Appendices and references

Appendix A. More on collisions

Narrowing wedges: When the middle ball is very light, the configuration wedge narrows. A billiard ball entering a wedge of angle $\beta$ bounces roughly $180^\circ / \beta$ times before escaping. As $\beta \to 0$, the walls become parallel ($g_1 = x - a \ge 0$ and $g_2 = a - x \ge 0 \implies g_1 + g_2 = 0$), forming an implied equality where pairwise impulse solvers chatter indefinitely.

Figure A1. Narrowing wedge. As angle $\beta$ decreases, bounces multiply; at $0^\circ$, the walls touch.

Three resolutions of simultaneous impact:

• Penalty: Soft contact springs $F = k \delta^p$. Specifies compliant contact dynamics; the outcome depends on the force law and damping.
• Constraints & Impulses: Solves a linear complementarity problem. When multiple contacts are active, some pairwise sequential solvers depend on contact ordering (e.g. Gauss–Seidel bubble-sorting velocities).
• Blowup: Replaces the corner with the space of incoming gap ratios, retaining the limiting outcomes for the stated pairwise model. Probability shares require a specified prior on gap ratios.

Figure A2. Comparing the three resolutions across mass ratios.

Appendix B. More on tensors

B.1 Projective duality and parameter monomialization

Background on tensor varieties and discriminants: Landsberg^[16] and Gelfand–Kapranov–Zelevinsky.^[17]

For general $2 \times 2 \times 2$ tensors, the boundary $\Delta = 0$ is the projective dual of the rank-one variety. At a smooth point of the boundary, its tangent hyperplane identifies a contact point on the rank-one variety. Biduality describes this correspondence; it does not by itself prove that the Nash blowup resolves every singular stratum of the boundary in one step.

On the parameter side, fitting the zero tensor with a single rank-one term $K = |a|^2 |b|^2 |c|^2$ has zero set equal to the union of the three codimension-two factor subspaces ($a = 0$, $b = 0$, or $c = 0$). Here $a,b,c\in\mathbb R^2$. Polar coordinates $a = u_a \hat{a}, b = u_b \hat{b}, c = u_c \hat{c}$ yield the monomial loss $K = u_a^2 u_b^2 u_c^2$ and Jacobian $dw = u_a u_b u_c du d\theta$. Each radial variable gives candidate $(1+1)/2=1$. Because the three minimizing divisors meet, the result is $\lambda=1$, $m=3$, rather than the sum of the three candidates. Assume compact factor domains with a prior positive near the all-zero factors.

B.2 Hyperdeterminant slices

Figure B1. 2D slices through 8D tensor space colored by rank (blue = rank 2, orange = rank 3, black = boundary $\Delta = 0$). Click any pixel to inspect its entries.

Appendix C. Thermodynamic integration and WBIC

Let $\varphi$ be a normalized prior and $K\ge0$ a centered loss. Write $Z_c(\beta)=\int e^{-n\beta K(w)}\varphi(w)\,dw$. Since $Z_c(0)=1$, differentiation gives the exact identity:

For a regular quadratic loss, the scaled quantity $\beta\mathbb E_\beta[nK]$ approaches $d/2$ in the concentrated regime. In a singular toy loss it approaches $\lambda$, with finite-concentration and possibly logarithmic corrections. The unscaled expectation behaves roughly as $\lambda/\beta$; it does not plateau at $\lambda$.

For statistical evidence, replace $K$ with average negative log likelihood $L_n$ and retain the baseline term $nL_n(w^*)$. Watanabe’s WBIC uses $\mathbb E_{\beta^*}[nL_n]$ at $\beta^*=1/\log n$ to approximate the leading free-energy asymptotics under its stated assumptions. It is a finite-sample approximation, not an exact quadrature identity. The main chapter’s temperature experiment displays the centered toy quantities separately.

Appendix D. Further learning-theory experiments

The main text follows one route from tensor collisions to the geometry of learning. These supplementary experiments explore energy-volume tradeoffs, two-neuron ReLU crossings, and two-well posterior competitions.

D.1 Energy versus volume: two wells tradeoff

Figure D1. Two wells: a sharp regular bowl of depth 0 ($\lambda = 1$) and a degenerate ring of depth $\delta$ ($\lambda = 1/2$). At small $n$, the ring wins on volume; at large $n$, the deeper bowl wins on energy.

D.2 Two-neuron ReLU crossing singularity

The target is $\operatorname{relu}(x)$ and the fitted function is $(1-a_2)\operatorname{relu}(x)+a_2\operatorname{relu}(x-b_2)$. With uniform inputs on $[-1,1]$, its mean squared error is $K=a_2^2D(b_2)$. For $|b_2|<1$, integrating the two linear pieces gives $D(b)=b^2(1/2-b/3)$ for $b\ge0$ and $D(b)=b^2(1/2-b/6)$ for $b\le0$. Thus the loss is piecewise polynomial and locally comparable to $a_2^2b_2^2$. Its two axes of perfect fits cross: either the second neuron disappears ($a_2=0$), or its kink merges with the first ($b_2=0$). Direct integration gives $\lambda=1/2$ and multiplicity $m=2$.

Figure D2. Left: loss landscape of a two-neuron ReLU crossing. Right: posterior average $n\langle K \rangle$ converging to $\lambda = 1/2$ at the slow rate $1/\log n$.

D.3 Two perfect fits: quadratic vs. quartic basin

Two candidate parameter settings both achieve zero training loss ($K = 0$). Near the blue point, the loss rises steeply as a quadratic bowl ($K \propto r^2$). Near the orange point, the loss stays flat, growing quartically ($K \propto r^4$).

Figure D3. Drag the concentration slider $n$ to the right. Nine hundred live posterior samples redistribute between the two basins. Notice the orange curve climb toward overwhelming dominance.

D.4 Priors and component emptying

Collision and emptying are different ways for an overfitted mixture to represent a smaller model. Rousseau and Mengersen’s result concerns posterior behavior in specified mixture models under identifiability and prior assumptions: the Dirichlet weight exponent can favor emptying or merging. It does not give a universal tensor RLCT formula merely from a vector dimension.

The explicit four-variable normal-crossing toy in Part IV has a computed $n^{-3/2}\log n$ evidence scale with a uniform prior. Changing that prior near a divisor can change the power or logarithm. State the model, domain, and prior before applying such an exponent elsewhere.

References

Bracketed numbers beside claims lead here. Each entry gives its source, publication status, and links back to the passages that use it. The original paper links in the text remain directly usable.

Resolution and Nash blowups

1. Heisuke Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero. Annals of Mathematics 79 (1964).

   The characteristic-zero resolution theorem; linked to Part I of the original paper.

   Used in Part I.

2. John Nash. Arc structure of singularities. Duke Mathematical Journal 81 (1995).

   Used in Part II.

3. János Kollár. Which are the simplest algebraic varieties? 2001.

   An introductory perspective on the geometric goal.

   Further reading.

4. Edward Bierstone, Dima Grigoriev, Pierre Milman, and Jarosław Włodarczyk. Effective Hironaka resolution and its Complexity (with appendix on applications in positive characteristic). 2012 preprint.

   See §3 for the marked-ideal algorithm used in Part I.

   Used in Part I.

5. Ana Bravo, Santiago Encinas, and Orlando Villamayor. A Simplified Proof of Desingularization and Applications. 2002 preprint; Revista Matemática Iberoamericana 21 (2005).

   Used in Part I.

6. Ana Bravo and Orlando Villamayor. A Strengthening of Resolution of Singularities in Characteristic Zero. Proceedings of the London Mathematical Society 86 (2003), 327–357.

   Additional control of total-transform factorization.

   Used in Part I.

7. Edward Bierstone and Pierre D. Milman. Functoriality in resolution of singularities. 2007 preprint.

   Used in Part I.

8. Jarosław Włodarczyk. Simple Hironaka resolution in characteristic zero. 2004 preprint.

   Homogenized ideals and canonical choices.

   Used in Part I.

9. Dan Abramovich, Michael Temkin, and Jarosław Włodarczyk. Functorial embedded resolution via weighted blowings up. 2019 preprint; Algebra & Number Theory 18 (2024), 1557–1587.

   Stack-theoretic weighted blowups; a different procedure from the displayed toric resolution.

   Used in Part I, Part II.

10. Michael Temkin. New methods for resolution of singularities. IAS lecture, December 8, 2025.

    A lecture, rather than a research-paper proof.

    Used in Part I.

11. Federico Castillo, Daniel Duarte, Maximiliano Leyton-Álvarez, and Alvaro Liendo. Non-normalized Nash blowup fails to resolve singularities in dimension three. 2025 preprint.

    Ordinary Nash iteration: a singular open chart returns after two steps. Does not settle normalized threefold iteration.

    Used in Part II.

12. Federico Castillo, Daniel Duarte, Maximiliano Leyton-Álvarez, and Alvaro Liendo. Nash blowup fails to resolve singularities in dimensions four and higher. Annals of Mathematics 203 (2026), 677–694.

    Failure for both ordinary and normalized iteration in the stated dimensions.

    Used in Part II.

13. David A. Cox, John B. Little, and Henry K. Schenck. Toric Varieties. American Mathematical Society (2011).

    The regular-fan smoothness and properness criteria used in the explicit resolution.

    Used in Part II.

14. Resolution. Formalizing resolution of singularities. October 2026 project report.

    The project’s own account; distinguish this report from the original theorem and from our numerical demos.

    Used in Overview, Part I, Part IV.

Tensors, collisions, and geometric fields

15. Vin de Silva and Lek-Heng Lim. Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM Journal on Matrix Analysis and Applications 30 (2008).

    Nonclosed low-rank sets and unattained approximations.

    Further reading.

16. J. M. Landsberg. Tensors: Geometry and Applications. American Mathematical Society (2012).

    Used in Appendix.

17. I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky. Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser (1994).

    Used in Appendix.

18. Sean McNamara and W. R. Young. Inelastic collapse and clumping in a one-dimensional granular medium. Physics of Fluids A 4 (1992).

    Further reading.

19. Etienne Vouga, Breannan Smith, Danny M. Kaufman, Rasmus Tamstorf, and Eitan Grinspun. All’s Well That Ends Well: Guaranteed Resolution of Simultaneous Rigid Body Impact. ACM Transactions on Graphics 36 (2017).

    A terminating contact algorithm, not an algebraic resolution theorem.

    Used in Part III.

20. Gregory Galperin. Playing pool with π. Regular and Chaotic Dynamics 8 (2003).

    Further reading.

21. A. Petrov, M. S. Nabizadeh, A. Dodik, and J. Solomon. Tangent blow-ups for processing non-manifold geometry. 2026 preprint.

    Tangent lifts for geometry processing; numerical data require estimated tangent information.

    Used in Part I.

22. Ryan Viertel and Braxton Osting. An approach to quad meshing based on harmonic cross-valued maps and the Ginzburg–Landau theory. 2017 preprint.

    Further reading.

23. A. Macq and colleagues. Ginzburg–Landau energy and placement of singularities in generated cross fields. 2020 preprint.

    Further reading.

24. Samuel Burer and Renato D. C. Monteiro. A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Mathematical Programming 95 (2003).

    Further reading.

25. David Palmer, Albert Chern, and Justin Solomon. Lifting directional fields to minimal sections. ACM Transactions on Graphics (2024).

    Further reading.

Singular learning theory

26. Sumio Watanabe. Algebraic Geometry and Statistical Learning Theory. Cambridge University Press (2009).

    The foundational treatment of resolution, learning coefficients, and Bayesian asymptotics.

    Used in Part III, Part IV.

27. Sumio Watanabe. Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory. Journal of Machine Learning Research 11 (2010), 3571–3594.

    See the distinctions among Bayesian, Gibbs, training, and generalization errors.

    Used in Part IV.

28. Sumio Watanabe. A Widely Applicable Bayesian Information Criterion. Journal of Machine Learning Research 14 (2013).

    Marginal-likelihood asymptotics and the WBIC approximation.

    Used in Part IV.

29. M. Aoyagi and S. Watanabe. Stochastic complexities of reduced rank regression in Bayesian estimation. Neural Networks 18 (2005).

    The population formula used for the factorized linear-network example.

    Used in Part IV.

30. Judith Rousseau and Kerrie Mengersen. Asymptotic behaviour of the posterior distribution in overfitted mixture models. Journal of the Royal Statistical Society, Series B 73 (2011).

    Prior-dependent emptying versus merging in the specified mixture setting.

    Further reading.

31. Edmund Lau, Zach Furman, George Wang, Daniel Murfet, and Susan Wei. The Local Learning Coefficient: A Singularity-Aware Complexity Measure. AISTATS, PMLR 258 (2025), 244–252.

    Definition and practical estimation of the LLC; empirical estimates have additional limitations.

    Used in Part IV.

32. J. Hoogland, G. Wang, M. Farrugia-Roberts, L. Carroll, S. Wei, and D. Murfet. Loss landscape degeneracy and stagewise development in transformers. 2024 preprint; TMLR (2025).

    An empirical application of local coefficient measurements.

    Used in Part IV.

33. D. Murfet, S. Wei, M. Gong, H. Li, J. Gell-Redman, and T. Quella. Deep learning is singular, and that’s good. 2020 preprint.

    Further reading.

Further reading: tensor decompositions

34. Stephan Rabanser, Oleksandr Shchur, and Stephan Günnemann. Introduction to Tensor Decompositions and their Applications in Machine Learning. 2017 preprint.

    See §§3.1.3–3.1.5 for outer products and rank-one terms, and §4.1 for CP decomposition. The collision model here is a restricted worked example.

    Used in Part III.

Definitions and further foundations

35. The Stacks Project Authors. Blowing up, Section 31.33 (Tag 01OF). The Stacks Project.

    Definition 31.33.1 defines the exceptional divisor as the inverse image of the center; Lemma 31.33.4 gives its Cartier-divisor property and the isomorphism away from the center.

    Used in Part II.