POC Bounded Classical – Law L10: Faraday’s Law of Induction – SSM

A bounded two-lane view of Faraday’s Law so the same induced EMF |eps| can be tagged calm, borderline, or stressed.

Domain. Electromagnetics / Generators / Lab Coil

Classical law.

In its simplest form for a coil with N turns and changing magnetic flux Phi:

• eps = -N * dPhi_dt

For this POC we focus on the magnitude of induced EMF (ignore the sign):

• |eps| = N * |dPhi_dt|

with a finite-difference estimate:

• dPhi_dt ≈ (Phi2 - Phi1) / dt

What this shows

In this law POC, we take Faraday’s Law of Electromagnetic Induction and show how:

• The Classical calculation gives the scalar EMF |eps| induced in a coil, and
• The Shunyaya Symbolic Mathematics (SSM) version keeps that EMF magnitude intact but adds a bounded alignment lane a in (-1,+1), indicating whether this particular eps = -N * dPhi_dt situation is calm, borderline, or stressed.

When:

• The coil speed is steady,
• Flux measurements are clean, and
• Timing is sharp,

Shunyaya Symbolic Mathematics (SSM) collapses to the Classical result and you effectively get the same answer.

But in real labs and machines, dPhi_dt is notoriously noisy:

• Jerky coil motion or shaft non-uniformity,
• Flux probe jitter,
• Coarse timing marks.

The alignment lane a and its band reveal posture that the classical scalar EMF alone cannot show.

POC display policy (simple).

For this POC, we use:

• a_semantics = "drift-positive"

so larger printed a means more drift / more risk. This is a display choice only; you can flip semantics without changing the math or phi((m,a)) = m.

1) Setup (inputs)

Imagine a lab induction coil near a changing magnetic field:

• A coil with known turns sits near a rotating magnet.
• You sample the magnetic flux through the coil twice, a short time apart.
• You estimate the induced EMF using a finite difference.

We model:

• Number of turns (reasonably well known):
  • N_m = 200, N_a = +0.02 — coil turns counted and archived; small uncertainty.
• Flux samples (two instants):
  • Phi1_m = 0.012 Wb, Phi1_a = +0.60 — first reading, coil still speeding up / probe noisy
  • Phi2_m = 0.004 Wb, Phi2_a = +0.25 — second reading, a bit steadier
• Sampling time interval:
  • dt_m = 0.040 s, dt_a = +0.10 — timing from scope or controller, modest jitter

In SSM:

• Each physical quantity is two-lane: (m, a) with a in (-1,+1).
• Collapse parity is phi((m,a)) = m — drop the alignment lane, keep the magnitude.

Our target:

• Compute classical EMF from the two flux samples, then
• Wrap it as (m_eps, a_eps) with a meaningful bounded alignment lane.

2) Classical calculation

Ignoring alignment, use a simple two-point estimate of dPhi_dt:

# classical illustration (no external packages required)

N    = 200      # turns
Phi1 = 0.012    # Wb
Phi2 = 0.004    # Wb
dt   = 0.040    # s

dPhi_dt = (Phi2 - Phi1) / dt
eps_mag = N * abs(dPhi_dt)

print(dPhi_dt)   # -0.200 Wb/s
print(eps_mag)   # 40.0 V

Classical result.

• dPhi_dt ≈ -0.200 Wb/s
• |eps| ≈ 40.0 V

A typical lab note might simply say:
“Induced EMF ≈ 40 V at this speed.”

There is no explicit distinction between:

• 40 V from a clean, smooth spin, and
• 40 V from a jerky, poorly timed, noisy flux measurement.

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

In Shunyaya Symbolic Mathematics (SSM), we:

1. Treat each flux sample as (m,a) and build a flux-change alignment for
   DeltaPhi = Phi2 - Phi1. Conceptually:
   • DeltaPhi = Phi2 - Phi1
   For posture, both samples matter. We combine their lanes via a sum of rapidities: a_DeltaPhi := tanh(atanh(a_Phi1_c) + atanh(a_Phi2_c))
2. Treat the time interval as (dt_m, dt_a) and build an alignment for
   dPhi_dt = DeltaPhi / dt using a division rule: a_dPhi_dt := tanh(atanh(a_DeltaPhi_c) - atanh(a_dt_c))
3. Combine the turns lane and derivative lane to get the EMF lane: a_eps := tanh(atanh(a_N_c) + atanh(a_dPhi_dt_c))
4. Keep the EMF magnitude as the classical value eps_mag_m: eps_mag_m = N_m * abs((Phi2_m - Phi1_m) / dt_m) phi((eps_mag_m, a_eps)) = eps_mag_m

So:

• The induced EMF magnitude remains ≈ 40 V.
• The bounded alignment lane a_eps tells you whether that 40 V comes from steady, well-resolved induction or noisy, jerky induction.

4) Tiny script (copy-paste)

# scenario_L10_faraday_induction.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_sum(a_list, eps=1e-6):
    """
    Sum of hyperbolic rapidities:
    a_out := tanh(atanh(a1_c) + atanh(a2_c) + ...)
    """
    U = 0.0
    for a_raw in a_list:
        a = clamp(a_raw, eps)
        U += 0.5 * math.log((1.0 + a) / (1.0 - a))
    return math.tanh(U)

def ssm_align_div(a_num_raw, a_den_raw, eps=1e-6):
    """
    Division for alignment lane:
    a_out := tanh(atanh(a_num_c) - atanh(a_den_c))
    """
    a_num = clamp(a_num_raw, eps)
    a_den = clamp(a_den_raw, eps)
    u_num = 0.5 * math.log((1.0 + a_num) / (1.0 - a_num))
    u_den = 0.5 * math.log((1.0 + a_den) / (1.0 - a_den))
    return math.tanh(u_num - u_den)

# 1) law-specific inputs: Faraday's law |eps| = N * |dPhi/dt|
# with dPhi/dt ≈ (Phi2 - Phi1) / dt

# number of turns (m, a)
N_m, N_a = 200, +0.02    # turns

# flux samples (m, a)
Phi1_m, Phi1_a = 0.012, +0.60   # Wb, noisy first sample
Phi2_m, Phi2_a = 0.004, +0.25   # Wb, calmer second sample

# time interval (m, a)
dt_m, dt_a = 0.040, +0.10       # s

# 2) classical magnitudes

dPhi_dt_m = (Phi2_m - Phi1_m) / dt_m
eps_mag_m = N_m * abs(dPhi_dt_m)

# 3) SSM alignments

# lane for DeltaPhi from two flux samples
a_dPhi = ssm_align_sum([Phi1_a, Phi2_a])

# lane for dPhi/dt = DeltaPhi / dt
a_dPhi_dt = ssm_align_div(a_dPhi, dt_a, eps=1e-6)

# EMF alignment from N * dPhi/dt
a_eps = ssm_align_sum([N_a, a_dPhi_dt])

print("Classical:")
print("  dPhi/dt ~", f"{dPhi_dt_m:.3f}", "Wb/s")
print("  |eps|   =", f"{eps_mag_m:.2f}", "V")

print("SSM (induced EMF lane):")
print("  |eps| =", f"m={eps_mag_m:.2f}, a={a_eps:+.4f}")

Later, your shared runner can assign bands, e.g.:

• |a| < 0.20 → A+ (calm induction snapshot)
• 0.20 <= |a| < 0.50 → A0 (borderline)
• |a| >= 0.50 → A- (stressed / noisy induction)

5) What to expect

Numerically:

• Classical:
  • dPhi_dt ≈ -0.200 Wb/s
  • |eps| ≈ 40.0 V
• SSM (induced EMF lane):
  • m = 40.0 V (same as classical)
  • a_eps will be fairly large positive (due to a very noisy first flux sample and a shaky dt).

Under the sample band policy, you will typically see:

• a_eps in A− (stressed) — indicating that although the EMF value is numerically reasonable, it comes from a noisy dPhi_dt estimate.

If you improved the setup:

• Smoother mechanical motion,
• More stable flux probe (smaller Phi1_a, Phi2_a),
• Better timing (smaller dt_a),

then:

• a_dPhi and a_dPhi_dt would shrink toward 0,
• a_eps would move toward 0,

and the same ~40 V would now appear as a calmer, more trustworthy EMF measurement.

6) Why this helps in the real world

• Generator designers can distinguish between EMF outputs that come from smooth, well-controlled operation and those that come from jerky, noisy operating points, even when the RMS voltages look similar.
• Lab experiments on induction can show students that dPhi/dt measurements are often the weakest link, and SSM makes that visible via a.
• Condition monitoring could use a_eps as a simple lane for operational posture, highlighting when induced voltages look fine but are rooted in unstable flux dynamics.

7) License and scope

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

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

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Disclaimer (summary):
All Shunyaya Symbolic Mathematics (SSM) POCs for popular laws are observation-only examples and must not be used for design, certification, or safety-critical decisions.