We study when a sound arithmetic theory $\mathcal S{\supseteq}S^1_2$ with polynomial-time decidable axioms efficiently proves the bounded consistency statements $Con_{\mathcal S{+}\phi}(n)$ for a true sentence $\phi$. Our main structural result characterizes the nonexistence of a length-optimal propositional proof system: this is equivalent to every such $\mathcal S$ failing to simulate $S^1_2{+}\mathsf{Exp}{+}\phi_{BB}(k)$ for all sufficiently large $k$, where $\phi_{BB}(k)$ asserts the exact value of the $k$-state Busy Beaver function and $\mathsf{Exp}$ asserts total exponentiation. We obtain this characterization by combining the Kraj\'i\v{c}ek--Pudl\'ak correspondence with our transfer theorem: for each fixed $\mathcal S$, the existence of a true extension not simulated by $\mathcal S$ is equivalent to this eventual Busy Beaver non-simulation property. A finite version transfers individual lower bounds to Busy Beaver instances with explicit parameter changes and proof-length bounds. For finitely axiomatized sequential $\mathcal S$, we also prove that $S^1_2{\vdash}Con_{\mathcal S}{\rightarrow}Con_{\mathcal S{+}\phi}$ yields polynomial-size $\mathcal S$-proofs of $Con_{\mathcal S{+}\phi}(n)$. We refute our earlier conjecture that $EA{\vdash}Con_{\mathcal S}{\rightarrow}Con_{\mathcal S{+}\phi}$ characterizes simulation.

In this finitely axiomatized sequential setting, a new Kolmogorov Blindness conjecture predicts hardness of random tautologies: true logarithmic-deficiency randomness axioms $x{\in}R^{\log}$ yield bounded-consistency statements requiring exponentially long proofs, at a rate depending on the fixed proof encoding, beyond an onset common to all strings of each sufficiently large length. It further predicts this hardness after adjoining a true randomness axiom $y{\in}R^{\log}$ with $|x|{=}|y|$, provided $x$ remains random conditional on $y$. For fixed $\Pi_1$ axiom families with computable proof-length bounds and a common computable onset, we prove a quantitative equivalence between dense hardness with polynomially small exceptional fractions and hardness above a corresponding Kolmogorov-complexity threshold.

This revision corrects Theorem~3.4 by using $S^1_2$ and Lemma~3.9 and Theorem~3.10 by adding total exponentiation to the Busy Beaver witness base, refutes Revision~1's Higher Relative Consistency conjecture (Assumption~5.1) using Hrube\v{s}'s construction, adds a Busy Beaver characterization of the nonexistence of a length-optimal propositional proof system (Corollary~3.13), a finite Busy Beaver transfer (Theorem~3.11), a quantitative density--randomness equivalence (Theorem~6.1 and Corollary~6.3), and alternative canonical witnesses over $\mathsf{PA}$ (Proposition~3.15), generalizes Revision~1's diagonal Busy Beaver family result (Theorem~3.11) to specified classes of theories (Corollary~3.14), and rationalizes the conjectural structure around Kolmogorov Blindness (Conjecture~5.1), replacing Revision~1's certificate-based Busy Beaver Hardness and Kolmogorov Hardness formulations (Conjectures~3.13 and~4.2), deriving reformulated pairwise hardness from Blindness (Theorem~5.3) and factual nonaccess from its universal form (Proposition~5.6), instead of assuming these properties separately (Revision~1, Conjectures~6.7 and~6.5).

We study when a sound arithmetic theory $\mathcal S{\supseteq}S^1_2$ with polynomial-time decidable axioms efficiently proves the bounded consistency statements $Con_{\mathcal S{+}\phi}(n)$ for a true sentence $\phi$. Equivalently, we ask when $\mathcal S$, viewed as a proof system, simulates $\mathcal S{+}\phi$. The paper's two unconditional contributions constrain possible characterizations. First, for finitely axiomatized sequential $\mathcal S$, if $EA{\vdash}Con_{\mathcal S}{\rightarrow}Con_{\mathcal S{+}\phi}$, then $\mathcal S$ interprets $\mathcal S{+}\phi$, implying $\mathcal S{\vdash^{n^{O(1)}}}Con_{\mathcal S}(p(n)){\rightarrow}Con_{\mathcal S{+}\phi}(n)$ for some polynomial $p$, and hence $\mathcal S{\vdash^{n^{O(1)}}}Con_{\mathcal S{+}\phi}(n)$. Second, if $\mathcal S$ fails to simulate $\mathcal S{+}\phi$ for some true $\phi$, then for all sufficiently large $k$ it also fails for $\phi_{BB}(k)$ asserting the exact value of the $k$-state Busy Beaver function. Informally, any argument showing that $\mathcal S$ fails to simulate some $\mathcal S{+}\phi$ also yields unprovable $\phi_{BB}(k)$ witnessing the same obstruction. These results suggest that relative consistency strength is a serious candidate for governing when simulation is possible, while leaving open whether it is the correct criterion.

The paper's central conjectural proposal is that the above sufficient condition is also necessary: if $EA{\not\vdash}Con_{\mathcal S}{\rightarrow}Con_{\mathcal S{+}\phi}$, then for every constant $c{>}0$, $\mathcal S\not{\vdash^{n^c}}Con_{\mathcal S{+}\phi}(n)$. Under this proposal, hardness follows in canonical cases where $\phi$ is $Con_{\mathcal S}$ or a Kolmogorov-randomness axiom. The latter yields further conjectural consequences and extensions.

KH and SETH-K-Finite are reformulated with the threshold EA?Con_S?Con_S+?, correcting the proof of HRC?KH; the no-mutual-help and density consequences are revised accordingly; figures and subsections on BB hardness and robustness are added.

We study when a sound arithmetic theory $\mathcal S{\supseteq}S^1_2$ with polynomial-time decidable axioms efficiently proves the bounded consistency statements $Con_{\mathcal S{+}\phi}(n)$ for a true sentence $\phi$. Equivalently, we ask when $\mathcal S$, viewed as a proof system, simulates $\mathcal S{+}\phi$. The paper's two unconditional contributions constrain possible characterizations. First, for finitely axiomatized sequential $\mathcal S$, if $EA{\vdash}Con_{\mathcal S}{\rightarrow}Con_{\mathcal S{+}\phi}$, then $\mathcal S$ interprets $\mathcal S{+}\phi$, implying $\mathcal S{\vdash^{n^{O(1)}}}Con_{\mathcal S}(p(n)){\rightarrow}Con_{\mathcal S{+}\phi}(n)$ for some polynomial $p$, and hence $\mathcal S{\vdash^{n^{O(1)}}}Con_{\mathcal S{+}\phi}(n)$. Second, if $\mathcal S$ fails to simulate $\mathcal S{+}\phi$ for some true $\phi$, then for all sufficiently large $k$ it also fails for $\phi_{BB}(k)$ asserting the exact value of the $k$-state Busy Beaver function. Informally, any argument showing that $\mathcal S$ fails to simulate some $\mathcal S{+}\phi$ also yields unprovable $\phi_{BB}(k)$ witnessing the same obstruction. These results suggest that relative consistency strength is a serious candidate for governing when simulation is possible, while leaving open whether it is the correct criterion.

The paper's central conjectural proposal is that the above sufficient condition is also necessary: if $EA{\not\vdash}Con_{\mathcal S}{\rightarrow}Con_{\mathcal S{+}\phi}$, then for every constant $c{>}0$, $\mathcal S\not{\vdash^{n^c}}Con_{\mathcal S{+}\phi}(n)$. Under this proposal, hardness follows in canonical cases where $\phi$ is $Con_{\mathcal S}$ or a Kolmogorov-randomness axiom. The latter yields further conjectural consequences and extensions.