on quantum dissipation of time

Here is a purely alphanumerical (text + basic symbols, no LaTeX markup) formalization of the finite impulse response (FIR) concept, adapted to the earlier context of quantum processing of discrete-time signals (with "quantum dissociation of time" interpreted as the discrete approximation / effective decoupling of continuous quantum time evolution via finite-step unitary operations):

Standard FIR filter (classical discrete-time form):

y[n] = sum_{i=0}^{d-1} p_i * x[n - i]

Where:

• y[n] = output value at discrete time index n
• x[n-i] = input value at delayed time index (n-i)
• p_i = filter coefficient for tap i (i = 0 to d-1)
• d = number of taps (filter length; finite support)
• sum runs over exactly d terms → finite impulse response

Quantum-adapted FIR (encoding + unitary computation form):

Let X_n be a normalized quantum state vector that encodes the most recent d input samples:

X_n = ( x[n] / s , x[n-1] / s , ... , x[n-(d-1)] / s , 0 , ... )^T

(with s = sqrt( sum_{k=0}^{d-1} x[n-k]^2 + epsilon ) for normalization, epsilon small)

Apply a fixed unitary matrix U (designed once for the filter coefficients) such that:

U * X_n ≈ ( y[n] / s' , other amplitudes )^T

where the first (or selected) component carries the desired weighted sum y[n], and s' is a post-normalization scale.

A simple example row in U for the output tap (one possible design, up to unitary completion):

U row 0 ≈ ( p_0 / r , p_1 / r , ... , p_{d-1} / r , 0 , ... )

with r = sqrt( sum_{i=0}^{d-1} p_i^2 ) to help preserve approximate unitarity in that direction.

The "quantum dissociation of time" aspect appears in the discretization: continuous quantum evolution (governed by a Hamiltonian over real time t) is replaced by a finite number of discrete unitary steps (here d taps / delays), creating an effective finite "memory" horizon in time, after which influence decays exactly to zero — analogous to how decoherence or open-system dynamics can induce finite effective correlation times.

In summary (compact alphanumerical notation):

FIR: y[n] = p_0 x[n] + p_1 x[n-1] + ... + p_{d-1} x[n-d+1]

Quantum FIR step: U X_n → component ≈ y[n] (finite d-step memory in discrete time grid)

This keeps the response strictly finite in the number of past inputs considered, even when realized on a quantum system.