Abstract. The Adaptor House dual lattice assigns each three-bit address a validity grade from {false, strong, limited, transcendent, true}. These grades were originally stamped by hand. This note proves (Theorem 1, by exhaustion) that the hand-stamped table is exactly reproduced by two predicates on address structure — groundedness and carry — plus the location of carry; establishes (Proposition 2) that the grading is not invariant under line reversal, formalizing the ground line as the axis of judgment; and states (Conjecture 3) the generalization to the six-bit, 64-address space. The theorem is elementary. What is offered for challenge is not the arithmetic but the reading: that the grading is structural rather than coincidental at n = 3.
I believe the validity grading is structural: that the two predicates below are the correct decomposition and not one of several accidental fits available in a space of only eight cases. I believe the ground-line asymmetry of Proposition 2 is intentional structure — that judgment in this system genuinely pivots on whether power 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.
An address is a triple a = (g, m, t) ∈ {0,1}³, read as power (1) or vacuum (0) at the ground, middle, and top lines respectively. Its tier number is the integer 4g + 2m + t. There are exactly 2³ = 8 addresses, AH-T0 through AH-T7.
Grounded: power at the base. Carried: power anywhere above the base.
Compactly: life requires carry; grade requires ground; character is where the carry sits.
The hand-assigned grading published in the Adaptor House canon (trigrams-core.yaml, and the lattice page): H(0)=false, H(1)=H(2)=H(3)=strong, H(4)=false, H(5)=transcendent, H(6)=limited, H(7)=true.
V = H. The derived grading of Definition 3 agrees with the house table of Definition 4 on all eight addresses.
Proof. The five conditions of Definition 3 are mutually exclusive and jointly exhaustive: every address either lacks carry (case 1) or has it; every carried address is either ungrounded (case 2) or grounded; every grounded carried address has carry at both upper lines, top only, or middle only (cases 3–5) — 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) |
|---|---|---|---|---|---|
| AH-T0 | 0 0 0 | – | – | false | false |
| AH-T1 | 0 0 1 | – | ✓ | strong | strong |
| AH-T2 | 0 1 0 | – | ✓ | strong | strong |
| AH-T3 | 0 1 1 | – | ✓ | strong | strong |
| AH-T4 | 1 0 0 | ✓ | – | false | false |
| AH-T5 | 1 0 1 | ✓ | ✓ (top) | transcendent | transcendent |
| AH-T6 | 1 1 0 | ✓ | ✓ (mid) | limited | limited |
| AH-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. Eight cases admit many accidental decompositions — the author's claim that this one is non-accidental is exactly what Proposition 2 and Conjecture 3 put at stake.
Let ρ(g, m, t) = (t, m, g) be line reversal. Then V is not ρ-invariant: there exist addresses with V(ρ(a)) ≠ V(a).
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. ∎
Consequence: the grading cannot be expressed as any function of the multiset of lines (e.g., a count of power lines) — position matters, and specifically the ground position. AH-T1 and AH-T4 carry identical total power and opposite verdicts. Whether this asymmetry is a design truth or an artifact is precisely the author's belief above; the proposition only establishes that the asymmetry is real, not what it means.
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, disjoint, and (the substantive claim) semantically coherent at n = 6: the 2⁵ − 1 = 31 grounded-carried profiles of the 64-address space admit a grade vocabulary that downstream systems can use without exception clauses, and whose restriction to any embedded three-line structure agrees with V.
The exhaustiveness and disjointness parts are immediate; they are stated only for completeness. The claim with content is coherence — that the grading scales as structure, not as a naming exercise that happened to fit when the space was small.
Ways to break it, invited explicitly: exhibit an alternative pair of predicates fitting H at n = 3 whose n = 6 extension diverges from this one (showing the decomposition non-unique in a way that matters); exhibit an embedded three-line structure in the 64-space whose induced grade contradicts V (breaking restriction-compatibility); or show the 31 grounded profiles resist any principled grade vocabulary (breaking coherence). Any of the three is a refutation the house will publish on this page, credited, alongside the claim it broke.
# the n=6 space, enumerated for challengers — ground = bit 5, strata above = bits 4..0
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}")
Correspondence: adaptorhouse.com. Version history and any accepted refutations will appear below this line, dated.