D-exp-MP01: PDL Structural Lacunae Confirmed by Materials Project Database Scan — Technetium and Promethium as Combinatorial Impossibilities (Projective Dynamic Logo Framework, Exploratory Document)

The Projective Dynamic Logo (PDL) framework derives the periodic table and the valley of nuclear stability from four combinatorial axioms (C1–C4) acting on finite signed graphs, with zero free parameters (D40, D47). Conjecture OP3 of document D40 identifies technetium (Z=43) and promethium (Z=61) as structural lacunae: elements for which no stable nuclear configuration is possible at the combinatorial level, independently of neutron number. These are the only two elements below bismuth with no stable isotope in nature. This exploratory document tests Conjecture OP3 against the Materials Project database (mp-api, ~160,000 inorganic materials) by scanning all Z=1–83 elements and counting experimentally observed compounds (theoretical=False, energy above hull < 0.05 eV/atom). The results are unambiguous: technetium shows a 10.5× deficit in observed compounds relative to its immediate neighbours (Mo, Ru), and promethium a 484× deficit relative to (Nd, Sm), with z-scores of −2.31σ and −5.08σ respectively against the distribution of 29 stable odd-Z elements above Z=20. Of promethium's 441 database entries, 439 carry the theoretical flag — pure DFT predictions with no experimental counterpart. The signal is specific to the PDL lacunae: the six magic-number elements (He, O, Ca, Ni, Sn, Pb) show percentage-observed values of 47–56%, indistinguishable from the general theorem group. These results constitute a strong empirical correlation with Conjecture OP3-D40. The causal mechanism connecting Block II (nuclear stability) to Block III (chemical compound richness) is identified as Open Problem OP4 of the PDL programme and remains to be established as a theorem. The document includes the annotated PDL periodic table as a figure, the full statistical analysis (z-scores, local ratios, aggregate statistics by PDL epistemic status), and the Python scripts used for data collection as supplementary material. Part of the Projective Dynamic Logo research programme. Full corpus: zenodo.org/communities/pdl-framework