Electronic pockets at a missing clamp in the PdAl plate of Pd₅AlI₂

Pd₅AlI₂ is a layered van der Waals metal of space group I4/mmm, a stack of slabs five atoms thick. In each slab a PdAl checkerboard plane sits between two square planes of palladium, capped with iodine, and the in-plane lattice constant is 4.05 Å. Thinel et al. image the cleaved iodine face by scanning tunnelling microscopy at about 7 K and find three kinds of point defect, about 70 of each in 3025 nm², or 2.3 × 10¹² cm⁻² per type. They identify the dark defect that sits above a palladium site as a palladium vacancy.

The clean surface has a nonzero density of states at every bias: the material is a metal. At the vacancy two peaks rise out of that continuum, at +475 and +620 mV. Mapped at +470 and +620 mV, the first is four-lobed, like an atomic d orbital, and the second circular, like an s orbital, and line spectra place both within about 1 nm of the site. Where two vacancies sit on neighbouring palladium sites each state becomes a pair, 440 and 510 mV for the four-lobed state and 590 and 650 mV for the circular one, the lower circular member with a node between the defects. Two vacancies two lattice spacings apart show only the lone states.

The authors model the PdAl plane as a decorated checkerboard: two palladium d orbitals, d_xz and d_yz, on each palladium site and one aluminium p_z orbital on each aluminium site, with nearest-neighbour hopping 0.395 eV. Destructive interference of the hopping gives one flat band and two dispersive bands. Round a missing palladium site the flat band’s compact localized states combine into states orthogonal to the dispersive continuum, and a local potential lifts them into it: bound states in the continuum. With a scalar term of 0.25 eV and a handed term of 1 eV on the surrounding sites, their 41 × 41 model places a −1 state at +400 meV and a +i state at +700 meV above the Fermi level. A density-functional calculation of a 4 × 4 × 1 vacancy supercell puts a vacancy peak at about 0.55–0.6 eV, against the measured 0.47 eV.

The PdAl plane is a plate of the one omnium, and its nuclei are the plate’s clamps: P-⁠3 names saturated nuclear cores the fixed pinning sites of a material, and P-⁠2 sets every mode geometry by those boundary conditions. A missing palladium site is a missing clamp. The electron is the closed three-strand pattern of I-⁠5, and it settles into the eigenmode the changed boundary offers; P-⁠2 reads localization as trapping in disconnected nodal pockets. A pocket is that mode’s own support, never an occupant held in a vessel (0.1b). The bias injects a pattern at energy eV above the Fermi reference, and a pocket’s level is a difference of two mode costs.

The record fixes what “settles into the nodal lines” means. The source defines its nodal lines by phase shifts of π, and the measured weight fills the lobes between them: the electron settles into the eigenmode whose nodal lines they are, never onto a zero of its amplitude. All three model orbitals are odd under the mirror of the PdAl plane, so every nucleus in that plane lies on the mode’s nodal set, and the π-lines are the zeros of the field’s across-plane derivative 𝓕. The surface maps match the model’s figure qualitatively; they do not measure the field in the plane itself.

The plate has an equation. A second time derivative of the free Schrödinger equation gives

so each real quadrature of Ψ obeys the Kirchhoff–Love plate equation, with that stiffness per areal mass. The two quadratures are the two transverse components of I-⁠4’s helix. In the crystal the lattice Hamiltonian replaces the free equation, and the lattice separates two kinds of pattern exactly. The compression operator 𝖢 acts on the palladium lobe field as a lattice divergence, the compression it feeds into the aluminium sites. A circulation, tangent lobes round one aluminium plaquette, feeds nothing into its own aluminium site or the diagonal ones and equal and opposite amounts into each of the four edge neighbours, so it feeds nothing at all: the circulations are the flat band. Compression patterns carry the two dispersive bands, and the inner-product identity makes each of them orthogonal to every circulation. The pocket is a circulation inside a compression continuum, and its width turns on when a band of its own kind reaches its level: I-⁠5 applied to the source’s hypothesis that the state delocalizes once its level overlaps the flat band.

Round a palladium site the four circulations combine by their quarter-turn eigenvalue 𝔢 = +1, −1, +i or −i. For 𝔢 = ±1 the lobes at the central site cancel; for 𝔢 = ±i they do not, and those two combinations run through the very site the vacancy removes. A missing clamp keeps the two real combinations and cuts the two handed ones. With the source’s scalar term alone on the eight surrounding palladium sites, both survivors are exact eigenstates 𝔲 = 0.25 eV above the flat band, for each of three placements of the term on the aluminium sites and on periodic cells of 9 × 9 and 13 × 13. The source assigns its four-lobed state to the −1 combination and its circular state to +i, which appears in its model only once the handed term is added. That term rotates the real lobe pair about the sheet normal, and on it a cut handed circulation is no exact eigenstate: with the source’s 1 eV handed term its weight spreads over several eigenstates, with spectral variance close to 1 eV², while separate bound poles also form in the finite cells.

The source places its two model states at +400 and +700 meV above the Fermi level, and the reconstruction places both real survivors at 𝔲 above the flat band. The source gives no alignment between the flat band and the Fermi level, so the two energy axes stay separate. On its own axis each survivor’s level is the local term itself.

Two neighbouring vacancies touch six circulations. The four central components at the two missing sites impose constraints of rank four and leave two survivors, and with one scalar term over the union of the two neighbourhoods both sit at 𝔲: the ideal pair does not split. The measured pairs split by ±35 and ±30 meV about centroids that coincide with the lone levels at the printed precision, so the splitting measures a coupling beyond the ideal model, and P-⁠2 reads the pair as one shared nodal channel settled system-wide (0.1a). A straight chain of 𝖭 vacancies keeps the same count, 2𝖭 constraints on 2𝖭 + 2 circulations, and two survivors at 𝔲 for any length.

The source relates its states to photonic and acoustic bound states in the continuum. By I-⁠2 the same nodal structure appears in every carrier window, and what the electronic pocket shares with those is the boundary-selected nodal geometry: the pocket itself is a mode of a closed pattern, outside the carrier identity, and no photonic lifetime transfers to it.

On the A-⁠1 chart the electron’s exterior compactness is small. With 𝓜 = m_e at a field point of 1 nm, α_ext = R_s/r_f ≈ 1 × 10⁻⁴⁸: a correct reading under I-⁠1, neither the binding strength of the pocket nor an address for the electron. The rest energy 510 998.950 eV keeps the period 8.09330 × 10⁻²¹ s and runs the Compton length 2.42631 × 10⁻¹² m (0.1a). The bias reading eV/ħ is a beat frequency between two mode costs, not the electron’s own, and the confinement operator that returns the levels is to be derived (O-⁠9).

The existing spectra fix level differences and figures. They measure no twist magnetism and no density-to-spectrum sensitivity, so each test below adds one FDT input to the conventional account. The twist tests compare neighbours, arranged so that the conventional response cancels and the comparator reads zero (Use 7); the isotope tests bound the conventional channels and subtract them. The three twist tests read P-⁠5: imposing a physical twist on a material is establishing a magnetic field. The two isotope tests read I-⁠7: a heavier isotope enters 𝓜 with more rest content, and for a purely density-coupled observable a larger R_s gives a larger response.

1. T-⁠1
   Static torsion of a single plate. Grip a cleaved crystal, 1–10 mm across and under 100 μm thick, and twist it statically to equal and opposite hands, with vacancies at mirror-related positions on both sides of the torsion axis. Read the circular state’s field doublet under reversed normal fields: the hand-and-field-odd combination 𝒟 = 𝔤𝓑_tw reads the added field for 𝓑 > |𝓑_tw|. FDT predicts a nonzero field whose sign reverses with the hand; the mirror-matched conventional comparator is 𝒟 = 0.
2. T-⁠2
   Relative twist of two flakes. Stack two flakes at matched positive and negative angles and read vacancies on the top slab, whose PdAl plane sits half a slab above the rotated interface. The prediction is a hand-odd normal field. A moiré contrast or a scalar shift of a resonance is not this observable.
3. T-⁠3
   Handed screw cores. Compare vacancies within three lattice spacings of left- and right-handed screw dislocations at equal radial and angular positions, against a flat region far from each core. An uncorrelated placement puts about 0.11 vacancies within that radius of a core, so the number of cores surveyed follows the measured availability.
4. T-⁠4
   Isotope shift of the lone four-lobed level. Grow crystals from enriched ¹⁰⁴Pd and ¹¹⁰Pd in one campaign and compare the four-lobed centre above the same calibrated Fermi reference. The logarithmic mass contrast is Λ_M = ln(𝔐_H/𝔐_L) = 0.0561502, and at the unit-sensitivity test point the residual beyond the conventional channels equals it. The assumed conventional bound on zero-point and lattice changes, to be validated on these samples, is |C| ≤ 0.005, and the design criterion is 0.010.
5. T-⁠5
   Isotope shift of the adjacent-pair coupling. In the same assayed crystals, compare the four-lobed pair coupling 𝔧 = (𝓔₊ − 𝓔₋)/2, 35 meV in the source. The test point again gives a residual of 0.0561502, with the two centres fitted jointly. About 1.5% of vacancies have a second vacancy on a nearest palladium site under random placement, so the area surveyed follows the measured clustering.

P-⁠5 fixes the existence and the hand-odd sign of the twist field; its size and absolute sign are to be derived, through a twist-to-field kernel. I-⁠7 fixes the sign of the density input; the sensitivity S_𝓜 of each level is to be derived, and the observable sign follows from the product of the sensitivities. A null result bounds this specimen’s field, or excludes the unit-sensitivity path, at the precision reached. Three further measurements test the ideal model itself with no FDT input added: a same-tip map between the two lone resonances for the second real survivor, a straight vacancy chain for the unsplit pair of survivors, and a flat band moved by strain or pressure for the onset of the pocket’s width when a band of its own kind reaches its level.

X-⁠1, exploratory. The measured pairs’ centroids coincide with the lone centres, 475 and 620 mV, and the coupling ratio is 0.86. The lower four-lobed member’s filled gap reads as a shared P-⁠2 channel and the lower circular member’s node through P-⁠4’s flow-direction rule: an in-sample reading of the measured order, with no phase-to-flow law derived, and never counted with the tests.

On the source’s own spectra, topography and dictionary the limiting map is the identity. It returns the lone and pair centres, the maps and line spectra, the lattice constant, the defect sites and counts with the density derived from them, the extent as estimated, and the pair phase rule as the source’s in-sample choice. On the model’s own hoppings and local terms it returns the simulated levels and finite-cell profiles, and on the density-functional inputs the periodic-cell density of states; the reconstruction makes the ideal cancellations explicit under its stated placements. The free Schrödinger equation is the slow-motion expansion of a closed pattern’s phase relation ω² = c²k² + (m_ec²/ħ)², and its time derivative gives the plate equation without changing the data. For the same stress tensor and data, I-⁠3 returns the curvature of general relativity. Setting the twist source and the density sensitivity to their conventional values restores both comparators. On the source’s own inputs the map is the identity, and the five tests above are where FDT’s added inputs meet new data.