Index
plain-language theorem explainer
Index is the type alias for natural numbers serving as abstract indices on tensors in the ILG action. Workers on relativistic field equations or tensor contractions in this framework cite it to keep index notation symbolic and dimension-independent. The declaration is a one-line type abbreviation with no computation or lemmas.
Claim. Tensor indices are elements of the natural numbers, denoted $I,J,K, L, M, N, P, Q, R, S, T, U, V, W, X, Y, Z, a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 1
background
The ILG.Action module re-exports geometry and field types to support construction of the action functional. Index supplies the symbolic convention that tensor indices are drawn from the natural numbers, keeping notation abstract rather than fixing a concrete dimension or range. Upstream, the Kronecker delta defined in Cost.Ndim.Connections.delta acts on Fin n, while the PeriodicTable.Index structure carries rail and block data for chemistry; the present Index remains a bare Nat alias to permit flexible index algebra in relativity.
proof idea
The declaration is a direct type abbreviation equating Index to Nat.
why it matters
This convention is referenced by ILGParams, kron, lowerIndex, and the antisymmetrize operation in Geometry.Tensor, as well as by kauzmann_lt_one in GlassTransition and r_prediction in PrimordialSpectrum. It supplies the index language needed for the ILG action that sits between the Recognition forcing chain and concrete predictions for the primordial spectrum and chemical ratios. The choice keeps tensor expressions dimension-independent, consistent with the framework's derivation of D=3 from the eight-tick octave.
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