UNIFIED DISPLACEMENT KERNEL — Semantic Compression

1 The Fragmentation Problem

Groove engines in digital audio workstations use branching logic to switch between timing models — linear offsets, curved drag, stochastic jitter, and hardware quantization. Each style requires its own code path, its own tests, and its own failure modes. Profiles cannot be interpolated. Hybrid feels cannot be composed. The timing behavior of these systems is typically unbounded and non-deterministic.

This fragmentation mirrors the broader problem Semantic Compression addresses: meaning is scattered across incompatible formats with massive structural overhead. In this domain, the “meaning” is musical feel — and the overhead is branching logic that grows linearly with every new groove style.

2 The Equation

A single additive displacement kernel replaces all branching logic. Groove style is controlled entirely by coefficient values.

T_final = Quantize_PPQN( T_grid + β · [ ΔL + Γ(m) · ( ΔC(n) + Ω(v) ) + Ψ(b) ] )

Unified Displacement Kernel — Canonical Form

Term	Name	Function	Layer
`T_grid`	Grid Time	Quantized step position from BPM and step index	Transport
`β`	BPM Scalar	90 / currentBPM — normalizes offsets to reference tempo	Transport
`ΔL`	Linear Offset	Static per-channel timing bias (milliseconds)	Event Scheduler
`ΔC(n)`	Curvature	Progressive drag: D_max · (n/N)^α	Groove Field
`Ω(v)`	Phase Coupling	Velocity-dependent timing shift: ρ · (v − v₀)	Physics
`Ψ(b)`	Macro Drift	Bar-periodic breathing: A · sin(2π · barIndex / period)	Phrase
`Γ(m)`	Harmonic Gravity	Mode multiplier — scales elastic components only	Harmonic
`Q_PPQN`	Hardware Quantize	Final transform — rounds to nearest hardware pulse	Hardware

The kernel is a pure function: no mutation, no side effects, no branching, deterministic for identical inputs. Harmonic gravity Γ(m) multiplies only the elastic components (ΔC + Ω), never linear offset or macro drift. Hardware quantization is applied last.

3 Global Boundedness

Each basis function in the kernel is individually bounded on compact domains.

Component	Domain	Bound
`β` (BPM scalar)	BPM > 0	0 < β ≤ 90 / BPM_min
`ΔL` (linear offset)	Per-channel constant	\|ΔL\| ≤ L_max
`ΔC(n)` (curvature)	n/N ∈ [0, 1]	\|ΔC\| ≤ D_max
`Ω(v)` (phase coupling)	v ∈ [0, 1]	\|Ω\| ≤ \|ρ\|
`Ψ(b)` (macro drift)	Periodic oscillator	\|Ψ\| ≤ M_max
`Γ(m)` (harmonic gravity)	τ ∈ [0, 1]	\|Γ\| ≤ Γ_max

1

Define the interior sum bound: C = L_max + Γ_max · (D_max + |ρ|) + M_max

2

Apply triangle inequality: |S| ≤ C (finite constant)

3

Bound total displacement: |Δ| ≤ (90 / BPM_min) · C = Δ_max

4

All basis functions are continuous on compact domains. By the Extreme Value Theorem, the displacement attains its bounds.

Δ ∈ [−Δ_max, Δ_max] for all valid configurations.

The final safety clamp is mathematically redundant but correct as an engineering invariant.

4 Style as Coefficient Vector

Every groove style maps to a coefficient vector. No branching. No style-specific code paths. The same kernel evaluates all profiles.

Style	ΔL	α	A	ρ	PPQN	Displacement Character
Boom Bap	10	1.0	0	0	96	Linear lean — static snare drag
Neo-Soul (Dilla)	0	1.25	4	0.2	96	Elastic pocket — progressive curvature
Timbaland	0	1.0	8	0	96	Phrase breathing — macro drift dominant
MPC60 Hardware	6	1.0	0	0	96	Grid + quantize — hardware emulation
G-Funk	8	1.0	3	0	96	Sub-bass lag with mild phrase sway
New Jack Swing	2	1.0	0	0.3	96	Tight grid with velocity-phase interaction

Six coefficient vectors. One equation. Decades of production knowledge compressed into structured data.

5 Significance

This case study extends the Semantic Compression thesis beyond enterprise documents into a new domain: deterministic temporal field modeling for digital audio.

The kernel demonstrates three properties central to the thesis:

Meaning Density. An entire production style is encoded in five numbers. The ratio of semantic content to representation size approaches the theoretical limit.

Format Elimination. Branching code paths, style-specific logic, and per-type test suites are replaced by a single additive equation. Presentation is eliminated; only structure remains.

Deterministic Validation. The boundedness proof guarantees that no configuration produces unbounded displacement. Same inputs produce the same output. The kernel satisfies four invariants: bounded, BPM-scaled, deterministic, and resettable at phrase boundary.

The Unified Displacement Kernel compresses decades of musical intuition into a coefficient vector.

Intelligence lives in data, not models.