Methodology & Mathematical Formulation

The central algorithmic challenge in adaptive neural inference is deciding when to allocate computation. Q-TensorFormer treats this as a dynamic constrained optimization problem governed by Lagrangian duality and PID control.

1. The Constrained Allocation Objective

Let a ∈ A denote an allocation decision (comprising tensor rank r, layer depth mode, attention pathway, and KV precision). We formulate the inference objective as maximizing expected prediction quality subject to hardware budgets:

Constrained Optimization Problem:

max_{a} ΔQ(a | z_t) s.t. C_k(a | z_t) ≤ B_k, ∀ k ∈ {Latency, Memory, Energy, Bandwidth}

Forming the Lagrangian dual with shadow prices λ = (λ_L, λ_M, λ_E, λ_B) ≥ 0:

L(a, λ | z_t) = ΔQ(a | z_t) - sum_k λ_k * [C_k(a | z_t) - B_k]

The unconstrained marginal utility of candidate action a is defined as:

U(a | z_t, λ) = ΔQ_hat(a | z_t) - sum_k λ_k * ΔC_k_hat(a | z_t)

2. Hysteresis Switching Deadband (τ)

Without stabilization, noisy fluctuations in token entropy cause rapid switching between different matrix ranks on consecutive tokens: a phenomenon known as routing chattering. Chattering destroys hardware cache locality and incurs CPU instruction pipeline stalls.

To prevent chattering, Q-TensorFormer implements a hysteresis deadband threshold τ = 0.15:

Let a* = argmax_{a} U(a | z_t, λ)

Action Decision Rule:
a_t = a*       if  U(a* | z_t, λ) - U(a_{t-1} | z_t, λ) ≥ τ
a_t = a_{t-1}   otherwise (retain previous configuration)

Empirical Impact: Over a sequence of 64 fluctuating tokens, introducing τ = 0.15 reduced rank-switching churn from 25.40% down to 0.00%, completely eliminating instability without quality loss.

3. Dual-Subgradient PID Controller

Because hardware execution conditions change dynamically (e.g. thermal throttling, OS background processes), static Lagrange multipliers fail over long horizons. Q-TensorFormer updates shadow prices using a Proportional-Integral-Derivative (PID) dual controller (src/resource_allocator.py):

e_k(t) = C_k_measured(t) - B_k_target
Integral term: I_k(t) = clamp(I_k(t-1) + K_i * e_k(t), -AntiWindup, +AntiWindup)
Derivative term: D_k(t) = K_d * (e_k(t) - e_k(t-1))

Multiplier Update:
λ_k(t) = max(0, λ_k(t-1) + K_p * e_k(t) + I_k(t) + D_k(t))

Anti-windup clamping prevents integral saturation during extended bursts of heavy inference, allowing immediate recovery when sequence demand relaxes.

4. Dynamic Information State Normalization

The raw 8D feature probe vector z_tilde_t contains signals spanning different physical units (e.g., entropy in nats, norms in Euclidean distance, latencies in milliseconds). Q-TensorFormer computes online Exponential Moving Average (EMA) statistics to normalize each dimension into a stationary [0, 1] range:

μ_t = α * μ_{t-1} + (1 - α) * z_tilde_t
σ^2_t = α * σ^2_{t-1} + (1 - α) * (z_tilde_t - μ_t)^2

Normalized State:
z_t = sigmoid( (z_tilde_t - μ_t) / sqrt(σ^2_t + 1e-6) ) * m_ablate