"""Executable examples for Part III chapters (17-19, 21, 22).

All numbers use v = 3/5, so gamma = 5/4 exactly and no rounding enters.

Run from the repository root:

    uv run python content/examples/math/part3/part3_kinematics_demo.py

ILLUSTRATIVE. These statements are not versioned claims and produce no
EvidenceResult. Each would need a claim contract, a fingerprint, and an adapter
run to count as engine evidence.
"""

import z3
from sympy import Rational, factor, simplify, sqrt, symbols

v = Rational(3, 5)
gamma = 1 / sqrt(1 - v**2)
assert gamma == Rational(5, 4)


def boost(t, x):
    return gamma * (t - v * x), gamma * (x - v * t)


def interval(t, x):
    return t**2 - x**2


print("gamma =", gamma, "; gamma^2 (1-v^2) =", gamma**2 * (1 - v**2))

print("\n== Ch 17: simultaneity is frame-dependent")
a, b = (0, 0), (0, 1)           # simultaneous in S: t = 0 for both
(ta, xa), (tb, xb) = boost(*a), boost(*b)
print("S' times of the two events:", ta, tb, "-> difference", tb - ta)
print("interval S:", interval(0, 1), " interval S':", interval(tb - ta, xb - xa))

print("\n== Ch 18/19: light-cone coordinates and the Doppler factor k")
t, x = symbols("t x", real=True)
tp, xp = boost(t, x)
print("t'-x' =", factor(simplify(tp - xp)), "   ; factor on (t-x):", (gamma * (1 + v)))
print("t'+x' =", factor(simplify(tp + xp)), "   ; factor on (t+x):", (gamma * (1 - v)))
k = sqrt((1 + v) / (1 - v))
print("k = sqrt((1+v)/(1-v)) =", k, "; k * (1/k) =", k * (1 / k))
print("gamma(1+v) * gamma(1-v) =", gamma * (1 + v) * gamma * (1 - v))
vs = symbols("v", positive=True)
ks = sqrt((1 + vs) / (1 - vs))
print("(k + 1/k)/2 - 1/sqrt(1-v^2) =", simplify((ks + 1 / ks) / 2 - 1 / sqrt(1 - vs**2)))
print("  ^ simplify() does NOT reduce this to 0 (heuristic limit; see Chapter 9).")
kp = symbols("k", positive=True)
v_of_k = (kp**2 - 1) / (kp**2 + 1)          # invert k^2 = (1+v)/(1-v)
print("with v = (k^2-1)/(k^2+1):  sqrt(1-v^2) =", simplify(sqrt(simplify(1 - v_of_k**2))),
      "; (k+1/k)/2 =", (kp + 1 / kp) / 2,
      "; 1/sqrt(1-v^2) =", simplify(1 / sqrt(simplify(1 - v_of_k**2))))

print("\n== Ch 21: time dilation and length contraction (3-4-5)")
# clock at rest in S' (x'=0), proper time tau = 4: the S-frame coordinates
tau = 4
t_S, x_S = gamma * (tau + v * 0), gamma * v * tau
print("S coordinates of clock reading tau=4:", (t_S, x_S), "; interval:", interval(t_S, x_S))
# rod of rest length L0 = 5 in S', ends at x'=0 and x'=5; measure both ends at t = 0 in S
L0 = 5
x_end_S = Rational(L0) / gamma      # x' = gamma (x - v t) at t=0 -> x = x'/gamma
print("rod: rest length", L0, "-> S-frame length at fixed t:", x_end_S)
# symmetry: S-clock seen from S'
tS_clock = 4
print("S-clock at x=0, t=4 -> S' coordinates:", boost(tS_clock, 0))

print("\n== Ch 22: velocity composition and causal order")
def compose(u, w):
    return (u + w) / (1 + u * w)
four_fifths = Rational(4, 5)
print("3/5 (+) 3/5 =", compose(v, v), "; 1 (+) 3/5 =", compose(1, v),
      "; 4/5 (+) 4/5 =", compose(four_fifths, four_fifths))
# timelike vs spacelike separation: time-order of two events under the boost
for dt, dx, kind in [(2, 1, "timelike"), (1, 2, "spacelike"), (1, 1, "null")]:
    dtp, dxp = boost(dt, dx)
    print(f"{kind:9s} (dt,dx)=({dt},{dx}) interval {interval(dt, dx):>2} "
          f"-> (dt',dx')=({dtp},{dxp}) interval {interval(dtp, dxp)}")

print("\n== Z3: timelike order is preserved by every subluminal boost")
T, X, V, G = z3.Reals("t x v gamma")
s = z3.Solver()
s.add(G > 0, G * G * (1 - V * V) == 1, V > -1, V < 1)   # boost hypotheses
s.add(T > 0, T * T - X * X > 0)                          # future timelike separation
s.add(G * (T - V * X) <= 0)                              # negated goal: t' > 0
print("negation of 't' > 0' for timelike (t,x):", s.check())
s2 = z3.Solver()
s2.add(G > 0, G * G * (1 - V * V) == 1, V > -1, V < 1)
s2.add(T > 0, X * X - T * T > 0)                          # spacelike
s2.add(G * (T - V * X) <= 0)
print("same for spacelike (can the order flip?):", s2.check(), s2.model())