POC Bounded Classical – Law L04: Ideal Gas Law – SSM

A bounded two-lane view of P * V = n * R * T so the same 2.5 bar pressure can be tagged calm, borderline, or stressed.

Domain. Thermodynamics / Lab Cylinder Test
Classical law. P * V = n * R * T

What this shows.
In this law POC, we take the Ideal Gas Law and show how:

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

When temperature and volume are well-behaved and n is well-known, Shunyaya Symbolic Mathematics (SSM) collapses to the classical result and you effectively get the same answer.

When readings are jittery (thermometer not settled, piston not perfectly fixed), the alignment lane a and its band surface posture that the classical scalar 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 the semantics (for example, make +a mean more stability) without changing the math or phi((m,a)) = m.

1) Setup (inputs)

Imagine a sealed vertical cylinder with a movable piston, containing a small amount of air:

  • The amount of gas n is known to be close to 1 mol.
  • Volume V is around 10 L, but the piston position has some uncertainty.
  • You gently warm the cylinder and take two temperature readings as it equilibrates.

We model:

  • Amount of gas (approximate, but stable):
    • n_m = 1.00 mol, n_a = +0.02 — almost exactly one mole.
  • Gas constant (treated as effectively exact here):
    • R_m = 8.314 J/(mol*K), R_a = +0.00
  • Volume (slightly uncertain piston position):
    • V_m = 0.0100 m^3 (10 L), V_a = +0.10 — piston not perfectly fixed.
  • Temperature measurements (instantaneous):
    • T1_m = 295 K, T1_a = +0.55 — thermometer still equilibrating
    • T2_m = 305 K, T2_a = +0.12 — later, calmer reading

In SSM:

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

Our target.

  • Use P = (n * R * T) / V.
  • Compare Classical pressure vs SSM pressure (m_P, a_P).

2) Classical calculation

Ignoring alignment, we just apply the ideal gas law with average temperature:

# classical illustration (no external packages required)

n  = 1.00      # mol
R  = 8.314     # J/(mol*K)
V  = 0.0100    # m^3  (10 L)

T1 = 295.0     # K
T2 = 305.0     # K

T_avg = 0.5 * (T1 + T2)
P     = (n * R * T_avg) / V

print(T_avg)  # 300.0
print(P)      # 249420.0 (approx)  ~2.49e5 Pa

Classical result.

  • T_avg = 300 K
  • P ≈ 2.49 × 10^5 Pa (about 2.5 bar)

A standard report might simply say: Pressure ≈ 2.5 bar. It does not say whether the thermodynamic state during measurement was really calm.

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

In Shunyaya Symbolic Mathematics (SSM), we:

  1. Treat each temperature sample as (m,a) and pool their alignments:
    • a_c := clamp(a, -1+eps, +1-eps)
    • u := atanh(a_c)
    • U += w * u with w := |m|^gamma
    • W += w
    • a_T_out := tanh( U / max(W, eps) )
  2. Treat n, R, and V as (m,a) too (some with nearly zero a).
  3. Combine alignments for n * R * T and then divide by V:
    • a_nRT := tanh(atanh(a_n_c) + atanh(a_R_c) + atanh(a_T_out_c))
    • a_P := tanh(atanh(a_nRT_c) - atanh(a_V_c))
  4. Keep the pressure magnitude as:
    • m_P = (n_m * R_m * T_avg) / V_m
    • phi((m_P, a_P)) = m_P

So:

  • Magnitude stays around 2.49 × 10^5 Pa.
  • Alignment a_P tells us whether this ideal-gas snapshot comes from a calm, well-behaved state or from noisy, borderline conditions.

4) Tiny script (copy-paste)

# scenario_L04_ideal_gas_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_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: Ideal Gas Law P = (n * R * T) / V

# amount of gas (m, a)
n_m, n_a = 1.00, +0.02     # mol

# gas constant (m, a)
R_m, R_a = 8.314, +0.00    # J/(mol*K)

# volume (m, a)
V_m, V_a = 0.0100, +0.10   # m^3 (10 L)

# temperature measurements (m, a) at two instants
T1_m, T1_a = 295.0, +0.55  # K
T2_m, T2_a = 305.0, +0.12  # K

# 2) classical magnitude: average temperature, then ideal gas law
T_avg = 0.5 * (T1_m + T2_m)
P_m   = (n_m * R_m * T_avg) / V_m

# 3) SSM alignments

# pooled temperature alignment
a_T = ssm_align_weighted(
    [(T1_a, T1_m), (T2_a, T2_m)],
    gamma=1.0,
    eps=1e-12,
)

# combined alignment for n*R*T
a_nRT = ssm_align_sum([n_a, R_a, a_T])

# pressure alignment from (n*R*T) / V
a_P = ssm_align_div(a_nRT, V_a, eps=1e-6)

print("Classical:", f"{P_m:.2f}")             # ~249420.00
print("SSM:", f"m={P_m:.2f}, a={a_P:+.4f}")   # a_P ~ +0.50 (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

You will see something like:

  • Classical: P ≈ 2.49 × 10^5 Pa
  • SSM: m = 2.49 × 10^5 Pa, a ≈ +0.2775

Under a simple band policy such as:

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

we get:

  • a ≈ +0.2775 sitting in A0 (borderline) — slightly above calm, but not in a stressed A− region.

So the ideal gas law fits (pressure is correct), but the bounded alignment lane tells you:

  • This particular P–V–T–n snapshot was taken under mildly noisy / borderline conditions (thermometer still drifting, piston position somewhat uncertain).

If both T1/T2 were very close and V_a small, you would instead see:

  • a_T ≈ 0, a_P ≈ 0, and SSM ≡ Classical in posture as well.

6) Why this helps in the real world

  • Lab instructors can distinguish a “nice ideal-gas point” from “numerically similar but noisy conditions”, using a as a compact stability signal.
  • Engineers can tell whether a measured P–V–T point is good enough to use as a calibration reference or should be re-measured under calmer conditions.
  • Dashboards in thermal test rigs can color pressure by a or by band, revealing at a glance which states were clean and which were nominally correct but posturally suspect.

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.