**The power-law scaling of FP32-BF16 loss-landscape mismatch (LMC) barriers (validated in surrogate Bayesian optimization) will constrain the interpretability of attention patterns in single-cell foundation models (e.g., scGPT), where co-expression signals degrade below a critical model size (~10M parameters).**

Two conjoined, separable claims must each be tested:

Claim A (numerical): The linear-mode-connectivity (LMC) loss barrier between an FP32-trained checkpoint and its BF16-trained (or BF16-cast) counterpart, measured as max interpolation loss increase along the linear path in weight space, scales as a power law B(N) ≈ a·N^(-b) with respect to model parameter count N, with b > 0, fit across at least 5 model sizes spanning 1M–500M parameters, R² ≥ 0.85 on log-log regression.

Claim B (biological/interpretability): In scGPT-style single-cell transformer models, the fidelity of attention-derived gene-gene co-expression signal (measured against ground-truth co-expression from matched scRNA-seq correlation/GRN benchmarks, e.g., BEELINE) degrades sharply below a critical parameter threshold N_c ≈ 10M, and this threshold co-occurs with (not merely correlates temporally with) the point where the Claim-A power-law-predicted LMC barrier exceeds a critical value B_c (operationalized as barrier > 10% of the loss range between initialization and convergence).

The compound falsifiable claim: N_c(interpretability collapse) is statistically indistinguishable (within 2× parameter count) from N_c(LMC barrier crosses B_c), across ≥3 independent training seeds and ≥2 dataset scales, implying a shared underlying mechanism rather than coincidence.

Minimum viable test (MVT):

• Train scGPT-architecture models at 5 sizes: 1M, 5M, 10M, 25M, 100M parameters, each with 3 seeds, on a fixed subsampled PBMC dataset (~50K cells, 2000 HVGs).
• For each model/seed: train in FP32 and in BF16-mixed-precision to matched validation loss (±2%).
• Compute LMC barrier: linear-interpolate weights between FP32 and BF16 checkpoints at 11 points (α = 0, 0.1, ..., 1.0), measure loss at each, report max(loss(α)) − max(loss(0), loss(1)).
• Extract attention maps from final layer(s), average over heads, compare top-k attended gene pairs against BEELINE ground-truth GRN edges (AUPRC/AUROC).
• Fit power law to barrier vs. N; identify N where barrier crosses B_c.
• Fit sigmoid/changepoint model to AUPRC vs. N; identify N_c(interp).
• Compare N_c values with bootstrap CIs (1000 resamples).

for size in [1M, 5M, 10M, 25M, 100M]:
    for seed in [1,2,3]:
        for precision in [FP32, BF16]:
            model = ScGPTEncoder(params=size, seed=seed)
            train(model, pbmc_data, precision=precision, until=converged)
            save_checkpoint(model, key=(size, seed, precision))

for size, seed in all_pairs:
    ckpt_fp32 = load(size, seed, FP32)
    ckpt_bf16 = load(size, seed, BF16)
    barrier = compute_lmc_barrier(ckpt_fp32, ckpt_bf16, alphas=linspace(0,1,11))
    store(size, seed, barrier)

    attn = extract_attention(ckpt_bf16, held_out_cells)
    coattn_matrix = aggregate_heads_and_cells(attn)
    auprc = score_against_beeline(coattn_matrix, ground_truth_grn)
    store(size, seed, auprc)

barrier_by_size = groupby_mean(barriers, key=size)
fit = loglog_regression(barrier_by_size, sizes)  # B = a * N^-b
N_c_lmc = solve_for_N(fit, B_c=0.10 * loss_range)

auprc_by_size = groupby_mean(auprcs, key=size)
sigmoid_fit = fit_changepoint(auprc_by_size, sizes)
N_c_interp = sigmoid_fit.inflection_point

compare(N_c_lmc, N_c_interp, method="bootstrap_CI_overlap")
run_ablation(FP32_only_models)  # control
replicate_on(tabula_sapiens_subset)