Truth-Table Quotients Do Not Close One-Tape MCSP Magnification: Small Fibers, Exact Cosets, and a Defect-Inflation Dichotomy
Year: 2026
Type: preprint
Abstract: McKay, Murray and Williams showed that an $N^{1.01}$ one-tape lower bound for MCSP at a small threshold $2^{\mu_* n}$ implies $P\ne NP$, while Cheraghchi, Hirahara, Myrisiotis and Yoshida proved nearly quadratic one-tape lower bounds only for thresholds $2^{\mu n}$ with $\mu>1/2$. We ask whether a truth-table encoding can transfer the latter to the former, and prove counting and additive-combinatorial barriers that hold for arbitrary, even uncomputable, maps. First, Lupanov's construction and circuit enumeration give $\log_2|\mathcal L_S|=\Theta(\min\{2^n,S\log S\})$ for the set $\mathcal L_S$ of size-$\le S$ functions; hence any map sending $\mathcal L_{2^{\mu_H n}}$ into the size-$2^{\mu_1 n'}$ functions on $n'=\kappa n+O(1)$ variables with $\mu_H>\kappa\mu_1$ has a fiber containing $2^{\Theta(n2^{\mu_H n})}$ easy functions. This excludes injective and small-fiber encodings, dummy padding, tensor lifts, symmetry canonicalisation and low-kernel affine maps, even with free composition. Second, iterated Harper isoperimetry shows that every large YES fiber is XOR-dispersed and that $\log_2(|\mathcal L_S+\mathcal L_S|/|\mathcal L_S|)\ge((1-\mu)/2-o(1))S$ for $S=2^{\mu n}$; therefore no exact coset-fiber, in particular no exact affine, quotient bridges the regimes. Third, for every remaining map, either the average target XOR-defect over fiber pairs or the average fiber inflation on $\mathcal L_S+\mathcal L_S$ is $2^{\Omega(S)}$, with a split shown to be optimal for cardinality arguments. Limits: the exact decision-bit map survives all counting arguments, so excluding predicate-specific quotients requires charging their one-tape computation; gaps $S_1\ge 2S_0$ remain open; two composition statements are conditional on an explicit model. Nothing here proves $P\ne NP$ or a new lower bound; small cases are checked exhaustively in the supplement. Archival copy of version 1 of this paper, published on rStation on 25 October 2026 (rStation p_vs_np paper 1, v1). paper.md is the manuscript exactly as published there; the PDF is typeset from it with the published title and abstract; supplement.zip is the supplement as published there. Files: paper.md — SHA-256 e9fd6232fd05136e3eb691fb9f0c14b922bdfbced16449f35b4cf7aa16c4d6b4 truth-table-quotients-do-not-close-one-tape-mcsp-magnification-small-fibers.pdf — SHA-256 9e00a0995339cac16acbbded976d363782ea2f30bdcbf9dfdc4759cb86168a43 supplement.zip — SHA-256 d79c8c7253106377ace82ce1c0472e20d4a8af99516f529662a1738166a2f97a
Author: Li, Hongmin
Institutions: Institute of Science Tokyo, The University of Tokyo
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225596017