From visual specifications to renderers and exports

A figure of a finite encoding should preserve the distinction that makes its mathematics useful. Two zero matrix entries can mean different things: one is forced by the parameter order, while the other is a coefficient that could have been nonzero. A missing resolution degree is different again. Without a termination certificate, its absence does not establish a vanishing term.

TamerOp prepares those distinctions before asking a rendering backend to draw. The input object and selected question produce a VisualizationSpec: literal coefficient text, semantic graphical layers, axes, annotations and retained mathematical metadata. CairoMakie and WGLMakie consume the same specification. Their job is layout and drawing; they do not construct a resolution, choose a basis or infer a mathematical certificate from a picture.

Separate the mathematical request from its presentation

The public path first checks that the object's recipe accepts the requested selection and options. Each object family declares its kinds and meaningful keywords. Building a specification may perform explicit mathematical work; rendering an existing specification reuses that prepared answer.

stored object + mathematical selection
                  |
           validate the request
                  |
       prepare coefficients and claims
                  |
           VisualizationSpec
             /           \
       Cairo figure     WGL scene
             \           /
         selected backend export

The specification has either layers or child panels. Its layers distinguish points, paths, support polygons, interval data, text and literal matrices. Semantic roles such as source, target and selected are resolved by the style at render time. This permits color or grayscale changes without changing a matrix, its labels or the membership of a support. Matrix coefficients stay textual; finite-field coefficients are not converted to magnitudes on a real color scale.

Panel metadata records selected indices, field, category, degree convention and computation scope. Matrix panels retain the exact selected matrix alongside the displayed block. Their target-row/source-column labels identify coordinates; a display limit is not a mathematical truncation. Empty matrices keep both original dimensions, so a map into the zero space remains distinguishable from one out of it.

Resolve storage conventions before drawing an arrow

The resolution guide's diamond module has projective multiplicities one, two and one in degrees zero, one and two. Those numbers describe the resolution terms; they do not decompose the resolved module. A support sheet therefore labels a selected principal summand of a term, not a purported persistent class.

Projective differentials run from degree k to k-1, whereas injective differentials run from k-1 to k. The selected degree consequently marks a source column in one view and a target row in the other. A one-step upset presentation stores its matrix transposed relative to that display convention. The preparation step normalizes these cases; the renderer sees only target rows and source columns. Injective coefficients are recovered at the recorded target socles, where they determine the maps between principal downsets.

For source label u and target label v, the principal-summand coefficient is allowed only when v ≤ u, on either side. A forbidden nonzero is rejected. The displayed forbidden zeros receive a dagger and a separate semantic role; permitted zeros retain ordinary coefficient text. At a selected finite vertex, preparation restricts the matrix to active summands, preserving their global IDs. An upset presentation produces a cokernel, a downset copresentation a kernel, and a fringe map an image. The visible construction and selected dimension follow that distinction.

The grade-plane view accepts a supplied planar order embedding. It compares order using the supplied coordinates before converting the drawing to floating point, and rejects distinct grades that collapse to one displayed point. It never uses Hasse-layout positions as grades. These remain finite-category multiplicities: RIVET's Betti convention concerns an ambient minimal free resolution and supplies mathematical context, not an automatic identification or an algorithm dependency.

A certificate is data, not a visual impression

The default table counts stored generators and declares its minimality and terminal status unchecked. Optional verification freshly checks the whole stored prefix. Besides the augmented equations, it checks vertexwise exactness: once consecutive composites vanish, adjacent ranks must sum to the dimension of the intermediate stalk. The projective augmentation must be surjective; the injective coaugmentation must be injective. Stored term maps must agree with the canonical principal-summand coordinates, and stored differential maps must agree with their extracted coefficient matrices.

Minimality uses each map's actual kernel or image. For a projective term at vertex v, let D be its map to the preceding object and let D_old retain columns born strictly below v. If b summands are born at v, the cover is minimal exactly when

\[\operatorname{rank}(D)-\operatorname{rank}(D_{\rm old})=b.\]

This says its kernel lies in the radical: no new generator can be removed using older generators. Dually, an injective hull is essential when its image contains the socle of the injective term. If E_soc consists of the standard basis columns for summands ending at v, the check is

\[\operatorname{rank}[D\ E_{\rm soc}]=\operatorname{rank}(D).\]

These checks work at the last retained term too. Absence of a next differential alone is insufficient to certify minimality. A nonzero terminal kernel or cokernel marks a truncated prefix; a zero one certifies termination. The view never pads uncomputed degrees with zero rows. Exact coefficients use field-aware rank and equality; numerical fields retain their declared tolerance semantics.

A supplied basis change undergoes a separate check. Its source and target matrices must be invertible and obey the grading, as must their inverses. With B = inverse(T)*A*S, the equation A*S = T*B certifies the selected incidence map in new coordinates. This does not find a minimal presentation or transform a whole complex. Adjacent differentials and the augmentation still use their original coordinates unless an independently constructed object supplies them.

Retained work, drawing costs and export

A table allocates a degree-by-vertex count array without elimination. Incidence selects one differential and optionally a stalk block. Explicit verification traverses all retained degrees and finite vertices and performs ranks; displaying only a few rows does not reduce that verification scope. Dense active stalks and basis-change inversions can dominate a view of a large sparse resolution. Matrices in specification metadata are copied so that later display edits cannot mutate the resolution. The graph and separate support panels add their own layout and scene costs; support-panel limits state how much is shown.

Neither resolution construction nor retained-result caching is hidden in these recipes. A caller can retain a specification and render it again with another style or backend. Rebuilding with verification recomputes the checks, avoiding stale claims after mutation. A different mathematical selection requires a new specification. Live inspection sessions have their own selection/lifecycle contract; loading WGLMakie does not add callbacks to static resolution views.

Backends register render and save functions through optional package extensions. An explicit backend must be active. Automatic inline rendering prefers WGL in a notebook when available, then Cairo; saving prefers an available static backend. Explicit filename exports use their declared format, while the simple stem-based export returns the actual path, backend and format. These choices change presentation and file production, not which mathematical object was prepared. Shared matrix/layout code keeps literal coefficients and semantic annotations consistent between Cairo and WGL.

The source boundaries are specification types, request/specification validation, resolution preparation, render/export dispatch, and shared Makie rendering. The executable guide develops the mathematical example; the reference owns keyword contracts.

See the bibliography.