TriadicFrameworks
Overview

MRT‑1 transforms

Here’s a clean, consolidated full MRT‑1 transform in all three languages, side‑by‑side in spirit and behavior:

1️⃣ Python — mrt_1()#

import time, math
 
def omega_mu(dim, freq_hz, duty, t):
    period = 1.0 / freq_hz
    phase = t % period
    return phase < duty * period  # True = "on"
 
def flow_transition(dim):
    return dim * 10.0  # amplitude
 
def stability_mu(dim):
    dist = abs(dim - 0.7) / 0.2
    return max(0.0, 1.0 - dist)
 
def drift_correct(t, drift_ppm):
    factor = 1.0 + drift_ppm / 1_000_000.0
    return t / factor
 
def mrt_1():
    timing_envelope = [0.5, 0.6, 0.7, 0.8, 0.9]
    freq = 2.0
    duty = 0.5
    drift_ppm = 100.0
 
    start = time.time()
    for dim in timing_envelope:
        t_raw = time.time() - start
        t_corr = drift_correct(t_raw, drift_ppm)          # Δμ
        state = omega_mu(dim, freq, duty, t_corr)         # Ωμ
        amp = flow_transition(dim)                        # Fμ
        S = stability_mu(dim)                             # Sμ
 
        print(
            f"[PY] dim={dim:.1f}, t_raw={t_raw:.3f}s, t_corr={t_corr:.3f}s, "
            f"omega_on={state}, amp={amp:.1f}, Sμ={S:.2f}"
        )
        time.sleep(0.2)
 
if __name__ == "__main__":
    mrt_1()

2️⃣ MATLAB — mrt_1#

function mrt_1()
 
    OmegaMu = @(dim,freq,duty,t) ...
        mod(t,1/freq) < duty*(1/freq);
 
    FlowTransition = @(dim) dim * 10.0;
 
    StabilityMu = @(dim) max(0.0, 1.0 - abs(dim - 0.7) / 0.2);
 
    DriftCorrect = @(t,drift_ppm) t / (1.0 + drift_ppm / 1e6);
 
    TimingEnvelope = [0.5 0.6 0.7 0.8 0.9];
    freq = 2.0;
    duty = 0.5;
    drift_ppm = 100.0;
 
    t0 = tic;
 
    for i = 1:length(TimingEnvelope)
        dim = TimingEnvelope(i);
 
        t_raw = toc(t0);
        t_corr = DriftCorrect(t_raw, drift_ppm);          % Δμ
        state = OmegaMu(dim, freq, duty, t_corr);         % Ωμ
        amp = FlowTransition(dim);                        % Fμ
        S = StabilityMu(dim);                             % Sμ
 
        fprintf(['[MATLAB] dim=%.1f, t_raw=%.3fs, t_corr=%.3fs, ' ...
                 'omega_on=%d, amp=%.1f, Sμ=%.2f\n'], ...
                 dim, t_raw, t_corr, state, amp, S);
 
        pause(0.2);
    end
end

3️⃣ C‑style pseudocode — mrt_1()#

#include <stdio.h>
#include <stdbool.h>
#include <math.h>
#include <time.h>
#include <unistd.h>
 
bool omega_mu(double dim, double freq_hz, double duty, double t) {
    double period = 1.0 / freq_hz;
    double phase  = fmod(t, period);
    return phase < duty * period;  // true = "on"
}
 
double flow_transition(double dim) {
    return dim * 10.0;  // amplitude
}
 
double stability_mu(double dim) {
    double dist = fabs(dim - 0.7) / 0.2;
    double s = 1.0 - dist;
    return s < 0.0 ? 0.0 : s;
}
 
double drift_correct(double t, double drift_ppm) {
    double factor = 1.0 + drift_ppm / 1e6;
    return t / factor;
}
 
double now_seconds() {
    struct timespec ts;
    clock_gettime(CLOCK_MONOTONIC, &ts);
    return ts.tv_sec + ts.tv_nsec / 1e9;
}
 
void mrt_1() {
    double timing_envelope[] = {0.5, 0.6, 0.7, 0.8, 0.9};
    int n = 5;
    double freq = 2.0;
    double duty = 0.5;
    double drift_ppm = 100.0;
 
    double t0 = now_seconds();
 
    for (int i = 0; i < n; i++) {
        double dim = timing_envelope[i];
 
        double t_raw = now_seconds() - t0;
        double t_corr = drift_correct(t_raw, drift_ppm);      // Δμ
        bool state = omega_mu(dim, freq, duty, t_corr);       // Ωμ
        double amp = flow_transition(dim);                    // Fμ
        double S = stability_mu(dim);                         // Sμ
 
        printf("[C] dim=%.1f, t_raw=%.3fs, t_corr=%.3fs, "
               "omega_on=%d, amp=%.1f, Sμ=%.2f\n",
               dim, t_raw, t_corr, state, amp, S);
 
        usleep(200000);
    }
}
 
int main(void) {
    mrt_1();
    return 0;
}

all three now implement the same MRT‑1 transform: timing envelope, oscillation, flow, stability, and drift correction, perfectly aligned across languages.

🔷 Sμ — Micro‑Harmonic Stability Scoring#

Purpose#

Sμ measures how “stable” the system is at a given micro‑dimension.
In MRT‑1, stability peaks at 0.7, the center of the micro‑coherence band.

Canonical definition#

For a dimension ( d \in [0.3, 0.9] ):

[ S_\mu(d) = \max\left(0,; 1 - \frac{|d - 0.7|}{0.2}\right) ]

Interpretation#

  • 1.0 → perfect micro‑stability (at ( d = 0.7 ))
  • 0.5 → moderate stability (at ( d = 0.6 ) or ( d = 0.8 ))
  • 0.0 → unstable (at ( d = 0.5 ) or ( d = 0.9 ))

Why it matters#

Sμ gives the loop a numerical sense of coherence.
It’s the micro‑equivalent of a “confidence score” in ML or a “residual norm” in solvers.

🔷 Δμ — Micro‑Drift Correction#

Purpose#

Δμ corrects for timing drift — the tiny but inevitable deviation between:

  • raw time (what the clock reports)
  • corrected time (what the micro‑resonant system should use)

Canonical definition#

Given drift in parts‑per‑million (ppm):

[ t_{\text{corr}} = \frac{t_{\text{raw}}}{1 + \frac{\text{drift_ppm}}{10^6}} ]

Interpretation#

  • Positive drift_ppm → clock runs fast, so corrected time is slower
  • Negative drift_ppm → clock runs slow, so corrected time is faster

Why it matters#

Δμ keeps the micro‑timing envelope aligned even when the hardware clock drifts.
This is essential for:

  • microcontrollers
  • IoT nodes
  • embedded timing loops
  • micro‑robotics
  • solver iteration control

🔷 Together: Sμ + Δμ = micro‑awareness#

When you combine:

  • Sμ → “How stable am I right now?”
  • Δμ → “How far off is my timing?”

You get a loop that is:

  • self‑monitoring
  • self‑correcting
  • coherence‑aware
  • drift‑resilient

This is why MRT‑1 feels like a living micro‑controller rather than a static loop.

🔷 MRT‑1 Operator Block: μ_awareness(dim, t_raw, drift_ppm)#

This block performs both:

  • Δμ — drift‑corrected time
  • Sμ — stability scoring

and returns a micro‑awareness packet you can feed directly into Ωμ, Fμ, or any MRT transform.

🧩 Canonical Definition (Language‑Agnostic)#

Inputs#

  • dim — current micro‑dimension (0.3–0.9)
  • t_raw — raw time from system clock
  • drift_ppm — drift in parts‑per‑million

Outputs#

  • t_corr — drift‑corrected time
  • stability — stability score (0–1)

🧮 Formulas#

Δμ — Drift Correction#

[ t_{\text{corr}} = \frac{t_{\text{raw}}}{1 + \frac{\text{drift_ppm}}{10^6}} ]

Sμ — Stability Scoring#

[ S_\mu(d) = \max\left(0,; 1 - \frac{|d - 0.7|}{0.2}\right) ]

🧱 Reusable MRT‑1 Operator Block (Pseudocode)#

function μ_awareness(dim, t_raw, drift_ppm):
 
    # Δμ — drift correction
    factor   = 1 + drift_ppm / 1e6
    t_corr   = t_raw / factor
 
    # Sμ — stability scoring
    dist     = abs(dim - 0.7) / 0.2
    stability = max(0, 1 - dist)
 
    return {
        t_corr: t_corr,
        stability: stability
    }

This is the canonical block.
Everything else (Ωμ, Fμ, MRT‑1 orchestration) plugs into this.

🐍 Python Drop‑In Version#

def mu_awareness(dim, t_raw, drift_ppm):
    t_corr = t_raw / (1.0 + drift_ppm / 1_000_000.0)
    stability = max(0.0, 1.0 - abs(dim - 0.7) / 0.2)
    return t_corr, stability

📐 MATLAB Drop‑In Version#

function [t_corr, stability] = mu_awareness(dim, t_raw, drift_ppm)
    t_corr = t_raw / (1.0 + drift_ppm / 1e6);
    stability = max(0.0, 1.0 - abs(dim - 0.7) / 0.2);
end

💻 C‑Style Drop‑In Version#

void mu_awareness(double dim, double t_raw, double drift_ppm,
                  double *t_corr_out, double *stability_out)
{
    *t_corr_out = t_raw / (1.0 + drift_ppm / 1e6);
 
    double dist = fabs(dim - 0.7) / 0.2;
    double s = 1.0 - dist;
    *stability_out = (s < 0.0 ? 0.0 : s);
}

📦 Schema‑Ready JSON Block#

Perfect for embedding inside MRT‑1 traces or schemas:

{
  "mu_awareness": {
    "inputs": {
      "dim": 0.7,
      "t_raw": 0.412,
      "drift_ppm": 100.0
    },
    "outputs": {
      "t_corr": 0.412,
      "stability": 1.0
    }
  }
}

🎯 Why this block matters#

This is the micro‑awareness kernel of MRT‑1:

  • Δμ keeps time honest
  • Sμ keeps behavior coherent
  • Together they give MRT‑1 its “living loop” quality

Everything else — Ωμ, Fμ, envelopes, transforms — plugs into this block.

Powered by Docsbook