Deficiency Is a Module: Channel-Resolved Closure, Vocabulary Granularity, and the Operator Gap on Token Arenas

Abstract

Deficiency Is a Module Channel-Resolved Closure, Vocabulary Granularity, and the Operator Gap on Token Arenas Driven by Dean A. Kulik October 2026 Abstract fH(V) = dim V module D = A/W, and every reader's channel deficit is a Fourier or isotypic coefficient of a single character χD = χA − χW; the projector apparatus used to measure them is eliminable. Second, which irreducible types are reachable as deficiencies depends on the granularity of the declaration vocabulary, not on the arena: the same 12-vertex witness realizes a trivial obstruction at five-family resolution and a three-dimensional sign-twisted obstruction at orbit resolution, with opposite channel behaviour. Third, reader saturation is not closure — an exhibited vocabulary has P0W = P0A while W ≠ A — and the criterion for when saturation does certify closure is a visibility condition that turns out to be Frobenius reciprocity. Two results concern dynamics. A field on the arena with equivariant local evolution reproduces the module statement exactly: a reader-null sector stays invisible under every admissible linear law, sources the visible channel through the lowest-order equivariant nonlinearity, and is then recoverable from a single visible observation up to sign. Pushing upstream, the arena does not select an evolution operator: equivariance, radius-one locality, self-adjointness and conservation leave a 22-parameter family whose dimension equals the number of edge orbits, and the standard graph Laplacian is the point where all 22 weights are equal. Requiring the transformation price to be generated by an equivariant potential cuts 22 to 7, excludes the Laplacian by a single edge orbit, and leaves the arena connected. Positioned against the literature, the representation-theoretic content of this paper is recovered rather than new; the direction of arrival, the arena's measured data, and one question — which deficiency types a given vocabulary lattice can reach — are what the work contributes. Full numerical records and non-claims are stated explicitly. 1. What Papers I–III Established, and the One Correction K = 2 token arena A2 G = Aut(X) ≅ C2 × A4, |G| = 24. Inside the 300-dimensional cochain space, the span of the 3- and 4-cell boundary vectors is the ambient relation module dim A = 235, dim AG = 15, m(A) = (15, 4, 14, 4, 24, 36), f Correction. (the scope of Papers I–III) The functional fH(V) = dim VH is the trivial-character coefficient of a complete decomposition, not the whole reading. Every numerical result of Papers I–III is correct as the α = 1H slice. What was too narrow is the quantifier: an unindexed predicate “this vocabulary closes” omits the reader and the channel. From here every closure claim carries (H, k) for a cyclic reader and (H, α) in general. That correction is not a repair of the earlier arithmetic. It reorganizes it, and the reorganization has consequences the original framing could not express. Sections 2 through 4 develop them; Sections 6 through 8 follow the same correction into dynamics. 2. The Deficiency Module and Its Character Let W ⊆ A be the span of a declared vocabulary. Papers I–III treated the shortfall as a scalar, δ = dim A − dim W. The first result of this paper is that the scalar is a dimension of a structured object and the structure is what answers every reader at once. 2.1 Spans are reader-stable, and the channel split is a character vocabulary r r₀ r₁ r₀+r₁ defect disj 216 105 111 216 0 spec4 168 84 84 168 0 rot4 150 72 78 150 0 spec3 174 87 87 174 0 rot3 20 10 10 20 0 disj+spec4 234 114 120 234 0 rot3+spec3 193 97 96 193 0 rot3+spec3+spec4 199 100 99 199 0 all five 235 115 120 235 0 Table 1. Split defect r − r₀ − r₁ across vocabulary unions. Zero in all 31 cases: every span is ρ-invariant, so the channels partition it with no loss. Selected rows shown; the three with r₁ < r₀ are the odd-dimensional spans. Proposition 2.1. (channel ranks from one trace) For a ρ-invariant span W with ρ² = I, r0(W) − r1(W) = Tr(ρ | W), hence r0 = (r + Tr)/2 and r1 = (r − Tr)/2. Computed independently as a trace, this reproduces every one of the 31 measured channel ranks. The explicit projectors are therefore a verification instrument, not the mechanism. 2.2 Deficiency is conserved across channels δ0 + δ1 = 235 − r = δ, δ0 − δ1 = −5 − Tr(ρ | W) show that the total shortfall does not change with the channel; the channel decides only where it sits. The vocabulary disj+spec4 is short exactly one dimension in the ambient. That dimension is z-even, so it lies entirely in k = 0, and the k = 1 reader does not see more — the hole is simply not in its channel. “Fails by one dimension” was never channel-free. Theorem 2.2. (the deficiency module) D = A/W is a G-module with character χD = χA − χW. For a cyclic reader H = ⟨g⟩ of order m, the channel deficit is the Fourier coefficient δH,k(W) = (1/m) Σj ζm−jk χD(gj), 2.3 Fifty channels, two outcomes With closure reduced to a character computation, the declaration tree at every cyclic channel follows with no projectors, no per-channel rank and no search. Across all 15 cyclic reader classes and 50 channels, with capacities ranging from 38 to 139: quantity k = 0 every k ≠ 0 closing vocabularies (of 31) 12 15 declaration paths to closure 82 68 branch points 76 66 path lengths {2:4, 3:18, 4:36, 5:24} {2:8, 3:12, 4:24, 5:24} Table 2. Channel-conditioned declaration trees. Two outcomes across fifty channels, split on whether the channel is the trivial character. The k = 0 row reproduces the trivial-reader counts of Papers I–III, which is the regression control. Only three of the 31 vocabulary subsets change closure status between k = 0 and k = 1, two of them minimal; patching the k = 0 predicate with exactly those two reproduces the k = 1 tree to the last length bucket, and removing them recovers k = 0. The mechanism is therefore an exact predicate-level identity rather than an inference from totals. 3. Granularity: the Reachable Deficiency Spectrum 3.1 A retracted refutation, and why it belongs in the record Hypothesis 3.1. (equivariant vocabularies) The character machinery requires ρ(g)W = W for all g. The admissible search domain is therefore unions of G-orbits of cells, whose spans are invariant by construction. This is not a technicality: it states what an equivariant vocabulary declaration is. 3.2 A 3⁻ obstruction in the same arena quantity value orbits in the union 33 dim W 232 (dim A = 235) W is G-invariant yes, verified dim Wᴳ 15 = dim Aᴳ dim D 3 m(D) { 3⁻ : 1 } Table 3. An orbit-level vocabulary with full invariant rank and nonzero deficiency. Verified exactly over ℚ, not modulo a prime. deficiency type δ at k = 0 δ at k = 1 what the invariant reader reports D ≅ 1⁺ (disj+spec4) 1 0 sees the hole; k = 1 appears closed D ≅ 3⁻ (33-orbit union) 0 3 reports closed; k = 1 sees all of it Table 4. Mirror obstructions in one arena, one group, one ambient module, one cell inventory. Only the declaration language differs. A second construction, filling every channel but one and preferring orbits that contribute least to the target, isolates 3⁻ with 15 orbits and m(D) = {3⁻ : 2}. For the other five irreducibles the same construction closed the target as well, which is a failure of that construction and not a proof of impossibility. Definition 3.2. (reachable deficiency spectrum) For a vocabulary family 𝒱 with admissible unions S and spans WS ⊆ A, set 𝔇(𝒱) = { [A/WS] : S admissible }, with irreducible support Supp(𝒱) = { λ : ∃S, mλ(A/WS) > 0 }. Measured on this witness: Supp(five families) = {1⁺} across all 31 unions, while Supp(90 orbits) ⊇ {1⁺, 3⁻}. The arena, the group and the rational irreducibles are unchanged. Only the admissible declaration language changes, so granularity is an algebraic control parameter — refining the vocabulary changes which quotients are reachable. Corollary 3.3. (three statements rescoped) Over the five-family lattice: every nonzero deficiency contains 1⁺; the trivial channel is the hardest, so k = 0 closure implies closure at every k ≠ 0 (zero counterexamples over 50 channels); and there are exactly two channelwise declaration trees. All three are false over the orbit lattice. They are family-lattice statements, not arena-wide statements. 4. Reader Saturation Is Not Closure 4.1 The criterion Theorem 4.1. (vocabulary-relative certification) Invariant saturation at reader H certifies genuine closure over a vocabulary class 𝒱 if and only if dim λH > 0 for every λ ∈ Supp(𝒱). A reader is not faithful or unfaithful in the abstract; it is certificate-faithful relative to the deficiency types its declaration language can realize. 4.2 The diagnostic inversion, which is Frobenius reciprocity H reader order obstruction types its invariant channel can see trivial 1 6 of 6 an involution 2 6 of 6 order 3 3 4 of 6 order 4 4 3–4 of 6 order 6 6 2 of 6 order 8 8 2 of 6 order 12 12 2 of 6 full Aut(X) 24 1 of 6 — only 1⁺ Table 5. Visibility by reader. dim λᴴ is non-increasing along subgroup containment — zero violations across all 37 containment pairs among the twelve readers of Papers I–III. Corollary 4.2. (the diagnostic ceiling) The apparatus of Papers I–III is the invariant channel of the full automorphism reader. It can see exactly one of the six rational obstruction types. It was correct about 1⁺ because 1⁺ is the only type it could ever have found. The