POC Bounded Classical – Law L01: Ohm’s Law – Shunyaya Symbolic Mathematics (SSM)

From a single voltage number to a visible posture lane `(m,a)`

Domain. Electrical / Lab Bench Test

Classical law. V = I * R

What this shows.

In this law POC, we take Ohm’s Law and show how:

The Classical calculation gives the correct scalar voltage V, and
The Shunyaya Symbolic Mathematics (SSM) version keeps that voltage magnitude intact but adds a bounded alignment lane a in (-1,+1), indicating whether this particular V = I * R situation is calm, borderline, or stressed.

When the supply and load are calm, SSM collapses to Classical and you effectively get the same answer.

When current and resistance are drifting or noisy, the alignment lane a and its band surface posture that the classical scalar V alone cannot show.

POC display policy (simple).

For this POC, we use:

a_semantics = "drift-positive"

so that larger printed a means more drift / more risk. This is a display choice only. You can flip the semantics (e.g., make +a mean more stability) without changing the math or phi((m,a)) = m.

1) Setup (inputs)

Imagine a 12 V DC bench supply driving a small device (for example, a motor or pump).

You take two current readings a few seconds apart; the device is slightly noisy on start-up, then calms down. The load resistance is approximately steady but not perfect.

We model:

Current measurements (instantaneous):
- I1_m = 1.92 A, I1_a = +0.72 — noisy moment (spin-up, higher jitter)
- I2_m = 1.98 A, I2_a = +0.05 — calmer moment (near steady-state)
Load resistance (approximate):
- R_m = 6.10 Ω, R_a = +0.10 — a real resistor with small tolerance & mild heating

For SSM:

Each quantity is represented as (m,a) with a in (-1,+1).
Collapse is phi((m,a)) = m (drop the alignment lane, keep the magnitude).

Our goal.

Use Ohm’s law V = I * R
Compare Classical voltage vs SSM voltage (m_V, a_V).

2) Classical calculation

First, ignore alignment and just do the usual Ohm’s law.
We average the two current readings and treat resistance as a scalar:

# classical illustration (no external packages required)

I1 = 1.92  # amps
I2 = 1.98  # amps
R  = 6.10  # ohms

I_avg = 0.5 * (I1 + I2)
V     = I_avg * R

print(I_avg)  # 1.95
print(V)      # 11.895

Classical result.

I_avg ≈ 1.95 A
V     ≈ 11.895 V

A typical lab report would say: "~11.9 V", with no extra information about how stable this situation is.

3) SSM calculation (same magnitude + bounded alignment lane)

In Shunyaya Symbolic Mathematics (SSM), we:

Treat each measurement as (m, a) with a in (-1,+1).
Compute an alignment for current using weighted pooling:
- For each sample, clamp and map into rapidity space:
  - a_c := clamp(a, -1+eps, +1-eps)
  - u := atanh(a_c)
- Accumulate with weights w := |m|^gamma (default gamma = 1):
  - U += w * u
  - W += w
- Collapse back to the alignment lane:
  - a_I_out := tanh( U / max(W, eps) )
Combine the current alignment with the resistance alignment using a product chaining rule for the law V = I * R:
- a_V := tanh(atanh(a_I_out_c) + atanh(a_R_c))
Keep the voltage magnitude as m_V = I_avg * R_m — exactly the classical result:
- phi((m_V, a_V)) = m_V = I_avg * R_m

So:

The magnitude is still 11.895 V.
The bounded alignment lane a_V tells us how stressed or calm the V = I * R relationship is in this situation, given one noisy current reading and one relatively calm one.

Intuitively:

Because I1 has high drift (a ≈ +0.72 under drift-positive semantics) and I2 is calmer (a ≈ +0.05), their pooled alignment a_I_out lands in a moderate range.
Combining this with a mildly drifting resistor (R_a ≈ +0.10) gives a voltage alignment a_V that is still moderate, not catastrophic, but certainly not “perfectly calm”.

4) Tiny script (copy-paste)

Below is a small script implementing this logic in ASCII-only Python:

# scenario_L01_ohms_law.py  (ASCII-only, top-level prints)

import math

def clamp(a, e=1e-6):
    return max(-1 + e, min(1 - e, float(a)))

def ssm_align_weighted(pairs, gamma=1.0, eps=1e-12):
    """
    pairs: iterable of (a_raw, m)
    weight w := |m|^gamma
    """
    U = 0.0
    W = 0.0
    for a_raw, m in pairs:
        a = clamp(a_raw)
        # atanh(a) = 0.5 * ln((1+a)/(1-a))
        u = 0.5 * math.log((1.0 + a) / (1.0 - a))
        w = abs(float(m)) ** gamma
        U += w * u
        W += w
    return math.tanh(U / max(W, eps))

def ssm_align_product(a1_raw, a2_raw, eps=1e-6):
    """
    Product chaining for alignment lane:
    a_out := tanh(atanh(a1_c) + atanh(a2_c))
    """
    a1 = clamp(a1_raw, eps)
    a2 = clamp(a2_raw, eps)
    u1 = 0.5 * math.log((1.0 + a1) / (1.0 - a1))
    u2 = 0.5 * math.log((1.0 + a2) / (1.0 - a2))
    return math.tanh(u1 + u2)

# 1) law-specific inputs: Ohm's law V = I * R

# current measurements (m, a) at two instants
I1_m, I1_a = 1.92, +0.72   # amps, noisy instant
I2_m, I2_a = 1.98, +0.05   # amps, calmer instant

# load resistance (m, a)
R_m,  R_a  = 6.10, +0.10   # ohms, mild drift

# 2) classical magnitude: average current, then V = I * R
I_avg = 0.5 * (I1_m + I2_m)
V_m   = I_avg * R_m

# 3) SSM alignments
a_I = ssm_align_weighted(
    [(I1_a, I1_m), (I2_a, I2_m)],
    gamma=1.0,
    eps=1e-12,
)

a_V = ssm_align_product(a_I, R_a, eps=1e-6)

print("Classical:", f"{V_m:.4f}")           # 11.8950
print("SSM:", f"m={V_m:.4f}, a={a_V:+.4f}") # a_V ~ +0.5173 (drift-positive)

Later, your shared runner can assign bands, for example:

|a| < 0.20          -> A+ (calm)
0.20 <= |a| < 0.50  -> A0 (borderline)
|a| >= 0.50         -> A- (stressed)

5) What to expect

Running the script gives roughly:

Classical: V ≈ 11.8950
SSM: m = 11.8950, a ≈ +0.52 (drift-positive semantics)

Under a simple band policy (for example, |a| ≥ 0.50 → A-):

band = A- (moderately stressed)

So:

Magnitude: same as classical (phi((m_V, a_V)) = m_V = 11.8950).
Posture: clearly not calm; one reading is noisy enough that we treat this Ohm’s law instance as “working, but in a stressed posture.”

If both current readings were calm and the resistor nearly ideal, we would see:

a_I ≈ 0, a_V ≈ 0, and SSM ≡ Classical (same number, posture neutral).

6) Why this helps in the real world

Lab technicians and students can see that two “valid” currents (1.92 A and 1.98 A) generating a seemingly neat voltage (~11.9 V) may still hide different levels of stress in the measurement setup.
Engineers can use a or its band to decide whether a given V = I * R condition is calm enough to trust for calibration, or whether they should re-measure, average longer, or adjust experimental conditions.
Dashboards can color identical voltage magnitudes differently using a or its band, improving trust and interpretability without changing any of the underlying arithmetic or the law itself.

7) License and scope

License. CC BY-NC 4.0 (non-commercial, attribution required).
Scope. Observation-only; not for critical use.

This POC is intended for thinking, experimentation, and education around bounded classical laws. It is not a safety case, design guarantee, or regulatory tool.

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All examples are observation-only and must be independently validated before any production or safety-critical use.

From a single voltage number to a visible posture lane (m,a)