[Verse 1] Place cell holding a promise, not just a spot on the floor, It's predicting where the feet are headed, counting steps before, Stachenfeld cracked the cipher, two-thousand-seventeen, Successor matrix whispers what future ground will mean, Row is where you're standing, column's where you'll roam, Discounted future footsteps, weighted paths back home, Place field's not a snapshot, it's a forecast drawn in gray, Predictive map of hallways you ain't wandered yet today. [Chorus] Eigen-vectors, eigen-vectors, spinning out the grid, Spectral bones inside the matrix, hexagonal amid, Successor map for basis, Fourier for space, Grid cells are the harmonics tiling up the place. [Verse 2] Take the transition matrix, let it settle, let it decay, Symmetrize, decompose it, watch the eigenmodes display, First few vectors low-frequency, blanket the whole terrain, Higher ones get choppier, hexagonal refrain, That's your grid cell spacing, scale by scale unfold, Smallest module tightest lattice, biggest spacing bold, Same trick that Fourier pulls on a signal in the wire, Graph Laplacian eigenbasis — structure, not desire. [Chorus] Eigen-vectors, eigen-vectors, spinning out the grid, Spectral bones inside the matrix, hexagonal amid, Successor map for basis, Fourier for space, Grid cells are the harmonics tiling up the place. [Verse 3] Matrix meets its spectrum, that's the data struct at play, Eigenvalues order harmonics like a scale you'd play, Sine and cosine on a line, but here the line's a maze, Graph Fourier transform turning corridors to phase, Every corner, every branch point folded into sum, Environment's own skeleton becomes the frequency drum. [Bridge] But hallways lie, two doorways looking the same, One observation, hidden context, aliasing's the game, George in twenty twenty-one split nodes into clones, Latent copies of a symbol, each one wearing different bones, Same corridor, three ghosts inside it, disambiguate the walk, Clone-structured graph remembers which specific door talked. [Verse 4] Zoom out further, cognitive map's a graph on graphs of graphs, Nodes are places, edges are the paths, Build your grid-world lattice, transition matrix wide, Solve for vectors, lay them next to real cell data side by side, Ridges match the rhythm, hexagons align, You just proved the theorem written in your own design. [Chorus] Eigen-vectors, eigen-vectors, spinning out the grid, Spectral bones inside the matrix, hexagonal amid, Successor map for basis, Fourier for space, Grid cells are the harmonics tiling up the place. [Outro] Diagonalize the hallway, let the spectrum speak, Predictive maps and clone states — same terrain, technique, Basis over structure, that's the lesson that remains, Grids are just the eigen-song your cortex sustains.
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