P1: the phase-space arena

P1 is the program’s first axiom: physics is formulated on phase space, the space whose points are a position together with a momentum, rather than on spacetime alone. The axiom gives phase space two structures. The symplectic form ω is the antisymmetric pairing of position with momentum that quantum mechanics fixes through Planck’s constant; the generalized metric H is a positive measure of length that treats position and momentum directions together. Much of the program is built from these two: the quarter-turn that exchanges position and momentum is generated by the combination I = ω⁻¹H, and the space of possible metrics H becomes the program’s modular field once only ω is held fixed. The lab notebook’s first version of P1 also listed a third structure, the neutral metric η; the current version lists only ω and H, and P2′ makes η a property of the vacuum.

Status: Alive — [P] — a postulate of #11 §4; postulates are assumed, not derived or machine-checked. The algebra built on its two structures is checked: #11 §6 tags I[Z]² = −1 for the allowed metrics [V] (battery test T2, the verification battery).

Statement#

P1  Physics on phase space with symplectic form ω and positive
    generalized metric H.                                        [P]

This is #11 §4’s wording. The program paper v1 (#9 §2) states the same axiom in nearly the same words, under the name “P1 (Arena)”. The axiom names an arena and two structures on it, and nothing more. Which structures are fixed and which can vary is the business of P2′; the symmetry is P3, matter is P4, and the vacuum is P5.

Phase space#

Phase space is the space of states of classical mechanics: each point is a position together with a momentum [T]. The lab notebook works “first on flat 𝒫 ≅ ℝ²ᵈ with coordinates (xᵃ, pₐ), d = 4” (#1 §2). So x and p each have d = 4 components, and x is a point of spacetime.

In quantum mechanics a state cannot sit at a point of phase space, because position and momentum cannot both be sharp (the canonical commutator) [T]. Quantum kinematics is still written in the same pair of variables, and the exchange the program is built on, Born reciprocity ρ: (x, p) → (p, −x), acts on the pair together (#11 §3). P1 places both variables in one space, which is the space ρ acts on. Spacetime carries only x.

The symplectic form ω#

The symplectic form is a bilinear form on phase-space vectors that is antisymmetric, ω(u, v) = −ω(v, u), and nondegenerate [T]. In the (x, p) block basis its standard form is

ω = [[0, 1], [−1, 0]]          ω(u, u′) = x·p′ − p·x′   for u = (x, p), u′ = (x′, p′)

(#1 §2 for the matrix; the second expression writes out the same form [T]). ω(u, u′) is the sum, over the coordinate pairs (xᵃ, pₐ), of the oriented areas that the two vectors’ projections span in each plane [T]. It is the structure in which Hamilton’s equations and the Poisson bracket {x, p} = 1 are written, and quantization turns that bracket into the commutator [x, p] = iℏ [T]. The linear maps that preserve ω form the symplectic group Sp(2d, ℝ) [T]; the Born rotation is one of them (#1 §2: it “preserves ω (it is symplectic)”).

The program ties ω to Planck’s constant. The lab notebook: “the symplectic structure is the constancy of ℏ — phase-space cells do not stretch” (#1 §1). Its cosmology section spells out the cell: “Rigidity of ω means the comoving phase-space cell has fixed volume” (#1 §6). P2′ keeps exactly this reading: “symplectic rigidity ≡ constancy of ℏ” (#11 §4).

The generalized metric H#

H is a symmetric, positive-definite form on phase-space vectors: it assigns every vector u a positive squared length H(u, u), with position and momentum directions measured together [T]. Its standard form is the identity,

H = [[1, 0], [0, 1]]

with “units chosen so a reference length λ and momentum ε = ℏ/λ are scaled out” (#1 §2). Some such pair of scales is needed: a length squared and a momentum squared can be added only after each is divided by a reference scale of its own kind [T]. The choice ε = ℏ/λ makes λε = ℏ, so only one free scale remains.

The name is the one used in double field theory, where a positive generalized metric on a doubled space combines the spacetime metric with the Kalb–Ramond B-field [T]. Born geometry uses H in the same role (the structures ω, η, H). In the program H holds the geometry of both halves of phase space. When H is block-diagonal, its position block is the spacetime metric g and its momentum block is the momentum-space metric g̃ (#5 §A); the general case is the family described below.

The complex structure I = ω⁻¹H#

From its two structures the program builds the linear map I = ω⁻¹H of phase space to itself (#11 §5). H is compatible with ω when (ω⁻¹H)² = −1, one of the two conditions of #11 §5’s compatibility theorem. I is then a complex structure [T]: it acts on phase space as multiplication by the imaginary unit acts on the complex plane, a quarter-turn that reverses every vector when applied twice. The metrics the program admits are the ω-compatible ones: “positive H compatible with ω alone form the Siegel domain Sp(2d,ℝ)/U(d)” (#5 §C).

This is where P1 meets the rest of the postulates. P3 states the symmetry through I: “The action is invariant under the Born rotation generated by I = ω⁻¹H” (#11 §4). So the generator of the program’s central symmetry is assembled from P1’s two structures. At the standard forms, I² = −1 holds by “direct matrix computation” (#1 §2).

What H can be#

The answer depends on what else is held fixed.

  • With η fixed as well (the original P2). The compatibility theorem [T][V] leaves only H = diag(g, g⁻¹), g ∈ Sym⁺(d) (#11 §5; battery test T1). The momentum metric is then slaved to the spacetime metric, g̃ = g⁻¹, and “no independent momentum-space geometry survives” (#5 §A; the corollaries).

  • With only ω fixed (P2′). “H ranges over Sp(2d,ℝ)/U(d): Z = X + iY (symmetric; Y > 0), H_Z the standard generalized metric (I[Z]² = −1 machine-verified [V])” (#11 §6). The family, from the Debt-1 note:

    H_Z = [[Y + XY⁻¹X, XY⁻¹], [Y⁻¹X, Y⁻¹]],     Z = X + iY,   X, Y symmetric,   Y > 0      (#5 §C)

    Here Im Z = g is the spacetime metric and Re Z = X is a second symmetric tensor, the momentum graviton (#11 §6). At X = 0 the family reduces to the rigid case diag(g, g⁻¹) (#5 §C). See the Siegel domain and the modular field Z = X + iY.

The machine check behind the [V] is the battery’s test T2, which builds H_Z for a randomly sampled X and Y with d = 4 and confirms that I[Z]² = −1, that H_Z is symmetric and positive definite, and that det H_Z = 1 (verify/battery.py; gate output T2 I[Z]^2 = -1, passing).

Superseded: the v0.1 arena with three structures#

The lab notebook’s first version of P1 (v0.1, September 2026) had three structures:

P1 (Arena). Physics is formulated on phase space 𝒫 carrying a Born geometry: a symplectic form ω, a neutral (split-signature) metric η, and a positive generalized metric H, mutually compatible in the sense of §2. [P] (#1 §1)

The neutral metric in standard form, beside the other two, is

η = [[0, 1], [1, 0]]          η(u, u′) = x·p′ + p·x′

(#1 §2 for the matrix). η is symmetric with signature (d, d), and every pure-position vector and every pure-momentum vector has zero η-length [T]. The Debt-1 note calls it “the structure that says which directions are position” (#5 §B). With all three structures the notebook defined I ≡ ω⁻¹H, J ≡ η⁻¹H and K ≡ η⁻¹ω, and “Direct matrix computation gives I² = −1, J² = +1, K² = +1, and IJ = K: a split-quaternion (para-quaternionic) algebra. A compatible triple (ω, η, H) realizing this algebra is precisely a Born geometry in the sense of Freidel, Leigh & Minic. [T]” (#1 §2; #11 §5 cites Born geometry as [3†]).

The third structure did not last. The notebook’s original P2 held η as rigid as ω. The compatibility theorem then froze H into diag(g, g⁻¹), and the nontriviality theorem showed that this left the program with nothing beyond general relativity plus one model of self-interacting dark matter. The v0.3 addendum made η emergent instead: “the order parameter itself, η = HJ” (#1 ‘Addendum (v0.3)’). #9 §2 and #11 §4 then list only ω and H in P1. η still appears in #11, but as a property of the vacuum (P2′; the order parameter), and #11 §5 still takes “ω, η standard” when it states the compatibility theorem.

How the other postulates use the arena#

  • P2′ states which of P1’s structures is rigid: ω only.
  • P3 takes its symmetry from I = ω⁻¹H.
  • P4 couples both kinds of matter universally to H.
  • P5 has the vacuum select a polarization of this phase space: a choice of Lagrangian subspace for ω (Lagrangian polarizations).

Sources#

  • papers/broken-born-checkpoint-v2.md §4 — the current statement; §5 — I = ω⁻¹H, the compatibility conditions and the theorem; §6 — the Siegel family of H and its [V] tag
  • notes/broken-born-theory.md §1 — the v0.1 statement, with η, and P2’s reading of ω as the constancy of ℏ; §2 — the standard forms of ω, η and H, the reference scales, and the split-quaternion algebra; §6 — the phase-space cell
  • notes/broken-born-theory.md ‘Addendum (v0.3)’ — η moved out of the arena
  • notes/debt1a-geometric-core.md §A — the slaved momentum metric; §B — η as the structure that says which directions are position; §C — the H_Z parametrization of the ω-compatible metrics
  • notes/broken-born-program-v1.md §2 — the v1 statement, “P1 (Arena)”
  • verify/battery.py T2 — the Siegel-parametrization checks
Backward Postulates P1–P5 Original P2: both structures rigid (superseded) Forward