# On the Naming Collisions in the Five-Valued Validity Grading: A Gloss and Coinage Proposal

**Adaptor House** (author of record)
adaptorhouse.com · July 2026 · v1.0 (AH-GRADE-GLOSS-1)

*Status: open questions, unreviewed. In response to* On the Derivability of a Five-Valued
Validity Grading over Three-Bit Addresses *(AH-PAPER-1). Answers and refutations will be
published, credited, at the address of record.*

---

## Abstract

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.

---

## 1. Why the words come 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.

## 2. Definitions

**Definition 1 (inherited).** 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].

**Definition 2 (grade string).** 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)

**Definition 3 (address string).** 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.

**Convention (citation of 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.

## 3. Question 1 — the Tarski collision

**Question.** 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.

## 4. Question 2 — the double "false"

**Question.** 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.

## 5. Question 3 — the threefold "strong"

**Question.** 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.

## 6. Predicate-compounds: readable handles as options

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:

- **S₅** (both upper lines carry). *grounded-both* is enumerative ("the two upper
  lines") and exactly parallel to the settled T3 handle *ungrounded-both* — note that
  settling T3 has already admitted *both* into the grammar, so this option adds no new
  lexeme. Its cost is criterion (iii): *both* denotes exactly-two and does not survive
  the six-bit space. *grounded-full* ("every upper line carries") generalizes to any
  stratum count — but **full is a live token of the system of record**: the pole
  vocabulary is the closed set {empty, full, unconfirmed} (the trope-record schema's
  machine-checked closed vocabulary; pole renaming — company internal citation by the
  author of record†). At T7
  the full pole and S₅
  in fact coincide, so the collision is also a convergence; whether fusing grade
  vocabulary with pole vocabulary at one address is felicity or exactly the
  cross-vocabulary collision criterion (ii) exists to forbid is a decree question, not a
  derivation.
- **T0** (no mark at any line). *uncarried-ungrounded* is pure negation-compound, zero
  new lexemes. *bare* is one word for total absence — it would be *defined* (no P at any
  line, criterion-(iii)-safe), but it is the only single-lexeme handle in the table,
  breaking the compound form every other address carries, and a one-word name for the
  empty address is where aura creeps back in: the word is admissible only if the grammar
  defines it, and suspect precisely because it sounds good undefined.
- **T4** (mark at ground, nowhere else). *uncarried-grounded* is the pure compound,
  parallel to T0's first option — decreeing the pair gives T0/T4 uniform negation-form.
  *ground-only* is the positive form (says where the mark *is*), reads better, and
  *only* generalizes to six-bit unchanged; the ontology's own working prose already says
  "top only / middle only" informally. Cost: admits *only* as an exhaustivity operator,
  a third quantifier lexeme if *both* or *full* is also admitted.

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.

## 7. Prose: the publishing prerogative

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.

## 8. Discussion

*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.

## 9. Author's Position

Held, not argued here. 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. **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.

## References

[1] Adaptor House, *On the Derivability of a Five-Valued Validity Grading over Three-Bit
Addresses* (AH-PAPER-1), adaptorhouse.com/works/papers/derivability-paper (2026).

[2] J. von Neumann, *Zur Einführung der transfiniten Zahlen*, Acta Litterarum ac
Scientiarum (Szeged) 1 (1923), 199–208; G. Spencer-Brown, *Laws of Form*, Allen & Unwin,
London (1969). Cited as inherited by AH-PAPER-1 [1][2].

[3] A. Tarski, *Pojęcie prawdy w językach nauk dedukcyjnych* (1933); English trans. in
*Logic, Semantics, Metamathematics*, Oxford (1956). Cited as inherited by AH-PAPER-1 [3].

[4] Adaptor House, canonical bindings (AH-CORE-2), `lattice-core.yaml` — the system of
record: doctrine, validity rule and cache, trope binding.

[5] Adaptor House, *The Trope Compendium*, tropes.adaptorhouse.com (2026) — the public
corpus; progenitor bindings — company internal citation by the author of record
(July 2026)†.

[6] J. Pearl, *Causality: Models, Reasoning, and Inference*, Cambridge (2000); J. Pearl
and D. Mackenzie, *The Book of Why*, Basic Books (2018) — the do-operator as the
precedent for formalism-defined naming.

[7] Adaptor House / Historical Present, *The Inherited Lattice* (AH-DUAL-1-H),
historicalpresent.com/shared/dual-lineage — concordance and classical readings.
**Archive-only**: interlocutor, never evidence; no derivation above reasons from it.

† *Company internal citation*: the decision is held in Adaptor House's internal register
of record. Inquiries and verification requests: adaptorhouse.com.

---

## Change log

- v1.0 (2026-07-17): first published edition. Participants, recorded as first-class
  metadata with the required provenance chain: Matthew L. Tyler (human; directed and
  refined) and Claude, Fable 5 (AI, provider Anthropic; drafted). Citation of record —
  company internal register†.
- v1.0-DRAFT-2 rev. f (2026-07-17): internal register identifiers removed from the
  public text on the author of record's review — decisions are cited as *company
  internal citation by the author of record* with date, marked † with a contact
  footnote (adaptorhouse.com); the register ids live in the house's internal register
  and in the citation metadata of record, not in the paper body.
- v1.0-DRAFT-2 rev. e (2026-07-17): §1 reframed on the author of record's review — the
  motivation is now the consumer-inheritance argument (any argument citing a grade as a
  premise inherits its label), not the house's own publication schedule; section retitled
  "Why the words come first"; abstract and §5's residue paragraph aligned. The series
  schedule survives only as a subordinate clause.
- v1.0-DRAFT-2 rev. d (2026-07-17): §6's four open handle cells restructured for
  one-pass decree — reduced to two grammar decisions (A: elision of Carried in the S₂
  handle; B: admission of quantifier lexemes *both*/*full*/*bare*/*only*), each option
  argued against criteria (i)–(iii). Two facts surfaced for the decree: *full* is a live
  pole token of the system of record (closed set {empty, full, unconfirmed}),
  and the settled T3 handle *ungrounded-both* has already admitted *both* into the
  grammar. No cell decided; consistent decree-shapes enumerated.
- v1.0-DRAFT-2 rev. c (2026-07-17): Author's Position moved to an archive-side companion
  (AH-GRADE-GLOSS-1-H, drafted at `drafts/grade-gloss-position.md`, for
  historicalpresent.com/shared) per the author of record; §9 is now a pointer declaring
  the Questions the paper's entire challengeable surface.
- v1.0-DRAFT-2 rev. b (2026-07-17): uniform retirement ruled by the author of record —
  *strong/limited/transcendent* retire into the compound grammar with the two colliding
  names; §6 aligned (no legacy aliases).
- v1.0-DRAFT-2 rev. (2026-07-17): authorship located correctly on the author of record's
  second correction — the house authors the IP *as* the string (§7 rewritten: authorship
  begins at the signature and the sentence, prose continues it; theorem/work boundary
  stated as a guard). Abstract aligned.
- v1.0-DRAFT-2 (2026-07-17): coinage design replaced on the author of record's
  correction — the Latin-participle candidate table (associations) withdrawn; names of
  record are now the FOL strings (§2 Convention), predicate-compound handles argued as
  options (§6), prose as publishing prerogative with corpus-scale refutation (§7).
  Pearl's do-calculus added as register precedent [6].
- v1.0-DRAFT (2026-07-17): first full draft.

*Correspondence: adaptorhouse.com. This document is released for challenge; §§3–5 state
what would close each Question.*