VOTKP, the vector of traits kept under peeling. Take an atom apart one electron at a time, name its family from the shape of the peel, thin a gold film to nothing, sweep a knot's phase around the circle, and watch which records change and which survive. Every step lands on one disk, and every step adds like a velocity.
Built from version 1.2 of the paper (28 September 2026) and its package. Ionization energies are NIST's, with NIST's own grades, via mendeleev 1.3.0; optical constants are Johnson and Christy's; nuclear values are NUBASE2020; Seifert matrices are KnotInfo's; the observer is CIE 1931 under D65. Everything is computed live in the page.
Γ = tanh(½ ln z)each peel step on the Smith disk
0.95 against 0.51family from the peel shape, and from the static record
171 / 171film labels from the first-order echo
ΔE*ab = 8.60a matched pair, after gold
21 → 0blind knot pairs from colour 3 to colour 4
−2 against +2one window separates a 13-crossing pair
|Γ| > 1the mass-5 and mass-8 gaps
Theorem 8.1 · Theorem 8.2 · Corollary 15.2
Peel an atom and watch each step land on the disk
Read each ionization ratio zₖ = IEₖ₊₁/IEₖ as an impedance. Its reflection Γₖ = (zₖ − 1)/(zₖ + 1) = tanh(½ ln zₖ) sits on the positive real radius of the Smith disk. Steps inside a shell reflect weakly. A step that breaks into a closed shell reflects strongly, and its shell part is fixed by n alone. As the steps add up, the composite walks toward the rim, and its hyperbolic distance from the centre is exactly ln(IEₖ/IE₁).
the ion being peeled
the Smith disk · steps and their composite
the ladder IEₖ · colour = NIST grade
ratio zₖ
reflection Γₖ
split: shell ⊕ screening
composite Γ₁→ₖ
hyperbolic distance
NIST grades of the two stages
measured · semi-empirical · theoretical
What the identity is: exact once the ratio is read as an impedance, and nothing more. No atomic interface or reflected wave is derived, and any increasing ladder has such a chart. The content is in the tabulated ratios: the peel shape recovers periodic group with held-out accuracy 0.95 against 0.51 for the static record (Run 01, pre-registered). The level, ln IE₁, carries the response instead.
§8.2 Run 01 · §8.3 the depth ladder · pre-registered
The shape of the peel finds the family
Hide every element's group. Read only its ladder of ionization energies, then ask its nearest neighbour which group it belongs to. The shape of the ladder, the step sizes ln(IEₖ₊₁/IEₖ), names the group of 95% of the elements at ten steps; the static record of the same numbers, ln IEₖ, names about half. Every number below is computed in your browser, leave one out, with the scaling fitted inside each fold. Switch to the depth ladder: train on the first 54 elements and name the heavy period 6. A short window misses exactly the elements whose outer shell it cannot reach yet.
the table, named from the ladder alone
accuracy as the peel window deepens
accuracy at this window
peel shape against static
missed
the paper
Reading a group from the ionization record ("count the electrons before the big jump") is textbook chemistry. What the paper adds is the measurement: the full-table, held-out accuracy of the log-ratio vector and its behaviour with depth. The level, ln IE₁, carries the response instead (Run 02: the peel shape loses on polarizability in 9 of 9 settings).
§8.4 · the ridge census · Z ≤ 103
The onion atlas: which closures survive the peel
Peel every element to the end and line the steps up by how many electrons are left. Each row is one element; each column is one electron count. A bright column is a closed shell reached from above: the step into it costs more than its neighbours. Watch the white peel front sweep down through deeper ions. The noble-gas closures 18, 36 and 54 fade; the nickel-like 28 grows with charge. This follows the Coulomb filling order of highly charged ions.
every element peeled, aligned by electrons left · colour = ln(IEₖ₊₁/IEₖ)
one closure, element by element
the census
prominence near · mid · deep
this closure
the peel front
noble against Janet
Shrink: the Ar-like and Kr-like closures stay ridges at every charge but shrink about tenfold. Vanish: the Xe-like closure vanishes in deep ions. Grow: the Ni-like closure grows with charge; the Pd-like one peaks at mid depth. Janet's left-step table ends its periods at the alkaline earths, and at those counts the ridge is gone: 0.004 against 0.477.
§8.8 · the column rhythm · francium, radium, 119 and 120
Down the column, and where the line breaks
Down each group, ln IE₁ is close to a straight line in the period. Fit the line on periods 2–6 of the alkali metals and it misses francium by 0.55 eV; for the alkaline earths it misses radium by 0.99 eV. The measured columns turn upward at the bottom. The same lines fall 1.55 and 2.14 eV below the relativistic predictions for elements 119 and 120. Where a line under-predicts, the step to that element has slipped a filled inner shell of a new kind, d or f, under the outer electron.
the column line, drawn on past the fit
where the line under-predicts: an inner shell of a new kind
the line at the last member
the line at 119 or 120
sign of the miss
the column rhythm
The sign pattern is that of the d-block and lanthanide contractions: a filled d or f shell screens the nucleus poorly, so the electron outside it is held more tightly than the column's trend. For Au, Hg, Fr and Ra the relativistic contraction of the outer s shell adds to it. The rule gives the sign, not the size of each miss. The two exceptions: Pt, whose ground configuration changes down the column, and Bi, whose 6p shell is split by spin–orbit coupling.
Theorems 6.1 and 6.2 · the thinning peel
Thin a metal film and watch its echo cross one circle
A film's reflection is r = Γ ⊕ x, with front surface Γ = (1 − N)/(1 + N) and back-surface echo x = r₁₂e^{4πiNd/λ}. The film reflects more than its bulk exactly when the echo lies outside one circle through the origin. That circle is fixed by the front surface alone; thickness and the back surface only move the echo. Play the peel from 200 nm down to nothing. For gold at 450 nm on glass the echo leaves the circle, so the film brightens before it fades: R = 0.408191 at 200 nm, 0.426742 at its maximum at 58.21 nm, and 0.163082 at 10 nm.
the echo x and the circle
R(d) − |Γ|²
the reflection r(d) on the disk
Gold path: the echo lies outside the circle, so reflectance is above bulk. Blue: inside, below bulk. Dashed line: the first-order boundary, tangent to the circle at the thick-film point.
index N
front surface |Γ|
as it thins
outside the circle
R(d) now
circle radius
Model films with bulk constants, not reflectance measurements. The first-order echo reproduces the rise-before-fall label in 171 of 171 cases, a result found after the run. Real gold films computed with their own measured constants failed both registered expectations of THIN-01. Gold samples also differ in bulk reflectance by more than the rise itself.
Theorem 6.3 · the thin end is a shunt admittance
The last nanometres: one shunt step from bare glass
To first order in thickness, a film acts on its substrate as a shunt admittance: Y = n_b + y, r = (n₀ − Y)/(n₀ + Y). At the substrate face y = −ikd(N² − n₀²) = Z₀σₛ, which is the familiar thin-sheet formula, exact to first order for any N. At the film's top face the same law reads y = −ikd(N² − n_b²). The usual mistake is to count all of N² as conductance. That is right only when |N²| ≫ n₀², n_b², and for gold it misses 7–35% of the change of r. The miss is all in the phase: the reflectance still agrees to first order.
r near bare glass, 3 nm → 0
errors against the exact film
where the whole-N² miss points
|N²|
shunt admittance y
Theorem 6.3 error
whole-N² miss
R − R_glass
What a prediction from the atom must supply at the thin end is the film's optical sheet response σₛ(ω), or N and d, at the optical frequency. A level spectrum does not fix it: the two-level sheet of §4 keeps its levels and metric while its optical strength runs from 0 to 1. A DC conductance does not give the optical σₛ either.
§6.2 · receiver geometry · the gold metamer
Two colours the eye matches, and gold tells apart
A colour record keeps three numbers of a whole spectrum. Two different spectra can share them exactly: to the CIE 1931 observer under D65 the two halves of the left disc are one colour. Reflect both from gold and the match breaks. The first observation threw away a difference that the next surface was still sensitive to. A neutral filter keeps the match: it is the control. The strongest pair is the top singular vector of CMP, the change a surface can make to what the first receiver could not see.
the two spectra, and the surface
what the observer records
XYZ before, spectrum A
XYZ after, spectrum A
ΔE*ab after the surface
ΔE*ab so far
Surviving matches (a theorem): a later linear map M keeps every match of the receiver C exactly when CM = TC for some 3 × 3 matrix T. Gold does not remember colour; the first observation discarded a distinction to which the next surface stayed sensitive. Whether people see the post-surface difference is a separate question. Silver and copper are computed here for comparison; the paper reports gold.
Theorem 15.1 · Corollaries 15.2 and 15.3 · order of peeling
Peel steps add like velocities
With u ⊕ v = (u + v)/(1 + uv), the reflection of a whole block of removals is the ⊕-sum of its steps: Γₖ→ₗ = Γₖ ⊕ … ⊕ Γₗ₋₁. This is Einstein's law for adding collinear velocities: Γ is a velocity, and τ/2 is a rapidity. As a radar picture, each peel ratio is the frequency ratio of an echo from a mirror approaching at Γc. The ⊕-sum forgets the order of its steps. A thin-film stack only remembers order through the phase between its additions.
⊕-adding the steps · the red tick is the telescoped block
rapidities add · Γ = tanh φ
the radar reading of the current step
order: swap gold and titania on glass
⊕-sum so far
telescoped block
difference
ln(IEₗ/IEₖ)
mirror speed β = Γ of this step
film ΔR on swapping
The radar picture is a representation: nothing in the atom moves. The relativistic reading unifies the atom and the film, but it predicts nothing new. The echo weight |1 − Γ²| separates rising films worse than |Γ| does (H-01, killed: 0.778 against 0.994).
A knot's layers vary independently. One knot, T(3,4), carries five different Hopf charges under five rational-map numerators. Distinct knots with identical HOMFLY, Alexander and Jones polynomials differ in volume. Where a record fails, the next rung resolves it: a longer peel window for atoms, a higher colour for knots.
one knot, several loadings
same fingerprint, different geometry (Table 7)
two resolution ladders
Hopf charge Q
knot
through 12 crossings
354 pairs
blind to HOMFLY, Alexander and Jones; volume separates 253
from 13 crossings
19 pairs
equal Alexander and Jones, different HOMFLY
No element-to-knot correspondence is claimed, and there is no scalar VOTKP. The layers are independent for an initial map, and relaxation couples them: the relaxed Skyrme–Faddeev energies rise with Q (Sutcliffe 2007).
§11.5 · §13.7 · the Levine–Tristram signature · ADM-01 and TIME-01
Sweep the phase: walls, windows, and one pair told apart
Turn a phase ω once around the unit circle and record the signature of (1 − ω)V + (1 − ω̄)Vᵀ from the knot's Seifert matrix. It can change only at a wall: a root of the Alexander polynomial on the circle. For a torus knot every representation carries the same sign, so the count of irreducible SU(2) representations is |σ|/2: T(3,4) climbs 0, 1, 2, 3 and back down at its six walls. Then open the 13-crossing pair. The two knots share HOMFLY, Alexander, Jones, three Khovanov homologies, knot Floer homology and classical signature 0. Between two walls, one reads −2 and the other +2.
the phase on the unit circle
the signature function
phase
signature σ_ω
count
walls
Of 2,525 thin, same-signature pairs blind to HOMFLY, Alexander and Jones through 13 crossings, Khovanov and knot Floer homology separate none; the signature function separates exactly one, certified in exact arithmetic at three rational points. Inside that window the two knots' longitudes also turn at different rates (numerically; not interval-certified). A phase becomes a time only when something drives it; no Hamiltonian here drives a knot's meridian phase.
Section 15.4 · the mass-5 and mass-8 gaps · NUBASE2020
A negative separation energy leaves the disk
No nucleus of mass 5 is bound, and none of mass 8 is stable. On the chart of Theorem 8.1, with z the next separation energy over the first and Γ = (z − 1)/(z + 1), every bound peel lies inside the disk. A negative separation energy gives |Γ| > 1. Slide the separation energy through zero and watch the point cross the rim.
the real axis of the Smith chart
the boundary reading
half-life against distance outside the rim
separation energy
ratio z
|Γ|
half-life
Among the three gap nuclei, the one nearest the rim lives longest, by five orders of magnitude. The distance outside the rim is close to twice the decay energy over the neighbouring step, so the ordering is essentially that of the decay energies. With three points and no escape barrier, it is an observation, not a law.
Section 15.5 · P-01 and G-02 · Athenodorou & Teper (2020), Table 17 as transcribed
Tight knots against the glueball spectrum
The tight-knot model sets a glueball's mass proportional to the length of a tight knot or link, in order of length. G-02 was registered before the table was used: the k-th shortest object against the k-th lightest state of the model's own J⁺⁺ sector, with a 5% kill on the median miss. Every arm fails. The masses spread faster than the lengths. Try the five one-parameter changes and see what a repair would need; those are post-hoc.
states and objects, over the lightest
the misses against the 5% kill
first ratio, predicted vs lattice
median miss, J⁺⁺ arm
later terms
0⁺⁺ tower arm
The registered verdict stands: the one-knot-per-state assignment is rejected in the form registered. With the flux-tube form fitted on the first ratio, the next J⁺⁺ ratio comes within 0.6% (uncorrected) and 1.5% (corrected), but no single change comes within 5% of both later terms. With five forms and two length sets tried, a 1% agreement is not rare. This is a lead for a registered test on a spectrum not yet compared. The published fit is to observed states, which this test does not reach.
In SU(2) lattice gauge theory, a canonical transformation maps the loop Hilbert space into the Hilbert spaces of hydrogen atoms. A loop of spin j sits at principal quantum number n = 2j + 1. The map is between state spaces, not spectra. On one plaquette the electric energy j(j+1) = (n² − 1)/4 grows, while the Coulomb levels −1/n² close up. On one SU(3) plaquette, the magnetic term connects the empty state to the fundamental loop.
hydrogen levels and one-plaquette loop levels
the magnetic term mixes loop labels
n = 3 after the map
⟨tr U|V|1⟩
A knotted Wilson loop is an operator whose overlaps with the energy eigenstates must be computed, not a particle label. The dynamical knot test is open (§17).