A central achievement of the program developed by de Mello Koch and collaborators is the identification of operator bases that diagonalize correlation functions in non-planar regimes of large-N gauge theories. These constructions, most notably the Schur and restricted-Schur bases, provide a precise mechanism for controlling operator mixing and interpreting finite-N effects in terms of emergent collective degrees of freedom.

Despite their success, such constructions are typically presented within a specific physical context, relying on matrix degrees of freedom, gauge invariance, and large-N limits. This raises a foundational question: which aspects of the emergence mechanism are genuinely physical, and which are consequences of more primitive algebraic structure?

In this paper we answer this question by deriving the full de Mello Koch emergence pipeline from the Motif Calculus Kernel of the Unified Topological Mass Framework (UTMF). We show that the representation-theoretic organization, orthogonality, and finite-size truncation of operator sectors follow solely from the kernel axioms. The gauge-theoretic interpretation is thus revealed to be one realization of a more general algebraic phenomenon.

We briefly recall the axiomatic structure introduced in the companion paper The Motif Calculus Kernel of the Unified Topological Mass Framework.

Let 𝒞 be a spherical braided dagger monoidal category with a generating object X. For each n ∈ ℕ, define the motif operator algebra:

The kernel provides:

1. A categorical trace Tr: 𝒜ₙ → ℂ
2. An adjoint †: 𝒜ₙ → 𝒜ₙ
3. A braiding-induced action of either the symmetric group Sₙ or the Hecke algebra Hₙ(q)
4. A positive semi-definite pairing ⟨a, b⟩ := Tr(a†b)
5. An emergent loop parameter N_eff := Tr(Id_X)

No additional assumptions are made.

In the symmetric regime, the braiding satisfies b² = Id. The induced generators bᵢ define a representation of the symmetric group Sₙ on X^{⊗n}, and hence an action of the group algebra ℂ[Sₙ] on 𝒜ₙ.

In the Hecke regime, the braiding satisfies:

The generators {bᵢ} define a representation of the Hecke algebra Hₙ(q). When q is generic, the representation theory parallels that of the symmetric group with q-deformed dimensions. When q is a root of unity, the representation theory becomes non-semisimple and induces an additional truncation of admissible sectors, yielding a level-type cutoff analogous to those arising in quantum group and fusion-category constructions. The symmetric case is recovered in the limit q → 1.

For each partition R ⊢ n, let eᵣ denote the minimal central idempotent in ℂ[Sₙ] (or its q-deformed analogue in Hₙ(q)). Define the corresponding projector:

The projectors satisfy:

These relations follow directly from the algebraic properties of the idempotents and do not depend on any physical interpretation.

Proof Sketch:

Cyclicity of the trace and dagger compatibility imply ⟨Pᵣ, Pₛ⟩ = Tr(PᵣPₛ). Idempotency yields vanishing for R ≠ S, while the diagonal value reduces to the trace of the central idempotent, producing the stated norm factor.

In the symmetric regime, the norm factors take the explicit form:

Consequently, sectors corresponding to Young diagrams with more than N_eff rows have vanishing norm and decouple. This reproduces the finite-N truncation central to the de Mello Koch emergence mechanism.

In the Hecke regime, the corresponding expression involves q-numbers and yields an analogous truncation, potentially of level type when q is a root of unity.

Consider two generating objects X and Y, and the mixed tensor space:

Restriction of an irrep R ⊢ n+m of S_{n+m} (or H_{n+m}(q)) to the subgroup Sₙ × Sₘ, corresponding in gauge-theoretic realizations to a U(N) × U(M) symmetry acting on X^{⊗n} ⊗ Y^{⊗m}, decomposes as:

where g(R;r,s) are Littlewood–Richardson multiplicities.

Using intertwiners resolving the multiplicity spaces, one constructs projectors:

These satisfy analogous orthogonality relations and provide a complete decomposition of the multi-species operator algebra.

The results above reproduce precisely the mathematical structure underlying the de Mello Koch emergence pipeline:

• representation-labeled operator bases,
• exact orthogonality,
• controlled non-planar mixing,
• and finite-N truncation.

Crucially, none of these results require gauge symmetry, spacetime, or holography. They arise entirely from the Motif Calculus Kernel. The de Mello Koch constructions are therefore contained as a special realization of this more general substrate.

We have shown that the full representation-theoretic emergence mechanism associated with Schur and restricted-Schur operators follows from the Motif Calculus Kernel of UTMF. This establishes a sharp separation between substrate-level algebraic structure and higher-level physical interpretation.

The results presented here position UTMF as a unifying framework in which known emergence mechanisms are derived rather than assumed. Subsequent work will address the derivation of kernel parameters and their physical interpretation.