Papers

In this paper, a new non-homogeneous linear second-order partial differential equation (PDE) is proposed and investigated, designed to describe the dynamics of correlated pair objects of the microworld (analogues of Cooper pairs or biphotons). The fundamental complex solution of the proposed equation initially contains a spatial background and a doubled phase, which allows for a natural localization of the pair condensate in space. In this work, the continuity and energy balance equations are derived, the discrete spectrum in potential fields is calculated, and specific effects of tunneling and nonlinear interaction of the pair conglomerate with electromagnetic radiation are described.

The main objective of this paper is to develop Krein-Milman-type theorems and Ulam-type stability results for graphs. To establish these results, we introduce several meaningful definitions of vertex-weighted convex graphs inspired by the concept of sequential convexity. We also present a close relationship between the two discrete structures, namely sequential convexity and perfect binary trees. We show that if a graph satisfies a certain convexity property approximately, then this property can be made exact by minimally perturbing the weights assigned to its vertices. Furthermore, we study several structural characterisations, formulate convex minorants for weighted graphs, and derive sandwich-type results. Special emphasis is placed on trees, and an investigation of extremal value problems is also carried out

A cycle of nine pitch classes in twelve-tone equal temperament can be arranged so that its nine consecutive three-note windows realize, exactly once each, every trichord type containing a semitone – types taken up to transposition but not inversion. Such cycles exist in abundance, and generalize to n-tone equal temperament, where the windows realize the n-3 semitone-containing types. In every one of the 1764 such cycles in twelve-tone equal temperament, the chromatic trichord 012 appears scalarly, as a direct chromatic run, and never in a broken "zigzag" contour such as 1-0-2. Why is the zigzag missing? We show that the zigzag presentation behaves as a conserved topological charge of an underlying interval-flow network: it occurs if and only if 3 divides n and n is neither 6 nor 12. The forward obstruction is a Z/3-valued conservation law, proved for all n by a weight-function argument; the two exceptional temperaments – one of them the common Western tuning – are excluded by finite certificates; and existence for every other multiple of 3 follows from an explicit construction. We also quantify rarity, and give exact censuses well beyond the reach of direct search. The classification is machine verified in Lean 4 in both directions. The paper is written for two audiences. Musical sections require no more than pitch-class set vocabulary; the mathematical development is self-contained and elementary, yet fully rigorous.

This paper introduces new characterizations, decomposition theorems, and stability results for convex sequences. We show that a sequence is convex precisely when its epigraph satisfies a midpoint convexity condition, thereby connecting discrete and geometric notions of convexity. A decomposition result proves that any sequence can be written as the difference of two convex sequences, with generalizations to higher-order convexity. We construct nontrivial convex minorants for bounded-below sequences and establish a Hyers-Ulam-type stability theorem showing that any approximately convex sequence can be uniformly approximated by a genuine convex sequence without significantly altering its values. Finally, for a concave sequence, we characterize those subsequences that are convex in it by providing slope inequalities and monotone auxiliary sequences. We explore the interplay among convexity, subadditivity, and periodically indexed subsequences.

We compute several new Laplace convolution integrals involving exponential functions, error functions, and parabolic cylinder functions. We extend our results using formulas for repeated integration and differentiation. We apply these convolution integrals to heat transfer and linear viscoelastic problems. As a by-product, we obtain new integral representations for the error function and the complementary error function.

Let k be a perfect field of characteristic p>0. We introduce an object-level construction for canonical strong embedded resolution and principalization over k. The construction program replaces monotonicity of pointwise numerical invariants by a well-founded history of addressed comparison factors. Starting from the differential-integral saturation of a marked Rees algebra, we construct total-Hasse activity packets, filtered coefficient cubes, semilinear Frobenius-Hasse sources, and literal transform data for ordinary permissible blowups. A six-row defect calculus routes local problems to certified surface, toroidal-monomial, binomial, and additive-type procedures, after which clean centre portfolios are serialized and descended on a global nerve. The central structure is a global replacement certificate transporting successor addresses, paid quotients, typed traces, displayed parents, reopening data, and terminal truth across macroblocks. We prove that a complete state equipped with this certificate admits a strict multiset replacement in a single dependent well-founded order; hence the iteration terminates and the exhausted state reconstructs a regular strict transform having normal crossings with the ordered boundary. We further formulate an object-level realization of the certificate through rigid generation, cross-generation no-reset, complete wild-capacity control, a centre-or-typed-exit alternative, structured cofibres, displayed-parent allocation, and literal terminal truth. The program page is also available at website https://sites.google.com/view/positive-char-resolution

Statistically Meaningful Geometry (SMG) is a differential-geometric and information-theoretic framework that lifts over-parameterized models into infinite-dimensional non-parametric Orlicz statistical fiber bundles with an Ehresmann connection, decoupling unobservable vertical gauge noise from horizontal statistically verifiable directions. We prove the First Edge Theorem: Amari's information geometry (IG) and conventional statistics (CS) are not autonomous statistical universes but degenerate boundary layers of the larger gauge-active SMG space. Taking the structural identifiability radius $R\to\infty$ breaks gauge symmetry, collapses vertical fibers, and forces the total space onto the IG manifold; subsequent local asymptotic normality as $N\to\infty$ flattens the remaining curvature, yielding \[ \mathrm{SMG}\xrightarrow{\;R\to\infty\;}\mathrm{IG}\xrightarrow{\;N\to\infty\;}\mathrm{CS}. \] The same fiber-bundle machinery transforms model non-identifiability from a singular collapse into a structured gauge space. Applications include resolving the deep-learning generalization paradox, constructing gauge-invariant gradient descent and holonomy-matched preference alignment for generative AI, and solving weak identification in structural econometrics via intrinsic horizontal geodesic search.

This work develops a general-order fractional differentiation operator on monomials from the discrete half-order coefficient structure established in Part I. Starting from the one-step transfer law of the integer coefficient family, the discrete coefficient relation is continued to a real argument and a canonical Gamma-functional form is selected under the stated normalization and regularity requirements. The operator order is then extended beyond one half by dividing the first derivative into equal operator steps, leading from orders 1/m and k/m to general rational and real orders. The resulting coefficient is independently verified through the addition law for operator orders on the monomial system, subject to the requirement that all intermediate expressions in the composition are defined. On the corresponding parameter domain, the construction yields the standard Gamma-ratio monomial formula for fractional differentiation. For nonnegative integer orders it reduces to ordinary repeated differentiation, while for positive non-integer orders it agrees at the monomial level with the left-sided Riemann-Liouville formula with lower limit 0. Its relation to the Caputo operator is also discussed, including the distinction for constants and low-degree polynomial terms. The aim is not to introduce a new classical fractional derivative, but to provide a constructive algebraic-operator route from a discrete coefficient law to the general fractional-order monomial formula, clarifying the roles of functional continuation, Gamma normalization, and operator composition.

We prove that a specified Lidman-Piccirillo piece $V$, a symplectic $4$-manifold with the homology of $S^2\times D^2$ built from a genus-$2$ surface bundle over a once-punctured torus by two Luttinger surgeries, is simply connected, for an explicit permitted choice of the two surgery parametrizations. Three consequences follow. The symplectic double $Z=V\cup_\sigma V$ is homeomorphic but not diffeomorphic to $S^2\times S^2$. The Lidman-Piccirillo manifolds $B$ and $W$ are homeomorphic. Since the figure-eight knot is slice in $B$ and not in $W$, they are the first pair of homeomorphic closed $4$-manifolds distinguished by unconstrained knot slicing, that is, by sliceness with no constraint on the homology class of the slice disk; detecting smooth structure this way goes back to Casson. Finally, the regluing of Lidman and Piccirillo's Theorem~2 applied to $Z$ yields a closed simply connected $4$-manifold homeomorphic but not diffeomorphic to $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. The consequences follow from the simple-connectivity statement by the classifications of Freedman and of Hambleton-Kreck, together with a rigidity analysis of the surgery parameters. The fundamental group is computed in the style of Baldridge and Kirk, from explicit based representatives of every meridian and Lagrangian push off, and the resulting relation system is decided by coset enumeration, after calibration on two configurations whose answers are known independently. The development calculations and finite-presentation decisions can be reproduced from the ancillary files.

In this paper, we introduce polynomial $\mathcal{Z}$-contraction by unifying the ideas of polynomial contraction and $\mathcal{Z}$-contraction and also provide demonstrative examples to show the effectiveness and novelty. Chiefly, we prove two fixed point results for polynomial $\mathcal{Z}$-contraction employing simulation functions. Lastly, we utilize our main result to discuss the existence and uniqueness of solution for boundary value problem with a concrete example in $L^p(0,\pi)$ under suitable assumptions.

Fix $\varepsilon>0$, let $x>1$ be a large real number and let $\text{rad}(n)=\prod_{p\mid n}p$ be the radical of an integer $n\geq1$. A triple $(a,b,c)$, with $a+b=c$ and $\gcd(a,b,c)=1$, such that $c>(\text{rad}(abc))^{1+\varepsilon}$, is called exceptional triple. Recent works have proved that the cardinality $\#\mathscr{E}(x)$ of set $\mathscr{E}$ of exceptional triples satisfies $\#\mathscr{E}(x)=O(x^{33/50})$. This note proves that the cardinality of the exceptional set $\mathscr{E}(x)$ of triples $(a,b,c)$ is an infinite set unconditionally for any $\varepsilon<2/35$.

This paper establishes the global proof of the Second Edge Theorem: as sample size tends to infinity, sample-dependent information-geometric manifolds—formed by the parameter space, sample-scaled Fisher metric, and dual alpha-connections—undergo metric-topological collapse onto the flat tangent space of Conventional Statistics at the true parameter. We first prove a universal tensor valence scaling law, under which tensor fields of valence one through four degenerate at rates determined by sample size. Score fluctuations stabilize, the Fisher metric freezes to its true-parameter value, affine connections dissolve, and Riemann curvature is annihilated. We then bridge geometric collapse with statistical decision theory, showing that Cheeger–Gromov flattening and Le Cam risk condensation are dual projections of the same asymptotic phase transition. A Fisher-compatible Ehresmann connection extends these results to over-parameterized and singular models, yielding horizontal leaf-space collapse and uniform local asymptotic normality. Unifying the First and Second Edge Theorems yields a nested dual-edge hierarchy: Conventional Statistics is the boundary of Information Geometry, which is itself the boundary of Statistical Mechanics and Geometry. Thus, Conventional Statistics is not a heuristic approximation but the unique zero-curvature thermodynamic attractor of regular parametric information manifolds. This redefines modern statistics as a dynamic non-equilibrium field theory of finite-sample fluctuations, phase transitions, and gauge-invariant interactions.

An Euler magic matrix is an integer matrix M with MM^t = gamma I for some gamma != 0, whose squared entries sum to gamma along both main diagonals; it is proper if its squared entries are pairwise distinct. Euler gave a proper example of order 4; Müller settled orders 3 (none exists) and 8; and Kominers settled order 5. We give an order-6 construction, a case not addressed by Müller or Kominers, exhibiting a proper Euler magic matrix of order 6 with gamma = 18500 together with a second, independent one with gamma = 43290. The proof is the explicit matrix and a finite exact verification. We also record an elementary counting bound gamma >= 2485 for proper order-6 examples.

Nicomachus' identity $\sum_{k=1}^{n} k^3 = \left(\sum_{k=1}^{n} k\right)^2$ is familiar, but the structure behind it is less often emphasized. We begin with Nicomachus' original pattern, in which successive cubes are expressed as successive blocks of consecutive odd integers, and show how triangular numbers naturally enter the resulting sum-of-cubes identity. We then reverse the usual triangular-number argument and start from a general difference of squares representing $n^p$ as a sum of $n$ consecutive odd integers. This construction is possible for every $p\geq 2$, but only for $p=3$ do the two squares correspond to consecutive triangular numbers, which explains why Nicomachus' odd-number blocks fit together without gaps or repetitions. Finally, using the leading terms of the Faulhaber polynomials, we show that, apart from the trivial case, Nicomachus' identity is the unique relation of the form $S_s(N)=S_1(N)^r$ for positive integer exponents.

The Awareness Logic of Ambiguity (ALA) is introduced as a formal framework for preserving the structure of judgment before scalar projection. An evaluative act is represented by a triadic interpretive state with three functionally distinct roles: interpretive valuation, contextual alignment, and epistemic restraint. ALA uses a cognitive order with reversed polarity for restraint and shows that the resulting state space forms a bounded distributive lattice. It develops weakly and strictly admissible scalar projections as declared interfaces, together with a role-preserving arithmetic core and canonical permissive and conservative aggregation operators. The Non-Collapse Theorem establishes that two states that are equal after scalar projection may become distinguishable under an ALA lattice operation; hence scalar equality does not imply operational equivalence. The paper also establishes the conditional semantic minimality of the triad and treats the unit cube as the normalized real-valued realization of a more general role architecture. A high-stakes clinical scenario illustrates why evidential support, contextual alignment, and restraint toward action should remain separate until a scalar reporting or decision interface is explicitly chosen. ALA therefore provides a structure-preserving calculus for reasoning under ambiguity before projection.

The FIDE Laws of Chess effective from 1 January 2023 terminate a game automatically upon fivefold repetition or after 75 consecutive moves by each player without a pawn move or a capture. Legal games of 17,697 plies were previously known, and a scheduling analysis of the pawn moves and captures gave an arithmetic upper bound of 17,699 plies, but games of 17,698 or 17,699 plies were not excluded. We close this gap. Partitioning play at pawn moves and captures yields at most 118 segments. Each segment has length at most 150 plies, and each change in the colour of successive segment endpoints reduces this bound by one ply. We prove that a game with all 118 segments must incur at least three such changes. Hence every game has at most $150 \cdot 118 - 3 = 17{,}697$ plies, and the known constructions are optimal. Moreover, in every maximum-length game, all sixteen pawns make six one-rank moves and promote.

Ground station handovers across Earth based Deep Space Network (DSN) terminals incur carrier lock reacquisition delays of 4 to 8 minutes per event, introducing periodic telemetry blackouts during cislunar flights. We address this limitation by formulating temporal handover sequences as perfect interval conflict graphs. For single spacecraft trajectories, we establish that adding a single reserve channel ($k=4$) enables static offline channel preallocation, eliminating spacecraft transponder retuning during station switches. To sustain link availability under unexpected atmospheric fades or station outages, we construct a Robust Space Communication Graph (R-SCG) model, demonstrating that four channels suffice to maintain uninterrupted coverage under single node failure conditions. For unmodeled link disruptions, we implement a distributed local recoloring scheme with $O(1)$ amortized complexity, augmented by Random Forest state classification for predictive channel reassignment. Furthermore, for colocated spacecraft clusters, we show that the joint ground space conflict graph retains chordality, bounding the required spectrum to $3+c$ channels for $c$ concurrent assets. A 10 day orbital simulation demonstrates that static channel preassignment recovers up to 2.7~hours of telemetry throughput relative to conventional reactive handover schemes.

Indefinite integrals are found for 63 half-integer Bessel and Struve functions, and incomplete gamma functions with integer indices, each multiplied by $Exp(-ax^{2})$ and divided by powers. A series solution is given for the individual terms (of any inverse power) in such functions, split into even and odd portions. Eight integrals are given involving these series.

A near Goldbach graph is a simple undirected graph whose vertex set consists of all positive even integers and there is an edge between two vertices $a,b$ if and only if $\frac{a+b}{2}, \frac{|a-b|}{2}$ are either odd primes or $1$. A finite near Goldbach graph $G(n)$ has the vertex set $\{x\in 2\mathbb{N}\, :\, x\leq 2n\}$ with the same adjacency rule. In this paper, we obtain two exact formulas for the degree of the even positive integer $x$ in $G(x/2)$. We compute a function $\eta(x)=\prod\limits_{p\mid x,\, p>2} \frac{p-1}{p-2}\, \frac{xe^{-0.183407}}{(\log\, x)^2}$ that approximates the degree of $x$ in $G(x/2)$ for a large even positive integer $x$. Finally, we introduce the concept of a nearly independent set of events and show that if the set of divisibility events for a large even integer $x$ is nearly independent, then $x$ can be expressed as the sum of two odd primes.

In this paper, we introduce and investigate a novel generalization of Zumkeller numbers termed $k$-IGMO numbers, inspired by a problem proposed in the International Gamma Mathematical Olympiad (IGMO) 2025. A positive integer n is defined as a $k$-IGMO number if its set of positive divisors can be partitioned into two disjoint subsets whose elements have sums differing by $k$. Under this definition, classical Zumkeller numbers correspond to the case where $k = 0$. The main result of the paper is the existence of infinitely many odd $k$-IGMO numbers for all non-negative integers $k$.

Trees of maximum $\sigma$-irregularity subject to a prescribed maximum degree $\Delta$ have been characterized for $\Delta=4$ and $\Delta=5$, with the extension to arbitrary $\Delta\geqslant 3$. This paper develops a unified framework for this class of problems built on three pillars. A majorization-theoretic reformulation that decomposes $\sigma$-extremization into a Schur-convex optimization over degree sequences and a rearrangement type optimization over tree realizations, valid for every $\Delta\geqslant 3$. We establish a duality between the maximization and minimization problems, alongside the general $\Delta$ closed form for $\sigma_{\max}(n,\Delta)$. A stability result showing that the optimality gap between extremal and near extremal trees is bounded independently of $n$ and grows as $\Theta(\Delta^3)$.

We study nonnegative solutions of the reciprocal cost law on the positive rationals and determine which of them admit regular extensions to the positive reals. Using the substitution $H=1+F$, we reduce the reciprocal composition law to d'Alembert's functional equation. We prove that every nonnegative solution on $\mathbb{Q}_{>0}$ is determined by one real weight $\alpha_p$ for each prime $p$, with only a global sign identification. Thus the rational solution space is infinite dimensional. We prove that every nonnegative rational solution has an algebraic extension to $\mathbb{R}_{>0}$, but a regular extension exists exactly when the prime weights satisfy $\alpha_p=\lambda\log p$ for some $\lambda\in\mathbb{R}$. In this case the extension is unique and belongs to the one-parameter family $F_\lambda(x)=\cosh(\lambda\log x)-1$. Otherwise the rational solution is unbounded on every nonempty open subset of $\mathbb{Q}_{>0}$, and the regular locus is closed and nowhere dense. We also extend the result to arbitrary nontrivial subgroups of $\mathbb{R}_{>0}$, where the alternative is governed by rational rank. Finally, we show that unit logarithmic curvature selects the canonical reciprocal cost $J(x)=(x+x^{-1})/2-1$, while, for a carrier $G$ whose logarithm is dense in $\mathbb{R}$, the single asymptotic condition $F(e^s)\sim s^2/2$ implies both regularity and calibration.

The basic result in the theory of linear ODEs is that the dimension of the fundamental set of linearly independent solutions is equal to the order of equation. This view is typically extended from integer-order to fractional differential equations. The purpose of this research is to demonstrate that this expectation is incorrect and explain why. We show that linear fractional differential equations may admit additional linearly independent solutions beyond their order. This phenomenon occurs when the coefficients of the equation cross certain threshold values. This phase transition produces a hyper-dimensional fundamental set of solutions and offers new insight into fractional differential equations and leads to the revisions of the statements of initial and boundary value problems.

This paper considers combinatorial and sampling approaches to deriving a priority vector from an incomplete pairwise comparison matrix. It is shown that the informationally significant combinations of matrix elements are in one-to-one correspondence with spanning trees of the graph induced by the matrix. The combinatorial approach is based on the enumeration of all spanning trees and the componentwise aggregation of the priority vectors derived from them. To reduce computational complexity, a sampling Monte Carlo approach with a sequential stopping criterion is proposed, ensuring a prescribed accuracy for estimating the components of the resulting vector.

For integers $1\le m\le n$, let $s(n,m)$ denote the number of $m$-tuples $(k_1,\ldots,k_m)$ of nonnegative integers satisfying \[ n=\sum_{j=1}^{m}\frac{k_j}{j}. \] We obtain a complete asymptotic expansion for $\log s(n,m)$, uniformly for all $n\ge m$ as $m\to\infty$. The coefficients are given explicitly in terms of limiting prime-block functions arising from the residue structure of the problem. We also determine the full hierarchy of multiplicative corrections in the sparse regime, identifying explicit constants at every fixed order and, in particular, the first correction constants $1/4$ and $(1-\log 2)/4$. The results give uniform two-parameter asymptotics as both the target and the number of allowed reciprocal parts grow.

At the end of 2025, a record computation of $\pi$ reached 314 trillion decimal digits, providing the largest numerical dataset ever generated for this constant. We exploit this unprecedented dataset to investigate whether the digits of $\pi$ themselves can serve as a source of pseudorandom numbers for estimating $\pi$ through the simplest Monte Carlo method. Our results go beyond the normality hypothesis by providing empirical evidence of a high degree of statistical randomness in the available digits, although not of digit independence, which cannot hold for a deterministic sequence. By optimizing the mapping of the digit sequence into Monte Carlo samples, we obtain the highest precision allowed by the dataset. As predicted, the method successfully reproduces the first sequence of decimal digits, demonstrating that the largest available dataset of $\pi$ digits can be used to recover $ \pi \approx 3.141593 $ through Monte Carlo simulation.

In this paper, we introduce the notion of crossed homomorphisms on associative superalgebras. We show that crossed homomorphisms are precisely the Maurer–Cartan elements of a naturally associated graded Lie algebra. This characterization enables us to construct a cohomology theory of crossed homomorphisms. We then investigate formal deformations of crossed homomorphisms and derive the corresponding deformation equations. It is proved that the infinitesimal part of a formal deformation is a 1-cocycle in the associated cohomology, while the obstruction to extending a deformation of finite order is a 2-cocycle.

Let $P^+(n)$ be the largest prime factor of $n$, and let $\chi=\chi_{-4}$ be $1$ on primes $1\pmod4$ and $-1$ on primes $3\pmod4$. For fixed $k\ge2$ we study \[ D_k(x)=\sum_{\substack{n\le x\\ \Omega(n)=k}}\chi(P^+(n)). \] Thus $D_k(x)$ compares the two residue classes according to the largest prime factor of integers having exactly $k$ prime factors, counted with multiplicity. Assuming RH for $\zeta(s)$ and $L(s,\chi)=\beta(s)$, we prove a pointwise explicit formula. The main term is a fixed negative contribution plus an absolutely convergent oscillating sum over the zeros $\rho=\frac12+i\gamma$ of $L(s,\chi)$. The ordering of the prime factors produces $k$ Perron denominators, and the principal coefficient of a zero is $O_k((1+|\gamma|)^{-k})$. We then show that the total size of all zero terms is strictly smaller than the fixed contribution. Hence $D_k(x)<0$ for all sufficiently large $x$. The analogous problem without fixing $k$ is still open.

Decision-making in complex systems often involves dealing with imprecise or uncertain information, frequently represented using fuzzy sets, particularly Triangular Fuzzy Numbers (TFNs). A crucial aspect of many fuzzy methods is the quantification of distance between TFNs. Many distance measures assume that all values are in the same scale, requiring a preliminary normalization stage when applied to heterogeneous attributes with different scales or units. This paper proposes the Triangular Fuzzy Rescaling Distance (d_{TR}), a metric designed to address this challenge. The d_{TR} uniquely integrates Linear Rescaling (LRE) directly into the distance calculation, ensuring normalization during the comparison of fuzzy numbers. We formally prove that d_{TR} satisfies the properties of a metric, including non-negativity, identity, symmetry, and the triangle inequality. Furthermore, we demonstrate that d_{TR} is bounded, scale-invariant, and origin-invariant. These properties, combined with a weighting vector for prioritizing dimensions, make d_{TR} suitable for applications involving heterogeneous fuzzy data, such as the construction of synthetic indicators, distance-based machine learning algorithms or multicriteria-decision aiding.

We present a novel microlocal analysis of beam hardening artifacts arising in cone-beam X-ray CT, where the set of X-ray sources is restricted to a 1D smooth curve $ \gamma\subset \mathbb{R}^3$. We show that, when the CT data is modeled in the standard way using the Beer Lambert law, the exponential term in the model creates singularities in the data that are not present in the linear X-ray transform. We assume that the attenuation coefficient $\mu$ (the reconstruction target) has a jump discontinuity across a surface $\mathcal{S}\subset\mathbb{R}^3$, and is smooth otherwise. We prove that these additional singularities in the data occur when the X-ray beam is tangent to $\mathcal{S}$ at two points simultaneously. To investigate how the singularities in the data propagate to the reconstruction space, we apply Filtered Back Projection (FBP) type reconstruction. We prove that the artifacts due to beam hardening are locally of conormal type, and lie on a 2-D surface which is the union of all double tangent rays which intersect $\gamma$. The artifacts are notably weaker than the reconstructed jumps of $\mu$ (i.e., the desired singularities), and we quantify this using the order of the corresponding conormal distributions. While our primary theory applies to the regions of $\mathcal{S}$ that are smooth, we also extend our theory to non-smooth $\mathcal{S}$ with "ridges." These arise where $\mathcal{S}$ is locally the intersection of two smooth surface patches meeting transversely along a curve (e.g., the edge of a cuboid). In addition, we present simulated reconstructions of metal objects in circular cone-beam CT to validate our theory.