arXiv subjects

Tristan Simas

Publications and source records attributed to Tristan Simas.

5 recordsLinked to original sources

arXiv2026

Descent Before Hardness: Orbit-Gap Obstructions in Exact Certification

Tractability tests are often computed from input syntax: support-graph treewidth, local coefficient patterns, backdoor tests, or action-count bounds. Before such a test can be lower-bounded or made algorithmic, it must define a predicate on the exact-certification problem itself. Equivalent presentations must receive the same verdict. The semantic object is the correctness quotient, whose classes are states with the same correct outputs. Correctness-preserving presentation moves generate closure orbits. A target that changes inside one closure orbit has an orbit gap and fails descent. Exact closure-invariant classification is possible exactly when the positive and negative orbit hulls are disjoint; the positive hull is then the least exact classifier, and computable orbit representatives make the classifier algorithmic. The results separate three layers. The descent layer gives orbit-gap obstructions for raw local syntax, raw action and coordinate counts, and raw support-graph predicates. The post-descent complexity layer applies ordinary reductions to descended objects: graph-predicate lower bounds transfer through action-gap graph extraction, and Action-Gap-Treewidth is NP-complete when the width bound is part of the input. The certification layer asks whether a proxy descends: for split proxies $b\wedge\varphi(z)$, SAT reduces to non-descent and UNSAT reduces to descent. Positive regimes use quotient-preserving normalizations or catalogues before model checking; bounded quotient size, bounded full Gaifman treewidth of the constructed quotient, sparse unary-gap certificates, and strict-margin perturbation balls give explicit cost bounds after quotient construction.

cs.CC↗

arXiv2026

The Optimizer Quotient and the Certification Trilemma

The optimizer quotient is the canonical object for exact decision-relevant information: it is the coarsest exact decision-preserving abstraction (Theorem 2.15). This paper proves that exact certification of this object's coordinate structure is subject to an impossibility trilemma: under $\mathrm{P} \neq \mathrm{coNP}$, no certifier can be simultaneously sound, complete on all in-scope instances, and polynomial-budgeted (Theorem 7.1). The cost of this impossibility varies by regime: coNP (static), PP-hard (stochastic decisiveness), PSPACE-complete (sequential). Six structural restrictions collapse certification to polynomial time. The finite reduction and verification core is mechanized in Lean 4.

cs.CC↗

arXiv2026

Coordinate-View Confusability Graphs and Matroid Rank Certificates

A coordinate-view presentation specifies a large confusability graph by coordinates rather than by an edge list. The problem is to certify zero-error recovery and Shannon capacity from the succinct presentation, before the exponentially large graph is materialized. For affine presentations over $\mathbb F_q$, Gaussian elimination gives a polynomial-time rank upper certificate from $O(rd\log q+Ld)$ input bits for a graph with $q^r$ vertices. Exactness of that certificate is equivalent to positivity of a Grassmannian avoidance count $N_{t^*}$. Positivity of $N_k$ is NP-complete for every fixed $k$, already over $\mathbb F_2$, while the ambient-rank parameter gives a fixed-parameter algorithm. The rank-one reduction is parsimonious for $\#SAT$. For arbitrary $t$, the finite decoder for $N_t$ reads the signed rank profile of the represented forbidden-point matroid; the kernel-intersection lattice alone does not determine the count. On the positive side, kernel sections give the exact formula $\Theta(G)=\log\alpha(G)=\min_{S\in\mathcal V}t(S)\log q$, with a projection-equality matrix optimal for Haemers minrank. A finite-field blocking theorem gives exactness when $q\ge L$, and Reed-Solomon/MDS codes give exact all-$k$-view families beyond that regime. Full-tuple coordinate-view graphs also have a polynomial-time cofinal-antichain normal form; transitive confusability is exactly intersection closure.

cs.IT↗

arXiv2026

Thermodynamic Limits of Proof

Every irreversible recorded distinction has a positive thermodynamic work floor. Landauer's principle supplies the ideal bound $\varepsilon\ge k_B T\ln 2$ per irreversible bit, experimentally verified to $\pm 10\%$. Proof available to an agent is checkable information for that agent: some substrate must produce, retain, and expose evidence that excludes answer-changing alternatives. A finite detector array operating at temperature $T$ for finite time has finite signal-acquisition capacity. Combining finite causal access, positive retained-record cost, and exact lower bounds on required records gives the Physical Counting Impossibility Theorem: no fixed-budget substrate can provide universal exact proof once the retained-record lower bound exceeds the declared budget. The theorem requires exactly $B<\infty$ and $\varepsilon>0$. An answer reports a value; proof supplies checkable grounds for accepting it. A reversible device may compute an answer and erase its scratch history, but proof requires retained, inspectable records. A global answer register, oracle response, entanglement witness, finite survey catalog, or trusted device output supplies proof only through an interface that exposes the relevant grounds to the verifier. A proposed interface must identify the retained-record lower-bound family $R(n)$ it induces. Sound operational claims about efficient solvability inherit the same finite-budget obstruction when their acceptance would license universal exact proof. Substrate-free derivability has proof status only when a physical verification event makes it available to an agent.

cs.CC↗

arXiv2026

Semantic Identity Compression: Zero-Error Laws, Rate-Distortion, and Neurosymbolic Necessity

Symbolic systems operate over precise identities: variables denote specific objects, pointers target precise memory locations, and database keys refer to singular records. Neural embeddings generalize by compressing away semantic detail, but this compression creates collision ambiguity: multiple distinct entities can share the same representation value. Exact identity recovery requires additional information precisely when representation fibers have size greater than one. The residual cost is controlled by a single combinatorial object: the collision-fiber geometry of the representation map $\pi$. Let $A_{\pi}=\max_u |\pi^{-1}(u)|$ be the largest collision fiber. The finite laws include a tight fixed-length converse $L \ge \log_2 A_{\pi}$, an exact finite-block scaling law, a pointwise adaptive budget $\lceil \log_2 |\pi^{-1}(u)|\rceil$, and an exact fiberwise rate-distortion law for arbitrary finite sources via recoverable-mass decomposition across representation fibers. The uniform single-block formula $D^\star(L)=\max(0,1-2^L/a)$ appears as a closed-form special case when all mass lies on one collision block, where $a = A_{\pi}$ is the collision block size. The same fiber geometry determines query complexity and canonical structure for distinguishing families. Because this residual ambiguity is structural rather than representation-specific, symbolic identity mechanisms (handles, keys, pointers, nominal tags) are the necessary system-level complement to any non-injective semantic representation. All main results are machine-checked in Lean 4.

cs.IT↗