POC Bounded Classical – Law L03: Hooke’s Law – SSM

A bounded two-lane view of F = k * x so the same 10 N force can be tagged calm, borderline, or stressed.

Domain. Mechanics / Materials Testing / Lab Bench
Classical law. F = k * x

What this shows.
In this law POC, we take Hooke’s Law and show how:

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

When the extension measurements are clean and the spring constant is well-characterized, Shunyaya Symbolic Mathematics (SSM) collapses to the classical result and you effectively get the same answer.

When the displacement readings are noisy or the spring is behaving non-ideally, the alignment lane a and its band surface posture that the classical scalar F 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 the semantics (for example, make +a mean more stability) without changing the math or phi((m,a)) = m.

1) Setup (inputs)

Imagine a simple spring test on a lab bench:

  • You hang a small weight on a coil spring and measure how far it stretches.
  • You take two displacement readings as the system settles: the first is a bit jerky (you just released the weight), the second is more calm.
  • The spring constant k is known from prior calibration but not perfectly exact.

We model:

  • Spring constant (approximate):
    • k_m = 200.0 N/m, k_a = +0.08
    • Reasonably well-known spring; small uncertainty in calibration and temperature.
  • Displacement measurements (instantaneous):
    • x1_m = 0.045 m, x1_a = +0.60 — slightly jerky reading (still settling)
    • x2_m = 0.055 m, x2_a = +0.10 — calmer reading (closer to steady-state)

In SSM:

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

Our target.

  • Use Hooke’s law F = k * x.
  • Compare Classical force vs SSM force (m_F, a_F).

2) Classical calculation

Ignoring alignment, we do the usual Hooke’s law with an average displacement:

# classical illustration (no external packages required)

k  = 200.0      # N/m (approx spring constant)
x1 = 0.045      # m (jerkier reading)
x2 = 0.055      # m (calmer reading)

x_avg = 0.5 * (x1 + x2)
F     = k * x_avg

print(x_avg)  # 0.05
print(F)      # 10.0

Classical result.

  • x_avg = 0.050 m
  • F = 10.0 N

A standard lab sheet would simply record: Force ≈ 10 N, with no sense of whether the spring behaviour during measurement was stable or twitchy.

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

In Shunyaya Symbolic Mathematics (SSM), we:

  1. Treat each displacement sample as (m, a) with a in (-1,+1).
  2. Compute an alignment for displacement using the weighted sum pooling rule:
    • 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_x_out := tanh( U / max(W, eps) )
  3. Combine the displacement alignment with the spring constant alignment using product chaining for the law F = k * x:
    • a_F := tanh(atanh(a_k_c) + atanh(a_x_out_c))
  4. Keep the force magnitude as m_F = k_m * x_avg — exactly the classical result:
    • phi((m_F, a_F)) = m_F = k_m * x_avg

So:

  • The magnitude of force is still 10.0 N.
  • The bounded alignment lane a_F tells us how “ideal” this Hooke’s law instance is, given one jerky displacement sample and one calmer one, plus a slightly uncertain k.

Intuitively:

  • One displacement reading is relatively noisy (x1_a ≈ +0.60 under drift-positive semantics).
  • The other is close to calm (x2_a ≈ +0.10).
  • Their pooled displacement alignment a_x_out lands in a moderate risk region.
  • Combining this with a mildly uncertain spring constant (k_a ≈ +0.08) yields a force alignment a_F that is clearly non-zero, but not catastrophic.

4) Tiny script (copy-paste)

Below is a small script implementing this Hooke’s law POC in ASCII-only Python:

# scenario_L03_hookes_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: Hooke's law F = k * x

# spring constant (m, a)
k_m, k_a = 200.0, +0.08   # N/m, mildly uncertain

# displacement measurements (m, a) at two instants
x1_m, x1_a = 0.045, +0.60  # m, jerkier reading
x2_m, x2_a = 0.055, +0.10  # m, calmer reading

# 2) classical magnitude: average displacement, then F = k * x
x_avg = 0.5 * (x1_m + x2_m)
F_m   = k_m * x_avg

# 3) SSM alignments
# pooled displacement alignment
a_x = ssm_align_weighted(
    [(x1_a, x1_m), (x2_a, x2_m)],
    gamma=1.0,
    eps=1e-12,
)

# force alignment from spring constant and displacement
a_F = ssm_align_product(k_a, a_x, eps=1e-6)

print("Classical:", f"{F_m:.4f}")            # 10.0000
print("SSM:", f"m={F_m:.4f}, a={a_F:+.4f}")  # a_F ~ +0.45 (drift-positive)

You can later add band assignment in your shared runner, 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: F ≈ 10.0000 N
  • SSM: m = 10.0000 N, a ≈ +0.4197 (under drift-positive semantics)

Under the sample band policy:

  • |a| < 0.20 → A+ (calm)
  • 0.20 <= |a| < 0.50 → A0 (borderline)

So:

  • a ≈ +0.4197 falls in A0 (borderline). The spring behaves acceptably, but not perfectly calmly; the initial jerkiness shows up in the bounded alignment lane, even though the final force value of 10 N looks perfectly reasonable.

If both displacement readings were calm and the spring constant alignment small, we would get:

  • a_x ≈ 0, a_F ≈ 0 and SSM ≡ Classical (both in magnitude and posture).

6) Why this helps in the real world

  • Lab instructors can use the alignment lane a to distinguish between “10 N from a clean, well-controlled stretch” versus “10 N from a jerky or poorly controlled setup.”
  • Test engineers can quickly see which Hooke’s law measurements are reliable for calibration and which ones should be repeated or averaged over longer windows.
  • Dashboards in materials testing rigs can color or band force values by alignment, so operators can see at a glance which measurements came from ideal spring-like behaviour and which came from noisy, marginal, or fatigued springs — all without changing the underlying law F = k * x.

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 L02: Newton’s Second Law – SSM
Next: POC Bounded Classical – Law L04: Ideal Gas Law – 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.