We define a minimal categorical "kernel" that underlies emergence in the Unified Topological Mass Framework (UTMF). The kernel is formulated as a strict spherical braided dagger monoidal category generated by a single object X. From these axioms we derive: (i) a canonical categorical trace on motif endomorphism algebras , (ii) a positive trace pairing , (iii) induced symmetry actions (symmetric group at q=1, Hecke algebra at generic q) from braiding and channel structure, (iv) representation-theoretic sector projectors, orthogonality, and truncation as vanishing of sector traces. We then supply a bridge theorem: any local recursive rewrite dynamics on kernel-diagram classes induces a derived metric ("rewrite distance") on physical equivalence classes, yielding an emergent coarse geometry and an effective dimension from ball-growth, with smooth manifolds arising only at infrared renormalization fixed points.
UTMF is organized in layers. At the base sits an algebraic substrate intended to be independent of geometric or physical assumptions. The purpose of this paper is twofold:
The kernel is not a "model of spacetime." It is a minimal substrate from which multiple higher-layer specializations (including physically interpreted ones) may be built.
2.1 Kernel Axioms
Let be a strict monoidal category with tensor product and unit object . We assume:
2.2 Generating Object and Motif Algebras
Fix a distinguished generating object . For , define:
Define the motif algebra:
3.1 Categorical Trace
For , define the (spherical) trace:
For generator powers write .
3.2 Cyclicity
Proposition 3.1 (Cyclicity):
For :
Reason: in a spherical setting, closing ab and sliding a around the closure yields ba.
3.3 Trace Pairing and Positivity
Define an inner-product-like pairing on :
Proposition 3.2 (Sesquilinear Hermitian form):
is sesquilinear and Hermitian:
Axiom (K5) (Positivity):
For all , (in under any faithful scalarization).
This is the kernel's "physical admissibility" constraint.
4.1 Braid Group Action
Define adjacent braids on :
Proposition 4.1 (Braid relations):
4.2 Symmetric Group Action (q = 1)
If the braiding is symmetric, i.e., , then .
Theorem 4.2 (Symmetric group action):
Under symmetric braiding, defines a representation of on .
4.3 Hecke Action (q ≠ 1) from Two-Channel Structure
Assume with orthogonal idempotents and braiding diagonal on channels:
Theorem 4.3 (Hecke algebra action):
The satisfy:
Hence define a representation of the Hecke algebra on and on .
5.1 Central Idempotents and Sector Projectors
Let denote the induced algebra homomorphism:
- in the symmetric case, or
- in the Hecke case (generic semisimple regime)
Let be the minimal central idempotents indexed by partitions . Define sector projectors:
They satisfy:
5.2 Orthogonality under the Trace Pairing
Theorem 5.1 (Sector orthogonality):
For all :
5.3 Norm Scalars and Admissibility
Define the sector trace weight:
By positivity, sectors with negative weight are forbidden; typically .
5.4 Truncation as Vanishing of Sector Weights
Proposition 5.2 (Truncation criterion):
A sector of n is admissible iff , equivalently in faithful trace realizations. Sectors with vanish from the kernel decomposition.
Theorem 6.1 (Sector factorization):
Any scalar functional that is central with respect to the symmetry/Hecke action decomposes as:
and the sectors are orthogonal under .
Interpretation for UTMF:
Higher-layer operators (mass/stability/corridor filters) may be defined so that their scalar readouts depend only on sector weights and sector-local invariants. Vanishing implements corridor exclusion; suppression of implements energetic penalties.
7.1 Configuration Space from Kernel Diagrams
Let Diag be the set of finite diagrams generated by X, β, ev, coev, tensoring, and composition. Let "≡" be the equivalence relation generated by categorical axioms, rigidity, spherical isotopy, and kernel relations.
No geometry is assumed.
7.3 Derived Metric: Rewrite Distance
Define rewrite distance between two physical configurations:
7.4 Recursive Emergence Theorem
Assume:
Theorem 7.1 (Geometry from recursive closure):
Under (R1-R4), rewrite distance d defines a well-posed metric on , and the restricted space (A, d) is a coarse geometry with geodesics (minimal rewrite paths).
Define balls and emergent effective dimension:
Key Insight:
Geometry is not primitive. It is an invariant of recursion on kernel-defined configuration classes. Smooth manifolds arise only at infrared renormalization fixed points.
The kernel isolates a minimal categorical substrate sufficient for:
Canonical trace and positivity
Symmetry/Hecke actions from braiding
Sector projectors and orthogonality
Truncation as vanishing sector weights
The bridge theorem shows that recursive closure on kernel configurations yields a derived metric and hence geometry. This provides an exact "recursion-before-geometry" foundation compatible with UTMF corridor and stability ideas.
Next Papers:
- A precise mass operator built from sector weights and rewrite-curvature invariants
- Root-of-unity specializations and their truncation thresholds
- Explicit renormalization flows Γ yielding an IR Lorentzian sector
A.1 Symmetric Case (q=1): C = Vect_ℂ, X = V = ℂ^N
Let permute tensor factors on . Then:
n=2:
n=3:
For N=2: , i.e., (truncation).
A.2 Hecke Case (q ≠ 1)
Define q-numbers:
n=2 projectors:
n=3 idempotents:
With :
At roots of unity where , these expressions signal the expected truncation/non-semisimplicity mechanism.