5 - Finite-Bias Epsilon and Dynamic-Range Analysis
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Scale parameters on this backend
The work plan’s five absolute parameters were written for a fixed-range machine. On this backend the driver applies, to every uploaded Q-matrix,
where is the driver’s normalization factor. The plan parameters collapse as follows:
Plan parameter
Fixed-range purpose
On this backend
maximum_bias_strength
caps clamps + barriers (diagonal terms)
none — Q auto-normalised at driver level by
maximum_coupling_strength
caps couplings + quadratization penalties
none absolute; entries encoded signed int16 with 15 fractional bits after normalisation
minimum_weight_resolution
smallest representable weight step
relative: of ; absolute step
effective_temperature
converts weights into probabilities
β register (uint32, 6 fractional bits): the compiler must emit with each matrix
saturation_behaviour
clipping beyond the range
Q entries cannot clip — by construction; β does: encoded values trimmed to 26 bits, cap , i.e.
Two structural facts:
The driver normalises coefficients, not energy tables. What is uploaded is a job — see note 1, Def. 7.
No job edit is local. Implementing or updating a job means writing exactly these two objects — and on this backend they are coupled through . Every entry of is divided by a factor fixed by , so an edit that moves rescales the whole matrix, hence the the entire job needs: an uncompensated run samples at effective , the wrong (usually hotter) distribution. Re-emit the -schedule with every write.
Clamp strength selection
is the only clamp parameter. It compiles as one diagonal edit of magnitude per clamped variable (sign set by the target ; constant offset dropped) — no couplings touched. Choosing it is a three-step check: a floor, a ceiling, and a stopping point.
Step 1 — Floor: clamp hard enough. By the leakage law, mass still escapes the clamp at rate . Requiring gives
Intuition: at the odds of satisfying the clamp are the base-model odds — you start the climb from — and each unit of multiplies those odds by . Likely evidence starts high and needs little push; rare evidence () starts units lower and needs extra units of to reach the same leakage.
Step 2 — Ceiling: don’t become the biggest number in the matrix. lands as an entry of , and the driver grids the whole matrix into levels of its largest entry: . While it doesn’t set and is free. Once it does, every extra unit of coarsens the grid for everything else: the small gaps encoding CPT ratios (a gap is a probability ratio ) get rounded away, and β headroom shrinks (). Intuition: the machine has 15 bits to spread between its largest energy and its finest one; a huge clamp spends those bits on itself.
Step 3 — Stop at the crossover. The total error against the true (un-smoothed) target has two independent parts: leakage, which decays as , and the smoothing bias of the target itself — the ε floor — which no clamp strength can touch. The triangle inequality makes this exact: . So the error falls exponentially until it meets the floor, then flattens; past that point extra improves nothing and, by Step 2, actively degrades the encoding. Intuition: you are conditioning , not — once the clamp is faithful to , the remaining error belongs to ε, not to .
Rule: take the smallest passing Step 1 (and the η check of the next section, if a conflict exists), verify Step 2, stop.
Scale interference: η
When clamped evidence contradicts a (near-)deterministic CPT, the sampler has two ways to resolve the contradiction, and it takes whichever costs less total energy: obey the clamps and sit on a smoothed-impossible state, paying its barrier; or cheat on a clamp, paying per mismatch to reach a legal state. The tipping point between the two is exactly the threshold of the sharp-threshold proposition: the base-energy gap between the cheapest clamp-consistent and the cheapest clamp-violating configuration, per mismatch.
η < 1 — cheating on the clamp is cheaper: mass leaks out of the clamp into valid states, exponentially in (constraint wins);
η > 1 — violating the constraint is cheaper: the system occupies the nominally impossible states, exponentially in (clamp wins);
the flip is sharp at η ≈ 1 up to the entropic window of the proposition.
Properties
Why a state-level gap, not the barrier entry. The naive comparison is “clamp strength λ versus barrier height ”. But the sampler will never compare two entries; it compares two complete configurations, each paying one term per node. is the difference of those two sums: the terms shared by the best obeying state and the best cheating state cancel, and every term in which they differ contributes — the barrier left behind, but also the CPT cost of the state the escape lands on. The barrier is normally the dominant contribution to , not the whole of it: if the cheapest escape flips a parent rather than the child, the gap picks up the parent-term and new child-term differences as well.
Leading order in ε. Splitting into its ε-dependent and ε-independent parts makes this quantitative. If satisfying the clamp forces the best -state through smoothed zeros, write with its ordinary finite CPT terms; with the best violator barrier-free at one mismatch ():
As the first piece blows up like while the second does not move, so is only the asymptote of η (for ). At any real ε it is off by three finite corrections — the multiplicity , the offset, the finite-entry difference — each a fixed number of energy units, hence a fixed shift of the flip point. That matters precisely because η is a threshold diagnostic: an error is harmless at η = 5, but certifying η > 1 is the diagnostic’s entire purpose, and near 1 is exactly where these shifts change the verdict. Use the exact ; keep as intuition only.
Routing. can only be large if is large — i.e. if paying a barrier is unavoidable while satisfying the clamp — and the free variables decide that. The minimum defining runs over all completions of the clamped values: if some assignment of the free variables makes the evidence legal, carries no barrier, is small or , and the mass simply routes through that completion. This is not the clamp failing — the exact conditional itself concentrates there; the escape route through the free variables is the inference being asked for. η therefore has a precondition: check whether any completion is legal. If yes, there is no clamp-versus-constraint fight to referee; if no — every completion barred — then necessarily carries a barrier and η governs which side wins.
Scale invariance and quantisation trust. and are ratios of energy differences, so leaves η fixed; in particular the driver rescale (, compensated through β) passes η through untouched — which is what makes it the right diagnostic on this backend. Post-quadratisation, the max over violators automatically includes the auxiliary-penalty escape channel (breaking a consistency constraint at cost equal to the quadratisation penalty): an under-sized penalty can lose the three-way fight between clamp, barrier and penalty.
λ is resolution-free while ( unchanged); beyond, and every extra unit coarsens
η
, with the critical clamp strength defined below
clamp push vs constraint resistance, at the level of whole configurations
Validation metrics (distribution-level, against the exact conditional): total variation distance (worst-case event error), KL divergence (mass-weighted), query-level error (single marginal cell), plus the two masses above; report per compiled model.
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