POC Bounded Classical – Law L05: Conservation of Energy – SSM

A bounded two-lane view of E_in = E_out + E_loss so the same 51 J loss can be tagged calm, borderline, or stressed.

Domain. Energy Balance / Small Machines / Lab Bench
Classical law. E_in = E_out + E_loss

What this shows

In this law POC, we take a simple Conservation of Energy balance and show how:

  • The Classical calculation gives a consistent scalar energy loss E_loss, and
  • The Shunyaya Symbolic Mathematics (SSM) version keeps that loss magnitude intact but adds a bounded alignment lane a in (-1,+1), indicating whether this particular E_in = E_out + E_loss situation is calm, borderline, or stressed.

When input power, output work, and loss estimates are all well-behaved, Shunyaya Symbolic Mathematics (SSM) collapses to the classical result and you effectively get the same answer.

When measurements are noisy (supply jitter, jerky motion, crude loss estimation), the alignment lane a and its band surface posture that the classical scalar E_loss 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 small DC motor on a lab bench lifting a weight:

  • A 12 V supply powers the motor through a simple driver.
  • You measure current twice over a short lift.
  • You know approximately how far a weight is lifted.
  • From that, you estimate input energy, useful output work, and losses (heat, sound, friction).

We model:

  • Supply voltage (approximate but steady):
    • V_m = 12.0 V, V_a = +0.10 — decent bench supply, mild uncertainty.
  • Current measurements (instantaneous, motor start and near steady-state):
    • I1_m = 1.80 A, I1_a = +0.70 — jerkier start, higher jitter
    • I2_m = 1.60 A, I2_a = +0.15 — later, more stable
  • Lift time:
    • t_m = 3.0 s, t_a = +0.05 — stopwatch and control timing are reasonably good.
  • Lifted weight and height (useful mechanical output):
    • m_load_m = 2.0 kg, m_load_a = +0.05 — small mass uncertainty
    • h_m = 0.50 m, h_a = +0.10 — tape measure plus eye estimate

We will look at:

  • Input energy from the supply: E_in = V * I_avg * t
  • Useful mechanical output: E_out = m_load * g * h (with g ≈ 9.81 m/s^2)
  • Energy loss (by conservation): E_loss = E_in - E_out

2) Classical calculation

First, ignore alignment and do an ordinary energy balance.

# classical illustration (no external packages required)

V   = 12.0      # volts
I1  = 1.80      # amps
I2  = 1.60      # amps
t   = 3.0       # seconds

m_load = 2.0    # kg
h      = 0.50   # m
g      = 9.81   # m/s^2

I_avg   = 0.5 * (I1 + I2)
E_in    = V * I_avg * t             # input electrical energy
E_out   = m_load * g * h            # useful mechanical energy
E_loss  = E_in - E_out              # everything else (heat, noise, friction)

print(I_avg)   # 1.70 A
print(E_in)    # 61.2 J
print(E_out)   # 9.81 J
print(E_loss)  # 51.39 J

Classical result.

  • I_avg = 1.70 A
  • E_in ≈ 61.20 J
  • E_out ≈ 9.81 J
  • E_loss ≈ 51.39 J

A normal lab sheet might simply say:
Input ≈ 61 J, output ≈ 9.8 J, losses ≈ 51 J (≈84% losses).

There is no explicit distinction between clean, trustworthy measurements and borderline, noisy ones.

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

In Shunyaya Symbolic Mathematics (SSM), we:

  1. Treat all relevant quantities as (m,a) with a in (-1,+1).
  2. Pool the current alignment from the two samples using weighted sum pooling: a_c := clamp(a, -1+eps, +1-eps) u := atanh(a_c) U += w * u with w := |m|^gamma (default gamma = 1) W += w a_I_out := tanh( U / max(W, eps) )
  3. Combine V, I_avg, and t to get input energy alignment a_Ein using a product-style rule (sum of hyperbolic rapidities): a_Ein := tanh(atanh(a_V_c) + atanh(a_I_out_c) + atanh(a_t_c))
  4. Combine m_load, g, and h to get output energy alignment a_Eout: a_Eout := tanh(atanh(a_m_c) + atanh(a_g_c) + atanh(a_h_c)) (Here we treat g as effectively exact: a_g = 0.)
  5. Compute the scalar loss classically: m_E_loss = E_in - E_out
  6. For this POC, we define a simple SSM rule for the loss alignment: a_E_loss := tanh(atanh(a_Ein_c) + atanh(a_Eout_c)) Intuition: if either side of the energy balance is shaky, the loss lane should carry that shakiness.
  7. Collapse parity remains: phi((m_E_loss, a_E_loss)) = m_E_loss

So:

  • The magnitude of losses is still about 51.39 J.
  • The bounded alignment lane a_E_loss reports how “firm” or “soft” that loss figure is, given the noisy supply and output measurements.

4) Tiny script (copy-paste)

# scenario_L05_conservation_of_energy.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_sum(a_list, eps=1e-6):
    """
    Sum of hyperbolic rapidities for multiple lanes:
    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)

# 1) law-specific inputs: Conservation of Energy
# E_in = E_out + E_loss, solve for E_loss

# supply voltage (m, a)
V_m, V_a = 12.0, +0.10     # volts

# current measurements (m, a)
I1_m, I1_a = 1.80, +0.70   # amps, noisy start
I2_m, I2_a = 1.60, +0.15   # amps, calmer

# time (m, a)
t_m, t_a = 3.0, +0.05      # seconds

# load mass and height (m, a)
m_load_m, m_load_a = 2.0, +0.05   # kg
h_m,      h_a       = 0.50, +0.10 # m

# gravitational acceleration (m, a)
g_m, g_a = 9.81, 0.0       # m/s^2, treated as exact

# 2) classical magnitudes

I_avg_m = 0.5 * (I1_m + I2_m)
E_in_m  = V_m * I_avg_m * t_m          # input electrical energy

E_out_m = m_load_m * g_m * h_m         # useful mechanical energy
E_loss_m = E_in_m - E_out_m            # classical loss

# 3) SSM alignments

# pooled current alignment
a_I = ssm_align_weighted(
    [(I1_a, I1_m), (I2_a, I2_m)],
    gamma=1.0,
    eps=1e-12,
)

# input energy alignment from V, I, t
a_Ein = ssm_align_sum([V_a, a_I, t_a])

# output energy alignment from m, g, h
a_Eout = ssm_align_sum([m_load_a, g_a, h_a])

# loss alignment as combined posture of input and output
a_Eloss = ssm_align_sum([a_Ein, a_Eout])

print("Classical:")
print("  E_in   =", f"{E_in_m:.2f}", "J")
print("  E_out  =", f"{E_out_m:.2f}", "J")
print("  E_loss =", f"{E_loss_m:.2f}", "J")

print("SSM (loss lane):")
print("  E_loss =", f"m={E_loss_m:.2f}, a={a_Eloss:+.4f}")

You can attach a band in the shared runner, for example:

  • |a| < 0.20 → A+ (calm energy balance)
  • 0.20 <= |a| < 0.50 → A0 (borderline)
  • |a| >= 0.50 → A- (stressed / shaky balance)

5) What to expect

Numerically:

  • Classical:
    • E_in ≈ 61.20 J
    • E_out ≈ 9.81 J
    • E_loss ≈ 51.39 J
  • SSM (loss lane):
    • E_loss = 51.39 J (same m)
    • a_Eloss ≈ +0.6810 under drift-positive semantics

Under the sample band policy:

  • |a| < 0.20 → A+ (calm)
  • 0.20 <= |a| < 0.50 → A0 (borderline)
  • |a| >= 0.50 → A− (stressed / shaky balance)

we get:

  • a_Eloss ≈ +0.6810 → A− (stressed).

Interpretation:

  • The energy balance closes numerically, but the bounded alignment lane reveals that it was computed from:
    • a noisy supply current,
    • approximate lift height,
    • approximate mass.

So the headline loss (≈ 51 J) should be treated as posturally shaky, not a crisp calibration number.

If the motor current, timing, and geometry were all measured with much lower drift, you would see:

  • a_Eloss move closer to 0, and the same classical loss would then appear as a calmer, more reliable figure.

6) Why this helps in the real world

  • R&D teams evaluating small machines can separate “losses we trust” from “losses that are numerically plausible but measurement-noisy,” using the bounded alignment lane.
  • Test benches can color identical loss values differently by a or band, pointing engineers toward trials that deserve deeper investigation or repeat runs.
  • Teaching labs can demonstrate that conservation of energy is not just about numbers adding up, but also about how stable and trustworthy the underlying measurements are — exactly what SSM makes visible.

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.

Navigation

Previous: POC Bounded Classical – Law L04: Ideal Gas Law – SSM
Next: POC Bounded Classical – Law L06: Conservation of Momentum – SSM

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.