Papers

Markets and quantitative finance

Reading guide: how the three papers relate, and the question that would join them.

• The Validation Bottleneck: Alpha Discovery When Hypotheses Are Free What happens to discovery when generating hypotheses costs nothing and only validation is scarce. Two pre-registered experiments on public data. July 2026 · replication code · SSRN:7180460
• The AI-Native Asset Manager: Institutional Cognition as Competitive Advantage As frontier AI commoditizes information and analysis, the scarce resource shifts from intelligence to institutional epistemology — the architecture by which a firm generates, challenges, sizes, and learns from its investment beliefs. Returns decompose into narrative beta, cross-sectional factor alpha, and idiosyncratic issuer alpha, each demanding a different way of knowing, and the durable advantage proves to be a velocity rather than a position. April 2026 · SSRN:6487798
• Beyond De Prado and Cotton: Hierarchical and Iterative Methods for General Mean-Variance Portfolios Hierarchical Risk Parity and its Schur-complement generalization are signal-blind: both solve only the minimum-variance problem and cannot carry an arbitrary expected-return forecast. Three methods lift the framework to the general case at the same cost — HRP-μ, HRP-Σμ, and CRISP, a Gauss–Seidel shrinkage solver proved unconditionally convergent. arXiv:2604.23833 · replication code · Beyond HRP Portfolio Lab

Mathematics

Geometric topology and Floer theory

Instanton homology of knots

Explore the pillowcase — a guided trefoil example, from cutting the knot to its Floer complex.

Five papers on the knot version of the Atiyah–Floer program, which compares singular instanton knot homology with Lagrangian Floer homology of immersed curves in the pillowcase. The first assembles the instanton side for the (3,n)-torus knots and two-bridge knots. The second and third treat the (−2,3,q) pretzel knots: the second, building on the fifth, proves that the two sides agree for a decomposition of every knot in the family, and the third studies the bounding cochains that a second decomposition requires, where the comparison remains conjectural. The fourth determines the instanton differential for every torus knot, and the fifth computes the pillowcase side for the whole (3,n) family and finds that it matches the instanton side in every grading and, over the field with two elements, at chain level. For the torus knots, the first, fourth and fifth are best read in that order. Reading guide · Papers, sources and exact computations.

• Traceless characters and instanton gradings for 2-bridge and (3,n) knots An account of the instanton chain complex of the (3,n)-torus knots (its generators, gradings and differential) together with the traceless character varieties of two-bridge knots. Each result is credited to its source in the work of Lewallen, Klassen, Hedden–Herald–Kirk, Nagasato–Yamaguchi, Daemi–Scaduto and Li–Ye. Comparing the size of the complex with the size of its homology gives the rank of the differential for every torus knot, and settles the (3,n) pattern for all n. A second proof of the homology rank runs through Khovanov homology. The one nonzero differential of the knot 8₁₉ is matched with a bigon in the pillowcase. arXiv:2607.26095 · current version (PDF), 6 October 2026, 15 pages · code
• Instanton and pillowcase homology of the (−2,3,q) pretzel knots Hedden, Herald and Kirk conjectured that a Lagrangian Floer homology of immersed curves in the pillowcase recovers singular instanton homology. This paper proves the conjecture for the hyperbolic pretzel knots P(−2,3,q). Regluing the tangle that Hedden, Herald and Kirk use for a torus knot gives a decomposition of every knot in the family; the regluing acts on the pillowcase by a shear, and the resulting Floer complex is computed by hand. It agrees with instanton homology in every grading and has nonzero differentials. The same method treats a family of twisted torus knots, and the instanton homology of the whole pretzel family is computed over the integers. arXiv:2607.26096 · current version (PDF), 6 October 2026, 37 pages · code
• Maurer–Cartan elements in the pillowcase and the Bar-Natan invariant for the (−2,3,q) pretzel knots For a second decomposition of the same pretzel knots, due to Smith, the pillowcase curves give the wrong answer unless they carry bounding cochains, as a conjecture of Cazassus, Herald, Kirk and Kotelskiy predicts. Working in the algebraic model of Kotelskiy, Watson and Zibrowius, the paper constructs explicit Maurer–Cartan deformations, each a smoothing of the curve at a single self-intersection, that give the correct answer for the knots P(−2,3,5) and P(−2,3,7), and a second closure of the tangle selects one of them, conditionally on the conjecture and a hypothesis on the support of the cochain. The selected object differs by a single term from the Bar-Natan invariant of the tangle, and rational closures cannot tell the two apart. The paper asks whether the object the conjecture assigns to the tangle is its Bar-Natan invariant. Research manuscript · current version (PDF), 6 October 2026, 33 pages · source and code · Zenodo:23643150
• The S-complex differential of torus knots For every torus knot, the instanton chain complex built from traceless characters is larger than its homology. The gap is governed by two classical invariants, the signature and the Alexander polynomial. This paper computes the rank of the differential through floor sums and proves a sharp lower bound for it. It classifies exactly which torus knots have no differential at all, which supplies the converse to a criterion of Daemi and Scaduto. It also shows how the differential splits between gradings, and this determines the graded rational instanton homology of every torus knot. A comparison with the pillowcase computations of Hedden, Herald and Kirk leads to a conjecture linking the two complexes at chain level, which the next paper proves for the (3,n) family. Research manuscript · current version (PDF), 6 October 2026, 12 pages · source · Zenodo:23643177
• Pillowcase Floer homology of the torus knots T(3,n) Hedden, Herald and Kirk conjectured that a Lagrangian Floer homology of immersed curves in the pillowcase recovers singular instanton homology. This paper computes the pillowcase side for every (3,n)-torus knot. For small perturbations in an explicit range, the complex has exactly the generators predicted by the instanton side. Depending on n modulo 6, it has either no differential or exactly one, so its homology agrees with instanton homology in every grading. This confirms the conjecture for an infinite family with nonzero differentials, and, combined with a result of Daemi and Scaduto, it shows that the pillowcase complex is chain equivalent to the instanton complex, proving the conjecture of the paper above for this family. A numerical computation suggests that, without a bounding cochain, the answer for the knot 8₁₉ depends on the chosen decomposition. Research manuscript · current version (PDF), 6 October 2026, 40 pages · code · Zenodo:23643181

Donaldson theory and the Atiyah–Floer program

Explore Atiyah–Floer cobordisms — a guided tour from handle attachments to the comparison of instanton and symplectic cobordism maps.

Five manuscripts in gauge theory. Four take up the Atiyah–Floer comparisons of Daemi, Fukaya and Lipyanskiy and show how they interact with point operations and with four-dimensional cobordism maps, the latter with mod-two coefficients, integral transported orientations, and the identification and normalization of the usual maps. The fifth proves a one-bubble neighborhood theorem in SO(3)-monopole moduli spaces, an analytic input to Feehan and Leness’s approach to Witten’s conjecture. Reading guide · PDFs, LaTeX sources, and project repository.

• Point operations and matching-line bubbles in the Atiyah–Floer comparison Daemi–Fukaya–Lipyanskiy’s admissible SO(3) Atiyah–Floer comparison intertwines the surface and loop operations; this paper supplies the point-class calculation. Matching-line bubbling produces a universal genus-dependent scalar correction — nonzero in genera one and two — and thereby gives the full corrected observable-algebra action. Research manuscript · current version (PDF), August 2026 · Zenodo:23643185
• Cobordism maps and Atiyah–Floer isomorphisms for framed instanton homology I Daemi, Fukaya and Lipyanskiy proved a version of the Atiyah–Floer conjecture: framed instanton homology agrees with the Lagrangian Floer homology of a Heegaard splitting. This paper asks whether the comparison also respects four-dimensional cobordisms, and proves that, with mod-two coefficients, it does: comparison maps of their type carry the Kronheimer–Mrowka cobordism maps to maps defined purely by counting holomorphic curves, for every connected cobordism. The symplectic side thereby becomes a functor in its own right. The proof rests on two new analytic results, a collapse of a compression body next to a holomorphic strip, realizing a degeneration proposed by Salamon and Wehrheim, and an excision theorem along a torus. The sequel on integer coefficients appears below. Research manuscript · current version (PDF), original 25 September 2026; this version 30 October 2026, 156 pages · Atiyah–Floer Cobordism Explorer · Zenodo:23811438
• Cobordism maps and Atiyah–Floer isomorphisms for framed instanton homology II The sequel proves the comparison over the integers. The mixed isomorphisms of Paper I carry the integral instanton cobordism maps, oriented by almost-complex structures, to signed counts of holomorphic curves, up to one overall sign for each cobordism, so the symplectic maps form a projective functor on homology. Orientations are transported from gauge theory through the mixed equation. The new input is that collapsing a compression body and excising along a torus preserve orientations, and that the sign of attaching a sphere to a strip is stable under small perturbations and under reversing the splitting. Paper III below identifies the usual singular Hopf-link maps and the calibrated DFL count; the nonsingular cap and prescribed relative-spin comparisons remain open. Not yet independently refereed. Scope and source. Research manuscript · current version (PDF), original 30 September 2026; this version 30 October 2026, 110 pages · Zenodo:23811453
• Cobordism maps and Atiyah–Floer isomorphisms for framed instanton homology III Integral chain homotopy equivalences identify the usual singular Hopf-link framed instanton maps with the gauge-calibrated DFL mixed count and the transported symplectic maps, up to one overall sign and an explicit grading conversion. For every closed connected oriented three-manifold, surgery generation turns an algebraic orientation normalization into finite integer combinations of products of geometric counts; a polynomial in forward and reflected mixed counts transfers it to the symplectic side. The proof retains the companion analytic results and the stated ordinary surgery and oriented gluing inputs. The nonsingular cap and prescribed relative-spin comparisons remain open. Research preprint · current version (PDF), 26 October 2026; revised 30 October 2026, 32 pages · scope and source · Zenodo:23811477
• One-bubble neighborhoods in SO(3)-monopole moduli spaces An oriented one-bubble neighborhood theorem supplies the analytic neighborhood input to Feehan and Leness’s level-one link calculation underlying their low-degree comparison for Witten’s conjecture. The theorem treats compact regular Seiberg–Witten moduli spaces of arbitrary dimension whose solutions have nonzero spinor. Strong approximation in critical norms and recovery of the gluing parameters give reverse gluing; comparison with Donaldson’s instanton-addition map identifies the orientation. The low-degree consequence retains abundance, effectiveness, and Seiberg–Witten simple type. Higher bubbling levels and collisions remain necessary further work toward the general conjecture. Research manuscript · current version (PDF), 9 September 2026; revised 27 September 2026, 58 pages · Zenodo:23643193

Algebraic geometry and mirror symmetry

Tangent bundle stability of toric Fano varieties

Three papers on tangent-bundle stability for smooth toric Fano varieties: an anticanonical census through dimension six, stability mechanisms for blowups and toric bundles, and local stability chambers as the polarization varies. The third paper gives an all-dimensional family with arbitrarily many independent chamber boundaries and determines the nearby stable loci of the indecomposable strictly semistable census examples. Reading guide · Papers, code, and data.

• Slope stability of tangent bundles of smooth toric Fano varieties A Klyachko-criterion sweep with exact certificates over all 8,630 smooth toric Fano varieties in dimensions three through six. Every Kähler–Einstein variety in the census has polystable tangent bundle, but not conversely — 102 of the 109 five-folds with stable tangent bundle are not Kähler–Einstein. arXiv:2608.20411 · current version (PDF) · certificates and data
• Tangent stability on toric Fano blowups and bundles For the specified invariant Fano blowups of products of projective spaces indexed by trees, anticanonical tangent stability reduces to one divisor-degree inequality per vertex. Smooth toric Fano bundles with degree-six del Pezzo-product fibres attain the sharp stable Picard maximum ⌊5n/4⌋ + 1 in this bundle class. A new exact census certifies all 6,281 Picard-number-ten eightfolds: 64 stable, 692 strictly semistable and 5,525 unstable; every strictly semistable fan is a nontrivial product. The unrestricted Picard bound remains open. Preprint · current version (PDF), original 20 September 2026; this version 29 October 2026, 75 pages · source and exact proof data · Zenodo:23774632
• Local stability chambers for toric tangent bundles The number of local stability-chamber boundaries for toric tangent bundles is unbounded. This paper proves the anticanonical stability trichotomy for projective-line bundles over products of lines with positive and negative unit twists, and determines the local chambers of the strictly semistable family. A geometric criterion also gives nearby stable polarizations for all 31 indecomposable strictly semistable examples in the existing census in dimensions three through six. Their chambers have one, two, or three independent boundaries. The paper identifies locally free, pairwise nonisomorphic stable factors for both the family and these finite examples. Stability walls and geometric degeneration continues the programme with nonsimplicial chambers, spectra and flows. Research manuscript · current version (PDF), 26 October 2026; revised 27 October 2026, 17 pages · source and exact computations · Zenodo:23682140

Smoothability of Calabi–Yau threefolds

Reading guide: the six papers and their physics companions · Explore conifold transitions — an interactive guide to exceptional curves, vanishing spheres, and the relations that govern smoothability.

Six papers on smoothings of singular Calabi–Yau threefolds. Papers 1–3 study anticanonical hypersurfaces in Gorenstein toric Fano fourfolds: local singularity types, smoothings induced by deforming the ambient toric pair, and isolated mirror pairs. Paper 4 develops a global obstruction from vanishing periods, Paper 5 proves a necessary and sufficient smoothing criterion for threefolds with nodes and del Pezzo cone points, and Paper 6 proves the mirror counterpart on nodal components meeting its near-limit construction: the vanishing three-spheres of the Batyrev mirror satisfy exactly the relations of the exceptional curves. For X₉ it also classifies the full codimension-three four-node component and identifies a geometric reverse transition up to birational equivalence with the Batyrev partner family. Database-wide counts are computer-assisted, over a Kreuzer–Skarke input copy verified for reflexivity and absence of repetitions up to lattice equivalence. Papers, code, and data.

• Non-smoothable Calabi–Yau threefolds from reflexive polytopes Explicit projective Calabi–Yau threefolds with a single nonsmoothable singular point: an anticanonical cone over F₁, or a pentagon cone that deforms but admits no smoothing. The local census classifies 217 isolated Gorenstein toric germs with bounded primitive edge vectors, and a scan of the Kreuzer–Skarke classification finds the local obstruction in 39,175,536 polytopes (approximately 8.27%) — a lower bound for global nonsmoothability. Paper 1 · arXiv:2609.28499 · current version (PDF), September 2026 · source and guide
• Deformations of toric pairs and the smoothing of Batyrev Calabi–Yau threefolds Trinomial deformations of the ambient toric pair smooth the singular points over a chosen smoothable two-face whenever its dual edge has an interior lattice point, and a one-node example shows that this dual-edge threshold is sharp. Exactly 3,774 of the 30,241 smoothings in the Batyrev–Kreuzer census meet the criterion face by face; the remaining 26,467 need relations among exceptional curves from different faces. Paper 2 · arXiv:2610.06884 · current version (PDF), August 2026 · source and guide
• Doubly isolated Batyrev mirror pairs and non-smoothable Calabi–Yau threefolds Exactly 590 reflexive four-polytopes have both associated Batyrev hypersurfaces with at most isolated singularities; their two-faces fall into twelve lattice classes. The F₁ cone occurs in a unique mirror pair, with resolution Hodge numbers (20,26) and (26,20): one member is locally nonsmoothable, while every germ of its mirror is smoothable. The self-dual 24-cell is the unique smooth/smooth pair. Paper 3 · arXiv:2610.03753 · current version (PDF), August 2026 · source and guide
• A vanishing-cycle obstruction to smoothing Calabi–Yau threefolds Any smoothing of a projective Calabi–Yau threefold with nodes and degree-six or degree-seven del Pezzo cone points forces a relation among link classes on a crepant resolution that is small over the nodes and blows up the cone vertices. The relation is nonzero at every node and degree-seven point, and at each degree-six point smoothed through the one-dimensional component. The vanishing-period argument applies to smoothing arcs of arbitrary contact order. It rules out a smoothing over a formal disc of the 26-node, four-cone mirror member from Paper 3, although every singularity is locally smoothable; on its mirror partner, two nodes cannot be smoothed by any analytic deformation. The obstruction also excludes 76 of the 77 polytopes with at most nine vertices in the stated pentagonal framework. Paper 4 · arXiv:2610.35911 · current version (PDF), August 2026 · source and guide
• Smoothing Calabi–Yau threefolds with nodes and del Pezzo cone points A necessary and sufficient smoothing criterion for projective Calabi–Yau threefolds with nodes and exact anticanonical cone points over del Pezzo surfaces of degrees 5, 6 and 7 and over ℙ¹×ℙ¹: a homological relation must avoid every local discriminant. Selected deformation bases are smooth under a nonzero degree-six projection hypothesis. A nonsmoothable example has a reduced deformation space with four smooth components, showing why a relation nonzero at every singular point is insufficient. X₁₉ is smoothable and has a smooth 30-dimensional deformation base, although no proper ambient family in the paper’s class of single-degree Minkowski decompositions induces a smoothing. Paper 5 · arXiv:2610.35915 · current version (PDF), September 2026 · source and guide
• Mirror Symmetry for Conifold Relations in Calabi–Yau Threefolds For a class of Batyrev mirror families, we establish Morrison's predicted codimension for nodal degenerations and strengthen it to an equality of integral relation lattices. The result holds on every connected component of the nodal coefficient locus meeting our construction near the maximally degenerate limit. After marking and orienting the nodes, the relations among exceptional curves of the original small resolution coincide with those among mirror vanishing three-spheres, which generate a torsion-free subgroup. Each such component is smooth of codimension equal to the relative Picard number. We realise every relation near the limit by an integral four-chain lifted from a tropical two-cycle relative to the discriminant on Gross's degeneration base. The comparison characterises smoothability of the nodal partial resolution and, when it is smoothable, yields a symplectic conifold transition on the mirror. We treat specified projective toric partial resolutions of hypersurfaces with nodes and degree-six or degree-seven del Pezzo cone points. In an explicit four-node family, every member of the unique codimension-three nodal component admits a projective small resolution. For general members, these resolutions are birational to Batyrev mirrors of smoothings of the original nodal threefold, giving the reversed transition predicted by Morrison in this family. Paper 6, research manuscript · current version (PDF), September 2026, 234 pages · source and guide · Zenodo:23643206

Universal curve counting

How do universal curve counts change under a conifold transition, and when do their product expansions have integer exponents? The first paper proves rational specialization and carries out a complete compact calculation; the second gives a compact counterexample to general integrality for the scalar local-curve factors. Read them in that order for the transition setting and its arithmetic boundary. Reading guide · Papers, sources and exact calculations.

• Universal curve counting under conifold transitions A rational universal specialization theorem for projective conifold transitions with smooth algebraic total space and torsion-free second homology at both endpoints. A geometric support bound makes each bounded-degree sum finite. In a compact four-node transition, all 37 candidate contributions over a primitive line class are computed: eight are nonzero and their sum agrees with an independent stable-pair calculation. The smoothing coefficient in twice the line class is also determined. Integrality transfer is conditional on the resolution. Paper 1, preprint · current version (PDF), 28 October 2026, 18 pages · sources and exact calculations · Zenodo:23750304
• A counterexample to integral product exponents in universal curve counting A compact smooth Calabi–Yau threefold has a genus-ten scalar local-curve product exponent at three times a primitive class with fractional part 2/3. An exact degree-three Frobenius-algebra calculation gives a divisibility obstruction, and a stable-pair moduli space isomorphic to projective thirteen-space has virtual count −14, violating it. Integrality holds in classes of divisibility at most two. The integer elementary expansion is an input to the counterexample. Paper 2, preprint · current version (PDF), 28 October 2026, 13 pages · sources and exact calculations · Zenodo:23750305

Wall crossing and scattering in open FJRW theory

Genus-zero open FJRW theory for the Landau–Ginzburg model xʳ + yˢ, following Gross, Kelly and Tessler: where wall crossing begins, how wall-crossing factors along rays of boundary degrees, and an annular wall structure for the quartic theory. Reading guide · Papers and exact verification code.

• The central-charge threshold for wall-crossing in two-variable open FJRW theory The genus-zero primary open FJRW invariants of Gross, Kelly and Tessler for xʳ + yˢ are independent of the choice of boundary conditions exactly when the central charge is less than one, that is, for the simple Fermat singularities, as Gross, Kelly and Tessler predicted. The paper determines the least number of interior insertions at which dependence appears, shows that a single descendent insertion produces dependence in every case, and identifies the wall generators with Hamiltonian vector fields. It also shows that the wall-crossing group acts simply transitively on systems of invariants once a torus direction is adjoined to its Lie algebra. Paper 1, research manuscript · current version (PDF), 11 October 2026, 15 pages · source · Zenodo:23643211
• Slope-ordered factorization of wall-crossing in two-variable open FJRW theory In the open FJRW theory of Gross, Kelly and Tessler, the wall of each critical disk lies on the ray spanned by its numbers of boundary marked points, and the transformation between the two extremal systems of invariants factors uniquely into automorphisms supported on these rays, ordered by slope. With descendents the factorization holds in every finite truncation, where it is realized by a homotopy of boundary conditions whose jumps occur ray by ray, which a single homotopy in the normal form of their extension lemma does not achieve in general. Between consecutive rays the potential is transported from constants analogous to theta-function coefficients, which can be nonzero on terms with several interior markings, where wall-crossing does not produce them. Paper 2, research manuscript · current version (PDF), 11 October 2026, 37 pages · source · Zenodo:23643215
• Annular scattering in quartic open FJRW theory For the quartic x⁴ + y⁴, the first-descendent wall-crossing generators of Gross, Kelly and Tessler force the disk-shaped Carl–Pumperla–Siebert base of the Landau–Ginzburg mirror to be replaced by an integral-affine annulus, a double cover with hyperbolic core holonomy. On the annulus the paper builds a wall structure, consistent at its two joints, that agrees with the Gross–Kelly–Tessler wall-crossing on the two generators but not on the first wall carrying one marking of each; the difference does not change periods and is removed by a normal form of unknown geometric origin. The Gross–Kelly–Tessler homotopies can be chosen compatibly as markings are added. Paper 3, research manuscript · current version (PDF), 11 October 2026, 85 pages · source · Zenodo:23643219

Moduli spaces and geometric analysis

Stability walls and geometric degeneration

How do algebraic stability walls govern the loss of analytic rigidity? These papers connect realizable stability chambers, small Dolbeault eigenvalues, Hermitian–Einstein metrics and Hermitian Yang–Mills flow. The trilogy proves the spectral mechanism, constructs genuine toric tangent examples, and establishes metric and flow results on the same family of bundles. A further paper extends stationary metric and spectral comparison to repeated stable factors, where matrix modes replace scalar graph directions. Reading guide · Papers, sources and reproducible calculations.

• Small Dolbeault eigenvalues along Hermitian Yang–Mills flow As Hermitian Yang–Mills flow carries a semistable bundle toward its graded limit, some eigenvalues of the Dolbeault Laplacian on endomorphisms tend to zero. The paper determines how fast each of them vanishes and with which leading coefficient, for bundles on curves with distinct stable factors and for two-step extensions in any dimension. Under a positivity condition on the weight grading of Haiden, Katzarkov, Kontsevich and Pandit, the eigenvalues that decay at the slowest rate have coefficients fixed by the bundle, one of them always one half, while at faster rates the coefficients remember the metric the flow started from. The paper also shows that the bounded correction in the known flow asymptotics converges, and that a related slope inequality is sharp in every rank. Research manuscript · current version (PDF), September 2026 · paper source and verification code · Zenodo:23643223
• Small Dolbeault eigenvalues near stability walls For an extension of several stable quotient factors by one stable subbundle, with all factors pairwise nonisomorphic and of equal limiting slope and every extension component nonzero, ranks and slope gaps determine the leading small spectrum of the Dolbeault Laplacian on trace-free endomorphisms at the Hermitian–Einstein metric. This paper gives an explicit matrix for every small eigenvalue, with relative error uniform even when the positive gaps vanish at arbitrarily different rates. For two factors it determines the second-order correction and, after rescaling and unitary identification with the graded limit, the first eigensection through first order. Toric tangent bundles supply arbitrarily many small eigenvalues with independently prescribed orders of vanishing. The metric-response and HYM-flow paper develops the corresponding dynamical theory in its stated two-step setting. Preprint · current version (PDF), 27 October 2026; abstract revised 27 October 2026, 13 pages · source and matrix computations · Zenodo:23717764
• Uniform Hermitian–Einstein spectra for bipartite two-step extensions Approaching intersecting stability walls can produce eigenvalues on widely separated scales. For fixed two-step extensions with distinct stable factors of equal limiting slope and a connected extension graph, this paper determines every small eigenvalue of the Dolbeault Laplacian on trace-free endomorphisms at a Hermitian–Einstein metric from a weighted graph Laplacian, with uniform relative error. It also gives a relative metric approximation and a spanning-tree expression for the leading vanishing factor of the zeta determinant. Ranks and slopes suffice on trees; cycles can retain dependence on the harmonic extension data. Preprint · current version (PDF), 27 October 2026; abstract revised 27 October 2026, 22 pages · source and reproducible calculations · Zenodo:23717660
• Uniform Hermitian–Einstein spectra with repeated stable factors For simple locally free two-step extensions with repeated stable factors within the layers, every small Hermitian–Einstein endomorphism eigenvalue is relatively asymptotic to an independently balanced matrix model, uniformly at arbitrarily separated positive scales. A projection estimate and degree argument compare the actual metric with the balanced metric without inverting the smallest finite Hessian eigenvalue. A fixed rank-five bundle on a product of four projective lines realizes six small modes. Preprint · current version (PDF), 29 October 2026, 18 pages · source and reproducible calculations · Zenodo:23780241
• Nonsimplicial local stability chambers of toric tangent bundles Smooth weak-Fano toric tangent bundles realize nonsimplicial stability chambers with every centered slope direction accessible. A general local normal form describes nearby Harder–Narasimhan filtrations without assuming the equal-slope subsheaves are locally free. Positive spanning gives full slope access, including a surface with a non-locally-free quotient. A positive-twist Bott family obstructs equal slopes for its natural filtration, while the fourfold example separates tangent walls from toric chambers and yields an eight-node Calabi–Yau transition from Hodge numbers (4, 60) to (2, 66). Preprint · current version (PDF), original 27 October 2026; this version 29 October 2026, 24 pages · source and reproducible calculations · Zenodo:23774770
• Metric response and Hermitian Yang–Mills flow for two-step extensions For fixed two-step extensions with pairwise nonisomorphic stable factors of equal limiting slope and a connected extension graph, this paper quantifies the response of Hermitian–Einstein metrics to changes of polarization. When the centered slopes give full local parameters, the singular values of the trace-free part of H⁻¹dH, for the Hermitian–Einstein metric H, are asymptotic to reciprocal graph eigenvalues in rank-weighted slope and normalized L² norms. A finite-dimensional invariant manifold gives late-time metric comparison up to a constant scalar and relative spectral comparison for Hermitian Yang–Mills flow at nearby stable, semistable and unstable parameters, with the comparison time chosen before the parameter neighborhood. An application connects chamber geometry, metric response and flow on the same weak Fano toric tangent bundles. Preprint · current version (PDF), 27 October 2026; abstract revised 27 October 2026, 27 pages · source and reproducible calculations · Zenodo:23717742

Singular quotient metrics

How do geometric quotient metrics behave when stabilizers change? This direction studies relative metric and spectral estimates through singular orbit types. Reading guide · Paper, source and finite calculations.

• Relative metric estimates near reducible Yang–Mills connections Near a central Yang–Mills connection on a closed surface, the classical quadratic model approximates the actual gauge quotient metric and every small gauge eigenvalue with relative error quadratic in the deformation parameter, uniformly across changing stabilizers and arbitrarily separated orbit scales. The paper computes the leading metric correction and gives explicit quaternionic projective and hyperbolic comparisons, including a common weighted four-cycle. Preprint · current version (PDF), 28 October 2026, 14 pages · source and exact finite calculations · Zenodo:23740619

Realization of moduli structures

Which prescribed gluing data and algebraic identities can compactified moduli spaces realize, and when can such a realization come from a Fredholm problem? Reading guide · Manuscript and source.

• Inverse Moduli Theory: Realizability of Degeneration Data by Compactified Moduli Spaces An inverse problem for Floer and related theories: realizing prescribed gluing data and algebraic identities by compactified moduli spaces, and determining when those realizations arise from geometric Fredholm problems. The principal result gives an obstruction to extending prescribed compatible stabilizing bundles across the pentagon associahedron: a reflected endpoint identification creates a nonorientable boundary bundle, while doubling the datum removes the obstruction. Chain and higher associativity identities become oriented filling conditions in dimension one, with higher extensions governed by smooth bordism. The analytic realization criteria remain conditional. Research manuscript · current version (PDF), August 2026 · paper source and finite calculations · Zenodo:23643225

Exotic four-manifolds

Fundamental groups and Luttinger fillings in Lidman and Piccirillo’s construction, with applications to the search for exotic smooth structures on S²×S² and ℂP²#ℂP̄². Reading guide.

• Toward an exotic S²×S² and an exotic ℂP²#ℂP̄² For a specified Luttinger filling, the paper derives the complete fundamental-group presentation and expresses the group as a one-relator quotient of an iterated HNN extension containing the genus-two surface group. It also determines the universal central extension with cyclic kernel, constructs geometric realizations, and extends the boundary involution over the filling. The working revision adds a proposed Lagrangian-torus capacity obstruction, exact tori detecting a transfer kernel, and a conditional symplectic-homology argument; these geometric additions await independent mathematical verification. A simply connected comparison explains their limitation, and a group-algebra criterion isolates a further nonvanishing problem. The final question is whether the filling relation kills a specified fiber commutator; it remains open, and the exotic homeomorphism conclusions are conditional. Partial-results paper · 56-page working revision · arXiv:2608.17267 · latest version (PDF), October 21, 2026 · paper source and exact algebra verification

Theoretical physics

Compact Higgs transitions

Four papers on how local Higgs branches change when their singular regions belong to one compact Calabi–Yau threefold. In M-theory, vector fields of the compactification can gauge flavor symmetries shared by several local sectors, so Higgs directions that exist in separate local models can be obstructed or become available only through a joint transition. Paper 1 develops the general gauging framework, Papers 2 and 3 study an explicit compact transition through its quantum couplings and its magnetic sheaves, and Paper 4 treats the Higgs branch at finite Planck mass. Reading guide · Papers, sources, and verification code.

• Global Constraints on Higgsing Localized Five-Dimensional Sectors in Compact M-Theory Vacua Which local Higgs directions survive when their singularities belong to one compact Calabi–Yau threefold? Wrapped M2-brane charges and compact divisor pairings determine a shared Abelian gauging matrix for conifold and E₁–E₄ sectors: degrees 8 (ℙ¹×ℙ¹), 7, 6 and 5. Its branchwise kernel is the invariant-coordinate image of the rigid moment-map zero locus and, under the stated exact-germ and projective crepant-resolution hypotheses, the global first-order deformation image. Combined with the mixed smoothing theorem, this gives an exact geometric existence criterion: the kernel must contain a vector outside every local discriminant. At E₄, an integral identification with the SU(5) diagonal coordinates gives five separate nonzero conic-pencil conditions. Degrees 1–4 remain outside this mixed comparison. Selected deformation bases are smooth when the relation space projects nontrivially to each selected E₃ summand. Examples distinguish cooperative Higgsing in X₉, sectors forced to remain undeformed, and nonzero E₃ deformations forced to retain a singularity. The finite-Planck scalar metric is not determined. Paper 1 · current version (PDF), revised 7 September 2026 · source and guide · Zenodo:23643227
• Bulk Gauging and Quantum Couplings across a Compact Higgs Transition In X₉, two conifold sectors and one E₂ sector can Higgs only together, through an explicit compact transition from Hodge numbers (6,122) to (3,123). Existence also follows from the exact mixed smoothing criterion; the explicit construction identifies the smooth phase needed for the charge and quantum calculations. The paper determines the primitive surviving charge lattice and cubic couplings, and proves that the shared flavor gauging becomes non-dynamical in every gravity-decoupling limit with bounded local Kähler scales. At finite auxiliary-circle radius, four conifold states span a primitive rank-three electric lattice. Laurent mutation and continuation of all eight regular periods identify the surviving genus-zero prepotential with that of the smooth phase on the marked branch, including its integral normalization and constant term. The four-state description is not uniform along the decompactification limit, which ends where the del Pezzo surface collapses; on an adjacent face of the nef cone the same states are massless five-dimensional hypermultiplets (Paper 4). Paper 2 · current version (PDF), revised 23 September 2026 · source and guide · Zenodo:23643235
• Magnetic Sheaves and Stability Walls in a Compact Calabi–Yau Transition Magnetic objects probe the same compact transition through sheaf counts and stability constraints. Fixed-support Hilbert schemes on the rigid F₁ and ℙ² surfaces give rank-one numerical Donaldson–Thomas contributions; two isolated sheaves on a reducible moving divisor each contribute +1 and remain Gieseker stable along ample polarizations approaching the contraction. Their vanishing deformation and obstruction groups also imply local persistence, with the same isolated contributions, in smooth polarized deformations of the resolved threefold. This does not transport the sheaves through the singular transition. The continued periods of Paper 2 constrain constituent phases and light electric-particle emission near the marked transition, while the probe charges carry confined magnetic flux after Higgsing. The physical BPS spectrum at the endpoint remains undetermined. Paper 3, supporting technical note · current version (PDF), revised 6 September 2026 · source and guide · Zenodo:23643240
• The Higgs Branch of a Compact Calabi–Yau Transition at Finite Planck Mass When a compact Calabi–Yau threefold degenerates, light states of M-theory can condense along a Higgs branch. Without gravity its local structure is a hyperkähler quotient; with gravity the geometry becomes quaternionic Kähler, and its metric is unknown. The paper shows that the local structure can be determined anyway. Quaternionic Kähler geometry has no Fayet–Iliopoulos parameter, and a new local model for quaternionic Kähler quotients near fixed points shows that gravity neither removes nor changes the orbifold singularity where the branch begins; it corrects only the metric, at second order in the distance. The result covers a whole class of transitions, including the cooperative X₉ transition of Paper 2, and a census of toric Calabi–Yau hypersurfaces finds thousands of instances. The physical conclusions rest on a stated regularity hypothesis. Paper 4 · current version (PDF), September 2026 · source and guide · Zenodo:23643242

Black hole information

Four papers on the Page curve, radiation spectra, thermodynamic fluctuations, and microscopic information recovery. Paper I identifies an order-one correction to the diagonal approximation; Paper II derives finite-size critical profiles in a specified canonical random-state model; Paper III gives exact examples of what entropy and spectral agreement leave undetermined; Paper IV develops the methods in a self-contained tutorial. Start with IV for an introduction, then read I–III. Reading guide · Papers, LaTeX sources, and guides. A separate philosophical companion considers information recovery and geometric singularities.

• The Page Curve Through the Transition: An Order-One Correction to the Diagonal Approximation In the west-coast model of black-hole evaporation, the exact planar radiation entropy exceeds the level-resolved diagonal approximation by a shape-independent mixing term as the area distribution spreads. At a symmetric Page point the correction is ½ log 2; its maximum occurs later in the transition. The proof reduces the exact spectral entropy to a single integral, with finite-width calculations and a Jackiw–Teitelboim example. The result depends on radiation being unable to resolve the relevant area sectors. Paper I · current version (PDF), 9 October 2026, 45 pages · source and guide · SSRN:6372958
• The Page Curve in Higher Dimensions: Phase Transitions, Critical Phenomena, and Finite-Size Broadening At a charged AdS critical point, a quartic fluctuation law replaces the Gaussian approximation. Within a specified canonical random-state model, the entropy-transition width scales as the three-quarter power of the critical entropy in every bulk dimension at least four. The paper derives the profile, temperature window, and finite-reservoir crossover, with regular neutral AdS and BTZ comparisons. It distinguishes spinodals from phase coexistence and thermodynamic entropy from modular energy; gravitational applications require the corresponding state and entropy-difference distribution. Paper II · current version (PDF), 10 October 2026, 41 pages · source and guide · SSRN:6372978
• The Page Curve Is Not Enough: Universality, Ensemble Dependence, and Microscopic Ambiguity in Black Hole Information Recovery Exact finite-dimensional examples show what a Page curve leaves undetermined. Two ensembles can share every averaged spectral moment yet have different entropy fluctuations; an isometry and an input-erasing channel can reproduce the entire radiation trajectory for one preparation while preserving different reference correlations. The paper also derives normalized replica-covariance identities and shows how fluctuation laws with equal variance can yield different transition profiles. Full Choi-state tomography supplies a positive channel-identification result. Paper III · current version (PDF), 10 October 2026, 44 pages · source and guide · SSRN:6521478
• From Hawking to Marchenko–Pastur: A Guide to the Page Curve and Beyond A self-contained tutorial from quantum entanglement and Hawking radiation through holographic entropy, islands, replica wormholes, and the Marchenko–Pastur law. Worked examples develop normalized replicas, Gaussian and critical fluctuations, finite reservoirs, ensemble comparisons, and quantum-channel diagnostics. The guide distinguishes established results and explicit model calculations from gravitational applications needing further assumptions. It provides the background for Papers I–III, with 25 figures, 16 tables, and seven reference appendices. Paper IV · current version (PDF), 10 October 2026, 131 pages · source and guide · Zenodo:23643250
• Informational Singularities and the Completion of Gravity: Replica Wormholes, Indeterminism, and Cosmic Censorship A philosophical companion examining the distinction between information recovery and resolution of geometric singularities. It extends Azhar and Namjoo’s taxonomy of indeterminism to quantum state maps and proposes Informational Cosmic Censorship as a quantum predictability principle. The essay considers what replica-wormhole and island calculations imply for these questions, including the difficulty of interpreting ensemble results as statements about a single theory. Philosophical companion · August 2026 · SSRN:7151179

AI and the machine economy

Reading guide: these papers and the two on international relations below, with the essays that accompany them.

• Prediction Markets as the Informational Substrate of the Machine Economy The three institutional gaps blocking autonomous machine-to-machine commerce — adjudication, reputation, and liability — share one root: there is no market price for agent performance risk, deliverable quality, or failure correlation. Five prediction-market mechanisms are designed to supply those signals, with the instrument delimited as carefully as it is developed: an informational deficit can be priced, but a primitive needing an agreed vocabulary rather than a resolving fact needs coordination instead. April 2026 · SSRN:6561300
• The Institutional Plumbing for a Machine Economy: A Readiness Assessment of Building Blocks, Missing Infrastructure, and Commercial Opportunities Software agents can now pay, but paying is not economic agency: open exchange between agents with no prior relationship also needs identity, delegated authority, bonding, verification, adjudication, settlement, portable reputation, and somewhere for residual loss to land. Auditing that stack against what actually shipped through 2026 — where roughly two-thirds of settlements moved inside linked clusters and a fifth were fictitious — gives the conclusion that machine commerce expands along a verification-and-risk frontier, not a payment frontier. March 2026 · SSRN:6372838

International relations and politics

Reading guide: read with the machine-economy papers above.

• Material Deployment Power in the Age of Artificial Intelligence: How Physical Constraints Reshape International Order Prevailing accounts treat AI as an informational technology; deployment at scale instead re-centres physical constraint, since it depends on energy, transmission, cooling, and critical materials that expand slowly and unevenly. The paper defines material deployment power — the capacity to mobilize those physical systems under rivalry — and shows it shapes trade policy and alliance behaviour independently of innovation leadership. April 2026 · SSRN:6481923
• Algorithmic Plausible Deniability: The Structural Erosion of Deterrence in the Age of Artificial Intelligence Delegating decisions to machines generates persistent uncertainty over authorship and intent, weakening deterrence even between rational states that prefer stability; unlike strategic ambiguity it is not chosen, but emerges from the delegation itself. The effects are asymmetric — hostile acts become more deniable while signals of restraint become less credible, which undermines escalation control. April 2026 · SSRN:6482039