THE FOUR LOST VIRTUES: Naming the Last Vacancies of the Periodic Table of the Virtues

One cell stays open. The Periodic Table of the Virtues leaves four permitted cells vacant: 1.D.AL.Φ, 1.R.FL.Φ, 1.N.FL.Φ, and 1.N.AL.Ψ. This paper names them. The naming is constrained rather than free. A cell’s coordinates fix the moral function required of whatever occupies it, and four admission criteria then determine whether a proposed occupant is an element or an artefact: the domain must be statable without reference to the coordinates that predicted it (E1); it must be attested independently of the system that requires it (E6); it must not be extensionally equivalent to a neighbouring element (E4); and its inversion must be single and nameable (E3). A name derived from an address satisfies none of these. It is a restatement of the prediction wearing the costume of its confirmation. Applying these constraints yields four elements, each attested in a tradition that did not possess the coordinate system: Panache at 1.D.AL.Φ, Sprezzatura at 1.R.FL.Φ, Hilaritas at 1.N.FL.Φ, and Curatorship at 1.N.AL.Ψ. Each carries a dispositional coding, an inversion channel derived in advance from the Capture Law, and an operational falsifier. Three of the four are Φ-rendered. The paper proposes that this is not a coincidence — that the vacancies clustered in the block where moral work is done through perceptible form, because the catalogue’s source traditions treat perceptible form as the ornament of virtue rather than its medium — and then states the confound that prevents the four namings from counting as evidence for it (§4.1). Naming Hilaritas also forces a clarification of the Capture Law: appropriability requires a separable output. A form whose product cannot be detached from its performance is not appropriable at any level of value, which is why such forms are commanded rather than harvested (§3.5). The provenances are then checked against the sources rather than recalled, and two do not survive intact (§5). Hilaritas is withdrawn to Tier 3: the Roman record treats Hilaritas and Laetitia as near-synonyms and supports no distinction between sustained and occasioned cheer, while the monastic sources attest cheer under command — which is this element’s inversion, not the element. Sprezzatura’s text is confirmed in every particular, but a substantial line of interpretation reads the practice as defensive, which would move it to the Defiance regime and possibly merge it with Panache. The table therefore closes at sixty-five of sixty-six permitted cells, with one pair contested. Closure is a claim, not an achievement, and §7 states what would have to be true for it to be premature. Keywords: virtue theory · moral taxonomy · Inversion Theorem · appropriability · institutional capture · Mendeleev prediction · sprezzatura · provenance verification Highlights 1. Four empty cells of the Periodic Table of the Virtues are named: Panache, Sprezzatura, Curatorship — and Hilaritas, provisionally. 2. A name taken from an address cannot fail, and so confirms nothing. Mendeleev predicted properties, not placeholders. The same standard is applied here, and it disqualifies most proposals. 3. Three of the four vacancies sat in the block where moral work is done through appearance — a bias in the catalogue's sources, not a flaw in the coordinate. 4. Appropriability requires a separable output. What cannot be detached from its enactment cannot be stolen, only commanded — which is why such virtues are hollowed rather than turned. 5. One element is withdrawn under audit. Rome treated Hilaritas and Laetitia as near-synonyms; the distinction was the author's, not the tradition's. The cell reopens. 6. The Rule of Benedict specifies hollowing in the sixth century: the act fulfilled, the cheer commanded, the unwilling heart penalised. The table closes at 65 of 66. One cell stays open — a stated prediction, and the paper's strongest claim that its criteria bind.