This is an authored mathematical synthesis and a set of worked derivations. It develops the artist's existing constrained-recoverability theory and transformer analysis, connects them to latent inversion, and places the operational-profile hypothesis under explicit quantifiers. Neither a familiar theorem wearing a new hat nor an unrun experiment receives a novelty certificate here.
Four kinds of sentence
- Classical result / worked consequence
- Continuous images of compact sets, Arzela-Ascoli, Brouwer/no-retraction, orbit stabilizers, quotient factorization, matrix conditioning, stochastic contraction, kernel composition. The chapters state the relevant hypotheses and provide the arguments used.
- Proposed definition
- Regime-relative irreverence and its recovery profile organize admissibility, departure, target, controls, information, cost and regularity. Their naming is proposed here; the underlying mathematics is established.
- Derived quantitative statement
- The all-word mixing bound is proved from a zero-sum estimate and decaying token diameter. No priority claim is made for this combination or its constants. The disk-to-ring costs are classical consequences, and the probability-mixture constant has explicit prior art in Basso.
- Unproved research hypothesis
- Adequate learned models share semantic kernels and differ only in costs; or a compact measured profile predicts unseen compositions. These are distinct hypotheses in the operations chapter. Neither follows from compactness or from a universal expressivity result.
A notation key with no admission exam
| symbol | meaning |
|---|---|
| E:X → Z | encoder / observed representation; E-1({z}) denotes a fibre, not an inverse function |
| q or pi | quotient map where specified; quotient topology matters for continuity |
| ||X||row | maximum of Euclidean norms of token rows, unless another row norm is declared |
| Delta(X) | largest distance between token rows |
| dinfinity(f,g) | supremum of output distance over a named compact input set |
| A(X) | row-stochastic attention matrix, dependent on input; not the entire transformer |
| eta(A) | minimum common probability mass of two rows; r=1-eta controls diameter contraction |
| closure | topological closure only in the named topology; composition closure is a separate assertion |
| inf empty set | infinity; an infimum need not be achieved |
We use native HTML formula text, with superscripts and subscripts, so the arguments do not wait for a remote typesetting service. Matrices in formula blocks read by rows. ASCII inequalities such as <= mean less than or equal. The only font is the site's own c0unt typeface.
Architecture and dynamical systems
- Vaswani et al., Attention Is All You Need (2017). Scaled dot-product attention, heads, residual/normalization arrangements and positionwise feed-forward maps. The bounds in this work are derived from stated formulas; they are not empirical certifications of trained weights.
- Ba, Kiros and Hinton, Layer Normalization (2016); Zhang and Sennrich, Root Mean Square Layer Normalization (2019). Our exact formulas retain a strictly positive epsilon. That choice matters for both continuity and inversion.
- Xiong et al., On Layer Normalization in the Transformer Architecture (2020). Context for the pre/post-normalization distinction. Training-gradient results are different from the invariant-domain claims proved here.
- Gaubert and Qu, Dobrushin's ergodicity coefficient for Markov operators on cones (2015). Established oscillation/diameter contraction from stochastic overlap. We give the finite-row proof directly.
- Kim, Papamakarios and Mnih, The Lipschitz Constant of Self-Attention (2021). Their negative global result concerns unbounded inputs; it does not contradict bounded-domain estimates. Castin, Ablin and Peyre, How Smooth Is Attention? (2024) further emphasizes norms, length and masks.
- Dong, Cordonnier and Loukas, Attention Is Not All You Need (2021). Rank collapse for specified pure-attention systems is relevant context, not permission to drop residuals, value projections or MLPs from a full-model claim.
- Brunton, Brunton, Proctor and Kutz, Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control (2016). Exact finite observable closure needs an invariant observable space; finite-dimensional states alone do not provide one.
Repair, quotients and symmetry
- Brouwer's fixed-point theorem and the no-retraction theorem are classical. The repair chapter gives the explicit fixed-point reduction proving the R+rho cost and distinguishes it from the stronger faithful-repair obstruction. Allen Hatcher's Algebraic Topology gives the disk, circle, degree and fixed-point background.
- Lawrence, Portilheiro, Zhang and Kaba (2025), section 2. Curie's principle states the stabilizer obstruction: equivariant outputs cannot have less symmetry than the input permits. Law-valued symmetry and sample symmetry require separate analysis.
- Dym, Lawrence and Siegel, Equivariant Frames and the Impossibility of Continuous Canonicalization (2024). Direct context for continuity obstructions and weighted-frame alternatives.
- Arsigny, Fillard, Pennec and Ayache, Log-Euclidean metrics for fast and simple calculus on diffusion tensors (2006). The log-shape metric is a declared geometry related to this framework; it is not automatically outline distance or physical work.
- Ambrosio and Puglisi, Linear extension operators between spaces of Lipschitz maps and optimal transport (2016). Probability-valued retractions and Wasserstein regularity are established mathematical objects.
- Basso, Lipschitz extension theorems with explicit constants, section 7.1, Lemmas 7.4-7.5. Prior art for the exact conical-mixture Wasserstein calculation and its sharp construction constant. Sharpness for that construction is not optimality over every possible repair kernel.
- Bennett, Logical Reversibility of Computation (1973). A classical precedent for retaining history to support reversal. A witness in a drawing does not invent reversible computation or establish a thermodynamic claim.
- McCarthy, Recursive Functions of Symbolic Expressions (1960), sections 3e-f, and the Maude manual on reflection and META-LEVEL. Symbolic functions, explicit evaluation and reified rewrite theories are established precedents. Underprint's actual bounded interpreter has a narrower, explicitly declared editable grammar.
The elementary fibre factorization, ambiguity bound and normalization inverse are proved in the inversion chapter. A choice of representative uses the usual set-theoretic choice convention when no constructive rule is given; measurable or continuous choice requires additional conditions. A learned decoder is not promoted to an inverse merely by being called a decoder.
Expressibility is not a performance receipt
- Feng et al., How Numerical Precision Affects Arithmetical Reasoning Capabilities of LLMs (2025). Formal precision regimes and learned task outcomes are distinct. A q8 weight file does not identify all intermediate arithmetic precision.
- Li, Liu, Zhou and Ma, Chain of Thought Empowers Transformers to Solve Inherently Serial Problems (2024). Expressivity under specified computational assumptions motivates separately recording dependency depth and generated tokens.
- Niu, Sterling, Grodin and Harper, A cost-aware logical framework (2021). Context for separating behavior from quantitative cost; this work does not claim a mechanization in their framework.
- Yun et al., Are Transformers Universal Approximators of Sequence-to-Sequence Functions? (2020); Perez, Barcelo and Marinkovic, Attention Is Turing-Complete (2021). Their architecture, precision, approximation and resource hypotheses matter. Neither result proves that a fixed trained browser model will solve the operations tasks, possess stable inverses or generalize universally.
What the diagrams calculate
The unminified mathematical kernel and the diagram code are local source files. They call no network service, read no private archive and execute no learned model. Native controls permit keyboard and touch use; reset returns each example to its declared initial inputs. Nothing depends on animation.
| instrument | calculation | proof boundary |
|---|---|---|
| two-token attention | row-softmax(xxT), matrix multiplication and a bounded-input inverse criterion | the exact triangle and unattainable diagonal are proved algebraically; plotted samples do not prove a closed region |
| normalization | exact iterate formulas evaluated at finite plot points | the sequence tending to zero proves non-equicontinuity; a large sampled slope alone would not |
| mixing | two-token pure identity-value mixing, diameters, product and exponential bounds | floating-point agreement is not exact finite-time consensus; the contraction proof supplies the bound |
| encoder fibres | E(x)=x2, two candidate sources and sign side information | the midpoint minimax radius and a fibre-constrained representative have different errors |
In a repository checkout, run node --test tools/mathematics.test.mjs, node tools/mathematics-audit.mjs and the Playwright runner tools/mathematics.browser.mjs against a local server serving docs/. The independent Python audit tools/mathematics-independent.py reproduces the numerical bounds using separate formulas; its dependency versions and measured results live in research/mathematics/AUDIT.md. These are repository-relative development instructions, not links to private materials.
Browser arithmetic is finite precision. At sufficiently extreme logits, a displayed probability may round to zero or one although exact finite-logit softmax stays strictly between them. A computed zero disagreement may similarly mean underflow or rounding. Equations and proofs describe real arithmetic unless explicitly qualified.