AH-PAPER-1 proved that the five-valued validity grading over the eight three-bit addresses — false, strong, limited, transcendent, true — is a theorem of two predicates, Grounded and Carried. Any argument that cites a grade as a premise inherits its label. This note asks whether the five labels can bear that weight, and finds three collisions: two grade names are also the working language's Tarski-style satisfaction values, and the system of record already disfigures them (false_, true_) to keep its own parser from reading them as booleans; one label ("false") is shared by the two addresses whose distinctness is exactly what AH-PAPER-1's Proposition 2 proves; and one label ("strong") covers three addresses that the grounded side of the lattice individuates into three words.
We pose each collision as a Question with explicit answer conditions, and propose a repair that relabels and extends but never revises the theorem. The repair is structural: the name of record for every grade and every address is its defining first-order string — and the string is not merely cited but authored: Adaptor House authors the intellectual property as the FOL string itself, a sentence composed in a signature of the house's own design. Readable predicate-compounds are then argued as options over the strings; and prose names are a third layer — the same authorship continued into the reading world, openly bound to its string, auditable by the corpus at scale. Names bound this way are assertions, refutable by data; names bound by evocative fit are associations, refutable by nothing. The tradition's vocabulary was the second kind. This series is the first.
A grade is cited as a premise: a consumer — a machine checker, a downstream argument, a paper building on the verdict — opens by stating V(a) and stands on it. A premise is only as safe as its name, and the collisions exhibited below are inherited by every consumer, present or future (this house's own downstream address papers among them). So the labels must be examined before further weight is placed on them. AH-PAPER-1 §7 states the license under which this note operates: the five grade names carry no mathematical content — they "are bindings, declared and versioned" [1]. Relabeling is therefore a version bump, not a revision of Theorem 1. Nothing below touches the grading itself: the predicates, the five-way partition, and the table are settled [1]. Only the words are on trial — and, it will turn out, the kind of thing a word is allowed to be.
As in AH-PAPER-1: an address is a triple a = (g, m, t) ∈ {0,1}³ at the ground, middle, and top lines; in the signature of the system of record (sorts A and L, one predicate P(a, ℓ)): Grounded(a) := P(a, ground); Carried(a) := P(a, middle) ∨ P(a, top). V is the derived five-valued grading of AH-PAPER-1 Definition 3, equal to the hand-fixed table H by Theorem 1 [1].
The grade string of a V-value is its defining condition from AH-PAPER-1 Definition 3, written in the signature. The five grade strings:
S₁ := ¬Carried(a) S₂ := Carried(a) ∧ ¬Grounded(a) S₃ := Grounded(a) ∧ P(a, middle) ∧ ¬P(a, top) S₄ := Grounded(a) ∧ ¬P(a, middle) ∧ P(a, top) S₅ := Grounded(a) ∧ P(a, middle) ∧ P(a, top)The address string of a is its characteristic conjunction — the full profile, e.g. for T4: Grounded(a) ∧ ¬P(a, middle) ∧ ¬P(a, top). Eight addresses, eight strings, pairwise inequivalent by construction. Every grade string is the disjunction of the address strings in its band; V is recovered from address strings by weakening. Nothing new is asserted by either definition — these are AH-PAPER-1's own conditions, promoted from defining the names to being the names.
Where a paper in this series names a grade or an address, the name of record is the string, cited literally — in prose as well as in proof. "V(a) = S₁" is a complete, unambiguous, machine-checkable sentence. Handles and prose names (§§6–7) may accompany the string; they never replace it in a derivation.
Should "true" and "false" be retired as grade names, on the ground that they collide with the satisfaction vocabulary of the very language in which the grading is derived?
The collision, exhibited. The working language of the system of record is a small first-order theory with Tarski's satisfaction relation as its account of truth [1][3]. In that setting, consider the sentence "T4 is false." Read as a grade assertion, it is a true sentence — H(4) = false [1, Def. 4]. Read as a satisfaction claim, it is ill-formed (an address is not a sentence and has no truth value), and a charitable reasoner repairing it will reach for the nearest well-formed neighbor — "Grounded(T4) is false" — which is false: Grounded(T4) holds. One surface string; two readings; opposite verdicts. The same construction lands on every address in the two affected bands, e.g. "T7 is true" (grade: true assertion) versus "'V(T7) = true' is true" (a satisfaction claim about a sentence containing the grade name as a constant — the exact use/mention slide Tarski's hierarchy exists to police [3]).
The collision is already live, not hypothetical. The canonical bindings file of the system of record cannot write these two grade names at all: it stores validity: false_ and validity: true_, trailing underscore, because its own parser would otherwise read the bare tokens as booleans [4]. The other three grades are stored unmangled. The system has thus already conceded, in its most authoritative file, that these two names collide with machine vocabulary — and is paying for the concession with a disfigured token that every downstream consumer must special-case.
Proposal. Retire both — and do not replace them with better words, because a better word repeats the underlying mistake at a safer distance. Under the Convention of §2, the grade currently called "false" is named S₁ = ¬Carried(a), and the grade currently called "true" is named S₅. Neither can collide with satisfaction vocabulary, because each is a sentence of the theory, and says exactly what it says. The relabel is a version bump under the §7 license [1]; what is new is that the binding now points at a string instead of a lexeme.
Answer conditions. This Question closes (a) against the proposal, by showing the ambiguity cannot arise in any live consumer — which requires, at minimum, showing the false_/true_ mangling in the bindings file is cosmetic rather than collision-driven; (b) for the proposal, by decree adopting the string bindings; or (c) by decree accepting the collision as a permanent feature. Decree belongs to the author of record; this paper cannot invoke it.
T0 (000) and T4 (100) share the label "false." Proposition 2 of AH-PAPER-1 proves the grading is not reversal-invariant precisely because position matters — its Corollary exhibits T1 and T4: identical mark counts, opposite verdicts [1]. The two "false" addresses differ in exactly the predicate the whole decomposition pivots on: Grounded. Does one label erase a distinction the theorem itself proves matters?
What a repair may not do. Coining two grade names for T0 and T4 would split the band {T0, T4} and make V six-valued. That is not a relabel; it is a different function, and Theorem 1 is about V [1]. The §7 license covers names, not arity. Any repair must leave the partition intact.
Resolution under the string convention — the question dissolves. The band's name of record is S₁ = ¬Carried(a): a verdict, and visibly only a verdict — the string contains no claim about Grounded at all, so it cannot be read as erasing a Grounded-distinction it never mentions. The distinction lives one layer down, where it always lived structurally: the address strings of Definition 3. T0's string is ¬Grounded(a) ∧ ¬P(a, middle) ∧ ¬P(a, top); T4's is Grounded(a) ∧ ¬P(a, middle) ∧ ¬P(a, top). Two different sentences, one entailing ¬Grounded and one entailing Grounded, both entailing S₁. The old label pair ("false"/"false") collapsed them because a lexeme has no internal structure to disagree in; the strings disagree in their first conjunct. A companion paper argues the philosophical weight of this — absence as a value makes both of these positive structural facts (Paper 4, forthcoming); here it suffices that the formal layer already holds the distinction, and the name of record, being the formal layer, holds it too.
Answer conditions. Closes against by exhibiting a live consumer that needs T0 ≠ T4 at grade level (which would force a six-valued refinement and a new theorem — a bigger event than this note); closes for by decree adopting the two-layer naming (grade strings + address strings); or closes sideways by decree assigning the distinction to the Progenitor prose layer (§7) alone.
T1 (001), T2 (010), T3 (011) share "strong." The grounded side of the lattice is individuated into three grade words by carry profile [1, Def. 3]. The ungrounded side has the same three profiles and one word. Is the asymmetry a bug or a claim?
The case that it is a claim. Definition 3 individuates by carry location only under ground: "grade requires ground; character is where the carry sits" [1]. The grading's live consumer behaves accordingly — the trope-binding rule of the system of record reads the band, not the profile: resolution at S₅ or S₄ earns payoff, S₂ sustains recursion, S₃ caps it, S₁ documents collapse [4]. No consuming code path distinguishes T1 from T2 from T3. On this reading the asymmetry is the theorem's own voice: an ungrounded address has character, but character without ground earns no grade distinction, because grade is a verdict about what recursion can stand on. Splitting the band would manufacture a distinction the proof does not make — and would change V's arity, which the §7 license does not cover.
The residue, and its resolution. Any per-address treatment of T1, T2, and T3 must cite the identical grade three times over. Under the string convention this is no longer a defect: each address is cited by its own string — T1, T2, and T3 are ¬Grounded(a) conjoined with three different carry profiles, three visibly different sentences — and cites S₂ as the shared verdict. The individuating word the old vocabulary lacked turns out to be a sentence the formalism had all along.
Answer conditions. Closes against by corpus contradiction — ATTC data showing the shared band verdict mis-validates a real traversal at T1/T2/T3; closes for by decree. A challenger may also attack the claim-reading directly: exhibit a principled grade vocabulary in which the ungrounded profiles earn grade distinctions, and say what "grade" then means without ground.
Strings are names of record, but prose has a cadence, and a paper that says "¬Grounded(a) ∧ P(a, middle) ∧ ¬P(a, top)" in every sentence will not be read. A handle is a readable compound built only from the signature's own morphemes — Grounded, Carried, the line names — so that the handle is derivable from, and exhausted by, its string. A handle imports no lexeme the theory does not define. This is the discipline of Pearl's calculus, which named its operator do — a word with no aura, defined entirely by the formalism that coined it [6].
Options, presented for decree and not decided here (single-option rows are proposed as settled; multi-option rows are argued below):
| String | Handle options |
|---|---|
| S₁ (band T0, T4) | uncarried |
| S₂ (band T1–T3) | carried-ungrounded · ungrounded |
| S₃ (T6) | grounded-mid |
| S₄ (T5) | grounded-top |
| S₅ (T7) | grounded-full · grounded-both |
| T0 address | uncarried-ungrounded · bare |
| T4 address | uncarried-grounded · ground-only |
| T1 address | ungrounded-top |
| T2 address | ungrounded-mid |
| T3 address | ungrounded-both |
The four open cells are two grammar decisions, not four word choices. Decreeing the grammar rules decides the cells; the rules are stated so that one pass settles all four.
Decision A — elision (decides S₂). May a grade handle drop a conjunct of its string? carried-ungrounded names both conjuncts; handle and string regenerate each other with no rule beyond concatenation — criterion (i) at full strength. ungrounded drops Carried: shorter, and it reads as the genus of the three address handles in its band (ungrounded-top, ungrounded-mid, ungrounded-both are exactly T1–T3, exactly S₂'s band) — the grade handle becomes the visible common prefix of its addresses. The costs: (1) regenerating the string from the handle now requires an elision rule in the grammar ("a grade handle omitting Carried asserts it") — criterion (i) holds only relative to that rule; (2) the bare word over-covers its string — T0's address string also entails ¬Grounded, so ungrounded read literally is true of an address outside S₂'s band, and only the grade/address layer distinction keeps the handle honest. Elision buys prefix economy at the price of a rule; no elision buys self-containment at the price of length.
Decision B — quantifier lexemes (decides S₅, T0, T4). The signature's morphemes are the two predicates and the three line names. both, full, bare, and only are none of these: each is a quantifier word over line positions, admissible only if the compound grammar defines it. The three cells stand or fall by which of these words the grammar admits:
The consistent decree-shapes, laid flat: admit no quantifier lexemes (S₅ then takes the mechanical pure form, grounded-mid-top — enumeration by line name, criterion-(iii)-safe but the longest handle in the table; T0/T4 take the uncarried- pair; and the settled T3 handle ungrounded-both would have to reopen as ungrounded-mid-top — this shape alone touches a settled row); admit both only (S₅ grounded-both, T0/T4 uncarried- pair — the minimal option, since T3 already paid for both); or admit lexemes per-cell, taking the readability of bare and ground-only at the cost of a richer grammar. Mixed decrees are coherent; the table records whichever pass is decreed.
Selection criteria, argued: (i) strict derivability — a machine holding the signature and the compound grammar regenerates the handle from the string, and vice versa; (ii) no collision with any live token of the system of record (the criterion did real work in drafting: an earlier candidate set drawn from Latin participles — inert, dormant, quiescent — was withdrawn wholesale, not for collision but for kind: they were associations, words attached by evocative fit, the exact method this house exists to retire; see §7); (iii) the compound grammar must extend to the six-bit space unchanged, so that AH-PAPER-1's Conjecture 3, if it survives, inherits its handles for free. The proposal is uniform: strong, limited, and transcendent do not remain as aliases — they retire into this grammar alongside the two colliding names. All five grades take their string as name of record and a compound as handle. The retired words persist only as versioned historical bindings [1, §7], readable in the prior literature, generative in none of this series: a vocabulary that names three bands by aura and two by collision is not half-repaired by fixing the collisions.
A third layer remains, and it is not decoration — but to say what it is, the paper must first say where authorship begins, because it does not begin here. Adaptor House authors the intellectual property as the FOL string. The signature is a designed language [1, §7's working guarantee]; a sentence composed in it — S₁, an address string — is an authored work: written, published, versioned, in the system of record, under the house's name. The mathematics inside the sentence is nobody's property: derivability is theorem, anyone may re-derive it, and every refutation condition in this paper stands open precisely because the reading is on trial. But refuting a reading never un-authors a work. What can be challenged is whether the string is true to the structure; what cannot be taken is that the house wrote it. Pearl does not own causality; he authored the do-calculus [6]. The distinction is load-bearing: it places the house's claim on the stable layer — the string survives every revision of the words above it — rather than on the prose, which this very section makes revisable.
Prose, then, is the same authorship continued into the reading world: a name the house publishes over its string, openly bound to it, revisable by evidence. This is the Adaptor House publishing prerogative — the right to say what an authored formal object shall be called in prose, exercised in public, on the record, with the string beneath it for anyone to check.
The form of such a claim is an assertion, not an association:
Absent is a useful word for the string cited, ¬P(a, ℓ): a true sentence about nothing being there. Research supports the binding [1][2][3] — the zero built upon (von Neumann), the unmarked state (Spencer-Brown), satisfaction of a negation (Tarski).
Dark Star is the prose the corpus methodology currently binds at address 010 — ¬Grounded(a) ∧ P(a, middle) ∧ ¬P(a, top), power held at the heart, unseen at either face [4][5]. The binding is decreed today (company internal citation by the author of record, July 2026†) and auditable tomorrow: the compendium's nearest-neighbor methodology, once its distances are FOL-derived, measures whether the records that cluster around this string are the records this prose predicts.
That last clause is the whole difference. A prose name bound to a string inherits the string's refutation conditions: at the current corpus scale (28 records) the binding is an assertion held mostly on argument; at 27,000 records it is solved — the distances either vindicate the name or retire it, and either outcome is a result. An associative name can never be wrong, which is why it can never be right. The classical tradition's vocabulary for these eight figures was associative — two millennia of readings with no string beneath them to disagree with; what association saw that assertion cannot is a question this house leaves open for the archive side to take up. The Progenitor papers (T0–T7, forthcoming) will exercise the prerogative once per address: derive the string, argue the handle, lay the prose claim, and hand the claim to the corpus to audit.
What adoption would touch. Adoption is a version bump on bindings [1, §7]: the canonical bindings file's validity cache tokens (and the trailing underscores of false_/true_ die with the collision that forced them), the ontology and proof pages' vocabulary, the grade tokens of the automated checks. Those are canon and execution strata; each change routes through its own authority (staged approval for canon). This paper rules on none of it; it argues.
Broader audiences. Nothing above requires machinery beyond the signature. For readers who want the strings as numbers, Gödel numbering gives every name of record an integer and makes "name" a computable function; for readers who want the lattice as sets, the ordinal construction already cited by AH-PAPER-1 [2] builds the addresses from ∅ up. Both are on-ramps, offered as invitation, not load-bearing: robustness is the key, and the robustness lives in the strings.
What is not claimed. No novelty in the instruments: the use/mention discipline is Tarski's [3], the zero is inherited [1][2], the operator-naming discipline is Pearl's [6]. No claim that the handle options of §6 are the only correct compounds — the grammar is the contribution; the cells are for decree. And no claim about the six-bit space beyond criterion (iii): whether the naming layers generalize is downstream of AH-PAPER-1's Conjecture 3, and waits on it.
The position — belief register, first person, per AH-PAPER-1 §6 — is published as this paper's companion document on the archive side: The Author's Position on the Grade-Vocabulary Gloss (AH-GRADE-GLOSS-1-H), historicalpresent.com/shared/papers/grade-gloss. Archive-only: the companion is never citable by live systems, and nothing in it is offered for challenge. The Questions of §§3–5 and their answer conditions are this paper's entire challengeable surface.