4 - Conditional Clamping

Proposition 2 - Conditional Clamping Limit

Let represent a BN-derived energy model, be a clamped index set with target . Then, provided .

Proof

Recall the notations for clamping, we have if and only if . Define the consistent set .

Splitting the Boltzmann weight
For : , so the weight is , independent of . For : , so as , the weight simply collapses to 0.

Fix a free block configuration , writing for the full state configuration, the free-variable marginal is with
The second sum is finite and every term , so it vanishes:
By assumption , and for a BN, (See Proposition 1), thus

In the numerator, every term with carries and vanishes; only survives:

Both limits calculated give

From the base model,

The two expressions are identical.

Interpretation: Clamping and conditioning coincide only in the limit.

A hard clamp () reproduces observational conditioning — it does not by itself implement Pearl’s . Whether a finite clamp on a BN-derived energy model reproduces interventional vs. conditional semantics is the open question flagged in the Intervention vs Observation section.

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Finite-bias leakage law

Proposition 2 makes clamping and conditioning coincide only in the limit; the following quantifies the distance at every finite . Keep and as above, and define the mismatch spectrum

the base-model probability of violating exactly clamps — in particular — and the leakage mass

Lemma — exact leakage law

With and the shorthand :

1. (Partition ratio and leakage)
2. (TVD identity)
3. (Bounds and asymptote)

Proof

1. Group the sum defining by the value of ; the slice has base weight by definition of , so

The consistent set is the slice, hence

2. For the identity on TVD:

• On , the clamped weight is -free, so pointwise, where (equivalently , clear from 1).
• Off , the conditional vanishes. TVD is the one-sided sum over the set where the conditional dominates, i.e. over :

3. From 1, put the leakage over a common denominator:

Since for every , and keeping only the term for the lower bound,

Every term of is nonnegative and , so keeping only gives Combining:

Asymptote: write with remainder . Then

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Critical clamp strength

The leakage law bounds how much mass escapes a clamp; the following identifies the strength at which a clamp overcomes the base landscape at all.

Let be the best clamp-consistent energy. For each violator define its critical strength — the at which the penalised violator ties with the best consistent state (the strength at which this violator stops being cheaper than the best consistent state) — and the threshold:

with a maximiser and its mismatch count.

Proposition — sharp threshold

Let be finite and . Then for every :

The mass on flips from to inside a window of width around : the clamp overcomes the landscape at , sharply up to entropic log-corrections.

Proof

Let a violator, by definition of :

Above threshold (): each , so with , so with ,

hence .

Below threshold (): keep only the violator in the sum and use :

Which side of the threshold a compiled clamp actually lands on — and how depends on the smoothing level and the hardware encoding — is discussed in the Phase 5 analysis.