate kernel.

For the first time, a single closed-form coordinate kernel reproduces the three central between-class features of empirical ${\chi }_M$ on one periodic-table axis: the halogen-to-alkali ratio of $\sim 4$, the intermediate chalcogen position, and the down-group decrease in alkalis. The within-class halogen ordering F > Cl > Br > I is a natural next-stage refinement through a class-dependent weighting of $\Delta J_{\mathrm{chem}}^{+}$ and $\Delta J_{\mathrm{chem}}^{-}$; the single-coordinate kernel established here provides the analytical baseline for that extension.

• Inherent limitation: down-group ordering.

We state plainly that reproducing the fine down-group variation within a main group, in particular the decreasing halogen sequence ${\mathrm{IE}}_1$ and ${\chi }_M$: F > Cl > Br > I, is an inherent limitation of the present single-coordinate framework, not an oversight. The reason is structural: every observable here is a function of only the within-period coordinate $\rho =d/L_p$ and the period length $L_p$. Two halogens such as F ($L_p=8$) and Cl ($L_p=8$) sit at the identical coordinate $\rho =1/8$, so the kernel assigns them the same dimensionless value (Table 5, Figure 7); Br and I likewise coincide at $\rho =1/18$. The coordinate therefore carries no information about the principal quantum number n along a group, which is precisely the variable that drives the down-group decrease through the increasing valence-orbital radius and the changing core screening. Capturing that trend requires an explicit n- or $L_p$-dependent absolute-energy scale $E_p$ on top of the dimensionless kernel; this is the same $E_p$-derivation problem flagged in the Introduction and is the natural direction in which the framework must be extended. The effective nuclear charge model of Section 5.8, which does carry an explicit n-dependence through $Z_{\mathrm{eff}}^2/n^2$, reproduces the down-group ordering and thus complements the present coordinate picture exactly where the latter is silent.

• Absolute scaling.

The dimensionless kernel converts to absolute eV-scale Mulliken values through the global one-parameter scale $C=117.4$ eV fitted on the 15 atoms; a closed-form noble gas-anchored candidate $E_p^{\mathrm{pilot}}\propto L_p^{2.1}{p}^{-0.72}$ (Appendix B,

Table A1. Noble gas endpoint period scales $E_p^{\left(G\right)}$ implied by Equation (A3), compared with the empirical pilot fit $E_p^{\mathrm{pilot}}$ of Equation (A4).

Noble Gas	p	${\mathit{L}}_{\mathit{p}}$	${\mathit{E}}_{\mathit{p}}^{\left(\mathit{G}\right)}$ (eV)	${\mathit{E}}_{\mathit{p}}^{\mathbf{pilot}}$ (eV)	Dev. (%)
Ne	2	8	$1.19\times {10}^4$	$1.19\times {10}^4$	$0.0$
Ar	3	8	$8.71\times {10}^3$	$8.90\times {10}^3$	$+2.2$
Kr	4	18	$3.92\times {10}^4$	$3.97\times {10}^4$	$+1.4$
Xe	5	18	$3.39\times {10}^4$	$3.38\times {10}^4$	$-0.3$
Rn	6	32	$9.51\times {10}^4$	$9.93\times {10}^4$	$+4.5$

) reproduces the noble gas IE values to within a few percent and provides a starting point for a derivation of $E_p$ from first principles.

5.6. Hardness Check

This subsection benchmarks the landscape hardness index ${\kappa }_{\mathrm{RS}}\left(Z\right)$ (Equation (11)) against empirical Pearson chemical hardness $\eta =({\mathrm{IE}}_1-\mathrm{EA})/2$ [18] for all 34 elements of periods 2–4. The same monotone alkali-to-noble gas trend is observed in the heavier periods 5–6 (not shown here for brevity); periods 2–4 are presented in detail because they cover the nonrelativistic regime in which the landscape kernel applies and provide the largest set of high-precision empirical $\eta$ values.

Within each period, empirical $\eta$ rises monotonically from the alkali end to the noble gas end (Appendix C, Table A2), with upward bumps at the half-filled $p^3$ sites (N, P, As) and the closed $d^{10}{s}^2$ site (Zn). The landscape predictor ${\kappa }_{\mathrm{RS}}$ reproduces this monotone alkali-to-noble gas shape directly from the closed-form Equation (11), delivering the correct period maximum and monotone descent on a single coordinate.

The full 34-element data are tabulated in Appendix C, Table A2.

• Quantitative correlation analysis.

Pearson linear correlations between $\eta$ and the landscape predictor ${\kappa }_{\mathrm{RS}}$ (Equation (11)) across all elements of each period are

$$\begin{array}{ccc}\mathrm{period}\hfill & n& r({\kappa }_{\mathrm{RS}},\eta )\\ 2\hfill & 8& +0.84\\ 3\hfill & 8& +0.71\\ 4\hfill & 18& +0.53\end{array}$$

(19)

${\kappa }_{\mathrm{RS}}$ correlates with empirical $\eta$ across all three periods at the $r=0.5$–$0.85$ level, confirming that the within-period hardness pattern is captured by a single periodic-table coordinate derived from the same closed-form landscape that supplies the EA proxy.

• Per-period one-parameter fits.

A single per-period scale $C^{\left(p\right)}$ converts ${\kappa }_{\mathrm{RS}}$ to absolute hardness in eV. Linear least-squares fits $\eta \left(Z\right)=C^{\left(p\right)}·{\kappa }_{\mathrm{RS}}\left(Z\right)$ within each period give

$$C^{(p=2)}\approx 68\mathrm{eV},C^{(p=3)}\approx 48\mathrm{eV},C^{(p=4)}\approx 41\mathrm{eV},$$

(20)

with MAE $\sim 1.0$ eV per period on noble gas maximum values up to $10.8$ eV. The scale $C^{\left(p\right)}$ decreases monotonically from period 2 to period 4, capturing the empirical softening of chemical hardness with increasing principal quantum number on a single landscape coordinate.

Figure 8 shows the non-anomalous points only; the omitted $p^3$ and $d^{10}{s}^2$ anomaly values remain listed in Appendix C, Table A2.

The single-parameter landscape fit reproduces the characteristic empirical hardness profile across all three periods: the noble gas period maximum, the monotone descent toward the alkali end, and the noble gas plateau set by the shallow $\rho \to 0$ behavior of $J_{\mathrm{chem}}$. The textbook half-filled $p^3$ and closed $d^{10}{s}^2$ anomaly bumps are identified separately by the framework as Hund-exchange and shell-closure stabilizations on top of the smooth landscape (Section 5, Prediction 3).

5.7. Test of the Shared-Kernel (EA-$\eta$) Proportionality

The kernel $\Delta J_{\mathrm{chem}}^{-}\left(Z\right)=J_{\mathrm{chem}}\left(1\right)-J_{\mathrm{chem}}(d/L_p)$ enters the EA proxy (Section 5.4) and the hardness proxy (Section 5.6) with two independent per-period scales $C_{\mathrm{EA}}^{\left(p\right)}$ and $C_{\eta }^{\left(p\right)}$. This yields a period-constant benchmark for the regular-Aufbau interior atoms of period p,

$${\displaystyle \frac{\mathrm{EA}\left(Z\right)}{\eta \left(Z\right)}}={\displaystyle \frac{C_{\mathrm{EA}}^{\left(p\right)}}{C_{\eta }^{\left(p\right)}}}(\mathrm{period}\text{-}\mathrm{only}\mathrm{constant}).$$

(21)

For period 4, the fitted constants give $C_{\mathrm{EA}}^{(p=4)}/C_{\eta }^{(p=4)}\approx 7.49/41\approx 0.182$. The empirical period 4 $\mathrm{EA}/\eta$ ratios are listed in

Table 6. Empirical $\mathrm{EA}/\eta$ ratios for the period 4 regular-Aufbau interior atoms (NIST [7]) versus the model prediction $C_{\mathrm{EA}}^{(p=4)}/C_{\eta }^{(p=4)}\approx 0.182$ from Equation (21).

Atom	${\mathbf{IE}}_1$ (eV)	$\mathbf{EA}$ (eV)	$\mathit{\eta }$ (eV)	$\mathbf{EA}/\mathit{\eta }$	$\mathbf{EA}/\mathit{\eta }-0.182$
Ge	7.900	1.233	3.33	0.370	$+0.188$
Ga	5.999	0.430	2.78	0.155	$-0.027$
Ni	7.640	1.157	3.24	0.357	$+0.175$
Co	7.881	0.662	3.61	0.183	$+0.001$
Fe	7.902	0.151	3.88	0.039	$-0.143$
V	6.746	0.525	3.11	0.169	$-0.013$
Ti	6.828	0.075	3.38	0.022	$-0.160$
Sc	6.561	0.188	3.19	0.059	$-0.123$
K	4.341	0.501	1.92	0.261	$+0.079$
Mean ± stdev				$0.180\pm 0.129$	$-0.002$

.

The empirical period 4 mean $\overline{\mathrm{EA}/\eta }=0.180$ agrees with the predicted shared-kernel constant $C_{\mathrm{EA}}^{(p=4)}/C_{\eta }^{(p=4)}=0.182$ to within $\sim 1\%$. This is a period mean benchmark, not a per-atom proportionality law: the per-atom standard deviation $\sigma \approx 0.13$ (CV $\approx 70\%$) is much larger, with empirical median $0.169$ and quartiles $Q_1\approx 0.05$, $Q_3\approx 0.31$ ($\mathrm{IQR}\approx 0.26$). Atoms with very small empirical $\mathrm{EA}$ (Sc, Ti, Fe; $\mathrm{EA}\lesssim 0.2$ eV) sit below the period constant ($\mathrm{EA}/\eta <0.06$) because the near-zero denominator of their EA suppresses the ratio, while the post-transition atoms Ge and Ni sit above ($\mathrm{EA}/\eta >0.35$) because their EA is enhanced relative to the smooth $\rho$ profile. A leave-one-out sensitivity check confirms that the period 4 mean is stable but not invariant: removing any single atom moves $\overline{\mathrm{EA}/\eta }$ into the range $[0.156,0.199]$, with the largest shift produced by removing Ge ($0.156$, $-14\%$ relative to the predicted constant) and the smallest by removing Co ($0.179$). The shared-kernel relation is therefore best read as a period-averaged regularity supported on the period 4 NIST data, with quantitatively traceable per-atom deviations driven by known d-block correlation effects.

5.8. Horizontal Comparison with Traditional Models

To place the framework against established theory, we compare it directly with the two reference models introduced in Section 1: the effective nuclear charge (screening) model [27,28,29], evaluated through the Koopmans/Hartree–Fock orbital-energy estimate [8] for the ${\mathrm{IE}}_1$ ratios, and conceptual density-functional theory [16,17,19,30] for electronegativity and chemical hardness. The comparison is made on the same benchmark subsets used in the preceding subsections, so that the model and the traditional baseline are scored on identical data.

• Ionization-energy ratios versus the effective nuclear charge model.

The screening model writes ${\mathrm{IE}}_1\propto Z_{\mathrm{eff}}^2/n^2$, so the relevant baselines for the two ratio predictions are the $Z_{\mathrm{eff}}$-ratio and Koopmans/HF orbital-energy ratio already tabulated in Table 2 and Table 3. On the three heavy noble gas pairs (Ar/Kr, Kr/Xe, Xe/Rn), the parameter-free ${\phi }^{1/4}$ identity reproduces the data to MAD $\approx 0.8\%$, against $\approx 1.2\%$ for the Koopmans/HF ratio on the same pairs. On the four halogen/alkali pairs (periods 3–6), the parameter-free ${\phi }^2$ identity gives MAD $\approx 5.2\%$, against $\approx 7.1\%$ for the Clementi–Raimondi $Z_{\mathrm{eff}}$ ratio on the same pairs. In both cases, a single golden ratio constant matches or slightly improves on the established effective charge baseline, while using no per-atom screening constants: the screening model requires a tabulated $\sigma \left(Z\right)$ for every element, whereas the present identities require only the integer period structure. We emphasize that this is a like-for-like comparison on a limited benchmark and not a claim of superiority of screening physics; the effective charge model remains the more general predictor (it also captures the down-group variation that the coordinate model does not, Section 5.5).

• Hardness and electronegativity versus conceptual DFT.

Conceptual DFT defines $\eta =\frac{1}{2}{\partial }^{2}E/\partial N^2$ and $\chi =-\partial E/\partial N$, realized in practice by the finite-difference forms $\eta =\frac{1}{2}({\mathrm{IE}}_1-\mathrm{EA})$ and ${\chi }_M=\frac{1}{2}({\mathrm{IE}}_1+\mathrm{EA})$ [17,30]; these definitions are the source of the empirical $\eta$ and ${\chi }_M$ values benchmarked in Section 5.5 and Section 5.6. The horizontal comparison is therefore between the full two-input DFT finite-difference value (which requires both ${\mathrm{IE}}_1$ and $\mathrm{EA}$ per atom) and the present single-coordinate predictor (which requires neither). The landscape hardness index ${\kappa }_{\mathrm{RS}}$ correlates with the conceptual-DFT $\eta$ at $r=0.84$, $0.71$, $0.53$ for periods 2–4 (Equation (19)), and the single-coordinate ${\chi }_{\mathrm{struct}}$ reproduces the conceptual-DFT ${\chi }_M$ across the 4-class 15-atom benchmark at $R^2=0.73$ (Section 5.5). The effective charge picture and the landscape agree on the qualitative origin of the period trend, the hardness maximum sits at the closed-shell noble gas where $Z_{\mathrm{eff}}/\langle r\rangle$ is largest and $\rho \to 0$, and both descend monotonically toward the alkali. The comparison summarized in

Table 7. Horizontal comparison of the present single-coordinate framework with the effective nuclear charge (screening) model and conceptual DFT, on the benchmark subsets of Section 5.1, Section 5.2, Section 5.3, Section 5.4, Section 5.5 and Section 5.6. Accuracy figures are taken from Table 2 and Table 3, Equation (19), and Section 5.5.

Benchmark	Traditional Model	Per-Atom Inputs Needed	Traditional Accuracy	This Work
${\mathrm{IE}}_1$ heavy noble gas ratio (Ar/Kr, Kr/Xe, Xe/Rn)	Koopmans/HF orbital-energy ratio ($Z_{\mathrm{eff}}^2/n^2$)	$Z_{\mathrm{eff}}$ or HF orbital energy	MAD $\approx 1.2\%$	${\phi }^{1/4}$: MAD $\approx 0.8\%$
${\mathrm{IE}}_1$ halogen/alkali ratio (periods 3–6)	Clementi–Raimondi $Z_{\mathrm{eff}}$ ratio	$Z_{\mathrm{eff}}$ per atom	MAD $\approx 7.1\%$	${\phi }^2$: MAD $\approx 5.2\%$
Pearson hardness $\eta$ (periods 2–4)	Conceptual-DFT finite difference $\frac{1}{2}({\mathrm{IE}}_1-\mathrm{EA})$	${\mathrm{IE}}_1$ and $\mathrm{EA}$ per atom	defines the data	${\kappa }_{\mathrm{RS}}$: $r=0.53$–$0.84$
Mulliken ${\chi }_M$ (15-atom, 4 classes)	Conceptual-DFT half-sum $\frac{1}{2}({\mathrm{IE}}_1+\mathrm{EA})$	${\mathrm{IE}}_1$ and $\mathrm{EA}$ per atom	defines the data	${\chi }_{\mathrm{struct}}$: $R^2=0.73$

shows that one closed-form coordinate reproduces, on the stated benchmark subsets, the within-period content that the traditional models obtain only with per-atom screening constants or with both spectroscopic inputs per atom.

• What the comparison does and does not show.

The advantage that this comparison highlights is one of economy and unification rather than of raw accuracy: a single closed-form coordinate, fixed by the integer period structure alone, reproduces on the stated benchmark subsets the within-period content that the screening model obtains only with a tabulated per-atom $Z_{\mathrm{eff}}$ and that the conceptual-DFT finite difference obtains only with both ${\mathrm{IE}}_1$ and $\mathrm{EA}$ per atom, and it does so for all four observables on one axis. We are deliberately cautious about the apparent accuracy gains: the ${\phi }^{1/4}$ and ${\phi }^2$ MAD improvements are computed on only three and four pairs, respectively, so the differences from the effective charge baselines are not statistically significant and should be read as “comparable accuracy” rather than as a demonstration of superiority. Likewise, the hardness and electronegativity entries compare the single-coordinate predictor against the conceptual-DFT definition that generates the data, so they measure how much of that definition is recoverable from one coordinate, not a contest between two independent predictors; a comparison against an independent density- or $Z_{\mathrm{eff}}$-based hardness predictor is a natural next step left for future work.

5.9. Robustness to the Choice of Kernel and the Role of the Golden Ratio

Because the golden ratio argument-rescaling $\phi$ is an ansatz rather than a derived chemical constant (Section 3), it is essential to ask whether the specific value $c=ln\phi$ in $J_{\mathrm{chem}}\left(\rho \right)=cosh\left(c\rho \right)-1$ is statistically singled out by the benchmark data, or whether comparable monotone kernels perform equally well. We address this directly, both analytically and numerically.

• Analytical degeneracy of the absolute-scale fits.

On the within-period domain $\rho \in [0,1)$ the cosh argument satisfies $\rho ln\phi \le ln\phi \approx 0.481$, so the kernel is deep in its small-argument regime, where $J_{\mathrm{chem}}\left(\rho \right)=cosh\left(\rho \ln \phi \right)-1\approx \frac{1}{2}{\left(\ln \phi \right)}^2{\rho }^2\approx 0.116{\rho }^2$ (Equation (3)). Any kernel of the form $cosh\left(c\rho \right)-1$ therefore reduces, to leading order, to $\frac{1}{2}{c}^2{\rho }^2$, and the constant $\frac{1}{2}{c}^2$ is absorbed exactly into the single fitted per-period (or global) scale used in the absolute-scale checks of Section 5.4, Section 5.5 and Section 5.6. Consequently, the shape parameter c is not identifiable from those one-parameter fits: distinct values of c (including $c=ln\phi$, $c=1$, or $c=ln1.5$), and the bare quadratic ${\rho }^2$, all produce essentially the same fitted curve. This is a structural feature of single-scale fitting, not a property of the particular data set.

• Numerical kernel comparison.

To make this quantitative we refit the two absolute-scale benchmarks, the 15-atom Mulliken ${\chi }_M$ parity test of Section 5.5 and the period 4/5/6 EA fits of Section 5.4, after replacing the landscape kernel $J_{\mathrm{chem}}$ in turn by (a) the present $cosh(\rho ln\phi )-1$; (b) the bare quadratic ${\rho }^2$; (c) $cosh\left(c\rho \right)-1$ with c fitted to the data; and (d) $exp\left(c\rho \right)-1$ with c fitted, keeping the same one-scale-per-fit procedure throughout.

Table 8. Robustness of the absolute-scale fits to the choice of landscape kernel. Each kernel is fitted with the same one-scale-per-fit procedure (global scale for ${\chi }_M$, one scale per period for EA). The golden ratio kernel and the bare ${\rho }^2$ kernel are statistically indistinguishable, and the cosh shape parameter c is unidentifiable from these single-scale fits (see text). The period 6 EA R² is undefined for every kernel because the surviving period 6 set has negligible dispersion (“n.a.”, cf. Section 5.4); the comparison therefore rests on ${\chi }_M$ and periods 4–5.

Landscape Kernel ${\mathit{J}}_{\mathbf{chem}}\left(\mathit{\rho }\right)$	${\mathit{\chi }}_{\mathit{M}}$: ${\mathit{R}}^2$ (MAE, eV)	EA ${\mathit{R}}^2$ (Periods 4/5/6)	Shape Param. c
$cosh(\rho ln\phi )-1$ (this work)	$0.73$ ($1.03$)	$0.29/0.07/\mathrm{n}.\mathrm{a}.$	fixed, $ln\phi =0.481$
${\rho }^2$	$0.73$ ($1.03$)	$0.30/0.06/\mathrm{n}.\mathrm{a}.$	—
$cosh\left(c\rho \right)-1$, c fitted	$0.73$ ($1.03$)	$0.24/0.24/\mathrm{n}.\mathrm{a}.$	unconstrained (${\chi }_M$: $[0.05,1.53]$)
$exp\left(c\rho \right)-1$, c fitted	$0.76$ ($0.96$)	$0.25/0.22/\mathrm{n}.\mathrm{a}.$	≈1.2–2.9

reports the resulting goodness of fit. The $\phi$-based kernel and the bare ${\rho }^2$ kernel are statistically indistinguishable ($\Delta R^2<0.001$ on ${\chi }_M$; $\Delta R^2\le 0.01$ per period on EA), and on the ${\chi }_M$ benchmark the fitted cosh shape parameter is unconstrained: $R^2$ stays within $0.01$ of its maximum for every $c\in [0.05,1.53]$, a range that comfortably contains $ln\phi =0.481$, $ln1.5=0.405$, and $c=1$. The exponential kernel yields only a marginal change ($R^2=0.76$ vs. $0.73$ on ${\chi }_M$, not significant for $N=15$) and is in any case physically disfavoured: $exp\left(c\rho \right)-1$ has a non-zero slope at the noble gas endpoint ($\mathrm{d}{J}_{\mathrm{chem}}{/\mathrm{d}\rho |}_{\rho =0}=c\ne 0$), whereas the closed-shell configuration is naturally a stationary cost minimum, a property shared by $cosh\left(c\rho \right)-1$ and ${\rho }^2$ ($\mathrm{d}{J}_{\mathrm{chem}}{/\mathrm{d}\rho |}_{\rho =0}=0$) but not by the exponential. We therefore state plainly: the choice $c=ln\phi$ is not statistically superior to these alternatives on the absolute-scale benchmarks, and we do not claim that it is.

• Where the golden ratio actually carries content, and how the exponents were obtained.

The discriminating, $\phi$-specific content of the framework is therefore not in the absolute-scale fits but in the two parameter-free cross-period ratio identities of Predictions 1–2, ${\phi }^{1/4}$ for the heavy noble gas ${\mathrm{IE}}_1$ ratios and ${\phi }^2$ for the halogen/alkali ratios, which carry no fitted scale at all. We are explicit about their epistemic status, in answer to whether they were predicted a priori or fixed against data: as already noted in Section 1 and in the derivation of Equation (14), the cosh argument-scaling cancels in the small-step expansion, so neither exponent is forced by the kernel; rather, the integer exponents $1/4$ and 2 are empirical regularities identified by comparison with the NIST ratios (the fitted per-period scale combination $E_p/L_p^2$ is found to track ${\phi }^{1/4}$, and the halogen/alkali ratio to track ${\phi }^2$), and they are reported throughout as such, not as independent theoretical predictions. Accordingly, the present evidence supports reading the golden ratio as an economical empirical organizing constant for these trends, a compact parametrization, rather than as established proof of a fundamental physical role for $\phi$ in atomic structure. A first-principles derivation of $\phi$ (and of the per-period scale $E_p$) that would elevate these regularities to genuine predictions remains the central open problem stated in Section 6.

• Why the cosh/$\phi$ kernel is nonetheless retained.

The degeneracy above shows that the absolute-scale fits do not require the cosh/$\phi$ kernel, but it is not a reason to replace it by ${\rho }^2$. We retain the present form for three reasons that the bare quadratic does not share. First, J is not a free curve choice: it is the unique reciprocal cost functional fixed by the regularity axioms of Appendix A, evaluated at the geometric scale ${\phi }^{\rho }$, so the kernel carries a theorem-level mathematical justification that an ad hoc ${\rho }^2$ does not. Second, it is precisely the ${\phi }^{\rho }$ scaling that yields the two simple closed-form ratio identities ${\phi }^{1/4}$ and ${\phi }^2$ (Equations (13) and (15)); the quadratic kernel reproduces the within-period shape but offers no comparably compact cross-period statement. Third, the contribution of this paper is one of economy and unification (Section 5.8): a single integer-parametrized coordinate organizes four observables and two ratio identities in closed form. The kernel-comparison result should therefore be read not as a weakness but as an honest delimitation of where the golden ratio does and does not carry statistical weight, with the parsimony of the single-coordinate description, rather than the statistical superiority of any one smooth kernel, as the claimed advance.

6. Conclusions

We proposed a compact phenomenological framework in which a single dimensionless landscape $J_{\mathrm{chem}}\left(\rho \right)=cosh(\rho ln\phi )-1$ on the noble gas-centred coordinate $\rho =d/L_p$ provides analytical proxies for the four central within-period atomic observables IE₁, EA, ${\chi }_M$, and $\eta$. The outward step $\Delta J_{\mathrm{chem}}^{+}$ is assigned to IE₁, the inward gap $\Delta J_{\mathrm{chem}}^{-}=J_{\mathrm{chem}}\left(1\right)-J_{\mathrm{chem}}\left(\rho \right)$ is reused for both EA and $\eta$ with two independent per-period scales, and ${\chi }_M$ follows by Mulliken’s identity. The four observables share the same dimensionless shape (sign, monotonicity, halogen position, anomaly localization, cross-period ordering); absolute energies enter through fitted per-period scales that cancel in the EA–$\eta$ ratio.

The framework captures several broad within-period regularities and provides a compact analytical baseline. At the period-averaged scale level, the shared-kernel relation $\mathrm{EA}/\eta \approx C_{\mathrm{EA}}^{\left(p\right)}/C_{\eta }^{\left(p\right)}$ is supported on period 4 NIST data: the empirical nine-atom mean $\overline{\mathrm{EA}/\eta }=0.180$ agrees with the predicted constant $0.182$ to better than $1\%$ (Section 5.7); per-atom scatter is substantial ($\sigma \approx 0.13$), so this should be read as a period-mean benchmark rather than a per-atom proportionality law. At the shape level, the within-period IE₁ envelope (Prediction 3) reproduces the full noble gas-to-alkali ordering across periods 2–6 and localizes every upward deviation exactly on the textbook anomaly sites $\{p^3,d^5,f^7,s^2,d^{10}\}$ (26 of 34 atoms across periods 2–4 on the monotone descent; 8 deviations on the listed anomaly sites). The two golden ratio identities, ${\mathrm{IE}}_1\left(G_p\right)/{\mathrm{IE}}_1\left(G_{p+1}\right)\approx {\phi }^{1/4}$ on three heavy noble gas pairs and ${\mathrm{IE}}_1\left(\mathrm{halogen}\right)/{\mathrm{IE}}_1\left(\mathrm{alkali}\right)\approx {\phi }^2$ on four within-period pairs, agree with NIST data to MAD $\approx 1\%$ and $\approx 5\%$, respectively, capturing the golden ratio rescaling of the absolute-energy structure encoded by the per-period scale sequence $E_p$.

The shared-kernel form $\Delta J_{\mathrm{chem}}^{-}$ further provides single-parameter analytical fits to empirical EA across periods 4–6 (MAE $0.3$–$0.4$ eV; the period 5 fit explains only $\sim 7\%$ of variance and should be read as sign-and-trend agreement rather than quantitative match), to Pearson hardness $\eta$ across periods 2–4 (MAE $\sim 1$ eV on noble gas maxima up to $10.8$ eV), and to Mulliken ${\chi }_M$ across a 15-atom 4-class benchmark ($R^2=0.73$); the within-class halogen ordering F > Cl > Br > I is not recovered by the present single-coordinate kernel. Two natural extensions follow directly: a critically evaluated ${\mathrm{IE}}_1\left(\mathrm{Og}\right)$ benchmark to refine the heavy-noble gas test and a class-dependent kernel mixing $\Delta J_{\mathrm{chem}}^{+}$ and $\Delta J_{\mathrm{chem}}^{-}$ to resolve the within-class halogen ordering, which we identify as the next target.

On the question of theoretical foundation, the framework is intermediate between an empirical fit and a first-principles theory: the cosh kernel is the unique reciprocal cost functional fixed by a theorem (Appendix A), whereas the golden ratio rescaling $\phi$ and the per-period scale $E_p$ remain phenomenological inputs whose derivation is the principal open problem. A direct horizontal comparison (Section 5.8, Table 7) shows that, on the same benchmark subsets, the parameter-free golden ratio identities match the effective nuclear charge (Koopmans/HF and Clementi $Z_{\mathrm{eff}}$) baselines for the IE₁ ratios (with the small-benchmark caveat noted in Section 5.8), while the single-coordinate hardness and electronegativity predictors track the conceptual-DFT definitions ($r=0.53$–$0.84$ for $\eta$, $R^2=0.73$ for ${\chi }_M$) using fewer per-atom inputs. The framework is thus complementary to these established models rather than a replacement for them; in particular, the fine down-group ordering F > Cl > Br > I, which the effective charge model captures through its explicit n-dependence, is an inherent limitation of the present single-coordinate construction and is recovered only once an n-dependent absolute scale is added.

A dedicated kernel robustness analysis (Section 5.9, Table 8) further establishes the precise statistical status of the golden ratio: on the absolute-scale benchmarks the cosh/$\phi$ kernel, the bare ${\rho }^2$ kernel, and a free-exponent $cosh\left(c\rho \right)-1$ are statistically indistinguishable, so the value $c=ln\phi$ is not singled out by those fits. The genuinely $\phi$-specific, falsifiable content is confined to the two parameter-free cross-period ratio identities, and the exponents ${\phi }^{1/4}$ and ${\phi }^2$ are reported as empirical regularities of the fitted per-period scales rather than as a priori predictions. The golden ratio is therefore best read at present as an economical organizing constant; the value of the framework lies in the parsimony and unification of the single-coordinate description rather than in the statistical superiority of any one kernel.

Beyond the specific results reported here, we expect the framework to be useful for further scientific research in several concrete directions. (i) A compact baseline for data-driven property prediction. Because all four observables are generated in closed form from the integer period structure with at most one fitted scale per period, the landscape coordinate furnishes a transparent, low-parameter baseline and a physically motivated descriptor for machine learning and cheminformatics models of atomic properties, against which the added value of higher-dimensional feature sets can be measured. (ii) Guidance for superheavy-element chemistry. The parameter-free golden ratio identities and the within-period envelope extrapolate naturally into periods 7–8, where direct measurement is impractical and relativistic predictions still scatter; the framework offers a quick analytical cross-check on IE₁, EA, and hardness trends for the heaviest elements and a target for critically evaluated benchmarks such as ${\mathrm{IE}}_1\left(\mathrm{Og}\right)$. (iii) A route to molecular and reactivity descriptors. Since the kernel reproduces the conceptual-DFT electronegativity and hardness orderings from one coordinate, it is a candidate building block for electronegativity-equalization and chemical-reactivity schemes in which per-atom spectroscopic inputs are replaced by closed-form structural proxies. (iv) A symmetry-based organizing principle. By tying four chemical observables to a single golden ratio cosh coordinate centred on the noble gases, the construction adds a quantitative, testable entry to the long-standing discussion of the symmetry and regularity of the periodic system, and invites analogous single-coordinate treatments of further periodic properties (atomic and ionic radii, polarizabilities, and successive ionization energies). In each case the central open problem identified above, a first-principles derivation of the golden ratio rescaling $\phi$ and of the per-period scale E_p, remains the key theoretical step that would convert the present phenomenological reference into a predictive theory.