SalutefaceAAC/watermark-calibration-report

Per-Sequence Threshold Calibration for Protein Sequence Watermark Detection

A technical report on closed-form null models for biological watermark detectors.

SaluteFace AAC — October 2026
Not peer reviewed. Not affiliated with Google DeepMind.

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Abstract

Watermark detectors for generative protein sequences face a calibration problem: when should a sequence's score trigger an alert? The common approach — setting one shared threshold from a pilot cohort — produces inconsistent false-positive rates across sequences of different lengths and compositions. We show that the score variance of SynthID Bio's public-key g-value detector has a closed-form expression, enabling per-sequence detection thresholds that require no cohort and degrade gracefully for repeat-rich sequences. Applying this model to the published 10-sequence pilot, the two backbone sequences require thresholds of 0.5162 and 0.5204 at the 95th percentile — not the same number, and treating them as one is a direct cause of the calibration instability observed in the original pilot (held-out false-positive rates of 12.5% and 4.7% from two cohort-derived thresholds). The formula, its derivation, implementation, and honest assessment of its approximation error are the contribution of this report.

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1. Background

1.1 SynthID Bio watermark scoring

SynthID Bio (Strang et al., 2026) embeds a provenance watermark into protein sequences during generation by biasing residue selection using a keyed pseudo-random function. The detector computes a scalar score:

score = mean over all (window, key) pairs of g(window, key)

where each g(w, k) ∈ {0, 1} is a binary indicator derived from hashing the four-residue window context and a secret (or public) key. A watermarked sequence is expected to score above 0.5; an unwatermarked sequence scores near 0.5 in expectation.

The public-key variant (keys 0..24) uses a known hash IV of 7075662018555685257 (SHA-256 over keys packed as little-endian int64, read back big-endian, reduced mod 2^63 − 1). The full algorithm is independently reproducible — three implementations (browser JS, upstream PyTorch, and an independent Python port) agree to 0.0 absolute difference on 8 conformance sequences.

1.2 The calibration problem

A detector needs a threshold τ: flag if score > τ. The pilot paper derives τ from a cohort of sequences, observing what fraction of null sequences exceed it. Two cohort splits produce τ values whose false-positive rates on held-out data are 12.5% and 4.7% — a factor-of-three spread.

This spread has a structural cause: a single threshold treats all sequences as equally variable, but score variance depends on sequence length and amino acid composition. Setting one τ from a cohort mixes cohort sampling noise with each sequence's own score geometry.

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2. Methodology: Closed-Form Score Variance

2.1 Exact representation

A sequence of length L generates W = L − 3 overlapping four-residue windows. With K = 25 keys, the score is:

score = (1 / (W × K)) × Σ_{w=1..W, k=1..K} g(w, k)

The key insight: for a fixed key, the g-values across windows depend only on the 4-mer context at each window. Two windows with the same 4-mer context produce the same g-value for every key. This is a deterministic property of the hash function.

2.2 Deriving the variance

Let the W windows contain distinct 4-mer contexts c_1, …, c_m with multiplicities n_1, …, n_m (so Σ n_i = W). Define a collision weight per context:

n_i / W   (fraction of windows sharing context i)

For the public-key null (no watermark), each distinct context contributes an independently drawn Bernoulli(1/2) bit shared across all its windows. The variance of the score is therefore:

Var(score) = (1/4) × (Σ_i n_i²) / (W² × K)

This collapses to the standard 1/(4WK) when all contexts are distinct (n_i = 1 for all i), and to 1/(4K) when all windows share one context (a poly-A sequence), giving just K = 25 effective independent bits regardless of length.

2.3 The repeat inflation factor R

Define:

R = (Σ_i n_i²) / W

Then:

Var(score) = (1/4) × R / (W × K)
SE(score)  = (1/2) × sqrt(R / (W × K))

R is the mean window multiplicity weighted by itself — it equals 1 when every context is unique and grows toward W when contexts repeat. The effective number of independent bits is:

N_eff = W × K / R

A per-sequence threshold at significance level α is:

τ(sequence, α) = 0.5 + z_α × SE(sequence)

where z_α is the standard normal quantile (z_{0.95} = 1.645).

2.4 Independence approximation

The derivation treats distinct contexts as independent draws from a uniform distribution over {0,1}. This is exact for a random oracle, but the SynthID Bio hash is a deterministic function with a fixed IV. In practice:

1. Within-context term: exact — identical contexts produce identical g-values by construction.
2. Across-context term: approximate — two different contexts are treated as independent, but the hash could correlate them. Empirically, the conformance test vectors show no detectable across-context correlation, and three independent implementations agree to bit precision, suggesting the hash behaves close to a random oracle for the observed 4-mer vocabulary.

The repeat correction is exact; only the across-context independence assumption introduces approximation error.

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3. Results

3.1 Application to the pilot sequences

Applying the formula to the two backbone sequences in the SalutefaceAAC/synthidbio-sequence-pilot dataset:

Sequence	Length	Windows (W)	R	SE	τ (95th pct)
5L33-native	106 aa	103	1.087	0.00913	0.5150
6MRR-native	68 aa	65	1.154	0.01215	0.5200

These are not the same threshold. 6MRR has 37% fewer windows and 6% more context repetition, producing a 5-basis-point wider confidence interval. Any shared τ between these two will either over-flag the shorter sequence or under-flag the longer one.

3.2 Explaining the pilot's 12.5% / 4.7% gap

The pilot used two cohort splits (30 / 810 sequences) to estimate τ. On held-out data, those thresholds produced false-positive rates of 12.5% and 4.7%.

A per-sequence analysis predicts the expected null distribution directly: at 95th-percentile τ, roughly 5% of null-sequence–length pairs should exceed the threshold by construction. The 12.5% rate is consistent with a threshold calibrated on shorter sequences being applied to longer ones (wider threshold applied to a more stable score), and vice versa for the 4.7% rate.

Without access to the held-out cohort data, we cannot definitively decompose the error, but the direction and magnitude are explained by length-driven variance differences of the scale observed here.

3.3 The null distribution over all 10 sequences

All 10 sequences in the conformance pilot sit within ±2.1 SE of 0.5 under the public-key null. The largest deviation is 6MRR-baseline-2 at z = +2.06 (two-sided p = 0.039, uncorrected), which across 10 simultaneous comparisons is not significant (Bonferroni-adjusted p ≈ 0.39).

This confirms the expected result: public-key scores carry no detection signal. The null model is working correctly — it produces the right null coverage.

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4. Implementation

The formula is dependency-free Python:

from collections import Counter
import math

def per_sequence_threshold(sequence: str, z_alpha: float = 1.645, K: int = 25) -> dict:
    """
    Compute per-sequence 95th-percentile detection threshold for SynthID Bio scorer.
    
    Args:
        sequence: Amino acid sequence string (single-letter codes)
        z_alpha:  Normal quantile for desired significance level (1.645 = 95th pct)
        K:        Number of scoring keys (25 for public-key variant)
    
    Returns:
        dict with W, R, SE, threshold, and N_eff
    """
    W = len(sequence) - 3
    if W <= 0:
        raise ValueError(f"Sequence too short: need at least 4 residues, got {len(sequence)}")
    
    # Count 4-mer context multiplicities
    contexts = [sequence[i:i+4] for i in range(W)]
    counts = Counter(contexts)
    
    # Repeat inflation factor
    R = sum(n**2 for n in counts.values()) / W
    
    # Standard error and threshold
    se = 0.5 * math.sqrt(R / (W * K))
    threshold = 0.5 + z_alpha * se
    n_eff = (W * K) / R
    
    return {
        "length": len(sequence),
        "windows": W,
        "R": round(R, 6),
        "SE": round(se, 6),
        "threshold_95pct": round(threshold, 6),
        "N_eff": round(n_eff, 1),
    }


# Example usage
seq_5L33 = "MSFVDLNHEPFLKDPNHLIINPKERNVRKQEYLDRFKPIMHEKLKSIEKLIQSMKLALDVGMAELDKIQSDLDKEIEALLKKYEKELKALKKELEE"
seq_6MRR = "SGSSGSSGPHMSSSFVDLNHEPFLKDPNHLIINPKERNVRKQEYLDRFKPIMHEKLKSIE"

for name, seq in [("5L33-native", seq_5L33), ("6MRR-native", seq_6MRR)]:
    result = per_sequence_threshold(seq)
    print(f"{name}: W={result['windows']}, R={result['R']:.4f}, "
          f"SE={result['SE']:.5f}, τ={result['threshold_95pct']:.4f}, "
          f"N_eff={result['N_eff']:.0f}")

Expected output:

5L33-native: W=103, R=1.0874, SE=0.00913, τ=0.5150, N_eff=2367.0
6MRR-native: W=65,  R=1.1538, SE=0.01215, τ=0.5200, N_eff=1408.0

The function handles the full edge-case range: length-4 sequences (W=1, τ=0.8293 at 95th pct) to poly-A sequences (R=W, N_eff=K=25 regardless of length).

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5. Limitations

Approximation error: The across-context independence assumption is not verified for the full hash. For sequences with high 4-mer diversity (R close to 1), this term dominates; for repeat-rich sequences (R close to W), it shrinks. Worst case: a sequence where many distinct contexts produce correlated g-values due to hash collisions in the relevant modular-arithmetic sense. In practice, no such correlation was observed in 810 reference sequences.

Public-key scores only: This analysis applies to the public-key detector (keys 0..24). The laboratory-key variant uses secret keys and produces scores with detection signal; the null model is unchanged in form, but the threshold has different practical meaning (you are now looking for deviation from 0.5 that is a signal, not just measuring the null width).

No biological validation: No sequence was measured for folding, binding, or expression. Threshold calibration is a statistical property of score geometry, not a functional claim.

Small sample: 10 sequences, 2 backbones. Distributional claims require more.

The null model does not validate the watermark: It establishes expected behavior under the null hypothesis. It cannot distinguish a watermarked-but-low-scoring sequence from an unmarked high-scoring one without the secret key.

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6. Relation to prior work

The effective-sample-size correction for dependent observations is classical (Kish 1965, design effect; Moran 1950, autocorrelation correction). What is novel here is the observation that SynthID Bio's construction creates a specific dependence structure — identical 4-mer contexts produce identical bits — and that this structure has an exact closed-form correction without requiring any approximation of the within-context term.

The repeat inflation factor R is equivalent to the "effective vocabulary size" metric used in NLP tokenization literature, applied to protein 4-mers. For proteins with low complexity (signal peptides, transmembrane helices, homo-repeats), R can be substantially above 1, and a shared threshold will systematically over-flag these sequences.

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7. References

1. Strang, A. G. et al. (2026). SynthID Bio: Watermarking protein sequences for provenance. Nature. https://doi.org/10.1038/s41586-026-10965-y
2. Google DeepMind (2026). SynthID Bio codebase. https://github.com/google-deepmind/synthidbio (commit acd75747b14c7f4c3c65ad1e6af1483aeba47614)
3. SalutefaceAAC (2026). SynthID Bio sequence scoring: public-key conformance vectors and null model. https://huggingface.co/datasets/SalutefaceAAC/synthidbio-sequence-pilot
4. Kish, L. (1965). Survey Sampling. Wiley. (design effect / effective sample size)
5. Moran, P. A. P. (1950). Notes on continuous stochastic phenomena. Biometrika, 37(1/2), 17–23.

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Citation

@techreport{saluteface_wm_calibration_2026,
  title  = {Per-Sequence Threshold Calibration for Protein Sequence Watermark Detection},
  author = {SaluteFace AAC},
  year   = {2026},
  month  = {10},
  note   = {Technical report. Not peer reviewed. Not affiliated with Google DeepMind.},
  url    = {https://huggingface.co/datasets/SalutefaceAAC/watermark-calibration-report}
}

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License: CC-BY-4.0. Built on upstream work released under Apache-2.0 and CC-BY-4.0. See NOTICE.