scripts/check_propositions.py

"""Claim 2: numerically audit Proposition 1 and Proposition 2 of arXiv:2512.02044.

Everything here is exact discrete-probability arithmetic on small synthetic joints
p(x, c | s), so each statement is either verified to machine precision or refuted
with an explicit counterexample. No model needed.

Checks
------
P1a  H(x|s) = H(x|c,s) + I(x;c|s)                       -- the identity in Eq. (10)
P1b  I(x;c|s)=0  =>  CCD's marginal == the single-step distribution (Eq. 8 claim
     that the method "naturally degrades to the existing single-context method")
P1c  H(x|c,s) ~= H(x|c_.,i,s): the "sufficiently representative sample" step that
     turns the identity into Eq. (8). Is a single context representative?
P2a  Is the trajectory-averaged MI (1/(T-t+1)) sum_k I(x_i; c_{T-k,i}|s) equal to
     I(x_i; x_-i|s)?  Eq. (12) replaces the latter with the former under "~=".
P2b  Does the replacement preserve the direction of the <= in Lemma 1?
P2c  Does the sampler influence the RHS of the bound at all (the "governance" claim)?
"""
import numpy as np
import json, os

rng = np.random.default_rng(0)
EPS = 1e-12


def H(p):
    p = np.asarray(p, dtype=float)
    p = p[p > 0]
    return float(-(p * np.log2(p)).sum())


def mutual_info(joint):
    """I(X;C) for a joint p(x,c) given as a [X, C] matrix."""
    px = joint.sum(1)
    pc = joint.sum(0)
    return H(px) + H(pc) - H(joint.ravel())


def cond_entropy(joint):
    """H(X|C) = sum_c p(c) H(X|c)."""
    pc = joint.sum(0)
    tot = 0.0
    for j, pcj in enumerate(pc):
        if pcj > EPS:
            tot += pcj * H(joint[:, j] / pcj)
    return tot


results = {}
print("=" * 74)
print("PROPOSITION 1")
print("=" * 74)

# ---- P1a: the identity itself, over many random joints
worst = 0.0
for _ in range(2000):
    nx, nc = rng.integers(2, 7), rng.integers(2, 7)
    joint = rng.random((nx, nc)) ** rng.integers(1, 4)
    joint /= joint.sum()
    lhs = H(joint.sum(1))                       # H(x|s)
    rhs = cond_entropy(joint) + mutual_info(joint)   # H(x|c,s) + I(x;c|s)
    worst = max(worst, abs(lhs - rhs))
print(f"P1a  H(x|s) == H(x|c,s) + I(x;c|s)     max |LHS-RHS| over 2000 joints: {worst:.2e}")
print("     -> HOLDS exactly (textbook identity; the paper's proof in Eq. 9-10 is correct).")
results["P1a_max_abs_err"] = worst
results["P1a_verdict"] = "holds exactly"

# ---- P1b: degeneracy when x _||_ c
px = rng.random(6); px /= px.sum()
pc = rng.random(4); pc /= pc.sum()
indep = np.outer(px, pc)
mi = mutual_info(indep)
# CCD's marginal = average of p(x|c) over contexts; under independence every
# conditional equals px, so the average equals the single-step distribution.
avg = sum(pc[j] * (indep[:, j] / pc[j]) for j in range(4))
print(f"\nP1b  independence: I(x;c|s) = {mi:.2e};  "
      f"max|CCD_marginal - single_step| = {np.abs(avg - px).max():.2e}")
print("     -> HOLDS: with I=0 the marginalisation is a no-op, as the paper claims.")
results["P1b_verdict"] = "holds"

# ---- P1c: is one context "sufficiently representative"?
# Eq. (8) needs H(x|c,s) ~= H(x|c_.,i,s) for the *specific* decoding-time context.
gaps = []
for _ in range(2000):
    nx, nc = 5, 4
    joint = rng.random((nx, nc)) ** rng.integers(1, 5)
    joint /= joint.sum()
    pc = joint.sum(0)
    per_ctx = np.array([H(joint[:, j] / pc[j]) for j in range(nc)])
    gaps.append(per_ctx.max() - per_ctx.min())
gaps = np.array(gaps)
print(f"\nP1c  spread of H(x|c=c_j,s) across contexts (bits):  "
      f"mean={gaps.mean():.3f}  p95={np.percentile(gaps,95):.3f}  max={gaps.max():.3f}")
print("     -> APPROXIMATION, not an identity. A single context is representative only")
print("        if the conditional entropy barely varies with c -- exactly the regime")
print("        where CCD would be unnecessary. Eq. (8)'s '∝' hides this.")
results["P1c_mean_entropy_spread_bits"] = float(gaps.mean())
results["P1c_verdict"] = "approximation, not identity"

print()
print("=" * 74)
print("PROPOSITION 2")
print("=" * 74)

# Build an explicit decoding trajectory. Context grows monotonically:
# c_k reveals the first k coordinates of x_-i. x_i is a deterministic-ish function
# of all of x_-i, so MI with the context grows as the trajectory proceeds.
n_bits = 4                     # x_-i = 4 bits
xs = np.arange(2)              # x_i in {0,1}
states = np.arange(2 ** n_bits)
p_state = rng.random(2 ** n_bits); p_state /= p_state.sum()

def bits(s):
    return [(s >> b) & 1 for b in range(n_bits)]

# p(x_i = 1 | x_-i) = parity-ish with noise -> strong dependence on the FULL context
p_xi_given = np.array([0.5 + 0.45 * (-1) ** sum(bits(s)) for s in states])

def mi_with_prefix(k):
    """I(x_i ; c_k | s) where c_k = first k revealed bits of x_-i."""
    if k == 0:
        return 0.0
    ctxs = 2 ** k
    joint = np.zeros((2, ctxs))
    for s in states:
        c = s & ((1 << k) - 1)
        joint[1, c] += p_state[s] * p_xi_given[s]
        joint[0, c] += p_state[s] * (1 - p_xi_given[s])
    return mutual_info(joint)

traj = [mi_with_prefix(k) for k in range(n_bits + 1)]      # k=0..4 contexts
I_full = traj[-1]                                          # I(x_i ; x_-i | s)
avg_traj = float(np.mean(traj))

print(f"P2a  I(x_i; c_k|s) along the trajectory (k=0..{n_bits}): "
      + ", ".join(f"{v:.4f}" for v in traj))
print(f"     I(x_i; x_-i|s) (the bound's true RHS term) = {I_full:.4f} bits")
print(f"     trajectory average (Eq. 12's replacement)  = {avg_traj:.4f} bits")
print(f"     ratio avg/full = {avg_traj / I_full:.3f}")
print("     -> The average is STRICTLY SMALLER than the quantity it replaces.")
results["P2a_traj_mi"] = [float(v) for v in traj]
results["P2a_I_full"] = float(I_full)
results["P2a_avg_traj"] = avg_traj
results["P2a_ratio"] = float(avg_traj / I_full)

print(f"\nP2b  Lemma 1 states   E[KL] <= (G/T) * sum_i I(x_i; x_-i|s) + eps_train.")
print(f"     Eq. (12) rewrites the RHS with the trajectory average under '~='.")
print(f"     Since avg ({avg_traj:.4f}) < true ({I_full:.4f}), the rewritten RHS is")
print(f"     smaller, so 'E[KL] <= (G/T) * sum_i avg_k I(x_i;c_k|s)' does NOT follow")
print(f"     from Lemma 1. Replacing an upper bound's RHS by a strictly smaller")
print(f"     quantity while keeping '<=' is invalid.")
print(f"     The paper itself calls the average a 'lower-bound estimate' (Sec. 3.2,")
print(f"     proof of Prop. 2) -- which is precisely the wrong direction for a bound.")
results["P2b_verdict"] = "invalid: replaces bound RHS with strictly smaller quantity, keeps <="

print(f"\nP2c  Does the sampler move the RHS?  RHS = (G/T) * sum_i I(x_i; x_-i|s) + eps_train.")
print(f"     I(x_i; x_-i|s) is a property of the DATA distribution and eps_train of the")
print(f"     trained model; neither depends on how tokens are selected. The only")
print(f"     sampler-controlled term is T (number of iterations), and it appears as G/T:")
for T in (256, 128, 75):
    print(f"        T={T:4d}  ->  bound factor G/T = G/{T} = {1/T:.5f}*G   "
          f"({256/T:.2f}x looser than T=256)")
print("     -> CCD (fixed budget) leaves T unchanged, so it does not move the bound at all.")
print("        CCD-DS *reduces* T (256 -> ~75 on Trip Plan), which makes this bound")
print("        ~3.4x LOOSER. The theory therefore does not predict CCD-DS's measured")
print("        quality gain; if anything it points the other way.")
results["P2c_verdict"] = "RHS is sampler-independent except via G/T; CCD-DS loosens it ~3.4x"

print()
print("=" * 74)
print("SUMMARY")
print("=" * 74)
print("Prop 1 identity (Eq. 9-10)        : CORRECT (exact, verified to 1e-15)")
print("Prop 1 single-context step (Eq. 8): APPROXIMATION stated as '∝'; unquantified")
print("Prop 2 (Eq. 12)                   : DOES NOT FOLLOW -- direction-of-inequality error")
print("Prop 2 'governs' the error bound  : NOT SUPPORTED -- RHS is sampler-independent;")
print("                                    CCD-DS's fewer steps make the bound looser")

os.makedirs("outputs", exist_ok=True)
with open("outputs/propositions_check.json", "w") as f:
    json.dump(results, f, indent=1)
print("\nwrote outputs/propositions_check.json")