The Noncomputability of Immune Reaction Complexity: Algorithmic Information Gaps under Effective Constraints

TL;DR

The paper introduces a validity-filtered, certificate-based framework for analyzing reactions as advice-driven computations constrained by a predicate $V(x,r)$. It defines the minimal realizer complexity $M(x)=\min_{r:V(x,r)=1} K(r)$ and the Normalized Advice Quantile $NAQ(x;\mathcal{T})$, a scale-free hardness measure that is robust to universal-machine choices and pool composition. A central result, the Exact Realizer Identity, shows that $C^{\mathrm{adv}}_{\mathrm{rxn},E_{\mathrm{univ}}}(x)=M(x)\pm O(1)$, with two-part bounds separating description cost $K(y)$ from selection cost $\lceil\log i_y^{\pi}(x)\rceil$, and several tightness regimes (finite ambiguity, fiber genericity). The work also develops resource-bounded variants $NAQ_t$, an NP-style linear bound, and an operational rate-distortion converse linking NAQ to information-theoretic limits, enabling data-driven calibration via compression proxies and empirical estimation via the DKW inequality. Collectively, these results illuminate fundamental limits on feasible advice in computation and offer a principled bridge to biological applications, notably a formal theory for adaptive immunity under computable constraints.

Abstract

We introduce a validity-filtered, certificate-based view of reactions grounded in Algorithmic Information Theory. A fixed, total, input-blind executor maps a self-delimiting advice string to a candidate response, accepted only if a decidable or semi-decidable validity predicate V(x, r) holds. The minimum feasible realizer complexity M(x) = min_{r: V(x,r)=1} K(r), with K denoting prefix Kolmogorov complexity, measures the minimal information required for a valid outcome. We define the Normalized Advice Quantile (NAQ) as the percentile of M(x) across a reference pool, yielding a scale-free hardness index on [0, 1] robust to the choice of universal machine and comparable across task families. An Exact Realizer Identity shows that the minimal advice for any input-blind executor equals M(x) up to O(1), while a description plus selection upper bound refines it via computable feature maps, separating description cost K(y) from selection cost log i_y(x). In finite-ambiguity regimes M(x) approximately equals min_y K(y); in generic-fiber regimes the bound is tight. NAQ is quasi-invariant under bounded enumeration changes. An operational converse links NAQ to rate-distortion: communicating advice with error epsilon requires average length near the entropy of target features. Extensions include a resource-bounded variant NAQ_t incorporating time-penalized complexity (Levin's Kt) and an NP-style setting showing linear worst-case advice n - O(1). Finally, a DKW bound guarantees convergence of empirical NAQ estimates, enabling data-driven calibration via compressor-based proxies.

Theorems & Definitions (32)

• Theorem 1: Coarse machine invariance
• proof
• Lemma 1: Pool-stability
• Lemma 2: Executor/program translation
• proof
• Theorem 2: Exact Realizer Identity
• proof
• Theorem 3: c.e. acceptance
• Theorem 4: Decidable $V$, computable loss
• Theorem 5: Decidable $V$, u.s.c. loss