POC Bounded Classical – Law L06: Conservation of Momentum – SSM

A bounded two-lane view of 1D momentum conservation so the same small imbalance delta_p can be tagged calm, borderline, or stressed.

Domain. Mechanics / Collisions / Lab Carts
Classical law. m1 * u1 + m2 * u2 = m1 * v1 + m2 * v2

What this shows

In this law POC, we take Conservation of Momentum in a 1D collision and show how:

  • The Classical calculation gives you scalar momenta before and after collision (and a small imbalance delta_p), and
  • The Shunyaya Symbolic Mathematics (SSM) version keeps that imbalance magnitude delta_p intact but adds a bounded alignment lane a in (-1,+1), indicating whether this particular momentum-conservation check is calm, borderline, or stressed.

When masses and velocities are measured cleanly and the track is nearly frictionless, Shunyaya Symbolic Mathematics (SSM) collapses to the classical result and you effectively get the same answer.

When the velocity readings are jittery (blurred marks, timing noise, slight track friction), the alignment lane a and its band surface posture that the classical scalar delta_p 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 two lab carts on a low-friction track:

  • Cart 1 moves to the right and collides gently with Cart 2, which is initially at rest.
  • After collision, both carts move to the right with different velocities.
  • You measure their masses and velocities with reasonable but not perfect accuracy.

We model:

  • Masses (fairly well known):
    • m1_m = 1.50 kg, m1_a = +0.05
    • m2_m = 1.00 kg, m2_a = +0.05
  • Initial velocities (before collision):
    • u1_m = 1.20 m/s, u1_a = +0.40 — cart 1 measured from marks + timer (a bit noisy)
    • u2_m = 0.00 m/s, u2_a = +0.05 — cart 2 nearly at rest
  • Final velocities (after collision):
    • v1_m = 0.70 m/s, v1_a = +0.35 — cart 1 slowed down
    • v2_m = 0.80 m/s, v2_a = +0.20 — cart 2 sped up

In SSM:

  • Each physical quantity is represented as (m,a) with a in (-1,+1).
  • Collapse is phi((m,a)) = m.

We will:

  • Compute classical momenta before and after.
  • Compute the classical momentum imbalance delta_p = p_before - p_after.
  • Wrap delta_p in a two-lane SSM form (m_delta_p, a_delta_p) where a_delta_p reports how trustworthy that conservation check feels.

2) Classical calculation

Ignoring alignment, we compute scalar momentum before and after collision:

# classical illustration (no external packages required)

m1 = 1.50   # kg
m2 = 1.00   # kg

u1 = 1.20   # m/s (before, cart 1)
u2 = 0.00   # m/s (before, cart 2)

v1 = 0.70   # m/s (after, cart 1)
v2 = 0.80   # m/s (after, cart 2)

p_before = m1 * u1 + m2 * u2
p_after  = m1 * v1 + m2 * v2
delta_p  = p_before - p_after

print(p_before)  # 1.80 kg m/s
print(p_after)   # 1.85 kg m/s
print(delta_p)   # -0.05 kg m/s

Classical result.

  • p_before ≈ 1.80 kg m/s
  • p_after ≈ 1.85 kg m/s
  • delta_p ≈ -0.05 kg m/s

So the law is almost satisfied — only a small mismatch, which might be explained by friction, slight misalignment, or timing errors. A typical lab might say:

Momentum is conserved within experimental error.

But this is qualitative. There is no compact, quantitative “posture” signal telling you how shaky or solid this conclusion is.

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

In Shunyaya Symbolic Mathematics (SSM), we:

  1. Treat each mass or velocity as (m,a) with a in (-1,+1).
  2. For each cart, build a per-cart momentum alignment using a sum of rapidities for mass and velocity: a_p1_before := tanh(atanh(a_m1_c) + atanh(a_u1_c)) a_p2_before := tanh(atanh(a_m2_c) + atanh(a_u2_c)) a_p1_after := tanh(atanh(a_m1_c) + atanh(a_v1_c)) a_p2_after := tanh(atanh(a_m2_c) + atanh(a_v2_c))
  3. Pool these cart-level alignments into global momentum alignments before and after, using a weighted sum rule: a_c := clamp(a, -1+eps, +1-eps) u := atanh(a_c) U += w * u with w := |p_cart|^gamma (p_cart = m * v) W += w a_before := tanh( U / max(W, eps) ) # over p1_before, p2_before a_after := tanh( U / max(W, eps) ) # over p1_after, p2_after)
  4. Compute the classical imbalance: m_delta_p = p_before - p_after
  5. Build an alignment lane for the imbalance by combining before and after alignments: a_delta_p := tanh(atanh(a_before_c) + atanh(a_after_c))
  6. Respect collapse parity: phi((m_delta_p, a_delta_p)) = m_delta_p

So:

  • The imbalance magnitude is still delta_p ≈ -0.05 kg m/s.
  • The bounded alignment lane a_delta_p tells you whether this “almost conserved” result came from calm, clean measurements or from noisy, borderline measurements that just happen to nearly cancel numerically.

4) Tiny script (copy-paste)

# scenario_L06_conservation_of_momentum.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_weighted(pairs, gamma=1.0, eps=1e-12):
    """
    pairs: iterable of (a_raw, m_term)
    where m_term is the magnitude associated with that lane (e.g., cart momentum).
    """
    U = 0.0
    W = 0.0
    for a_raw, m_term 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_term)) ** gamma
        U += w * u
        W += w
    return math.tanh(U / max(W, eps))

# 1) law-specific inputs: Conservation of Momentum in 1D

# masses (m, a)
m1_m, m1_a = 1.50, +0.05   # kg
m2_m, m2_a = 1.00, +0.05   # kg

# initial velocities (u, a)
u1_m, u1_a = 1.20, +0.40   # m/s, cart 1 before
u2_m, u2_a = 0.00, +0.05   # m/s, cart 2 before

# final velocities (v, a)
v1_m, v1_a = 0.70, +0.35   # m/s, cart 1 after
v2_m, v2_a = 0.80, +0.20   # m/s, cart 2 after

# 2) classical momenta

p_before_m = m1_m * u1_m + m2_m * u2_m
p_after_m  = m1_m * v1_m + m2_m * v2_m
delta_p_m  = p_before_m - p_after_m

# 3) SSM alignments

# per-cart momentum alignments (before)
a_p1_before = ssm_align_sum([m1_a, u1_a])
a_p2_before = ssm_align_sum([m2_a, u2_a])

# per-cart momentum alignments (after)
a_p1_after  = ssm_align_sum([m1_a, v1_a])
a_p2_after  = ssm_align_sum([m2_a, v2_a])

# overall momentum alignment before and after
a_before = ssm_align_weighted(
    [(a_p1_before, m1_m * u1_m),
     (a_p2_before, m2_m * u2_m)],
    gamma=1.0,
    eps=1e-12,
)

a_after = ssm_align_weighted(
    [(a_p1_after, m1_m * v1_m),
     (a_p2_after, m2_m * v2_m)],
    gamma=1.0,
    eps=1e-12,
)

# imbalance alignment as combined posture of both sides
a_delta_p = ssm_align_sum([a_before, a_after])

print("Classical:")
print("  p_before =", f"{p_before_m:.3f}", "kg m/s")
print("  p_after  =", f"{p_after_m:.3f}", "kg m/s")
print("  delta_p  =", f"{delta_p_m:.3f}", "kg m/s")

print("SSM (imbalance lane):")
print("  delta_p =", f"m={delta_p_m:.3f}, a={a_delta_p:+.4f}")

A shared runner can attach bands, for example:

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

5) What to expect

Numerically (with the chosen values):

  • Classical:
    • p_before ≈ 1.80 kg m/s
    • p_after ≈ 1.85 kg m/s
    • delta_p ≈ -0.05 kg m/s
  • SSM (imbalance lane):
    • delta_p = -0.05 kg m/s (same magnitude)
    • a_delta_p ≈ +0.6744 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 / noisy conservation test)

we get:

  • a_delta_p ≈ +0.6744 → clearly in A− (stressed).

Interpretation:

  • The classical law appears almost satisfied (delta_p small).
  • But the bounded alignment lane shows that this “good-looking” conservation check is actually based on noisy velocity measurements and modest mass uncertainty.
  • In other words, the numbers look good, but the posture is shaky.

If all velocities and masses were measured more calmly (smaller a values), you might see:

  • a_before ≈ 0, a_after ≈ 0, a_delta_p ≈ 0
  • The same small delta_p would then be marked as a calm, reliable conservation check.

6) Why this helps in the real world

  • Teaching labs can go beyond “within experimental error” and quantify how trustworthy a conservation check is, using a or its band.
  • Researchers running many collision trials can automatically distinguish between clean and noisy runs, even when the scalar momenta appear similarly close.
  • Dashboards in robotics or experimental rigs can flag conservation tests by band, guiding attention toward trials where physics seems fine numerically but the posture is clearly stressed.

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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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.