The modular field

The modular field is the program’s name for a single field, Z = X + iY, whose value at each point is a symmetric matrix of complex numbers and which describes every phase-space metric compatible with the fixed symplectic form ω. Those metrics form a space well known in mathematics, the Siegel domain, the matrix version of the upper half of the complex plane, whose points the symmetries of ω move by fractional-linear (Möbius) maps. The distinguished part of the domain’s edge, its Shilov boundary, is the Lagrangian Grassmannian: the space of all ways of splitting phase space into position and momentum. The program reads the imaginary part Y as the metric of spacetime and the real part X as a new field, the local tilt of the split between position and momentum. Reciprocity acts on Z as Z → −Z⁻¹, and the dark-energy field τ is an angle on the domain’s boundary. The mathematics is standard; the physical readings are the program’s derived claims, and one question about the field content is open.

Stands: the standard mathematics of the Siegel domain (its parametrization machine-checked at one sampled point, battery test T2) and of the Lagrangian Grassmannian; the program’s derived claims [D] the modular field Z, reciprocity as modular inversion (machine-checked at d = 1) and τ as the Maslov phase (τ’s lightness a postulate [P]) · Died: none · Open: the field content of the broken vacuum, and on the modular-field page the Re Z sector of the geometric action (OP1) · Superseded: a folded section on the three-way classification of X · Conjectures: chat-only, folded: massive X waves

Map#

The Siegel domain Sp(2d, ℝ)/U(d) comes first, as standard mathematics. It is the space of complex symmetric matrices Z = X + iY with positive-definite Y, the matrix version of the upper half-plane, and the standard H_Z parametrization turns each point into a phase-space metric compatible with ω. The page shows why the parametrization’s identities hold, how the symplectic group moves the points (the translations Z → Z + B and the inversion Z → −Z⁻¹ among them), the invariant target metric that a kinetic term would use, and the bounded form of the domain whose distinguished (Shilov) boundary is Λ(d). Battery test T2 checks the identities at one sampled point, and the round-3 audit sampled the family’s ω-compatibility with its own code.

The modular field Z = X + iY is the program’s reading of that space as a field, alive and derived [D]. It follows the argument step by step: the postulate repair frees the metric; at X = 0 the metric is diag(g, g⁻¹), so Im Z = g and the vacuum is ⟨Z⟩ = i·g; off that slice Re Z = X mixes position and momentum directions and is the momentum graviton. It records #5’s claim that one field carries the whole geometric program, what the Debt-1 notes made of X, the open first-principles route from the action S_H, and the round-3 auditor’s caution that a dynamical H and a second spin-2 field X are assumed, not derived.

Reciprocity as modular inversion Z → −Z⁻¹ shows how the founding symmetry acts on the field: the quarter-turn is one of the domain’s Möbius maps, checked symbolically at d = 1 and argued for general d. At the fixed point Z = i, reciprocity is one element of a continuous U(1), which is why the polarization is a Goldstone angle. On the vacuum slice the map is g → g⁻¹, scale-factor duality, and the circle |Z| = 1 is identified, structurally, with the constraint a·ã = 1 (see flag 23 in review/wiki-build-flags.md). The same matrix generates S-duality on charges.

The Lagrangian Grassmannian Λ(d) returns to standard mathematics: the compact space U(d)/O(d) of all position/momentum splittings, of dimension 10 at d = 4, its Maslov class π₁ = ℤ through the map det², and its place as the Shilov boundary of the Siegel domain. It is the space of vacuum polarizations once only ω is rigid (under the original P2, which also froze η, calc note #5 calls the compatible polarizations a discrete set; see flag 1 in review/wiki-build-flags.md), and its d = 1 case is the toy model’s compact Goldstone angle. It comes before τ because τ is defined on it.

τ as the Maslov phase of the boundary identifies the dark-energy field with the angle τ ≡ f·arg det² around the Maslov circle. The identification is derived [D], the topology that makes τ periodic is literature [T], and τ’s lightness among the ten moduli is a postulate [P]. The page explains why the period is topological, how calc note #5’s “arg det of the boundary variable” relates to the lab notebook’s f·arg det², why τ is not eaten when X becomes massive, and where its potential, the open questions about that potential, and its observational test are treated. The toy models support the structure’s consistency only.

Open: field content of the broken vacuum closes the section with what is not settled. #5 listed the fields left on the broken vacuum: the graviton, X, the ten moduli of Λ(d) and matter. X was later resolved as a massive spin-2 field, which supersedes #5’s three-way classification; the handoff’s chat-only remark on massive X waves is folded in there too. Whether inverse-Higgs constraints remove the nine non-Maslov moduli was never worked out, and the lab notebook’s polarization-genesis sketch, which needs those moduli, is unreconciled with #5’s expectation.

Boundaries#