The Clickable Grid
Seven matrices all claim to describe one 7×7 grid of unit resistors. Each one is telling the truth — but a matrix on a graph is not a spreadsheet, it is a field. Click an edge or a vertex below and every row and column renders where it actually lives, in ten clickable views.
One arena, seven matrices, ten views
The arena is the free (ungrounded) 7×7 grid graph: $n = 49$ vertices in row-major order ($v = 7r + c$), $m = 84$ unit-conductance edges (42 horizontal + 42 vertical), degree histogram 4 corners of degree 2, 20 boundary vertices of degree 3, 25 interior of degree 4 — total $168 = 2m$. The incidence matrix $B \in \mathbb{R}^{84\times49}$ uses Vishnoi's sign convention from mathematica/electrical_networks.wls: $+1$ at the tail, $-1$ at the head, and the orientation is fixed once, lexicographically — every edge points rightward or downward (tail index < head index). Free means no vertex is grounded, so $L = B^\top B$ is singular with exactly the constants in its kernel, every inverse below is a Moore–Penrose pseudoinverse, and every potential field lives in the mean-zero gauge. The grid has exactly 19,872,369,301,840,986,112 spanning trees — OEIS A007341(7), verified by an exact integer determinant, and §4 will spend that number.
Everything below is one widget with ten modes — eight distinct matrices between them: $B$ and $X$ each get a row mode and a column mode, and the walk mode builds the eighth, the grounded Green function $G_g$, out of partial sums rather than baking it whole. Edge modes ask you to click an edge, vertex modes a vertex; the gold interpretation strip says what you are looking at, one sentence per mode. Seven matrices are baked into this page whole — $B$, $X$, $L$, $L_{\mathrm{edge}}$, $L^+$, $L_{\mathrm{edge}}^+$, $\Pi$ — plus the walk-mode partial-sum fields, all from results/grid_viz.json (35 machine checks, all PASS, 0.2 s), and the badge re-verifies eight of the page's identities from the raw baked entries every time the page loads. The prose sections after the widget are keyed to its modes: §1 rows and columns of $B$ and $X$, §2 the two Grams, §3 the two pseudoinverses, §4 the projection $\Pi$, §5 the random-walk sum.
Rows measure, rows respond — the 1-D story goes 2-D
The dual-basis explainer built this pair on a 1-D chain: row $e$ of $B$ is a measurement — read $u$ against it and you get the potential drop across one edge — and row $e$ of $X := \mathrm{pinv}(B^\top) = BL^+$ is the matching response, the field you get by driving that same edge with a unit dipole. In 1-D the responses were centered steps. Here they become genuine 2-D dipole fields: click an edge in X row mode and the potential bulges positive at the tail, negative at the head, and wraps around the grid. The two definitions of $X$ agree to 1.8×10⁻¹⁵ (the pinv computed independently of $L^+$), each row solves its dipole problem $L^+ b_e$ to 1.4×10⁻¹⁵, and — the free-graph signature — every response row is balanced: row sums zero to 3.0×10⁻¹⁵, because with no ground the field can only be known up to a constant, and the mean-zero gauge spends that freedom.
Columns tell the transposed story. Column $v$ of $B$ (B col) is the incident-edge stencil: $\pm1$ arrows on the $\deg(v)$ edges touching $v$, nothing else — the divergence stencil at one vertex. Column $v$ of $X$ (X col) is the internal current pattern of a unit injection at $v$: the current every edge carries when one unit enters at $v$ and the gauge extracts $\tfrac1{49}$ everywhere. That is Kirchhoff on the free graph, $B^\top X = I - \tfrac1{49}\mathbf{1}\mathbf{1}^\top$ to 1.1×10⁻¹⁵ — and it is Maxwell's unit-load picture from the dual-basis page's §3, now visible edge by edge.
The Gram bookkeeping carries over whole: $X^\top X = L^+$ (covariance = Gram of the dense responses) to 3.4×10⁻¹⁵, and biorthogonality holds modulo the gauge, $X^\top B = I - \tfrac1{49}\mathbf{1}\mathbf{1}^\top$ to 1.1×10⁻¹⁵ — exactly the free-path $P_8$ footnote of the dual-basis page, promoted to the full grid: freedom costs one direction, and the centering projector is what "identity" means once you've paid it.
Does $BB^\top$ have a name? Usually: the edge Laplacian
Multiply $B$ by itself in the two possible orders and you get two Grams over two different index sets. $B^\top B = L$ lives on vertices and everyone knows it: click L row and row $v$ is $\deg(v)$ at $v$, $-1$ at each neighbor, $0$ elsewhere — verified entrywise, exactly 0 error, integers against integers. It is the conditional (stencil) reading: two measurement columns overlap iff the vertices are adjacent.
$BB^\top$ lives on edges, and it is most often called the edge Laplacian — sometimes just "the down part of the first combinatorial Hodge Laplacian $L_1$" (the up part, the curl term $\partial_2\partial_2^\top$, needs 2-cells we didn't include). Click BBᵀ row: the diagonal is $2$ (unit conductances, two endpoints), and the off-diagonal entry for two edges sharing a vertex $v$ is the product of their signs there, $s_e(v)\,s_f(v)$: $+1$ when the edges converge or diverge at $v$ (tail–tail or head–head), $-1$ when they form a directed path through it. On this orientation the grid has 72 pairs at $+1$ and 142 at $-1$, and $72 + 142 = 214 = \sum_v \binom{\deg v}{2}$ — every adjacent pair accounted for, structurally exact. Strip the signs and the name becomes legible another way: $|B||B|^\top = 2I + A_{\mathrm{line\ graph}}$, exactly — the edge Laplacian is $2I$ plus a signed line-graph adjacency.
And here is why the name is deserved rather than decorative: $B^\top B$ and $BB^\top$ share their nonzero spectrum — always, for any matrix — so the edge Laplacian sings the same 48 notes as $L$, with multiplicity, max gap 6.2×10⁻¹⁵. What differs is the silence: $L$ has one zero (the constants), $BB^\top$ has $84 - 48 = 36 = m - n + 1$ zeros — the cycle space of the grid, one dimension per independent square. The shared 48 even have a closed form, because the grid is the Cartesian product $P_7\,\square\,P_7$: $\lambda_{ij} = 4\sin^2\!\tfrac{i\pi}{14} + 4\sin^2\!\tfrac{j\pi}{14}$, verified to 4.4×10⁻¹⁵, with $\lambda_2 = 0.198062$ and $\lambda_{\max} = 7.60388$ (the 1-D factors of that closed form are the eigenmode ledger's modes).
Two Green functions, and a beautiful identity
Invert the vertex Gram (pseudo-ly) and you get the free grid's Green function. L⁺ col renders column $v$ of $L^+$: the potential field of $+1$ injected at $v$ while the gauge extracts $\tfrac1{49}$ everywhere — $LL^+ = I - \tfrac1{49}\mathbf{1}\mathbf{1}^\top$ to 2.6×10⁻¹⁵, symmetric with $L^+\mathbf{1} = 0$ to 1.0×10⁻¹³. This is the grounding explainer's pinv row grown into a 2-D field, and the 2-D cousin of the green tents: a bump at the source, sagging into a negative far field because a free graph must balance its books.
Invert the edge Gram and something lovely happens: $$L_{\mathrm{edge}}^+ \;=\; X X^\top,$$ to 6.8×10⁻¹⁵ — the pseudoinverse of the edge Laplacian is the Gram of the responses. Entry $(e,f)$ is $\langle$dipole field of $e$, dipole field of $f\rangle$: how strongly a unit dipole driven across $e$ is felt across $f$, in the resistance metric. Click (BBᵀ)⁺ row and you see it as an edge field: dense, sign-striped by the orientation, decaying with distance — dipole-to-dipole transfer. It mirrors §1's $X^\top X = L^+$ perfectly: one Gram ordering inverts the vertex Laplacian, the other inverts the edge Laplacian. And the two stories are already entangled: $L_{\mathrm{edge}}L_{\mathrm{edge}}^+ = \Pi$ to 1.5×10⁻¹⁵ — inverting the edge Laplacian hands you exactly the projection of §4. Modes BBᵀ row, (BBᵀ)⁺ row and Π row are one story told three times.
$\Pi$ knows the spanning trees and ignores the cycles
The cross-Gram of measurements and responses is Vishnoi's projection, $\Pi = BL^+B^\top = XB^\top$ (1.1×10⁻¹⁵): symmetric, idempotent to 1.1×10⁻¹⁵, spectrum exactly $\{0\times 36,\ 1\times 48\}$ to 7.2×10⁻¹⁵. It projects edge space onto the cut space (currents that come from potentials) along the cycle space — $48 + 36 = 84$, the whole edge space accounted for. In Π row mode, row $e$ is the current a unit dipole across $e$ drives through every other edge.
Its diagonal is a small miracle of overloading: $\Pi_{ee} = b_e^\top L^+ b_e$ is the effective resistance of edge $e$, equals the leverage score of row $e$ of $B$ (1.7×10⁻¹⁵), and — Vishnoi's Thm 4.5 — equals the probability that a uniformly random spanning tree contains $e$. The page checks that last claim the honest way: exact integer Matrix–Tree deletion ratios, $P[e \in T] = (N(G) - N(G - e))/N(G)$ against float $R_{\mathrm{eff}}$, all 84 edges, 1.4×10⁻¹⁵, on the exact tree count 19,872,369,301,840,986,112. The corner edges are the most indispensable ($R_{\mathrm{eff}} = 0.697977 = \frac{633635}{907816}$ exactly), central edges the most redundant ($0.511745$), and the 84 marginals sum to exactly 48 in Fractions — every spanning tree has $n - 1 = 48$ edges, which is also Foster's theorem, $\sum_e R_{\mathrm{eff}}(e) = \operatorname{tr}\Pi = 48$ (7.1×10⁻¹⁵).
And what $\Pi$ kills is as telling as what it keeps. Press feed it a cycle: the widget overlays the face cycle $c$ around the square $24 \to 25 \to 32 \to 31 \to 24$ — divergence-free, $B^\top c = 0$ — and recomputes $\Pi c$ from the baked matrix live: $\max|\Pi c| =$ 5.6×10⁻¹⁷, $(I - \Pi)c = c$. A cycle current is invisible to $\Pi$: it moves charge around a loop, changes no potential anywhere, and no dipole measurement can see it. All 36 face cycles pass the same test and together span $\ker\Pi$ (1.1×10⁻¹⁶) — they are the 36 zeros of §2's edge Laplacian.
Ground a corner, release a walker, watch the inverse assemble
The last mode grounds a vertex — the grounding move, delete row and column $g$ — and inverts what is left: $G_g = (L_{\setminus g})^{-1}$. The Neumann / path expansion of report 11 §2 then says $$G_g \;=\; \sum_{k\ge0}\,(D^{-1}W)^k\,D^{-1}\Big|_{\text{off }g},$$ convergent because $\rho(D^{-1}W|_{\text{off }g}) = 0.994702 < 1$; summed to $K = 6597$ it matches the exact inverse to relative 1.2×10⁻¹⁴, and the fitted decay rate of the partial-sum error is $\rho$, 0.994702 to the printed digit (an even-$K$ fit — the grid is bipartite, so the error only steps down every other term), giving $\tau_{\mathrm{relax}} = -1/\ln\rho = 188.3$ steps. In walk Σ mode you watch this happen: click a source, drag the dashed ring to move the ground, and slide $K$ through $0, 1, 2, 4, \dots, 2048$ as the partial-sum field fills in the grounded Green column — a spike of $\tfrac1{\deg s}$ at first, spreading outward, sharpening into the exact potential (residual and live rate in the readout). The widget recomputes partial sums for any source and ground on the spot, and at load it verifies all 13 baked fields for the default pair $s = v_{24}$ (center), $g = v_0$ (corner), plus the exact column via the four-point formula $G_g[i,j] = (e_i - e_g)^\top L^+ (e_j - e_g)$ from the baked $L^+$.
The sum has a face: it is a walker. Precisely — verified in exact Fractions, no Monte Carlo — $G_g[i,j]$ is the expected number of visits to $j$ by the simple random walk started at $i$, before absorption at $g$ (the start counts as a visit when $i = j$), divided by $\deg j$: the absorbing-chain recurrence closes exactly on all 48×48 pairs, and the identity $G_g[i,j] = \mathbb{E}[\text{visits}]/d_j$ holds to 2.2×10⁻¹⁵ against the float inverse. At the default pair: the walker from the center visits the center $\frac{1389607}{226954}$ times in expectation, and $G_0[24,24] = \frac{1389607}{226954}\cdot\tfrac14 = \frac{1389607}{907816}$. Reversibility is visible in the bookkeeping — $d_i N[i,j] = d_j N[j,i]$ exactly, which is why $G_g$ is symmetric — and the row sum of the fundamental matrix prices the whole journey: the expected hitting time from center to corner is exactly $\frac{3083931}{16211} = 190.236938$ steps.
Where the suite has already touched this grid
As dual bases — one space, two bases: this page's $X$ rows are that page's centered steps gone 2-D, and the free-graph biorthogonality $X^\top B = I - \frac{1}{49}\mathbf{1}\mathbf{1}^\top$ is its $P_8$ gauge footnote, now the main text.
As grounding — where you put the zero: §5's ground $g$ is that page's row/column deletion; drag the ring and you are choosing where the zero lives, and the physics (the walker) does not care.
As impulse responses — the green tent: columns of $L^+$ are 2-D tents in the mean-zero gauge; the $X$ rows are their differences, dipole fields, same identity as in 1-D.
As walk expansions — report 11 §2: the series §5 truncates at the slider's $K$ is the same path expansion, and the fitted rate = $\rho$ story repeats verbatim, bipartite caveat included.
As stiffness — report 09 §4: $L = B^\top B$ is the Gram-of-measurements statement that started the whole precision reading; the edge Laplacian is what that statement looks like from the other side of the product.