Theoretical grounding: Liu & Gore, Neural Scaling Universality: If Exponents Are Fixed, Time to Understand Coefficients (arXiv:2606.25008, Jun 2026). Distilled in
katgpt-rs/.research/315_Neural_Scaling_Universality_Exponents_Fixed_Coefficients_Actionable.md.
The AGENTS.md rule "Use sigmoid not softmax" is enforced pervasively in this codebase — fast_sigmoid / simd_sigmoid_inplace in katgpt-core/src/simd/, direction_vector_decode in the analytic lattice decoder, select_action in the bridge layer, the CUCG G6 gate (benches/cucg_goat.rs, statically asserts 0 softmax hits), CommittedFieldBlend and PersonalityWeightedComposition blend weights, and the LM head's log-softmax over vocabulary (the one place softmax is correct, since the LM head must produce a proper distribution). Liu & Gore argue that scaling-law exponents are universal and fixed by three generic mechanisms: the 1/3 time exponent is fixed by softmax's partition-of-unity nonlinearity; the inverse-width exponent by superposition; the inverse-depth exponent by ensembling. Coefficients, not exponents, are the actionable lever. Per their argument, our pervasive use of sigmoid gates (gentler, per-coordinate, no partition constraint) places our runtime blending gates in a different universality class than softmax-gated blending — not better, just structurally distinct, with different effective exponent behavior. This is the missing theoretical justification that Research 295 deferred for the sigmoid-not-softmax rule. The runtime canary in CUCG G6 (benches/cucg_goat.rs) demonstrates the structural difference quantitatively: under large scale T, softmax output entropy collapses to 0 (one-hot) while sigmoid normalized entropy plateaus at log(n_positive) — a nonzero floor set by input sign structure, not by partition dynamics.
Liu & Gore's complementary claim — coefficients, not exponents, are where performance is won — endorses the codebase's existing committed-blend stack as the correct inference-time instantiation of their recommendation. Four primitives navigate coefficient space without touching the (universal) exponent structure: PersonalityWeightedComposition (Plan 297, per-tick sigmoid-gated personality drift), CommittedFieldBlend (Plan 321, BLAKE3-committed archetype blend logits), the Spectral Budget Router (Plan 254, NS-depth layer allocation), and the MerkleFrozenEnvelope freeze/thaw swap in riir-neuron-db. Resources spent improving blend granularity, commit frequency, or drift stability are spending on the actionable axis the paper identifies — not on a workaround around scaling limits.
- Not a claim that sigmoid is better than softmax. The paper makes no such claim and neither do we. The claim is structural: sigmoid gates live in a different universality class. Which class is better for which inference task is empirical (riir-train territory).
- Not a Chinchilla-style compute recipe. The paper argues coefficients matter but does not prescribe how to navigate them at training time. Our coefficient navigation is inference-time (blend, route, freeze) and already shipped.
- Not a constraint on the LM head. The LM head softmax is exactly where the 1/3 exponent should be fixed (it is the canonical softmax nonlinearity). The escape applies to blending gates, not to the output distribution.