Modern multi-tenant AI clusters are increasingly communication-bound, driven by high-volume and multi-round GPU-to-GPU collective communication. Consequently, the GPU dispatcher’s choice of a physical GPU subset for each tenant largely determines the job’s effective collective bandwidth and thus its performance ceiling. Existing dispatchers predominantly rely on static, topology-aware heuristics that prioritize GPU resource compactness, assuming that minimizing physical distance maximizes communication bandwidth. However, we reveal that this assumption often fails due to complex system-level bottlenecks, such as non-linear NIC saturation and inter-node link heterogeneity. This paper presents <inline-formula><tex-math notation="LaTeX">${\sf BandPilot}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">BandPilot</mml:mi></mml:math><inline-graphic xlink:href="tang-ieq2-3716225.gif"/></alternatives></inline-formula>, a performance- and contention-aware GPU dispatching primitive that optimizes effective collective bandwidth for multi-tenant AI clusters. Specifically, <inline-formula><tex-math notation="LaTeX">${\sf BandPilot}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">BandPilot</mml:mi></mml:math><inline-graphic xlink:href="tang-ieq3-3716225.gif"/></alternatives></inline-formula> learns a data-efficient bandwidth model from sparse NCCL measurements via a hierarchical design. Guided by the model, <inline-formula><tex-math notation="LaTeX">${\sf BandPilot}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">BandPilot</mml:mi></mml:math><inline-graphic xlink:href="tang-ieq4-3716225.gif"/></alternatives></inline-formula> uses an equilibrium-driven heuristic as a fast front end, and invokes a pruned elimination search when a controller predicts that further refinement is worthwhile. To account for multi-tenant interference, <inline-formula><tex-math notation="LaTeX">${\sf BandPilot}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">BandPilot</mml:mi></mml:math><inline-graphic xlink:href="tang-ieq5-3716225.gif"/></alternatives></inline-formula> virtually merges a candidate allocation with co-located cross-host jobs to conservatively estimate shared bottleneck capacity and predict contention-degraded bandwidth. Across a 32-GPU H100 cluster and heterogeneous simulations, <inline-formula><tex-math notation="LaTeX">${\sf BandPilot}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">BandPilot</mml:mi></mml:math><inline-graphic xlink:href="tang-ieq6-3716225.gif"/></alternatives></inline-formula> achieves 90 – 97% bandwidth efficiency relative to the best-found reference, improving average efficiency by 20–30% over topology-compactness heuristics.
<p> Our objective in the present work is to develop a fairly complete arithmetic theory of critical <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -functions on the eigencurve. To this end, we carry out the following tasks: </p> <p> We give an “étale” construction of Bellaïche’s <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -functions at a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta"> <mml:semantics> <mml:mi> θ </mml:mi> <mml:annotation encoding="application/x-tex">\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -critical point on the cuspidal Coleman–Mazur–Buzzard eigencurve. </p> <p> We introduce the algebraic counterparts of these objects (which arise as appropriately defined Selmer complexes) and develop Iwasawa theory in this context, including a definition of an Iwasawa theoretic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript upper I w Superscript c r"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>I</mml:mi> <mml:mi>w</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>c</mml:mi> <mml:mi>r</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathscr {L}^{cr}_{Iw}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . </p> <p>We formulate the (punctual) critical main conjectures, and study its relationship with its slope-zero counterpart. Along the way, we also develop descent theory (paralleling Perrin-Riou’s work).</p> <p> We introduce what we call <italic>thick</italic> (Iwasawa theoretic) fundamental line and the <italic>thick</italic> Selmer complex to counter Bellaïche’s secondary <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -functions. This allows us to formulate an infinitesimal thickening of the Iwasawa main conjecture, and we observe that it implies both slope-zero and punctual critical main conjectures, but it seems stronger than both. </p> <p> We establish an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript script upper X"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {O}_\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic leading term formula for the two-variable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -function over the affinoid neighborhood <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X equals upper S p m left-parenthesis script upper O Subscript script upper X Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> <mml:mi>m</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow>
<jats:p> We explore numbered Boolean algebras over classes <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="normal">Ξ</mml:mi> </mml:math> </jats:inline-formula> of arithmetical and analytical hierarchies. We show the existence and uniqueness (up to computable isomorphism) of universal Boolean <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="normal">Ξ</mml:mi> </mml:math> </jats:inline-formula> -algebras, determine the classes in which such algebras exist, and classify the universal algebras up to isomorphism. As applications, we characterize Tarski–Lindenbaum algebras of four semantic classes of models of a given finite rich signature, namely, the class of all countable saturated models having decidable <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>ω</mml:mi> </mml:math> </jats:inline-formula> -stable theories, the class of prime models with decidable theories and first-order definable elements, the class of models with decidable non-finitely axiomatizable theories, and the class of models with finitely axiomatizable theories; it is shown that the first three of these algebras are universal Boolean algebras over hierarchy classes <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> </jats:inline-formula> , <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>2</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> </jats:inline-formula> , and <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> </jats:inline-formula> , respectively, and the fourth one is a countable atomic Boolean <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Π</mml:mi> <mml:mn>3</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> </jats:inline-formula> -algebra whose quotient modulo the Frèchet ideal is a <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>4</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> </jats:inline-formula> -universal Boolean <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>4</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> </jats:inline-formula> -algebra. </jats:p>
Cryptographic evidence systems face fundamental trade-offs between confidentiality (Conf), reliability (Rel), and opposability (Opp), with Conf<inline-formula> <tex-math notation="LaTeX">$\cdot $ </tex-math></inline-formula>Rel<inline-formula> <tex-math notation="LaTeX">$\cdot $ </tex-math></inline-formula>Opp products ranging from 0.27 to 0.57 in deployed systems. The Conditional CRO Trilemma recently established that institutional capacity <inline-formula> <tex-math notation="LaTeX">$\Delta $ </tex-math></inline-formula> fundamentally constrains achievable trade-offs through the bound <inline-formula> <tex-math notation="LaTeX">$\text {Conf}\cdot \text {Rel}\cdot \text {Opp} \leq \Delta /\lambda + \mathcal {O}(\lambda ^{-2})$ </tex-math></inline-formula>. However, the imminent deployment of post-quantum cryptographic systems raises critical questions: do quantum adversaries and quantum channel effects modify these bounds? This paper presents the Post-Quantum Conditional CRO Trilemma, establishing that both institutional capacity and quantum interpretability loss fundamentally constrain achievable trade-offs. Under explicit assumptions about quantum-aware institutional limits, we prove<disp-formula> <tex-math notation="LaTeX">\begin{equation*} \text {Conf}^{q} \cdot \text {Rel}^{q} \cdot \text {Opp}^{q} \leq \frac {\Delta _{q} + \eta _{q} \cdot H(C)}{\lambda } + \mathcal {O}(\lambda ^{-2}) + \text {negl}(\lambda)\end{equation*} </tex-math></disp-formula>where <inline-formula> <tex-math notation="LaTeX">$\Delta _{q}$ </tex-math></inline-formula> quantifies the quantum-aware institutional extraction capacity, <inline-formula> <tex-math notation="LaTeX">$\eta _{q}$ </tex-math></inline-formula> is the quantum interpretability loss coefficient, <inline-formula> <tex-math notation="LaTeX">$H(C)$ </tex-math></inline-formula> is the contextual entropy, and <inline-formula> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> is the security parameter. Our main contributions are as follows: 1) a quantum extension of the institutional capacity model with parameters <inline-formula> <tex-math notation="LaTeX">$\Delta _{q}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\eta _{q}$ </tex-math></inline-formula>; 2) complete information-theoretic proofs using Quantum Pinsker’s inequality, the Gentle Measurement Lemma, and quantum Fano bounds; 3) an internal coherence analysis showing that NIST PQC finalists (Dilithium, Kyber, and SPHINCS+) imply <inline-formula> <tex-math notation="LaTeX">$\Delta _{q} \in [{28, 67}]$ </tex-math></inline-formula> bits under quantum adversaries, consistent with quantum-aware bounded rationality; and 4) a framework for matching post-quantum protocols to quantum-aware institutional capacity. We demonstrate the internal coherence of our framework against the same deployed systems analyzed in the Conditional CRO work, Helios voting, zk-SNARKs compliance, and ECDSA signatures, showing how quantum effects modify their implied institutional capacities. This trilemma shifts post-quantum protocol design from purely computational security to quantum-constraint-aware engineering, with applications to GDPR-compliant quantum-safe systems and eIDAS quantum readiness.
<jats:p> The thermal partition function, <jats:inline-formula> <jats:alternatives> <jats:tex-math>Z</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Z</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> , of a <jats:inline-formula> <jats:alternatives> <jats:tex-math>CFT_d</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>F</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> on <jats:inline-formula> <jats:alternatives> <jats:tex-math>S^{d-1}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> is parameterized by the inverse temperature <jats:inline-formula> <jats:alternatives> <jats:tex-math>\beta</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> along with <jats:inline-formula> <jats:alternatives> <jats:tex-math>\lfloor d/2\rfloor</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">⌊</mml:mo> <mml:mi>d</mml:mi> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">⌋</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> angular velocities <jats:inline-formula> <jats:alternatives> <jats:tex-math>\omega_i</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> . In this paper, we investigate the behaviour of this partition function when <jats:inline-formula> <jats:alternatives> <jats:tex-math>n</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> of the <jats:inline-formula> <jats:alternatives> <jats:tex-math>\omega_i</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> are scaled to unity (the largest allowed value) at fixed values of the other <jats:inline-formula> <jats:alternatives> <jats:tex-math>(\lfloor d/2\rfloor-n)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mo stretchy="false" form="prefix">⌊</mml:mo> <mml:mi>d</mml:mi> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">⌋</mml:mo> <mml:mo>−</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> angular velocities. We argue that <jats:inline-formula> <jats:alternatives> <jats:tex-math>\ln Z</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">ln</mml:mi> <mml:mo></mml:mo> </mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> develops a simple pole in <jats:inline-formula> <jats:alternatives> <jats:tex-math>(1-\omega_i)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> for each <jats:inline-formula> <jats:alternatives> <jats:tex-math>\omega_i</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> that is scaled to unity. The residue of this product of poles is a theory-dependent (so non-universal) function of <jats:inline-formula> <jats:alternatives> <jats:tex-math>\beta</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> and the fixed angular velocities. The inverse Laplace transformation of this partition function constrains the functional form of the field theory entropy as a function of charges in a limit in which angular momenta and the twist are scaled as follows. While <jats:inline-formula> <jats:alternatives> <jats:tex-math>n</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> special angular momenta <jats:inline-formula> <jats:alternatives> <jats:tex-math>J_1\ldots J_n</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>…</mml:mi>
A set <inline-formula> <tex-math notation="LaTeX">$S\subseteq V(G)$ </tex-math></inline-formula> is called a <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-total dominating set of a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> if every vertex of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> lies within distance at most <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> of some other vertex in <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$k\ge 1$ </tex-math></inline-formula>. The minimum cardinality of such a set is called the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-total domination number of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">$\gamma _{t,k}(G)$ </tex-math></inline-formula>. In this paper, we investigate <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-total domination from a coverage-based perspective. We establish a new lower bound for <inline-formula> <tex-math notation="LaTeX">$\gamma _{t,k}(G)$ </tex-math></inline-formula> in terms of the diameter of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>. To analyze neighborhood coverage in <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-total domination, we extend the concepts of shadow and share to <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-neighborhoods and introduce the neighborhood coverage number. We also extend the concept of redundant domination to the setting of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-neighborhoods. Together, these concepts quantify both the coverage provided by <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-neighborhoods and the overlap among them. These concepts yield new insights into the structure of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-total dominating sets and are applied to obtain results for circulant graphs. We show that the shadow graph operation preserves the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-total domination number; that is, <inline-formula> <tex-math notation="LaTeX">$\gamma _{t,k}(D_{2}(G))=\gamma _{t,k}(G)$ </tex-math></inline-formula> for every graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>. For strong product graphs, we establish general upper bounds and prove that <inline-formula> <tex-math notation="LaTeX">$\gamma _{t,k}(H\boxtimes H')=\gamma _{t,k}(H)$ </tex-math></inline-formula> whenever <inline-formula> <tex-math notation="LaTeX">$r(H')\le k$ </tex-math></inline-formula>. As a consequence, we obtain exact values of the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-total domination number for several classes of shadow and strong product graphs, including shadow graphs of paths and cycles, and strong products of paths and cycles.
