Born geometry

Born geometry is the study of three structures that can sit together on phase space: the symplectic form ω, the pairing of each position with its momentum that underlies the quantum commutator [x, p] = iℏ; a neutral metric η, a measure of length with as many positive as negative directions, whose zero-length (null) directions mark out which coordinates count as position and which as momentum; and a positive generalized metric H, which measures lengths in phase space. When the three fit together, the operators built from them in pairs multiply by a fixed rule, the split-quaternion algebra: one of them squares to −1 and the other two to +1. The section’s central result, the compatibility theorem, says that holding both ω and η fixed leaves only metrics in which momentum-space geometry is the exact inverse of position-space geometry. That result left the program with no content of its own and forced it to let η vary. The compatibility theorem and its corollaries are proven only for metrics in which every length comes out positive, the Euclidean case; extending them to spacetime, where time enters with the opposite sign to space, is open.

Stands: the structures ω, η, H, double field theory, the Kalb–Ramond B-field, the Born connection, the compatibility theorem and its corollaries — literature or standard mathematics, with the program’s derived action of ρ on the triple [D] and the theorem’s block proof machine-checked here [V] · Died: nothing yet · Open: Lorentzian dressing of the Euclidean skeleton (OP3); connection uniqueness under P2′, folded into the Born connection; a mechanism that selects one section, folded into double field theory and open in the literature too · Conjectures: the slaved layer is algebraic (chat-only), folded into the corollaries

Map#

The structures ω, η, H and the split-quaternion algebra comes first because every later page uses its definitions: the standard forms of the three structures, the operators I = ω⁻¹H, J = η⁻¹H and K = η⁻¹ω with I² = −1, J² = K² = +1 and IJ = K, Freidel, Leigh and Minic’s name “Born geometry”, and the para-Hermitian literature. It also holds the program’s derived result [D] that ρ preserves ω and H and reverses η, and the unreconciled naming of the real structure (J in #11’s postulate P2′, K in the toy-model note).

Double field theory and the generalized metric is string theory’s doubled geometry, built on η alone: the O(d, d) generalized metric with its B-field, and the section condition and the three routes to it, none of which picks out one section. It is established physics and serves as the contrast case. Doubling with η gives the metric an antisymmetric partner, doubling with ω gives it a second graviton, and holding ω rigid as well removes the B-field. A mechanism that selects one section is open, in the literature as in the program.

The Kalb–Ramond B-field is that antisymmetric partner, the 2-form gauge field of closed string theory [T]: its gauge symmetry δB = dλ, a 3-form field strength, strings as its charges, its mixing with the metric under T-duality, and its place in double field theory’s generalized metric. The program uses it three ways: corollary 2 of the compatibility theorem freezes it out once ω is rigid as well as η; calc note #6 sets it against the symmetric momentum graviton X, which is a second graviton while the B-field is not; and its healthy propagation beside the metric and the dilaton is calc note #5’s precedent for doubled kinetic terms free of ghosts.

The Born connection is the unique connection compatible with all three structures under the conditions of a theorem of Freidel, Rudolph and Svoboda [T]. Its curvature R_B[H] gave the lab notebook’s action S_H, and its place on the first-order tier is part of the case for the frame formulation of the tuning test. Whether the connection stays unique when only ω and H are fundamental is an open item on the same page.

The compatibility theorem is the section’s central result: with ω and η both rigid, the positive metrics compatible with both are exactly H = diag(g, g⁻¹). The statement is Freidel–Leigh–Minic’s [T]. The block proof, battery test T1 [V] and round 3’s symbolic, Gröbner-basis and numerical re-derivation are the program’s record, including the finding that the proof’s positivity step is not needed.

Corollaries: unimodularity, no B-field, slaved momentum metric follows the theorem with its three consequences: det H = 1, which the program reads as ℏ-rigidity; no B-field; and the slaved momentum metric g̃ = g⁻¹. det H = 1 is checked at one point of the modular family (battery test T2), so the family-wide claim is [D]. The third corollary contradicted the program as it stood, reducing it to “Branch R”, and forced the change from P2 to P2′, after which the block form holds only on the vacuum slice X = 0.

Open: Lorentzian dressing of the Euclidean skeleton comes last because it qualifies the theorem and the results built on it. The theorem, the Siegel parametrization and the chain that makes X massive are proven for a positive-definite metric; in Lorentzian signature the stabilizer becomes U(1, d−1) and the target metric indefinite. It is #11’s open problem 3, nothing has been computed, and #11 counts it among the items on which the word “theory” waits.

Boundaries#