Tensorgami archive

Preserved full study · Exposition · Galois theory · Fourier duality

Fourier, Symmetry, and Reconstruction

Derived coordinate rules turn root motions, tensor chains, kernel composition and local ramification into explicit calculations. The broader Fourier frameworks are retained as background.

Reading version: Revised November 3, 2026

Preserved full study · Exposition · Graph theory · Homological algebra · Motives

What a Boundary Map Can Prove

Calculate with incidence matrices, cancel edge subsets, reduce graph transport by a tree, match nodal branches, and eliminate integration variables. The Hodge and motivic theorems are a separate reference layer.

Reading version: Revised November 3, 2026

Preserved full study · Exposition · Differentials · Derived geometry · Homotopy theory

What a Derivative Retains

Increment arithmetic, elimination, cyclic coordinates and path spaces expose the operations behind derivative and higher-interaction calculations. General spectral identifications are retained as references.

Reading version: Revised November 3, 2026

Preserved full study · Exposition · Quotients · Jordan–Hölder · Reconstruction

Quotients, Refinement, and Reconstruction

Exchange and quotient comparison lead to concrete calculi for transporter graphs, commuting-operator couplings, and attachments reconstructed from relations. Higher universal frameworks are background.

Reading version: Revised November 3, 2026

Preserved full study · Teaching companion · C₆ · Quotients · Worked examples

Six Elements, One Comparison

Keep every output of a comparison relation, remove its two ambiguities, and match the factors of two decompositions of C₆. Seven exercises and worked answers test the construction.

Reading version: Revised November 3, 2026

Preserved full study · Exposition · Algebra · Galois theory

A Galois-Theoretic and Frobenius-Descent Viewpoint on a Polynomial Composition Law

Read a polynomial composition condition through generic fibres and descent. The positive-characteristic argument keeps the role of Frobenius explicit.

Preserved full study · Exposition · Algebraic geometry

Normality, Coherent Extension, Tensor Operations, and Affineness

Choose algebras, modules, and local sections that make the constructions in Hartshorne II.5.14–18 explicit, then verify the gluing back to geometry.

Research record · Preprint · 2026 · Representation theory · Tensor-triangular geometry

Finite generation of permutation twisted cohomology

Henry Shin

Representation spheres give multigraded section rings for permutation modules. This paper proves a finite-module theorem over arbitrary commutative Noetherian bases and uses the doubled regular representation to construct an explicit Noetherian model for the tensor-triangular geometry of permutation complexes.

Research record · Published · 2020

Curvature growth of some 4-dimensional gradient Ricci soliton singularity models

Bennett Chow, Michael Freedman, Henry Shin, Yongjia Zhang

Advances in Mathematics 372 (2020), 107303

Curvature estimates for four-dimensional gradient Ricci soliton singularity models, using geometric analysis and topological arguments.

Research record · Preprint · 2026 · Additive combinatorics

Iterated-sumset spectra: The complete exponent law and its rank geometry

Henry Shin

Determines the growth exponent of the number of possible h-fold sumset sizes for fixed-size integer sets, together with sharp bounds for integer models preserving finite additive relations. Also proves the formula conjectured in OEIS A227589 for minimal labelings of identical four-sided dice that maximize the number of distinct sums.

Research record · Preprint · 2026 · Geometric analysis

Open full-metric formation and local marked first-order asymptotic moduli of FIK blowdown singularities

Henry Shin

Formation of FIK singularities under Ricci flow and the local parameters governing their first-order asymptotic behavior.

Research record · Preprint · 2026 · Combinatorial group theory

A global girth obstruction for Garg–Mineyev taiko product structures

Henry Shin

A global obstruction to the Garg–Mineyev triple-girth route to zero-divisor and unit counterexamples in group rings over F₂, proved through incidence geometry.

Research record · Preprint · 2026 · Analysis · Partial differential equations

From rough local mass to logarithmic spectral ladders for damped waves

Henry Shin

Local damping mass, resolvent bounds, and decay, including spectral asymptotics near logarithmic damping wells.

Research record · Preprint · 2026 · Algebraic combinatorics

Exchange identities and symmetric slices of the valley Delta conjecture

Henry Shin

Exchange identities and symmetry for the area-zero and area-one slices of the valley Delta generating function.

Research record · Preprint · 2026 · Extremal combinatorics

One extra edge forces Berge pancyclicity

Henry Shin

Berge cycles of every length from a Hamiltonian Berge cycle and one additional edge, in the stated large-order uniformity regime.

Research record · Preprint · 2026 · Algebra · Arithmetic geometry

Hilbert-90 quotient maps, torsion defects, and symmetric monodromy

Henry Shin

Rational maps from Hilbert-90 descent: cancellation, torsion, and characteristic-zero monodromy.

Research record · Preprint · 2026 · Number theory · Algebraic geometry

Coset-refined trace statistics, nodal characters, and affine branches in cubic norm tori

Henry Shin

Trace distributions on cosets in cubic norm-one tori, including singular fibers and local branch geometry.

Research record · Preprint · 2026 · Arithmetic geometry

Logarithmic source curves in polynomial fiber products

Henry Shin

Geometry at infinity of polynomial fiber products, rational lifts, and their arithmetic counting behavior.

Exposition · Fixed Tensorgami edition 2026-10-20.1

Fourier, Symmetry, and Reconstruction

Preserved package from this dated edition. Its text and sources retain their original publication context.

Supplied edition label: Revised and expanded, October 2026

Revised October 11, 2026: the introduction and conclusion connect the study to its companion through their shared motivic inquiry. The web edition, PDF, and TeX source agree.

Exposition · Fixed Tensorgami edition 2026-10-20.1

What a Boundary Map Can Prove

Preserved package from this dated edition. Its text and sources retain their original publication context.

Supplied edition label: October 2026 · revised

Revised October 11, 2026: clarified the retained boundary data, metric probability law, Jacobian extension, and period construction; corrected two subsection headings. The web edition, PDF, and TeX source agree.

Exposition · Fixed Tensorgami edition 2026-10-20.1

What a Derivative Retains

Preserved package from this dated edition. Its text and sources retain their original publication context.

Supplied edition label: October 2026

First published October 17, 2026. Revised October 17, 2026 with clarifications in sections 5, 6, 12 and 13. The matching PDF edition of 39 pages and downloadable Markdown reflect the revised reading edition.

Exposition · Fixed Tensorgami edition 2026-10-20.1

A Weighting Argument for Pointwise Convergence to Convergence in Measure

Preserved package from this dated edition. Its text and sources retain their original publication context.

Revised September 22, 2026: clarified positive indexing, the convention for empty suprema, and the domain of the measurable envelope. The PDF and TeX source agree.

Exposition · Fixed Tensorgami edition 2026-10-20.1

IBM Ponder This July 2010

Preserved package from this dated edition. Its text and sources retain their original publication context.

Exposition · Fixed Tensorgami edition 2026-10-20.1

Divisibility of Separated Polynomials

Preserved package from this dated edition. Its text and sources retain their original publication context.

Revised September 22, 2026: added a full fiber-constancy criterion over arbitrary fields, using Hasse derivatives to retain inseparable multiplicities. The original composition theorem and its free-module proof are unchanged. The PDF and TeX source agree.

Exposition · Fixed Tensorgami edition 2026-10-20.1

A Galois-Theoretic and Frobenius-Descent Viewpoint on a Polynomial Composition Law

Preserved package from this dated edition. Its text and sources retain their original publication context.

Revised September 22, 2026: clarified the nontriviality conditions in the closing reducibility discussion and the remainder argument when the outer polynomial is zero. The composition theorem and its hypotheses are unchanged. The PDF and TeX source agree.

Exposition · Fixed Tensorgami edition 2026-10-20.1

Normality, Coherent Extension, Tensor Operations, and Affineness

Preserved package from this dated edition. Its text and sources retain their original publication context.

Revised September 22, 2026: clarified the nonempty-scheme and rank-zero conventions, independence of lifts in the associated graded, and the local arguments underlying section-ring finiteness, coherent extension, relative Spec, and bundle sections. The PDF and TeX source agree.

Exposition · Fixed Tensorgami edition 2026-10-20.1

Extracting the First Prime Greater than m

Preserved package from this dated edition. Its text and sources retain their original publication context.

Revised October 7, 2026: clarified the introduction while preserving the problem’s IRC provenance. The theorem, proof, and appendix are unchanged.

Exposition · Fixed Tensorgami edition 2026-10-20.1

A Note on Disjoint Decompositions of [0,1) into Closed Sets

Preserved package from this dated edition. Its text and sources retain their original publication context.

Revised September 22, 2026: explicitly established that the closed remainder is nonempty before applying the Baire category theorem.

Exposition · Fixed Tensorgami edition 2026-10-20.1

Lebesgue's Criterion for Riemann Integrability

Preserved package from this dated edition. Its text and sources retain their original publication context.

Revised September 22, 2026: included the zero-length interval, corrected two strict oscillation bounds to non-strict bounds, and supplied a direct Darboux proof of the composition corollary that uses neither direction of Lebesgue’s criterion. Revised October 7, 2026: rebuilt the editable source, clarified overlapping covers and closed-cell estimates, and developed the Thomae and Cantor-set examples. The PDF and TeX source agree.

Exposition · Fixed Tensorgami edition 2026-10-20.1

Products Preserve Riemann Integrability

Preserved package from this dated edition. Its text and sources retain their original publication context.

Revised September 22, 2026: the product proof chooses positive bounds for the factors, making its strict estimate valid also when both functions vanish. Nonempty intervals are explicit, and the opening proof outline has been reflowed. The PDF and TeX source agree. Revised October 7, 2026: condensed the explanatory prose and made the hypotheses of the concluding Lipschitz principle explicit.

Problem set · Fixed Tensorgami edition 2026-10-20.1

Selected Problems in Discrete Mathematics and Analysis

Preserved package from this dated edition. Its text and sources retain their original publication context.

Revised September 22, 2026: Problem 2 specifies a disjoint partition with distinct moduli greater than one; Problem 5 has the corrected series value; and Problem 6 distinguishes three recorded outcomes from an exact-onset timing variant. The web edition, PDF, and TeX source agree.

Essay · Fixed Tensorgami edition 2026-10-20.1

Mathematical Life in the Age of AI

Preserved package from this dated edition. Its text and sources retain their original publication context.

Essay · Fixed Tensorgami edition 2026-10-20.1

When Accountability Stops Being Reciprocal

Preserved package from this dated edition. Its text and sources retain their original publication context.

Essay · Fixed Tensorgami edition 2026-10-20.1

The Performance of Depth

Preserved package from this dated edition. Its text and sources retain their original publication context.

Essay · Fixed Tensorgami edition 2026-10-20.1

What a New Proof Makes Possible

Preserved package from this dated edition. Its text and sources retain their original publication context.

Essay · Fixed Tensorgami edition 2026-10-20.1

When a Description Becomes a Verdict

Preserved package from this dated edition. Its text and sources retain their original publication context.

Essay · Fixed Tensorgami edition 2026-10-20.1

The Authority of a Vision

Preserved package from this dated edition. Its text and sources retain their original publication context.

Essay · Fixed Tensorgami edition 2026-10-20.1 · Version 75

Mathematical Understanding in the Age of AI

Preserved package from this dated edition. Its text and sources retain their original publication context.

Essay · Fixed Tensorgami edition 2026-10-20.1 · Version 75

The Culture of Mathematical Judgment

Preserved package from this dated edition. Its text and sources retain their original publication context.

Exposition · Fixed Tensorgami edition 2026-10-20.1 · October 2026 (retained predecessor)

A Tree Is a Discrete Poincaré Lemma

Preserved package from this dated edition. Its text and sources retain their original publication context.

Supplied edition label: October 2026

Complete edition · 29 October 2026 UTC

Tensorgami edition 2026-10-29.2

Complete website and public sources from this edition, including version 2 of Finite generation of permutation twisted cohomology.

Complete edition · 20 October 2026 · UTC

Tensorgami edition 2026-10-20.1

Complete public collection, available sources, and retained earlier editions. The individual packages in this catalogue come from this release.