# On the Derivability of a Five-Valued Validity Grading over Three-Bit Addresses

**Adaptor House** (author of record)
adaptorhouse.com · July 2026 · v1.0

*Status: standing challenge, unreviewed. Refutations will be published, credited, at the address of record.*

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## Abstract

We consider a finite structure arising in a working symbolic system: the eight three-bit addresses, each assigned one of five validity grades — *false*, *strong*, *limited*, *transcendent*, *true* — by a table originally fixed by hand. We prove (Theorem 1, by exhaustion) that the hand-fixed table is exactly reproduced by two predicates on address structure, groundedness and carry, together with the location of carry. We establish (Proposition 2) that the grading is not invariant under line reversal, and consequently cannot be expressed as any function of the multiset of lines: position matters, and specifically the ground position. We then state (Conjecture 3) a generalization to the six-bit, 64-address space, and give explicit conditions under which the conjecture — and with it the structural reading of the grading — should be considered refuted. The theorem is elementary; what is offered for challenge is the reading, not the arithmetic.

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## 1. Introduction

A symbolic infrastructure in active use assigns to each of eight primitive states a validity grade governing whether recursive constructions over that state are admitted, and with what character. The grades were assigned by judgment before any derivation was attempted, and were treated for some time as axioms of the system.

The present note records three findings. First, the grades are theorems: a two-predicate decomposition reproduces the full table without exception. Second, the decomposition is genuinely positional — the table provably cannot be recovered from any count or multiset of the lines, which formalizes an asymmetry the system's designers had asserted informally (that one line, the *ground*, is the axis of judgment). Third, the decomposition suggests a canonical generalization to larger address spaces, stated here as a conjecture with explicit refutation conditions.

The author's epistemic position is stated plainly in §6 and hedged in the first person. A fit across eight cases admits accidental decompositions; whether this one is structural is exactly what the conjecture puts at stake.

The philosophical background of the system — a construction from a single primitive act of distinction, in the spirit of the ordinal construction 0 = ∅, 1 = {∅} [1] and of the calculus of indications [2] — and the model-theoretic framing of its working language [3] are documented at the address of record and are not required for the results below.

## 2. Definitions

**Definition 1 (Address).** An *address* is a triple a = (g, m, t) ∈ {0,1}³, read as a mark (1) or its absence (0) at the *ground*, *middle*, and *top* lines respectively. Its *tier number* is the integer 4g + 2m + t. There are exactly 2³ = 8 addresses, T0 through T7.

**Definition 2 (Predicates).**

> Grounded(a) := (g = 1)
> Carried(a) := (m = 1) ∨ (t = 1)

Grounded: a mark at the base. Carried: a mark anywhere above the base.

**Definition 3 (The derived grading V).**

> V(a) = **false** iff ¬Carried(a)
> V(a) = **strong** iff Carried(a) ∧ ¬Grounded(a)
> V(a) = **true** iff Grounded(a) ∧ m = 1 ∧ t = 1
> V(a) = **transcendent** iff Grounded(a) ∧ m = 0 ∧ t = 1
> V(a) = **limited** iff Grounded(a) ∧ m = 1 ∧ t = 0

Compactly: life requires carry; grade requires ground; character is where the carry sits.

**Definition 4 (The fixed table H).** The hand-assigned grading of the system of record:

> H(0) = false, H(1) = H(2) = H(3) = strong, H(4) = false,
> H(5) = transcendent, H(6) = limited, H(7) = true.

## 3. Theorem

**Theorem 1 (Derivability).** V = H on all eight addresses.

**Proof.** The five conditions of Definition 3 are mutually exclusive and jointly exhaustive: every address either lacks carry or has it; every carried address is either ungrounded or grounded; every grounded carried address has marks at both upper lines, at the top only, or at the middle only, and no other carry profile is non-empty. It therefore suffices to evaluate V on each address and compare with H.

| Tier | (g m t) | Grounded | Carried | V(a) | H(a) |
|------|---------|----------|---------|------|------|
| T0 | 0 0 0 | – | – | false | false |
| T1 | 0 0 1 | – | ✓ | strong | strong |
| T2 | 0 1 0 | – | ✓ | strong | strong |
| T3 | 0 1 1 | – | ✓ | strong | strong |
| T4 | 1 0 0 | ✓ | – | false | false |
| T5 | 1 0 1 | ✓ | ✓ (top) | transcendent | transcendent |
| T6 | 1 1 0 | ✓ | ✓ (mid) | limited | limited |
| T7 | 1 1 1 | ✓ | ✓ (both) | true | true |

Agreement on all eight rows. ∎

**Remark.** The proof is trivial once the predicates are named; the content of the result is that *these* predicates name it. The system's pre-existing informal descriptions of the tiers — e.g., T4 as "force at the root with nothing above to carry it," T5 as "marks at both faces, absence at the heart" — are restatements of the defining conditions, recorded years before the decomposition was stated. The names knew the rule before the rule was written down.

## 4. The Asymmetry

**Proposition 2 (Ground asymmetry).** Let ρ(g, m, t) = (t, m, g) be line reversal. Then V is not ρ-invariant.

**Proof.** V(1,0,0) = false but V(ρ(1,0,0)) = V(0,0,1) = strong. Likewise V(1,1,0) = limited while V(0,1,1) = strong. ∎

**Corollary.** V is not a function of the multiset {g, m, t}; in particular it is not a function of the number of marks. T1 and T4 carry identical total marks and receive opposite verdicts.

The proposition establishes that the asymmetry is real; it does not establish what the asymmetry means. Whether the privileged ground line is design truth or artifact of the small case is the author's belief in §6, and the conjecture in §5 is its test.

## 5. The Conjecture and Its Refutation Conditions

**Conjecture 3 (Six-bit generalization).** For addresses a ∈ {0,1}ⁿ with a designated ground line and n−1 upper strata, define Carried and Grounded as before. Then the trichotomy —

> **false** iff ¬Carried; **strong** iff Carried ∧ ¬Grounded; **graded** iff Carried ∧ Grounded, with grade determined by the carry profile of the upper strata

— remains exhaustive and disjoint (immediate), and, substantively, **semantically coherent** at n = 6: the 2⁵ − 1 = 31 grounded-carried profiles of the 64-address space admit a grade vocabulary usable by downstream systems without exception clauses, whose restriction to any embedded three-line structure agrees with V.

**Refutation conditions, invited explicitly.** Any of the following breaks the conjecture or the reading behind it:

1. **Non-uniqueness that matters.** Exhibit an alternative predicate pair fitting H at n = 3 whose n = 6 extension diverges from the one above.
2. **Restriction failure.** Exhibit an embedded three-line structure in the 64-space whose induced grade contradicts V.
3. **Coherence failure.** Show the 31 grounded profiles resist any principled grade vocabulary.

Refutations will be published at the address of record, dated and credited, alongside the claims they broke.

**Enumeration for challengers** (ground = bit 5, strata above = bits 4..0):

```python
for n in range(64):
    bits = [(n >> i) & 1 for i in range(6)]
    ground, upper = bits[5], bits[4::-1]
    carried = any(upper)
    grade = ("false" if not carried else
             "strong" if not ground else
             f"graded:{''.join(map(str, upper))}")
    print(f"{n:02d} {n:06b} {grade}")
```

## 6. Author's Position

Held as belief, not theorem. I believe the validity grading is structural: that the two predicates above are the correct decomposition and not one of several accidental fits available in a space of only eight cases. I believe the ground asymmetry of Proposition 2 is intentional structure — that judgment in this system genuinely pivots on whether a mark is grounded. And I believe Conjecture 3 is the right generalization test: if the grading survives the jump to 64 addresses without exception clauses, my reading is ratified; if it requires patches, the community will have shown me the table was stamped after all. Either outcome is a result.

## 7. Discussion

Three observations, briefly.

*The direction of discovery.* The table preceded the rule by years. The result is therefore an instance of a familiar and always slightly uncanny event: a vocabulary chosen for expressive reasons turning out to encode a decomposition its authors had not articulated. We make no larger claim about this than the record supports; we note only that the informal tier descriptions and the formal defining conditions are, on inspection, the same sentences.

*What the formalism buys.* The system of record operates as a small first-order theory whose signature contains addresses, lines, and one predicate. Assertions outside the signature are not false in the theory; they are not well-formed. This is the working guarantee that motivated the formalization: semantic drift in downstream reasoners is demoted from a temptation to a detectable syntax error [3].

*What is not claimed.* No claim is made that the five grade names carry mathematical content; they are bindings, declared and versioned. No claim is made that n = 3 is distinguished; the conjecture exists precisely because it may not be. And no claim of novelty is made for the instruments — the ordinal construction [1], the primitive distinction [2], and satisfaction-based truth [3] are used exactly as inherited.

## References

[1] J. von Neumann, *Zur Einführung der transfiniten Zahlen*, Acta Litterarum ac Scientiarum (Szeged) 1 (1923), 199–208.

[2] G. Spencer-Brown, *Laws of Form*, Allen & Unwin, London (1969).

[3] A. Tarski, *Pojęcie prawdy w językach nauk dedukcyjnych* (1933); German trans. *Der Wahrheitsbegriff in den formalisierten Sprachen*, Studia Philosophica 1 (1935), 261–405; English trans. in *Logic, Semantics, Metamathematics*, Oxford (1956).

[4] Adaptor House, *The Dual Derivation* and *The First-Order Ground*, adaptorhouse.com/ontology (2026) — the system of record.

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*Correspondence: adaptorhouse.com. Version 1.0, July 2026. This document is released for challenge; see §5 for the terms.*