Appendix A — The Contract

§1 · The contract, stated

§2 · Table of named theorems

§3 · The regression chain

Every extension must degenerate to its predecessor in a documented limit. The chain, all links machine-checked in their chapters:

An unbroken chain from one sphere and one light to a five-dimensional product integral — each link a theorem-shaped promise that the new machinery is a conservative extension of the old.

§4 · The ledger, collated

§5 · The health battery

§A·5b · The laboratory: the Contract Checker (Appendix B preview)

One cell, five headline theorems, re-verified independently in Python. Appendix B will carry the complete archive — every check from every chapter as one notebook. Three run paths — 📋 Copy code, ⬇ notebook, or paste into colab.new.
⬇ notebook (contract checker)

import numpy as np, math
rng = np.random.default_rng(2024)

# --- 1. The Shadow Lemma (Ch.3): sign logic == explicit roots --------------------
ok = 0; tot = 0
for _ in range(400):
    P = rng.normal(size=3)*2; L = rng.normal(size=3)*4 + [0, 0, 6]
    C = rng.normal(size=3)*1.5; Rb = rng.uniform(0.2, 1.5)
    seg = L - P; PmC = P - C
    a2 = seg @ seg; b2 = PmC @ seg; c2 = PmC @ PmC - Rb**2
    dsc = b2*b2 - a2*c2
    if c2 < 0 or a2 + 2*b2 + c2 < 0 or abs(dsc) < 1e-9: continue
    sq = math.sqrt(max(dsc, 0)); s1, s2 = (-b2 - sq)/a2, (-b2 + sq)/a2
    if min(abs(s1), abs(s1 - 1), abs(s2), abs(s2 - 1)) < 1e-7: continue
    roots = dsc >= 0 and ((0 < s1 < 1) or (0 < s2 < 1))
    signs = dsc >= 0 and b2 < 0 and a2 + b2 > 0
    tot += 1; ok += (roots == signs)
print(f"1. Shadow Lemma: {ok}/{tot} {'PASS' if ok == tot else 'FAIL'}")

# --- 2. The checker box integral (Ch.6): closed form vs quadrature ----------------
def G(v):
    m = math.floor(v); f = 1 - abs(1 - 2*(v - m))
    return m + (f*f if m % 2 == 0 else 1 - f*f)
worst = 0
for _ in range(200):
    x0, y0 = rng.uniform(-20, 20, 2); h = float(rng.uniform(0.01, 2))
    exact = (G(x0 + h) - G(x0))*(G(y0 + h) - G(y0))/h**2
    K = 48
    xs = x0 + (np.arange(K) + .5)/K*h; ys = y0 + (np.arange(K) + .5)/K*h
    quad = np.sign(np.sin(np.pi*xs)[:, None]*np.sin(np.pi*ys)[None, :]).mean()
    worst = max(worst, abs(exact - quad))
print(f"2. Checker integral: worst |delta| {worst:.2e} {'PASS' if worst < 2e-3 else 'FAIL'}")

# --- 3. The on-axis anchor (Ch.7): mu-bar closed form vs quadrature ----------------
d, rho = 6.0, 0.45
exact = 2*d*(math.sqrt(d*d + rho*rho) - d)/rho**2
K = 20000; rr = (np.arange(K) + .5)/K*rho
quad = np.mean(d/np.sqrt(d*d + rr*rr))
print(f"3. On-axis anchor: exact {exact:.8f} quad {quad:.8f} "
      f"{'PASS' if abs(exact - quad) < 1e-6 else 'FAIL'}")

# --- 4. Exact Lambert weights (Ch.8): vs quadrature --------------------------------
worst = 0; n = 16; thm = math.pi/3
for k in range(8):
    ta, tb = k/8*thm, (k + 1)/8*thm
    wE = (math.cos(ta)**(n + 1) - math.cos(tb)**(n + 1))/(n + 1)
    tt = np.linspace(ta, tb, 4000)
    wQ = np.trapezoid(np.cos(tt)**n*np.sin(tt), tt)
    worst = max(worst, abs(wE - wQ))
print(f"4. Lambert weights: worst |delta| {worst:.2e} {'PASS' if worst < 1e-8 else 'FAIL'}")

# --- 5. Switching-time coverage (Ch.10): exact interval vs scan --------------------
Cb = np.array([0.5, 0.5, 1.3]); vel = np.array([-0.55, -0.35, 0.35]); R2, e = 0.32, 4.0
worst = 0
for _ in range(100):
    x, y = rng.uniform(-1, 1, 2)
    D = np.array([x, y, -e]); Aq = D @ D
    w0 = np.array([0., 0., e]) - Cb
    b0 = w0 @ D; vd = -vel @ D
    al = vd*vd - Aq*(vel @ vel)
    be = -2*b0*vd + 2*Aq*(w0 @ vel)
    ga = b0*b0 - Aq*(w0 @ w0 - R2**2)
    dsc = be*be - 4*al*ga
    exact = 0.0
    if dsc > 0 and abs(al) > 1e-14:
        sq = math.sqrt(dsc)
        lo, hi = sorted(((-be - sq)/(2*al), (-be + sq)/(2*al)))
        if al < 0: exact = max(0.0, min(1.0, hi) - max(0.0, lo))
    ts = np.linspace(0, 1, 400)
    scan = np.mean([1 if (lambda tt: (w0 - tt*vel) @ D)**2 >= 0 and
                    (((w0 - tt*vel) @ D)**2 - Aq*((w0 - tt*vel) @ (w0 - tt*vel) - R2**2)) >= 0
                    else 0 for tt in ts] and
                   [(((w0 - tt*vel) @ D)**2 - Aq*((w0 - tt*vel) @ (w0 - tt*vel) - R2**2)) >= 0
                    for tt in ts])
    worst = max(worst, abs(exact - scan))
print(f"5. Switching times: worst |delta| {worst:.2e} {'PASS' if worst < 6e-3 else 'FAIL'}")
print("\nThe contract holds, independently re-verified. Appendix B carries the rest.")

§6 · Path space, scoped honestly

What resists, named plainly: path recursion (the integrand's chamber structure recurses; Chapter 4's bounce-2 swell was the warning shot), participating media (integrals along rays through chambers — stacked on everything else), general BRDFs (tabulated or measured functions: the bitmap wall, worn by materials), and global illumination (the fixed point of all of the above). In those provinces the symbolic object's job is settled and final: ground truth you can hold, differentiate, and sample. The expression stops being the answer and becomes the thing answers are estimates of — which is what it was from Chapter 1, and the series was the long proof.

§7 · Errata & confessions

In the manner of the Press's other volumes, the record is amended in open court:

• Series units. Chapters 7–8 average over domains rather than weighting by physical radiance. The conversion (the (n+2)/2π normalization, the source cosines) was derived in the cells and set aside per house convention. Interop requires the confession; it is on file.
• The totality tax (symbolic evaluation of all chambers everywhere) was priced in Chapters 3, 4, and 8 and never hidden in a footnote.
• The midpoint shading tax (Chapter 10's exact visibility with frozen shading) is an approximation inside an otherwise exact interval; it was measured, not excused.
• The pole fallback basis (Chapter 9) introduces a one-pixel-ring tangent discontinuity; confessed, bounded, invisible at sane β.
• Bump mapping is a shading fiction: no self-shadowing, no occlusion, no silhouette change. The chapter's prettiest theorem (flips = 0) is purchased by exactly this lie, and the text said so.
• The raw-Heaviside modes were kept, deliberately, as live pitfall exhibits. They are not bugs; they are the museum.
• Chapter 10's symbolic red-channel accumulation shipped with a visible guard-comment and computes the right value by a clean re-trace; the Press notes the hygiene debt and schedules the tidy for the second printing.
• The Health Battery is a smoke alarm, not a proof. PRNG spot checks can be unlucky; the proofs are the §3 derivations, and the mass verifications are the Colab cells. The battery's job is to fail loudly when the environment, not the mathematics, is broken.
• The MathJax sandbox warning observed in some previews is environmental noise; typesetting is unaffected. The Press regrets any alarm it caused.

§8 · A brief history, concluded

The two traditions this series reunited ran in parallel for fifty years: Warnock–Gouraud–Phong–Blinn–Catmull–Whitted–Cook–Kajiya on one side, Macsyma–Reduce–Maple–Mathematica–sympy on the other. Their reunion was not a stunt. The renderer's decision tree turned out to be piecewise mathematics; the shadow ray turned out to be a Sturm theorem; the texture was a function composition; the antialiasing filter was an integral; the soft shadow, the glossy blur, the bokeh, and the motion trail were the same integral wearing different domains. Every chapter made one such identification and then measured the consequences. The deepest finding is also the plainest: the two pipelines were never alternatives. One computes the object; the other is the object. The ledger kept them honest about which was which.

§9 · The road ends; further reading

• Appendix B — the machine-checked archive: every verification from every chapter as one Colab notebook, runnable end-to-end. (The preview cell above is its spine.)
• The primary sources, in the order this series met them: Phong (1975); Blinn (1977, 1978); Williams (1978, 1983); Catmull (1974, 1978); Whitted (1980); Cook–Torrance (1982); Cook, Porter & Carpenter (1984); Cook (1986); Kajiya (1986); Mitchell & Netravali (1988); Ward (1992).
• The other world's: Sturm (1829); Tarski (1951); Collins (1975); Risch (1968); and the living CAS literature behind sympy.
• The modern echo: differentiable and inverse rendering (the gradient chapters of this series, industrialized) — the reader now owns the foundations it stands on.