Langues Weighting System (LWS)

Status: active reference
Updated: February 19, 2026
Scope: Layer 3 (Langues metric) + Layer 6 (flux breathing)

Purpose

This document defines the mathematical contract for Langues weighting in SCBE. It is code-aligned to:

packages/kernel/src/languesMetric.ts
src/symphonic_cipher/scbe_aethermoore/cli_toolkit.py (LWS/PHDM weight profiles)

If this document conflicts with code, code is canonical.

1. Core Metric

Let:

x = (x1..x6) be current state
mu = (mu1..mu6) be trusted reference state
d_l = |x_l - mu_l| >= 0
w_l > 0 be langue weights
beta_l > 0 be growth factors
omega_l > 0 and phi_l be temporal frequency/phase

Canonical metric:

L(x, t) = sum_{l=1}^6 w_l * exp(beta_l * (d_l + sin(omega_l*t + phi_l)))

Kernel implementation form takes d directly as a 6D point:

L(d, t) = sum_{l=1}^6 w_l * exp(beta_l * (d_l + sin(omega_l*t + phi_l)))

2. Weight Profiles

This repo currently uses two explicit profiles.

2.1 LWS-Linear (base operations)

Used in toolkit paths (cli_toolkit.py) for base operational weighting:

KO: 1.000
AV: 1.125
RU: 1.250
CA: 1.333
UM: 1.500
DR: 1.667

2.2 PHDM-Golden (governance/crisis scaling)

Used in kernel/harmonic paths and crisis-oriented weighting:

w_l = phi^(l-1) for l = 1..6
approximately: 1.000, 1.618, 2.618, 4.236, 6.854, 11.090

2.3 Rule

Always label which profile is active (lws or phdm) in experiments and claims. Do not mix profile results without explicit conversion.

3. Mathematical Properties

Below, assume w_l > 0, beta_l > 0.

3.1 Positivity

L(x,t) > 0 for all x,t.

Reason: each summand is positive (w_l > 0, exp(.) > 0).

3.2 Monotonicity in Deviation

For each dimension:

dL/dd_l = w_l * beta_l * exp(beta_l * (d_l + sin(...))) > 0

Any increase in d_l increases cost.

3.3 Bounded Temporal Breathing

Because sin(.) in [-1,1]:

L_min(x) <= L(x,t) <= L_max(x)

where:

L_min(x) = sum_l w_l * exp(beta_l * (d_l - 1))
L_max(x) = sum_l w_l * exp(beta_l * (d_l + 1))

3.4 Convexity in `d`

For each d_l:

d^2L/dd_l^2 = (beta_l^2) * w_l * exp(beta_l * (d_l + sin(...))) > 0

So L is strictly convex in d.

3.5 Smoothness Clarification

In variables d_l (distance inputs), L is smooth (C^infinity).
In variables x_l with d_l = |x_l - mu_l|, L is not differentiable at x_l = mu_l due to absolute value.

This is expected and does not break runtime behavior.

3.6 Gradient (distance form)

nabla_d L = [w_l*beta_l*exp(beta_l*(d_l + sin(...)))]_{l=1..6}

Steepest descent for alignment is -nabla_d L.

4. Fluxing / Fractional Dimension Extension

Define flux coefficients:

nu_l(t) in [0,1]
nu(t) = (nu_1..nu_6)

Flux metric:

L_f(x,t) = sum_{l=1}^6 nu_l(t) * w_l * exp(beta_l * (d_l + sin(omega_l*t + phi_l)))

ODE used by kernel-style implementation:

dot(nu_l) = kappa_l*(nu_bar_l - nu_l) + sigma_l*sin(Omega_l*t)

with clipping:

nu_l <- clip(nu_l, 0, 1)

Effective dimensionality:

D_f(t) = sum_{l=1}^6 nu_l(t) (range [0,6])

Semantic states (from getFluxState):

Polly: nu >= 0.9
Quasi: 0.5 <= nu < 0.9
Demi: 0.1 <= nu < 0.5
Collapsed: nu < 0.1

5. Cycle-Averaged Energy (Fixed d)

For one dimension with fixed d_l:

E_l = (1/T) * integral_0^T w_l*exp(beta_l*(d_l + sin(omega_l t + phi_l))) dt

Over one full sinusoidal cycle:

E_l = w_l * exp(beta_l*d_l) * I0(beta_l)

where I0 is the modified Bessel function of order 0.

Total cycle-averaged energy:

E = sum_{l=1}^6 w_l * exp(beta_l*d_l) * I0(beta_l)

6. Reference Implementation (Python)

import numpy as np

def langues_metric(x, mu, w, beta, omega, phi, t, nu=None):
    d = np.abs(x - mu)
    s = d + np.sin(omega * t + phi)
    nu = np.ones_like(w) if nu is None else nu
    return float(np.sum(nu * w * np.exp(beta * s)))

def flux_update(nu, kappa, nu_bar, sigma, Omega, t, dt, nu_min=1e-6):
    dnu = kappa * (nu_bar - nu) + sigma * np.sin(Omega * t)
    return np.clip(nu + dnu * dt, nu_min, 1.0)

7. Testing Contract

Minimum tests for any LWS changes:

Positivity: L > 0 for random valid inputs.
Monotonicity: increasing one d_l increases L.
Flux bounds: nu_l remains in [0,1].
State classification: thresholds map to Polly/Quasi/Demi/Collapsed.
Profile labeling: outputs identify lws vs phdm.

Existing related test surfaces:

tests/harmonic/languesMetric.test.ts
src/symphonic_cipher/tests/test_harmonic_scaling.py

8. Integration Notes

Layer 3: L or L_f provides weighted geometry cost.
Layer 6: flux ODE controls breathing intensity via nu.
Layer 12 coupling should declare regime:
- wall enforcement (H(d,R)=R^(d^2) family),
- bounded scoring (1/(1+d+2*pd) family),
- or another explicit formula.

Do not claim one universal harmonic formula across all modules without regime tags.