arXiv subjects

Shalender Singh

Publications and source records attributed to Shalender Singh.

8 recordsLinked to original sources

arXiv2026

Repairing the refined-decoupling proof of the 5/4 planar pinned Falconer theorem

We show that the literal arbitrary-packet form of the refined-decoupling estimate printed in the Guth--Iosevich-Ou-Wang proof of the planar pinned Falconer theorem is false, even after its tube dimensions are normalized. An explicit collar construction gives a fixed-power counterexample: all packets are active on one square while none of their smaller labelled tubes meets that square. We then give a non circular repair of the original proof route. The analytic input is an enlargement-stable arbitrary-packet theorem in which a packet is concentrated on an $a$-dilate and multiplicity is counted with a strictly larger $b$-dilate. We prove this theorem directly from weighted $\ell^2$ decoupling by an induction that tracks the dilation margin through parabolic rescaling. The corrected theorem applies directly to the original Falconer parent packets after an exact frequency truncation and an absolute small-packet cutoff; no parent-to-canonical decomposition is needed. We then rebuild the good-tube incidence estimate, retain the neighborhood forced by local constancy, and supply a uniform regularization and limiting argument. The principal frequency exponent remains $-(\alpha+1)/3$, so the energy argument closes exactly for $\alpha>5/4$. The pinned theorem itself is not contradicted and is also known through later microlocal methods. A later canonical wave-packet treatment of refined decoupling overlaps with the activity-tube viewpoint but not with the counterexample or the repaired Falconer proof chain.

math.CA↗

arXiv2026

From Liar Paradox to Incongruent Sets: A Normal Form for Self-Reference

We introduce incongruent normal form (INF), a structural representation for self-referential semantic sentences. An INF replaces a self-referential sentence with a finite family of non-self-referential sentences that are individually satisfiable but not jointly satisfiable. This transformation isolates the semantic obstruction created by self-reference while preserving classical semantics locally and is accompanied by correctness theorems characterizing when global inconsistency arises from locally compatible commitments. We then study the role of incongruence as a structural source of semantic informativeness. Using a minimal model-theoretic notion of informativeness-understood as the ability of sentences to distinguish among admissible models-we show that semantic completeness precludes informativeness, while incongruence preserves it. Moreover, incongruence is not confined to paradoxical constructions: any consistent incomplete first-order theory admits finite incongruent families arising from incompatible complete extensions. In this sense, incompleteness manifests structurally as locally realizable but globally incompatible semantic commitments, providing a minimal formal basis for semantic knowledge. Finally, we introduce a quantitative semantic framework. In a canonical finite semantic-state setting, we model semantic commitments as Boolean functions and define a Fourier-analytic notion of semantic energy based on total influence. We derive uncertainty-style bounds relating semantic determinacy, informativeness, and spectral simplicity, and establish a matrix inequality bounding aggregate semantic variance by total semantic energy. These results show quantitatively that semantic informativeness cannot collapse into a single determinate state without unbounded energy cost, identifying incongruence as a fundamental structural and quantitative feature of semantic representation.

cs.AI↗

arXiv2026

An Algebraic Rigidity Framework for Order-Oblivious Deterministic Black-Box PIT of ROABPs

Deterministic black-box polynomial identity testing (PIT) for read-once oblivious algebraic branching programs (ROABPs) is a central open problem in algebraic complexity, particularly in the absence of variable ordering. Prior deterministic algorithms either rely on order information or incur significant overhead through combinatorial isolation techniques. In this paper, we introduce an algebraic rigidity framework for ROABPs based on the internal structure of their associated matrix word algebras. We show that nonzero width-$w$ ROABPs induce word algebras whose effective algebraic degrees of freedom collapse to dimension at most $w^2$, independent of the number of variables. This rigidity enables deterministic witness construction via intrinsic algebraic invariants, bypassing rank concentration, isolation lemmas, and probabilistic tools used in previous work.Thus, we obtain the first order-oblivious deterministic black-box PIT algorithm for ROABPs, running in quasi-polynomial time $n\cdot(wd)^{O(w^2)}$. This establishes that algebraic rigidity alone suffices to derandomize PIT in this model, without assuming ordering information. The framework further isolates a single remaining obstacle to full polynomial-time complexity. We formulate a Modular Stability Conjecture, asserting that width-$w$ ROABPs are stable under hashing into cyclic quotient rings $\mathbb{K}[\lambda]/< \lambda^r-1 >$ once the modulus exceeds a polynomial threshold in $w$ and the individual degree. This conjecture arises naturally from the low-dimensional coefficient structure revealed by rigidity and is supported by extensive empirical evidence. Assuming the conjecture, our methods yield a fully polynomial-time deterministic black-box PIT algorithm for ROABPs, matching the complexity of the best-known white-box algorithms and reducing the black-box problem to a concrete algebraic stability question.

cs.CC↗

arXiv2026

Inverse Falconer Distance Theorems over the Integer Residue Rings $\mathbb{Z}_n$

We establish an ideal-theoretic rigidity principle for quadratic distance images over integer residue rings. Specifically, we prove that near-extremal collapse of the distance set in $\mathbb{Z}_n^d$ forces strong algebraic structure supported on annihilator submodules arising from the arithmetic of $n$. As a consequence, we obtain the first inverse theorem for the Falconer distance problem over $\mathbb{Z}_n$ for composite moduli. We show that if a set $E \subset \mathbb{Z}_n^d$ of size $|E| \asymp n^{(d+1)/2}$ determines only $O(n)$ distinct squared distances, then $E$ must be supported on a coset of an annihilator submodule on which the distance form is algebraically degenerate. The proof introduces a divisor-depth decomposition intrinsic to $\mathbb{Z}_n$, together with a lifting mechanism that transfers local degeneracies at prime moduli into global ideal-theoretic constraints. This yields a complete classification of near-extremizers for the Falconer distance problem in the ring setting, revealing a rigidity phenomenon with no analogue over fields.

math.NT↗

arXiv2026

Inverse Theorems for Point-Sphere Incidences over Finite Fields

Let Q be a nondegenerate quadratic form on $\mathbb F_q^d$, with q odd & $d\ge2$. We prove a two-scale inverse theory for point-sphere incidences. At the maximal-incidence endpoint, put $\mu=1-I(P,S)/(|P||S|)$. If $\mu<(d+1)/(d+2)^2$, then deleting at most a proportion $\sqrt{(d+1)\mu}$ from each side places the retained configuration in one explicit lifted-linear model: the sphere parameters $\Phi(S_Q(c,r))=(2c,r-Q(c))$ lie in an affine space L of dimension at most d, & the retained points lie in its common affine-quadric base. The natural deletion distance satisfies $\mu\le dist_{lin}(P,S) \le2\sqrt{(d+1)\mu}$ and the square-root exponent is optimal. Under codimension-two nonconcentration, & provided at least two spheres survive, the model reduces to one coaxal pencil. This classification has a sharp lower-scale boundary. A polynomial-root chart on one lifted rational normal curve realizes every admissible d-uniform hypergraph; consequently any list of individual point-set templates approximating all resulting constant-K systems within $o(q^d)$ edits has size $\exp(\Omega_d(q^d))$. Exact fixed-nonzero-radius centered direct sums occur even when $|S|\ge4q/K^2$, & a prescribed-radius twisted-cubic system in dimension three has a complete rich-pair graph but neither a heavy pair section nor a quantitatively nontrivial centered block decomposition. At the fixed-radius deviation scale, what remains universally true is sharp adaptive extraction. If $\sigma(P,S)\ge Kq^{(d-1)/2}\sqrt{|P||S|}$ & $K^2q^{d-1}|S|\ge4|P|$, then at least $K^2|S|/4$ ordered pairs have a common P-section of size at least $K^2q^{d-1}/(4|S|)$; for $d\ge3$ the witness count improves to $K^2q|S|/8$. We prove mixed-radius profile stability for both signs, a log-free refinement, a cap-free positive-surplus decomposition, & applications to affine hyperplanes, pinned distances, dot products.

math.CO↗

arXiv2025

Operational entanglement of collective quantum modes at room temperature

Quantum entanglement is commonly assumed to be fragile at ambient temperature and over macroscopic distances, where thermal noise and dissipation are expected to rapidly suppress nonclassical correlations. Here we show that this intuition fails for collective quantum modes whose dynamics is governed by reduced open-system channels rather than by microscopic thermal equilibrium. For two spatially separated collective modes, we derive an exact entanglement boundary based on the positivity of the partial transpose, valid in the symmetric resonant limit. From this result we obtain an explicit minimum collective fluctuation amplitude, expressed entirely in measurable noise, bandwidth, dissipation, and distance-dependent coupling parameters, required to sustain steady-state entanglement at finite temperature. We further show that large collective occupation suppresses but does not eliminate quantum phase diffusion, so the steady state remains phase symmetric and does not collapse to a classical mean-field despite macroscopic signal amplitudes. Stochastic simulations of the reduced open-system dynamics, together with matched classical correlated-noise null models analyzed through an identical pipeline, confirm that entanglement witnesses are violated only in the quantum regime. Our results establish a minimal, platform-independent framework connecting collective-mode dynamics, noise injection, distance, and operational certification of macroscopic entanglement.

quant-ph↗

arXiv2025

Partial Collapse and Ensemble Invariance under Continuous Quantum Measurement

Wavefunction collapse is commonly associated with unavoidable physical disturbance of the measured system. Here we show that in driven-dissipative quantum systems, continuous measurement can induce strong trajectory-level collapse while leaving the ensemble-averaged steady state strictly invariant. We identify measurement-invariant steady states whose unconditional density matrix remains unchanged under continuous monitoring, despite pronounced measurement-induced localization in conditioned quantum trajectories. This separation between trajectory-level collapse and ensemble invariance defines a regime of partial collapse, in which measurement-induced localization is continuously counteracted by dissipative dynamics. We derive a necessary and sufficient condition for steady-state invariance under continuous measurement and identify Liouvillian symmetry as a concrete dynamical mechanism enforcing it. Our results clarify the distinction between conditional collapse and physical disturbance in open quantum systems and provide a framework for non-invasive continuous monitoring in driven-dissipative settings.

quant-ph↗

arXiv2025

Bell-Inequality Violation for Continuous, Non-Projective Measurements

Many solid-state quantum platforms do not permit sharp, projective measurements but instead yield continuous voltage or field traces under weak, non-demolition readout. In such systems, standard Bell tests based on dichotomic projective measurements are not directly applicable, raising the question of how quantum nonlocality can be certified from continuous time-series data. Here we develop a general theoretical framework showing that Bell-CHSH inequality violation can be extracted from continuous, non-projective measurements without assuming any specific collapse model or phase distribution. We show that sufficiently long continuous measurements of a single entangled pair sample its internal phase-probability structure, enabling effective dichotomic observables to be constructed through phase-sensitive projections and coarse-graining. The resulting Bell correlator is governed by two experimentally accessible resources: intrinsic single-qubit phase spread and nonlocal phase locking between qubits. We benchmark the resulting estimator against conventional projective-measurement CHSH tests implemented via quantum-circuit simulations using Qiskit, finding quantitative agreement in the Bell-violating regime without parameter fitting. Classical deterministic correlations cannot violate the CHSH bound, whereas quantum phase-locked systems recover the nonlinear angular dependence characteristic of entanglement. Our results provide a practical route to demonstrating Bell nonlocality in platforms where measurements are inherently continuous and weak.

quant-ph↗