Paper I: figure and atlas descriptions
These descriptions explain the information carried by Paper I’s ten chapter figures and twenty-two visual-atlas plates. They identify the arrows, axes, matrix entries and comparisons without relying on colour. They are reading aids for the retained manuscript, with its hypotheses and proof scope.
Read Paper I · Accessibility and reading options
Figure 1: from seeds to a realization
Six boxes form a directed sequence. Across the top row, read Choose seeds (generators in the object), then Address them (indices for the seeds), then Apply operators (words in the fixed algebra). An arrow descends to Record relations (kernel and quotient data). The lower row runs from right to left: Form the hull (span or closure), then Reconstruct (a realization map). A separate box below gives an optional packet view: project to expose blocks when projectors are available. This is a guide to the presentation data, rather than a claim that every object comes with all these choices automatically.
Figure 2: transfer blocks of a sampled operator
This three-by-three heatmap gives the Frobenius norm of each block of a sampled operator. Rows are output packets and columns are input packets. Select input packet j, apply the operator, then select output packet i. The packet dimensions are 5, 6 and 7. Numbers are rounded to one decimal place; darker cells indicate larger norms.
| Output packet | Input 1 | Input 2 | Input 3 |
|---|---|---|---|
| Output 1 | 5.0 | 5.0 | 8.1 |
| Output 2 | 5.7 | 6.2 | 6.0 |
| Output 3 | 5.7 | 6.3 | 7.3 |
Diagonal blocks act within a packet. Off-diagonal blocks transfer between different packets. A norm gives the size of a block; it does not retain the block’s signs or its complete linear action.
Figure 3: one seed can span the space or stall
The horizontal axis counts Krylov vectors, from 1 through 8. The vertical axis is the dimension of their span in an eight-dimensional space. The vectors are the fixed seed v followed by Tv, T squared v, and subsequent powers. The cyclic companion operator has ranks 1, 2, 3, 4, 5, 6, 7, 8. The operator with two equal Jordan blocks has ranks 1, 2, 3, 4, 4, 4, 4, 4. The curves coincide through the fourth vector; one then rises to the full dimension while the other remains at four. The repeated blocks cannot be separated by this single polynomial orbit.
Figure 4: semisimple multiplicity and seed rank
The horizontal axis is multiplicity m, from 1 to 24. The vertical axis is the simple-module dimension d, from 1 to 8. Every cell displays, by shading, the least integer greater than or equal to m divided by d. The legend ranges from one to twenty-four seeds. Along a row, the required count increases by one whenever the multiplicity exceeds another multiple of d. Highlighted examples are d = 8, m = 8: one seed; d = 4, m = 12: three seeds; d = 2, m = 16: eight seeds; and d = 1, m = 24: twenty-four seeds. This is the displayed split simple block formula under the manuscript’s semisimple hypotheses.
Figure 5: one seed for path adjacency
A six-vertex path illustrates a seed at its first endpoint. Beneath it, a plot has number of vertices on the horizontal axis and adjacency seed rank on the vertical axis. Every plotted path size from 2 through 20 has rank 1, so all points lie on a horizontal line. Repeated application of the adjacency operator to an endpoint reaches a new furthest vertex at each step. The finite plot illustrates the manuscript’s general endpoint argument; the plotted cases alone are not its proof.
Figure 6: two seeds for cycle adjacency
A six-vertex cycle accompanies a plot of cycle sizes 3 through 20. The horizontal axis is number of vertices; the vertical axis is adjacency seed rank. Every plotted point has rank 2. The annotation identifies a largest eigenspace of dimension two, which a single adjacency-polynomial orbit cannot span. Read this with the complex-field hypotheses in the accompanying cycle result. The figure concerns the undirected adjacency operator, not a directed cyclic shift.
Figure 7: periodic shift through six residue packets
The left panel is a six-by-six support matrix for a periodic shift on twenty-four coordinates. Both axes are numbered 0 through 5; columns are input packets and rows are output packets. The only ones are at output/input pairs (1,0), (2,1), (3,2), (4,3), (5,4), (0,5). All other entries are zero. A one means a nonzero block, rather than a numerical value for the whole block. The right panel draws the same routing as the directed cycle 0 → 1 → 2 → 3 → 4 → 5 → 0. This finite periodic model is distinct from the infinite bilateral shift, whose phase-boundary block retains an address shift; see atlas plate 19 below.
Figure 8: multiplication by y exchanges two packets
The coordinate algebra is described by the relation y squared = x cubed − x. Its two coefficient lanes are the even packet k[x] and the odd packet yk[x]. The transfer matrix has first row 0, x cubed − x and second row 1, 0; columns are inputs and rows are outputs. A directed arrow from even to odd has weight one. The return arrow from odd to even has weight x cubed − x. There are no within-packet arrows for one application of multiplication by y. Two applications multiply each packet by x cubed − x, matching the relation.
Figure 9: projector products after a change of basis
This numerical heatmap records Frobenius norms of products of transported projectors Q_i Q_j. There are three four-dimensional packets in a twelve-dimensional space. Diagonal entries are 12.16, 16.02 and 17.31. The off-diagonal entries, by output row, are:
- Row 1: column 2 is 1.5 × 10⁻¹⁴; column 3 is 2.2 × 10⁻¹⁴.
- Row 2: column 1 is 1.2 × 10⁻¹⁴; column 3 is 2.3 × 10⁻¹⁴.
- Row 3: column 1 is 1.5 × 10⁻¹⁴; column 2 is 1.7 × 10⁻¹⁴.
The displayed exact identities are Q_i = U P_i U inverse, Q_i squared = Q_i, and Q_i Q_j = zero for different indices. A nonorthogonal change of coordinates can give large projector norms. The tiny off-diagonal numerical values are roundoff in this illustration, rather than nonzero terms in the exact identity.
Figure 10: two proof families meet in the worked models
Two left-to-right chains converge on one Worked realizations box. The generator chain is Fixed operator grammar → Hulls and presentations → Generator counts and bounds. The packet chain is Finite projector system → Analysis and synthesis → Blocks and transport. Each final box points to the shared worked-realization box. Three formulas below identify coordinate transport of projectors, an intertwining relation, and a block obtained by selecting input and output packets around an operator. This is a structural overview of the arguments, not an exhaustive dependency graph of Lean declarations.
Visual atlas: twenty-two plates
The original atlas occupies PDF pages 83–104, labelled 75–96 in the printed manuscript. The descriptions below follow its plate order. Page links open the original PDF at the corresponding page. The atlas’s surrounding explanations and hypotheses remain part of the reading.
Plate 1: what can one seed actually do?
Four rows compare minimum seed counts for the same six-dimensional rational space. Scalar multiples of the identity require 6 seeds. Polynomials in two equal nilpotent chains require 2. Adding the two chain projectors to that grammar lowers the count to 1. The full six-by-six matrix algebra also requires 1. Filled squares count seeds; they do not measure runtime. The point of the comparison is that the admissible algebra changes while the ambient space stays fixed.
Plate 2: an atlas of the objects
A three-column table pairs each symbol with its role and a concrete example. The space X supplies possible target vectors; S supplies seeds; the fixed algebra supplies allowed operations; and the hull is their invariant linear span. Projectors extract packet spaces, while a transfer block specifies an input packet, an operator and an output packet. A word quotient records seed relations as a module. An invertible U changes coordinates; an intertwiner need not be invertible. A topology specifies which limits are allowed. The reading order for P_beta T P_alpha is right to left: extract alpha, apply T, then extract beta.
Plate 3: two chains can stay linked
Two horizontal three-vertex chains run from e_0 to e_1 to e_2 to zero, and from f_0 to f_1 to f_2 to zero. Vertical dotted pairings join corresponding positions. They mean equal coefficients, not additional permitted arrows. Starting from e_0 + f_0 gives e_1 + f_1, then e_2 + f_2, then zero. Thus the orbit spans three directions. Allowing the chain projectors would separate the two heads and produce all six directions, but that is a larger operator grammar.
Plate 4: a quotient proves two seeds are necessary
A two-by-six matrix has its only nonzero entries in the first row’s e_0 column and the second row’s f_0 column. Both entries are one. It keeps the two chain heads and forgets the advanced positions. The induced action on this two-dimensional quotient is scalar, so one seed’s image remains on a line. The quotient therefore supplies a lower bound of two seeds. The two separate chain heads attain that bound in the original space.
Plate 5: relations identify words, not just outputs
Three two-by-two matrices show C = E_12, B = E_21 and their product CB = E_11. C has a one only in its upper-right cell, B only in its lower-left cell, and CB only in its upper-left cell. C kills the first basis vector; CB sends that vector to itself. Multiplying a seed-annihilating relation on the right can therefore destroy it. The presentation quotient is a quotient of left modules; a seed annihilator is not automatically a two-sided ideal.
Plate 6: a reached type can still hide a direction
Five slots are grouped as two directions in U and three in V. In the first row, a seed supported at U fills both U slots and only the first two V slots: the generated dimension is 4. Adding the missing V direction fills all five slots. A third row shows that one mixed seed also fills all five when vertex projectors are admissible. Reaching a vertex type, filling its space, and minimizing the unrestricted number of seeds are different questions.
Plate 7: exact does not mean orthogonal
In a coordinate plane, one packet lies on the horizontal axis and the other on the diagonal line. The vector x = (3,2) is the sum of Px = (1,0) and Qx = (2,2); dashed sides complete their parallelogram. The packet directions are not perpendicular. The squared norm of x is 13, whereas the sum of the two component squared norms is 9. The picture illustrates exact algebraic reconstruction without an energy-preserving orthogonal splitting.
Plate 8: a transfer block has an input and an output
Two three-by-three matrices show an operator and its square. The rows of T are (1,2,0), (0,1,3), (4,0,1). The rows of T squared are (1,4,6), (12,1,6), (8,8,1). Columns identify inputs and rows identify outputs. The square sums products over intermediate packets. For example, entry (1,2) is 1 times 2 plus 2 times 1, giving 4. Entry (1,3) is 2 times 3, giving 6. These are actual signed coefficients, not a matrix of norms or a support graph.
Plate 9: two visible routes can cancel exactly
A diamond graph starts at e_1, branches to e_2 and e_3, then rejoins at e_4. The first two arrows have weight +1. The e_2-to-e_4 arrow has weight +1, while the e_3-to-e_4 arrow has weight −1. The two length-two contributions cancel. The neighbouring Krylov matrix has columns e_1, e_2 + e_3 and zero, so its rank is two. Every graph vertex is reachable, but the single-operator hull does not span the four-dimensional space. Inserting a packet projector between steps would change the allowed grammar.
Plate 10: which zero patterns survive multiplication?
Three matrices compare an allowed support pattern, an offending product and the transitive closure. The first allows the diagonal, 1-to-2 and 2-to-3, but omits 1-to-3. Multiplying E_32 by E_21 produces E_31, outside that pattern. Adding the missing position gives the lower-triangular support pattern. Here a one means “allowed position,” not a fixed coefficient. For the universal supported algebra on nonzero packet spaces, diagonal positions supply the identity and transitive closure supplies all compositions.
Plate 11: move the packets with the coordinates
The top row shows T = diag(1,2), the shear with rows (1,1) and (0,1), and the conjugated operator with rows (1,1) and (0,2). The transformed matrix gains an off-diagonal entry. Beneath it, the transported projectors have rows (1,−1), (0,0) and (0,1), (0,1). Relative to these oblique packets the cross-packet blocks are still zero. The packet maps must move with the coordinates; a change in ordinary matrix sparsity alone does not describe a change in packet routing.
Plate 12: shared control and independent control differ
Both panels start from the mixed seed (1,1). Shared scalar action reaches only the diagonal line, represented by points (a,a). Independent coordinate action reaches the whole plane, represented by (a,b); the shading is only a finite window of that plane. The corresponding seed counts are two for shared scalars and one for independent scalars. In the independent direct-sum formula, component seed lists can occupy the same positions, so the count is their maximum rather than their sum.
Plate 13: how much row space can one seed supply?
Three two-by-five seed matrices cover columns 1 and 2, then 3 and 4, then 5. Their cumulative orbit dimensions under left multiplication by two-by-two matrices are 4, 8 and 10. One seed has at most two independent row-space directions, while the multiplicity space has dimension five. Three seeds are therefore needed in this model. The displayed target matrix has rows (1,2,3,4,5) and (6,7,8,9,10); its successive column pairs determine coefficients that synthesize it from the three fixed seeds.
Plate 14: distinct labels let one polynomial separate chains
Two six-by-six Krylov matrices compare equal and distinct primary labels. Two equal nilpotent three-chains give rank 3 for the mixed seed. Replacing the second operator block by two times the identity plus the nilpotent block gives determinant 512, hence full rank six. This changes the operator, not just its coordinates. The plate supplies a polynomial that is zero modulo t cubed and one modulo (t−2) cubed; it separates the two chain heads using operations already in the polynomial grammar.
Plate 15: walk counts make an endpoint cyclic
A five-vertex path marks vertex 1 as the seed. Under it, the five Krylov columns count walks of lengths 0 through 4. The matrix rows are (1,0,1,0,2), (0,1,0,2,0), (0,0,1,0,3), (0,0,0,1,0) and (0,0,0,0,1). Every diagonal entry is one, and every entry below the diagonal is zero. A vertex cannot be reached sooner than its distance from the endpoint, and there is one shortest path. This is the unit-triangular structure used by the general endpoint proof.
Plate 16: closing a path changes the seed obstruction
A six-vertex cycle highlights adjacent vertices 0 and 1 as a generating pair. A bar chart gives eigenvalue multiplicities: −2 has 1, −1 has 2, 1 has 2, and 2 has 1. These are exact dimensions, not estimated spectral peaks. A single vertex seed spans four dimensions; the adjacent pair spans all six. The plate’s two-row scalar quotient certifies why one arbitrary seed cannot suffice, and the adjacency recurrence constructs the remaining vertices from the pair.
Plate 17: a relation becomes a packet product rule
Two infinite coefficient lanes are shown through degree four. The top lane is 1, x, x squared, x cubed, x to the fourth, …. The lower lane is y, xy, x squared y, x cubed y, x to the fourth y, …. Higher powers of y are rewritten using y squared = x cubed − x; they are not discarded. Each element has the form a(x) + yb(x). Multiplication by y swaps the lanes, with the return transfer weighted by x cubed − x. The diagram is about normal forms and multiplication, not the genus or smoothness of a drawn algebraic curve.
Plate 18: density is not finite generation of each vector
A stem plot shows the sequence x[n] = 2 to the power −|n|. It peaks at one at zero and halves at each step away from zero in either direction. The shaded window contains indices −3 through 3. Outside every finite window, entries remain nonzero; the picture is not periodic wraparound. Truncating to indices −N through N gives squared error two-thirds times 4 to the power −N. The displayed errors for N = 1, 3 and 5 are 1/6, 1/96 and 1/1536. Shifted impulses give finite-support approximants whose norm limit is the infinite sequence.
Plate 19: residue packets retain an address carry
Three rows divide a window of integer addresses into residue classes modulo three. Row 0 contains −6, −3, 0, 3 and 6; row 1 contains −5, −2, 1 and 4; row 2 contains −4, −1, 2 and 5. Each address belongs to exactly one row. Under one shift, packet 0 moves to packet 1, packet 1 to packet 2, and packet 2 to packet 0 with a shift of the coarse address. In formulas, the new packet 0 at k uses the old packet 2 at k−1. Replacing that boundary shift by a scalar would lose information about the infinite sequence, even if the phase graph looked the same.
Plate 20: stop when the subspace stops growing
The horizontal axis counts hull-expansion iterations; the vertical axis is subspace dimension. Both curves start at dimension one for the same mixed seed. The dashed curve allows only the nilpotent operator and plateaus at dimension 3. The solid curve also allows the chain projectors and reaches dimension 6. Repeated plateau values indicate the stop. Equality of consecutive subspaces certifies invariance only after images under the whole declared generating family have been included. The dimensions are exact, rather than results of a numerical rank tolerance.
Plate 21: tensor products require every cross term
A two-by-two coefficient rectangle expands the tensor product of 2e_1 + 3e_2 with 5f_1 + 7f_2. Rows correspond to e_1 and e_2; columns correspond to f_1 and f_2. The first row is 10, 14, and the second is 15, 21. All four basis tensors occur. Keeping only the diagonal entries would omit both cross terms, 14 and 15. The full rectangular family gives the stated product upper bound on algebraic tensor seed rank. It is not a universal equality or a theorem about completed topological tensor products.
Plate 22: what the pictures certify
A three-column table connects seven topics to their mathematical content and the hypotheses that must remain attached. Presentations require a fixed grammar. Finite packets require idempotence, annihilation and completeness; orthogonality is additional. Semisimple counts require the specified decomposition data. Primary and graph counts retain their field and multiplicity conventions. Path and cycle proofs are distinct from finite tests. Coordinate normal forms do not by themselves establish smooth projective geometry. Completion requires its topology and boundedness hypotheses. The accompanying text distinguishes written proofs, formal contracts and executable records.
Sources and coverage
Descriptions were checked against all ten rendered chapter figures, the twenty-two rendered atlas plates, their captions and the retained atlas source. They do not replace the manuscript’s arguments or add a mathematical review. The existing PDF remains the reference for the original visual layout.