The periodic table as a ground-state ladder

1 · Why the periodic table

The periodic table is a model almost everyone has met, and it comes with an answer key. The National Institute of Standards and Technology publishes the ground-state electron configuration of every neutral atom it lists: 108 of them, hydrogen to hassium. That makes it a fair test of what our engine is for, finding the lowest-energy arrangement of a model it is handed, on a model nobody had to invent for the occasion.

What this paper contributes, losses first, each with its basis and the section that carries it:

1. Measured, one run. Held out, rubidium to hassium, the fitted model matches 46 of 72 elements against 54 for the filling order alone, and its exchange term gains four elements and loses ten (sections 5 and 8).
2. Measured, one run. A control: a table of orbital energies computed from Slater's rules matches 6 of the first 20 elements, with a tie counted as a miss (section 4).
3. Prediction. That an element's unpaired electrons and measured magnetic susceptibility rise with the halo of its margin: registered, not run (section 9).
4. Theorem in the model. An objective over spin-orbitals whose lowest filling is a configuration, with the exclusion principle holding by construction (section 3).
5. Measured, one run. On the fit set, hydrogen to krypton, 36 of 36; every answer up to twenty spin-orbitals is the best of all 1,048,576 arrangements, and all 118 answers are the objective's own lowest filling, which a reader can check by arithmetic (section 5).
6. Reading. The table heard as chords: a mapping of computed levels for the reader, not a result (section 12).

2 · The claim, stated as a model

This paper puts forward a modelling statement, which reports no new fact about atoms. Each element's ground state is read as the lowest-energy filling of its Z electrons over spin-orbitals, under the exclusion principle, given a table of orbital energies and two small interaction terms. The periodicity of the table is read as the order in which that filling proceeds.

Nothing here computes an atom. The filling order is an input to the model, not a result of it, and section 9 lists what the model leaves out.

3 · The variables and the objective

Every spin-orbital is one variable: a subshell such as 3d, one of its orbitals, and one of the two spin values. The variable is +1 when the spin-orbital is empty and −1 when it holds an electron. A spin-orbital holds one electron or none, so the exclusion principle holds by construction.

E  =  Σᵢ εᵢ nᵢ   +   λ (Σᵢ nᵢ − Z)²   −   K Σ nᵢ nⱼ   +   U Σ nᵢ nⱼ
                                          pairs in one      pairs in one
                                          subshell, with    orbital, with
                                          the same spin     opposite spins

Here nᵢ is the occupation of spin-orbital i: it is 1 when the spin-orbital holds an electron and 0 when it does not, so nᵢ = (1 − sᵢ)/2 for its ±1 variable sᵢ. εᵢ is the energy of the spin-orbital's subshell. Z is the atomic number. λ is the weight that holds the electron count at Z. K is the exchange energy that two electrons of the same spin gain in the same subshell, which is Hund's first rule (Hund 1925). U is the cost of two electrons sharing one orbital; the two names follow the standard models of electrons on a lattice, where U is the on-site repulsion and K the exchange. The element's ground state is the assignment with the lowest E.

Every εᵢ is counted from the energy of the Z-th spin-orbital in the table's own order. That changes no filling of Z electrons, and it keeps λ small. Each element's objective carries every spin-orbital through the end of its own period: 20 for hydrogen to calcium, 1s to 4s, then 36, 54, 86 and 118 for the later periods.

4 · The orbital energies, and what was fitted

The orbital-energy table is Madelung's rule written as numbers (Madelung 1936; Janet 1928). Subshells fill in order of n + l, and for equal n + l the one with the smaller n comes first, so the k-th subshell in that order is given the energy k: 1s is 1, 2s is 2, 2p is 3, and so on to 7p at 19. One unit of energy in this note is one step of that order.

K and U were chosen on hydrogen to krypton only, the first 36 elements. Of the pairs on a grid of hundredths that reproduce all 36, we took the pair that reproduces them with the widest margin: K = 0.27 and U = 0.00. λ is 4, twice the smallest weight at which no answer changes. Those constants were then applied, unchanged, to rubidium and every element after it.

As a control, we computed a table of orbital energies from Slater's rules (Slater 1930), element by element, without their standard refinement (Clementi and Raimondi 1963), and ran the same objective on it in one run, once with the fitted exchange and once with K and U at zero. Slater's rules give 2s and 2p one energy, and from scandium on they put 3d far below 4s, so a filling objective built on them reproduces 6 of the first 20 elements either way, with a tie counted as a miss. A table of computed orbital energies is not the order in which the shells fill, and this model needs the order (Pilar 1978; Vanquickenborne and others 1989).

5 · What we measured

The results run losses first, every count beside the same objective with K and U at zero, which is the filling order alone, and every count is one deterministic run of 30 September 2026. Our engine is described on our public pages, and we describe it here in those words and no others: up to twenty moving parts, it gets there by visiting every point, all 1,048,576 of them; above twenty, it tells you how far from the best it could be.

Above twenty, every answer states how far from the best it could be, and those margins are wide: between 399 and 13,934 units, element by element in the next section. The electron-count term links every pair of spin-orbitals, and a margin that must hold over every arrangement of such an objective is loose. So every answer was also checked a second way, with arithmetic a reader can repeat. Apart from the count, the objective scores each subshell on its own, so its lowest filling follows from the best number of electrons in each subshell. All 118 answers are that lowest filling.

Same observation, three vocabularies. Three fields (information theory, systems and graph theory) describe the same situation in different words, and an insight in one usually has a twin in the other two. This series names a twin only where its own model has the object, and says so where a twin is missing or runs the other way. Each twin is our reading, not an identity, unless a result the paper states makes the two one number. The objective of section 3 is one such observation. In information theory each spin-orbital is one bit, empty or holding an electron, and each element's filling up to calcium is one of the arrangements of twenty such bits. Read as a system, once each subshell's count of electrons is fixed, each subshell settles on its own. Read as a graph, the pair terms K and U join spin-orbitals only inside one subshell, while the electron-count term joins every pair, which is likely why a margin that must hold over every arrangement is loose.

6 · Element by element

The table below is every element, hydrogen to oganesson, from the same one run of 30 September 2026. Hydrogen to calcium were solved by scoring every arrangement of twenty spin-orbitals. Every later answer states how far from the best it could be, in steps of the filling order.

7 · The ladder

Every count in this section is from the same one run. From element Z to element Z + 1 the objective changes in one number: the electron count that λ holds the answer to. On 113 of the 117 steps nothing else changes. On the other four, into scandium, rubidium, cesium and francium, the objective also gains the spin-orbitals of the new period: 16, 18, 32 and 32 of them.

The answers climb the same ladder. On 109 of the 117 steps, the answer for Z + 1 is the answer for Z with one more electron. The eight exceptions are the steps into chromium, copper, molybdenum, silver, neodymium, holmium, uranium and einsteinium. The table itself steps by one electron on 95 of NIST's 107 steps.

8 · Where the model and the table part: the exchange term gains four elements and loses ten

In the same one run, the exchange term, fitted on the first 36 elements, gains four exceptions (chromium, copper, molybdenum and silver) and loses ten ordinary elements: 82 of 108 with it, 88 of 108 without. The table has 20 elements whose configuration breaks the filling order, and they are where the model is tested. The count is NIST's own: it includes lawrencium, which NIST lists with its last electron in 7p, while some other references list it in 6d, in the filling order. The misses come first.

• Ten ordinary lanthanides and actinides are broken by the exchange term: neodymium, promethium, samarium, holmium, erbium, thulium, plutonium, einsteinium, fermium and mendelevium. The model moves outer s electrons into the f shell. Within the model, the likely cause is one exchange constant, fitted where a shell holds ten electrons, set against the one-step spacing between 4f and 6s, which is a chosen input; the two were never separated. With K and U at zero, all ten are reproduced.
• Lanthanum, cerium, gadolinium, actinium, thorium, protactinium, uranium, neptunium, curium and lawrencium are missed. Each puts an electron into 5d, 6d or 7p earlier than the filling order says. An ordered table places each subshell once: a fitted table cannot see what a fitted table does not carry.
• Niobium, ruthenium, rhodium and palladium are missed. Their row leaves the filling order in more places than chromium's row does, and constants that keep vanadium and nickel in the filling order keep niobium and rhodium there too.
• Platinum and gold are missed. In this table 4f sits between 6s and 5d, so the gap from 6s to 5d is two steps rather than one, and the fitted exchange does not close it.
• Chromium and copper are reproduced, as a fit, not as an explanation. With K = 0.27, five same-spin 3d electrons beside one 4s electron are lower than four beside two, and a full 3d shell beside one 4s electron is lower than nine beside two. The chemistry literature explains these two atoms by the small energy difference between 3d and 4s and the repulsion between d electrons, and it rejects the extra stability of a half-full or full shell as the cause (Vanquickenborne and others 1989; Melrose and Scerri 1996; Schwarz 2010).
• Molybdenum and silver follow without being fitted. The same constants give 4d5 5s1 and 4d10 5s1 on the next row, as NIST lists them.

Where the model's filling is right and NIST gives a term to compare with, its spin multiplicity agrees with NIST's ground level every time, 78 of 78. For the other 4, lead, seaborgium, bohrium and hassium, NIST gives no term, so they were not compared.

9 · Limits, and what this paper does not say

The open limits come first, each with what would close it.

• The held-out count is below the filling order alone, 46 of 72 against 54, and the misses are reported, not tuned away. It would close with a fit on the first 36 that also reproduces the rest, or a stated reason why each miss is out of reach.
• The on-site term U is not identified on the fit set: every value from 0.00 to 0.17 fits the first 36 equally, and 0.00 was taken for the widest margin. It would close with a fit that separates the values, such as U held positive, or with measured data that only one value fits.
• The one-step energy spacing between 4f and 6s is a chosen input, and the ten misses among the lanthanides and actinides likely come from it; it was never separated from the exchange term. It would close with a measured or computed energy table in place of the chosen spacing, re-run on the same elements.
• Ten answers have no measured reference. NIST lists no neutral ground level above hassium, so the answers for meitnerium to oganesson are shown with no measured reference to compare them with; the published values for these elements are calculations, not measurements (Desclaux 1973; Fricke 1975; Pyykkö 2011). It would close with a published ground level for any of the ten.
• The NIST answer key's redistribution terms are pending. The ground levels are used for research under the source's published terms, with NIST credited; written confirmation of redistribution is pending. It would close with NIST's written terms, recorded in the data register.
• The timing is one session on one processor. The search at twenty was timed once, 60 calls on one desktop processor. It would close with the same calls read on a second machine with the load recorded.
• Not yet a test of magnetism. If the halo of a margin is the turning side of a two-valued reading, an element's count of unpaired electrons and its measured magnetic susceptibility should rise with the halo of its margin on the ladder. That prediction is registered, not run: the public table it needs is not yet part of our record, and its power is not stated.

The standing statements, which no measurement closes:

• Not electronic structure. The objective has no off-diagonal terms: nothing moves an electron from one orbital to another, and there are no X or Y Pauli terms. A classical ±1 objective does not hold them.
• No molecules and no bonding. One atom, and only its filling.
• No absolute energies. E is a score in steps of the filling order, not in electron-volts.
• No derivation of the filling order. Madelung's rule is an input.
• Nothing about excited states, term energies, spin–orbit coupling, or which total angular momentum the ground level takes.

This note describes a model of electron filling with a fitted table, not a calculation of atoms. The engine's method is not described here and will not be. Every number is as of 30 September 2026.

10 · How this was made: the method and the chain of records

The question was set on 30 September 2026 as a written brief: can the lowest-energy answer of a stated objective be read up the periodic table from hydrogen, one element after another, and does it reproduce NIST's ground configurations? Three scope rules held throughout: the periodic table is the reader's model and is described in full; how the engine searches is not described (a limit of section 9); and nothing here claims to calculate atoms.

Day solved the model for every element, checked each answer against the NIST table and drew the ladder from that record. Claude (Anthropic), the consultant named in the byline, designed and built the rest from the engineering side: the objective, the one pull of the NIST reference saved unchanged, the measurement, the checks that pin every number on this page to its record, and the draft of this note.

The chain of records, in order:

• the NIST ground levels, pulled once on 30 September 2026 and saved byte for byte with their address, access time and sha256;
• the measured record, deploy/elements_ladder.json, every number on this page printed from it by its emitter and re-derived by its check mode;
• the magnetism prediction, registered on its own before any data, and not yet run;
• the tests that hold this page equal to its source and every number to its record, and this page, generated from its source.

11 · Companion papers

Seven papers published beside this one take the same approach: each traces its figures to a recorded measurement and says what it did not measure.

• The Operational Unit: the model behind the engine, and how it was built.
• Closing the gaps: four sustainability problems an engine can schedule, and what it cannot do about them.
• Sense, remember, act: what the engine senses, remembers and learns.
• The halo of a decision: a two-way choice, the partner it is tied to and the event that tied them, in textbook terms, with what an experiment could measure.
• The Earth as a two-state recorder: the Earth's dipole read as a two-way choice from public records, with a prediction written before the data, and the planet's clocks.
• HOPE: recall in a library of Hilbert spaces: recall as settling to the nearest stored state, in a library of small spaces kept one per model, with the measured answer that they do not make one large space.
• The unit of work: a receipt for one settled problem, booked on a hash-chained ledger, and what it is not, with the economy's measured losses first.

12 · Appendix · Hearing the table

This appendix is a mapping for the reader, not a result.

The standard way to turn a spectrum into sound keeps its ratios and moves it into the range of hearing (Hermann and others 2011). An energy level E is a frequency ν = E/h. An octave is a factor of two, so halving frequencies keeps their intervals up to whole octaves. Each level is halved until it lies in the octave just below the A at 440 hertz, and one element's levels then sound together as a chord.

We applied that mapping to the levels Slater's rules give for each element's filled subshells, from the same one run. These are computed levels, not the measured lines that sonification usually plays, so the chords carry Slater's approximation with them: where Slater's rules give two subshells one level, the chord sounds one note twice, though the real atom has two (Zanella and others 2022; Smith 2024). Slater's rules make a poor filling table (section 4), and no measured table was compared with these levels, so the chord says nothing the numbers do not.

Acknowledgements

• NIST Atomic Spectra Database (Kramida, Ralchenko, Reader and NIST ASD Team), DOI 10.18434/T4W30F. The answer key: the neutral ground configurations NIST lists, read on 30 September 2026 and saved byte for byte. NIST asks for attribution and reserves its rights in the compilation. Used for research under the source's published terms; written confirmation pending.
• PubChem Periodic Table, National Center for Biotechnology Information: read for lawrencium, whose configuration it lists differently from NIST.

Replicate this paper, or tell us what is wrong

We invite replication and correction. The measured results come from records committed in our repository. The records are not yet published; until they are, write to us through the contact page, naming the paper and the number you want to rebuild or the error you found. We correct an error we can reproduce on the paper's web page, with its date.

The tests this paper registered before reading their data are frozen, each with the rule that would fail it, in docs/PAPERS_DUALFIELD_1_2026_10_04.md.

How to cite this paper

Please cite this paper as:

Agile Integrations LLC. 2026. "The periodic table as a ground-state ladder." Version 1, 30 September 2026. https://agileintegrations.net/blog/the-periodic-table.html

Citation file: CITATION.cff.

13 · References

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• Clementi, E. and Raimondi, D. L. 1963. "Atomic Screening Constants from SCF Functions." Journal of Chemical Physics 38, 2686–2689. The standard refinement of Slater's rules, not used here. 10.1063/1.1733573
• Desclaux, J. P. 1973. "Relativistic Dirac-Fock expectation values for atoms with Z = 1 to Z = 120." Atomic Data and Nuclear Data Tables 12, 311–406. 10.1016/0092-640X(73)90020-X
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• Hermann, T., Hunt, A. and Neuhoff, J. G., editors. 2011. The Sonification Handbook. Logos Verlag, Berlin. ISBN 978-3-8325-2819-5
• Hund, F. 1925. "Zur Deutung verwickelter Spektren, insbesondere der Elemente Scandium bis Nickel." Zeitschrift für Physik 33, 345–371. 10.1007/BF01328319
• Janet, C. 1928. La classification hélicoïdale des éléments chimiques. Privately printed, Beauvais.
• Klechkovskii, V. M. 1962. Soviet Physics JETP 14, 334.
• Kramida, A., Ralchenko, Yu., Reader, J. and NIST ASD Team. 2024. NIST Atomic Spectra Database. National Institute of Standards and Technology, Gaithersburg, MD. 10.18434/T4W30F, accessed 30 September 2026
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• Smith, W. 2024. "Interactive Musical Periodic Table: Sonification of Visible Element Emission Spectra." In Proceedings of the Sound and Music Computing Conference 2024.
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• Zanella, A., Harrison, C. M., Lenzi, S., Cooke, J., Damsma, P. and Fleming, S. W. 2022. "Sonification and sound design for astronomy research, education and public engagement." Nature Astronomy 6, 1241–1248. 10.1038/s41550-022-01721-z