Notation

This page lists the mathematical symbols of the Broken-Born program as its documents and this wiki use them: what each symbol means, which document defines it and which wiki page explains it, grouped by subject in the order of the wiki’s sections. The program’s central objects are written the same way in all the current documents: the reciprocity map ρ, the symplectic form ω, the neutral metric η, the generalized metric H, the modular field Z = X + iY and the dark-energy field τ. Many of their letters also serve other purposes. H is also the Hubble rate, ρ an energy density and X a parameter of the dark-matter calculations; g is a spacetime metric, a coupling constant and a magnetic charge; μ is a width in momentum in the dark-matter sector and a coupling constant in the toy models. The Collisions section near the end lists every letter with more than one meaning or convention, together with the suspected source errors that bear on them.

Status: Record — untagged — a register compiled from the finished wiki pages and the documents they cite. It adds no results and no numbers of its own; each entry names the document that defines the symbol, and where an entry and its document differ, the document governs.

How to read this page#

Entries. Each entry gives the symbol in bold, its meaning, the document that defines it and the wiki page that explains it, linked to the section where that page introduces the symbol. A letter with several meanings appears under each subject that uses it and again under Collisions.
Documents. References such as “#11 §5” use the artifact numbers of HANDOFF.md §3: #1 is the lab notebook, #2–#4 the Debt-3 calc notes (3a, 3b and 3c), #5–#8 the Debt-1 calc notes, #9 the superseded program v1, #11 the checkpoint, #13 conjugate special relativity and #14 the withdrawn soliton manuscript. “The toy note” is papers/reciprocity-toy-models.md, “the worklog” its working record notes/toy-lagrangian.md, and “the v3 worklog” review/v3-worklog.md. Documents maps every number to its file.
Flags. “Flag n” is item n of review/wiki-build-flags.md, a suspected error in a source document found while this wiki was built and awaiting Paul’s ruling. Entries quote the source as written and name the flag; none is corrected here.
What is left out. Symbols that live inside a single derivation, such as integration variables, summation indices and the matrices of one proof, are left to their pages. Words are on the Glossary, cited works on References.

Conventions the wiki follows#

Writing. The wiki writes mathematics in Unicode, as the checkpoint does: H = diag(g, g⁻¹), Z → −Z⁻¹, βₙ = β₄₋ₙ, w ≥ −1. Display equations go in text blocks. The lab notebook, the calc notes and #13 write their formulas in LaTeX, and the wiki transcribes them; where a note writes v_* the wiki writes v*.
The three products of the structures. With ω and η in standard form, #11 §5 defines three products of ω, η and H that obey a split-quaternion algebra, and the lab notebook (#1 §2) uses the same definitions (the operators I, J, K):
```
I = ω⁻¹H,   J = η⁻¹H,   K = η⁻¹ω        I² = −1,   J² = K² = +1,   IJ = K
```
The real structure. #11’s postulate P2′ calls J the real structure and writes η = HJ; the toy note and #11 §14 item 11 call K the real structure and write η = ωK. Pages on #11’s postulates use J, pages on the toy models use K, and the wiki does not choose between them (Collisions).
Reused letters. Where a page uses a letter in one of its secondary meanings (ρ as a density, H as the Hubble rate, μ as a coupling), it names the meaning in words at first use, and several pages carry a note on their own collisions.
GW on this wiki means Grosse–Wulkenhaar, the second toy model. Gravitational waves are always written out.
† outside a formula is the citation mark: a reference cited from model memory. Inside the toy models’ formulas (φ†φ, ψψ†) it is the Hermitian adjoint.

Phase space#

Coordinates, reciprocity and the symplectic form#

d — the number of position coordinates, so that phase space has 2d dimensions: d = 4 physically, d = 1 in the toy models. — #1 §2; toy note §1 — Phase space and its three structures
𝒫 ≅ ℝ^{2d}; (xᵃ, pₐ) — flat phase space, and its coordinates: d positions xᵃ and their momenta pₐ. — #1 §2 — Phase space and its three structures
ρ — Born reciprocity, the Born rotation (x, p) → (p, −x): a quarter-turn of phase space. As a matrix it is [[0, 1], [−1, 0]], symplectic, with ρ² = −1 and ρ⁴ = 1; on the modular field it acts as Z → −Z⁻¹. — #11 §3, §6; toy note §1 — The map
R — the matrix [[0, 1], [−1, 0]] by which ρ acts on tangent vectors in #1. “R preserves ω (it is symplectic), preserves H (it is orthogonal), and flips the sign of η”. — #1 §2 — How ρ acts on the triple
ρ²; ℤ₄ — the phase-space parity (x, p) → (−x, −p), which preserves ω, η and H, and the group of four elements that ρ generates. The lab notebook’s breaking pattern ℤ₄ → ℤ₂, with the vacuum keeping {1, ρ²}, is superseded by the Goldstone picture. — #1 §2 — Superseded: the discrete breaking pattern
ψ(x), ψ̃(p); F — a state’s position and momentum wavefunctions, and the Fourier transform that relates them. F implements ρ on wavefunctions: it conjugates x̂ into p̂ and p̂ into −x̂, and applied twice it is parity. — [T]; toy note §5 — Definition; Twice is parity
x̂, p̂; [x, p] = iℏ — the position and momentum operators, and the canonical commutator, which #11 §3 reads as ruling out joint locality in x and p. — #11 §3 — What the commutator says
ℏ — Planck’s constant, set to 1 in the toy models’ checks. #11 §4 reads its constancy as the rigidity of ω: “symplectic rigidity ≡ constancy of ℏ”. — #11 §4 — ℏ as the one rigid structure
ω; ω(u, v) — the symplectic form, [[0, 1], [−1, 0]] in d × d blocks, the one structure held rigid under P2′. On vectors u = (u_x, u_p) and v it is ω(u, v) = u_x · v_p − u_p · v_x, a signed area in each position–momentum plane. — #11 §4; #1 §2; toy note §1 — Definition
Sp(2d, ℝ); S — the group of linear maps that preserve ω; Sp(2, ℝ) in the toy models, where it contains ρ. verify/toy_coset.py uses the generic element S = [[s₁, s₂], [s₃, (1 + s₂s₃)/s₁]]. — #11 §1, §6; toy note §2 — The symplectic group
W, W_ψ — the Wigner function of a state ψ, a function on phase space whose two marginals are |ψ(x)|² and |ψ̃(p)|². — [T]; toy note §5 — Definition

Born geometry#

η — the neutral (split-signature) metric, [[0, 1], [1, 0]] in the standard frame: a symmetric pairing with as many positive as negative directions. Rigid under the original P2; under P2′ it is emergent and serves as the order parameter. — #1 §1–§2; #11 §4 — Phase space and its three structures
H — the positive generalized (phase-space) metric of postulate P1. — #11 §4 — P1: the phase-space arena
I, J, K — the three products I = ω⁻¹H, J = η⁻¹H and K = η⁻¹ω. I is a complex structure (I² = −1), itself a quarter-turn of phase space, and the lab notebook calls ρ’s group “the ℤ₄ generated by the complex structure I” (P3); J and K are real structures (J² = K² = +1). In #1, J is the order parameter: “Reciprocity is broken iff ⟨J⟩ ≠ 0 in the vacuum”. — #11 §4–§5; #1 §2 — The operators I, J, K
g; g̃ — the spacetime metric, g = Im Z under P2′, and the momentum-space metric, slaved to g̃ = g⁻¹ when ω and η are both rigid. — #11 §5–§6; #5 §A — Corollary 3: the slaved momentum metric
diag(g, g⁻¹); Sym⁺(d) — the only positive H compatible with both rigid ω and rigid η, and the set of real symmetric positive-definite d × d matrices g. — #11 §5 — Statement
A, B, C; E1, E2 — the blocks of H = [[A, C], [Cᵀ, B]] in the proof of the compatibility theorem, and the round-3 referee’s two residual matrices, E1 = (η⁻¹H)² − 1 and E2 = (ω⁻¹H)² + 1. — #5 §A; #11 §5; review/round3/referee/audit-notes.md — The proof; Round 3’s independent re-derivation
det H = 1; B-field — the first corollary of the theorem, unimodularity, and the antisymmetric Kalb–Ramond field of string theory and double field theory, which the second corollary excludes: “no Kalb–Ramond analog”. — #5 §A; #11 §5 — Corollary 1; Corollary 2; The Kalb–Ramond B-field
R_B[H] — the curvature of the Born connection, in the lab notebook’s geometric action. — #1 §2–§3 — The lab notebook’s geometric action
U(1, d−1) — the stabilizer when g has Lorentzian signature, written U(1,3) in #9. — #6 §3; #9 §10 — What changes in Lorentzian signature
x̃ᵢ; η^{AB} — in double field theory, the dual coordinate paired with each xⁱ and conjugate to winding, and the neutral metric of the doubled space, the pairing of each coordinate with its dual, which O(d, d) preserves and from which the section condition is built. It has the same standard form as the program’s η. — [T] — The doubled space; The section condition

The modular field and τ#

Z = X + iY — the modular field: a complex symmetric d × d matrix with Y > 0, the coordinate on the Siegel domain Sp(2d, ℝ)/U(d). — #5 §C; #11 §6 — The modular field Z = X + iY
H_Z; H(Z) — the generalized metric in Siegel form, [[Y + XY⁻¹X, XY⁻¹], [Y⁻¹X, Y⁻¹]]; for d = 1 the toy note writes H(Z) = [[Y + X²/Y, X/Y], [X/Y, 1/Y]]. — #5 §C; toy note §2 — The domain as a space of phase-space metrics
X — Re Z, the momentum graviton. — #5 §C; #11 §6 — Step 4: Re Z = X; The momentum graviton X
Y — Im Z, identified with the spacetime metric g. — #5 §C; #11 §6 — Step 3: Im Z = g
⟨Z⟩ = i·g — the vacuum, which sits at X = 0. — #6 §3 — Step 2: the vacuum sits at X = 0
Z → −Z⁻¹; Z′ — ρ acting on Z as modular (Möbius) inversion. At d = 1 the image Z′ = −1/Z has components X′ = −X/(X² + Y²) and Y′ = Y/(X² + Y²). — #5 §C; #11 §6 — How reciprocity acts on Z
Z = i; |Z| = 1 — the fixed point of the inversion, the self-dual vacuum, and the circle that #11 identifies with the cosmological constraint a·ã = 1; flag 23 questions that identification. — #5 §C; #11 §6, §10 — The fixed point; The circle |Z| = 1
Z → Z + B — the shift symmetry, with B a constant symmetric matrix, contained in the rigid Sp(2d, ℝ). — #6 §1; #11 §7.1 — Shift symmetry
I[Z] — the complex structure ω⁻¹H_Z on the Siegel family; battery test T2 checks I[Z]² = −1. — #7 §2; #11 §6 — The domain as a space of phase-space metrics; Battery test T2
Λ(d) — the Lagrangian Grassmannian U(d)/O(d), the space of all position/momentum splittings: compact, of dimension d(d+1)/2 = 10 at d = 4, and the Shilov boundary of the Siegel domain. Λ(1) is the circle that model I’s compact Goldstone angle parametrizes. — #1 §2; #11 §6; toy note §2 — The space of them: U(d)/O(d); The toy model’s Λ(1)
π₁ = ℤ; det² — the Maslov class of Λ(d), its fundamental group, carried by the map det² to the circle. — #1 §2; #11 §6 — The Maslov class
σ-modes — the ten moduli of Λ(4): τ plus nine others, which #1 §2 assumes heavy [P] and #5 §D expects inverse-Higgs constraints to eliminate in part [O]. — #5 §D; #1 §2 — The list; Inverse-Higgs constraints
τ — the dark-energy field, the Maslov phase of the boundary variable: τ ≡ f·arg det² in #1 §2, “arg det of the boundary variable” in #5 §C, periodic with period 2πf. — #1 §2; #5 §C; #11 §6 — The definition; Two wordings
f; τᵢ; V₀ — τ’s period scale, its initial value and the height of its potential: the three free numbers of the dark-energy sector, of which the measured dark-energy density fixes V₀. — #1 §2–§3, §7 — The count, step by step
V(τ) = V₀[1 + cos(τ/f)]; S_τ — τ’s potential and its action, S_τ = ∫√−g [½(∂τ)² − V(τ)]. As written the potential is never negative for V₀ > 0, while the same documents need V < 0 for the turnaround (#1 §7: “A cosine potential is negative over half its range”); no document resolves the conflict (flag 27). — #1 §3; #11 §10 — S_τ; The periodic potential; Open: the sign of V(τ) and the turnaround
θ_pol — the vacuum tilt angle in the lab notebook’s sketch Ω_χ/Ω_x^(0) ~ tan²θ_pol. — #1 §8 — The notebook’s sketch

Symmetry breaking#

Spontaneous breaking in general#

G; G₀; Φ₀ — a symmetry group, the subgroup that leaves the vacuum in place (its stabilizer), and the vacuum; the vacuum manifold is the coset space G/G₀. — [T] — Vacua as a coset; The vacuum manifold and the stabilizer
M²ᵢⱼ — the Hessian of the potential at its minimum, the mass-squared matrix whose zero eigenvalues count the flat directions. — [T] — Counting with the Hessian
π; ∇_μπ; P_μ — a Goldstone field, on the inverse-Higgs page a membrane’s displacement; its covariant derivative in the coset construction; and the generators of translations. An inverse-Higgs constraint sets ∇_μπ = 0 and solves it for another would-be Goldstone field. — [T] — The commutator criterion
E = −J Σ⟨ij⟩ cos(θᵢ − θⱼ) — the XY model of angles θᵢ on a lattice with coupling J, the textbook case of the Mermin–Wagner–Coleman theorem. — [T] — Why: spin waves in two dimensions
S[n]; n^a; g_ab; f — a sigma model’s action S[n] = (f²/2) ∫ d^D x g_ab(n) ∂_μ n^a ∂_μ n^b, with fields n^a valued in a curved target space of metric g_ab, a coupling f and a base of dimension D. — [T] — What a sigma model is

The order parameter, the section condition and the matter coupling#

η = HJ; η = ωK — the order parameter as #11’s P2′ writes it, with J the real structure, and as the toy note and #11 §14 item 11 write it, with K the real structure. The two give the same η; see Collisions. — #11 §4, §14; toy note §1 — Three notations for one order parameter
L±; L, L̃ — the two eigenlines of K, which are η’s null lines and are Lagrangian; ρ exchanges them. The worklog writes them L and L̃, following Freidel–Leigh–Minic, with L = ker(η + ω) and L̃ = ker(η − ω). — toy note §1, §5; worklog — The toy setting
η^{AB}∂_A f ∂_B g = 0; η^{AB}∂_A∂_B f = 0 — the section condition the toy note imposes on matter fields f and g (d = 1), with A and B running over the two phase-space coordinates. — toy note §5 — The condition built from η
ρ∘(K → −K) — the twisted element: ρ combined with the flip K → −K, which maps model I’s vacuum (1, K₀) to itself. — toy note §2 — The twisted element fixes the vacuum
I[W]; I_K; I_{+K}, I_{−K} — the (H, K)-built coupling I[W] = ∫{L+} du (∫{L−} dv W)², local in x in one vacuum and local in p in the other; I_K in the round-4 records, with I_{+K} and I_{−K} for the two signs of K. The toy note calls I “odd in K”; flag 2 notes that I ≥ 0, so it is only not invariant under K → −K. — toy note §5 — The coupling

Model I: the coset sigma model#

Model I is the sigma model of the pair (H, K) on a flat two-dimensional base (toy note §2).

f_H, f_K, λ; L — the couplings of model I’s Euclidean Lagrangian L = (f_H²/8) Tr(H⁻¹∂H H⁻¹∂H) + (f_K²/4) Tr(H⁻¹ ∂Kᵀ H ∂K) + (λ/8) Tr(H⁻¹ Cᵀ H C). — toy note §2 — The Euclidean Lagrangian
C — the combination C = KI + IK, with I = ω⁻¹H, in model I’s potential; the potential vanishes exactly when K is compatible with H. — toy note §2 — The compatibility potential
S — an element of Sp(2, ℝ), acting as H → SᵀHS, K → S⁻¹KS. — toy note §2 — The symmetries
dS₂ — two-dimensional de Sitter space, K’s own target space, whose indefinite metric “would give a ghost”. — worklog ‘Chunk 1 results’ — Model I: the de Sitter target of K; The K kinetic term and the ghost; Two-dimensional de Sitter space as the target of K
K(φ); φ — the circle of model I vacua at H = 1, K(φ) = [[cos φ, sin φ], [sin φ, −cos φ]], and the polarization angle φ; ρ maps K(φ) to −K(φ) = K(φ + π). — toy note §2 — At H = 1: a circle of real structures
K₀; K_ref — a reference vacuum (toy note §2) and the reference axes of the ρ-preserving potential (§4); verify/toy_coset.py sets both to diag(1, −1). — toy note §2, §4 — At H = 1; The bounded potential
O = K₁₁ — model I’s order parameter, the upper-left entry of K, which flips sign under ρ (K_11 in the gate’s output). — toy note §2 — The order parameter (criterion S3)
ℤ₂ (K → −K) — model I’s separate symmetry, the flip of K. — toy note §2 — Model I and its two symmetries
μ; V_μ; V_λ — the coupling of the ρ-preserving potential; the naive potential −μ[Tr(K K_ref)]², written V_μ in the round-4 records, which is unbounded below once H is dynamical and is dead; and V_λ, the compatibility term, in those records’ total λV_λ + V_μ. — toy note §4; review/round4-adjudication.md F1–F2 — Obituary: the naive ρ-preserving potential
V_b — the bounded potential V_b = −μ[Tr(K K_ref)]²/(2 Tr KᵀK), written V′ in the round-4 referee’s notes. — toy note §4; review/round4/referee/audit-notes.md — Definition
S(t) — the round-4 referee’s path S(t) = [[cosh t, sinh t], [sinh t, cosh t]], with H = SᵀS and K = S⁻¹K₀S, along which the compatibility term stays zero. Flag 10 notes that the referee’s notes use “boost” for two different families. — review/round4/referee/audit-notes.md Step 2(a) — F1 and F2
E; J; h; T; L — the frozen-H lattice model E = −J Σ⟨ij⟩ cos(φ_i − φ_j) − h Σ_i cos 2φ_i, with stiffness J and potential strength h (h = 1 in model A, h = 0 in model B), simulated at temperature T on lattices of size L = 16, 32, 64. — toy note §4 — The lattice model
m; M; U — the ρ-odd order parameter m = ⟨cos φ⟩; the U(1) order parameter of the run without potential, ⟨|M|⟩ = |⟨e^{iφ}⟩|; and the Binder cumulant U = 1 − ⟨m⁴⟩/(3⟨m²⟩²). — toy note §4; verify/toy_s4.py — What the diagnostics show
ℤ₂ (ordered phase) — in “ordered ℤ₂ phase”, the discrete order the frozen-H Monte Carlo finds, with an order parameter odd under ρ. — toy note §4; #11 §14 item 11 — With the potential: an ordered ℤ₂ phase

Model II: the Grosse–Wulkenhaar model#

Model II is the self-dual Grosse–Wulkenhaar model on the noncommutative (Moyal) plane (toy note §3).

θ; ⋆; x̃ — the Moyal noncommutativity [x₁, x₂] = iθ (θ = 1 in the note), the star product, and the dual coordinate x̃ = 2θ⁻¹x. — toy note §3 — The Moyal plane
Ω — the model’s oscillator frequency; at Ω = 1, the self-dual point, the model is invariant under the Langmann–Szabo duality. — toy note §1 — The self-dual point
f_mn; H = ½x² — the matrix base of the Moyal plane, with f_mn ⋆ f_kl = δ_nk f_ml, built from the states of the oscillator H = ½x². — toy note §3 — The matrix base
D — the Langmann–Szabo duality transform D f(x) = (πθ)⁻¹ ∫ d²y e^{−i x̃·y} f(y), which acts on the matrix base as D f_mn = (−1)^m f_mn (D₋ in the round-4 referee’s notes). — toy note §3 — Statement
Π; N — parity on the matrix base, Π = diag(1, −1, 1, −1, …) = e^{−iπN}, with N = diag(0, 1, 2, …) counting oscillator levels; the duality acts on a field as Dφ = Πφ, “the α = π element of e^{−iαN}”. — toy note §3; worklog; verify/toy_gw.py — What “left multiplication by parity” means
φ, φ_mn; μ²; λ — the field and its matrix components, the mass parameter and the quartic coupling in S[φ] = Σ_mn (m + n + 1 − μ²)|φ_mn|² + λ Tr((φ†φ)²); the note works at μ² = 4 and λ = 1. — toy note §3 — The action
N (truncation); U(1)^N × U(1)^N — the number of matrix levels kept, and the symmetry under which every condensate phase of the complex model is a flat direction. — toy note §3–§4 — Step 1
q_m; S_min; β; φ_vac(β) — the occupations q_m = |φ_mm|² of a diagonal vacuum, the least action S_min, and the relative phase β of φ₀₀ and φ₁₁, which labels the complex vacua φ_vac(β) = diag(√q₀, √q₁ e^{iβ}, 0, …). — toy note §3 — The vacua; The flat angle β
Δ(β) — the polarization Δ(β) = ⟨x²⟩_vacuum − ⟨x²⟩_dual = Δ(0) cos β, with Δ(0) > 0: the β = 0 vacuum is broader in x than in p. The worklog’s first version had the sign reversed. — toy note §3 — The polarization Δ(β); Superseded: the reversed sign of the polarization
ψ = Πφ — the alternating form in which the real model’s quartic term Tr((ψψ†)²) is invariant. — toy note §3 — Real fields: only a covariance
q₀, q₁; diag(a₀, −a₁); F(±) — the occupations in the real vacua diag(±√q₀, ±√q₁); the preferred real vacuum diag(a₀, −a₁), the β = π configuration, sharper in position; and the one-loop Gaussian free energies of the two relative signs, with F(+) − F(−) = 2.634. — toy note §3 — The statement
G_ij; T₀(N) — the propagator 1/(i + j + 1 + M²) in the round-4 referee’s tadpole check, and the referee’s level-0 tadpole T₀(N) = Σ_k [G₀ₖ + Gₖ₀] at M² = 1, which grows as 2 ln N where the toy note says ln N (flag 11). — review/round4/referee/audit-notes.md Step 2(c) — The coefficient: ln N or 2 ln N

Gravity#

Spin-2 fields#

g_μν; η_μν; h_μν — the spacetime metric, the Minkowski metric (unrelated to the program’s η) and a small perturbation, g_μν = η_μν + h_μν. — [T] — Linearized gravity and the graviton
Γ^λ_μν; R_μν, R; G_μν; T_μν — the Christoffel symbols, the Ricci tensor and curvature scalar, the Einstein tensor and the stress-energy tensor, in Einstein’s equations G_μν ∝ T_μν. — [T] — The metric and its curvature
S_EH; M_P; G — the Einstein–Hilbert action, S_EH ∝ M_P² ∫ d⁴x √−g R, with M_P the Planck mass, which sets the strength of gravity through Newton’s constant G. Its two conventions are under Collisions. — [T] — The Einstein–Hilbert action and the field equations; The coefficient of the curvature term
ξ^μ — the vector field that generates an infinitesimal change of coordinates, δh_μν = ∂_μξ_ν + ∂_νξ_μ. — [T] — Linearized gravity and the graviton
m_g — the graviton mass; gravitational-wave bounds give “m_g ≲ 10⁻²³ eV territory” (#7 §4), which #8 §3 applies to the exactly massless even combination. — [T]; #7 §4; #8 §3 — The graviton’s mass; Row 6: the exactly massless graviton
f_μν — in dRGT massive gravity, a fixed reference metric; in bimetric gravity, the second dynamical metric f. — [T] — The potential
e^a_μ; η_ab; ε_abcd — the vielbein (frame) field, from which the metric is built as g_μν = e^a_μ e^b_ν η_ab; the fixed matrix with respect to which the frame axes are orthonormal; and the totally antisymmetric symbol from which the wedge potentials are built. — [T] — The frame field; Wedge products
M_g, M_f — the Planck masses of the two metrics g and f; reciprocity forces them equal. — #8 §3; review/v3-worklog.md R7 — From one dynamical metric to two; Two metrics, two Planck masses; Open: stability at equal Planck masses
V = 2m²M² Σₙ₌₀⁴ βₙ eₙ(𝕏); 𝕏 = √(g⁻¹f) — the Hassan–Rosen potential, with five constant coefficients βₙ and the frame ratio 𝕏. #8 defines neither m nor M; in Hassan and Rosen’s papers m sets the mass scale and M is a Planck-mass scale built from M_g and M_f (from model memory †). — #8 §1 — From one dynamical metric to two
Sₙ(g, f) = S₄₋ₙ(f, g) — the exchange identity of the potential’s terms, which #8 states without defining Sₙ. — #8 §1 — The exchange identity
eₙ; λᵢ; pₖ — the elementary symmetric polynomials of the eigenvalues λᵢ of √(g⁻¹f), and the power sums pₖ = λ₁ᵏ + … + λ_dᵏ = Tr(Mᵏ). Battery test T7 checks that p₄ lies outside the linear span of the eₙ. #7 §2 identifies Tr(g²) with “p₄ of the square-root eigenvalues”; flag 14 notes that this holds for √(f⁻¹g), not √(g⁻¹f). — #7 §2; #8 §1 — Definition; Power sums and Newton’s identities
α (partially massless point) — the parameter of the partially massless point β₀ = 3α⁻²β₂, β₁ = β₃ = 0, β₄ = 3α²β₂; at α = 1 the point lies inside the palindromic branch. — #11 §7.4 — The parameter point; At α = 1
H; H₀; Λ — on the spin-2 pages, the constant Hubble rate of de Sitter space, the present Hubble rate, and the positive cosmological constant, related in de Sitter space by Λ = 3H². — [T] — De Sitter space; Cosmological solutions of bimetric gravity
l_P, t_P; M_Pl c; c⁴/G — the Planck length and Planck time, the Planck momentum, and the Planck force, all built from ℏ, c and G; l_P · M_Pl c = ℏ. — [T] — Units from ℏ, c and G
m_inert, m_grav; Φ — a body’s inertial and passive gravitational masses, whose ratio the weak equivalence principle makes the same for every body, and the Newtonian gravitational potential. — [T] — The weak form: universality of free fall
e^(−Mr)/(4πr); ℏ/(Mc); 1/(q² + M²) — the Yukawa potential of a mediator of mass M, its range, and the same law in momentum, a Fourier pair. — [T] — The potential of a massive mediator; The same law in momentum
V = −(G m₁ m₂/r)(1 + α e^(−r/λ)) — a fifth force written as a Yukawa term beside Newton’s potential, with strength α relative to gravity and range λ; neither is a symbol of the program (λ is not the Born length). — [T] — Fifth forces: a Yukawa term beside gravity

The momentum graviton X#

X_μν — the momentum graviton as a symmetric tensor field beside the metric. — #6 §1–§3; #11 §7.1 — The momentum graviton X
λ_μ; δX_μν = ∂₍μλ_ν₎ — the dual-diffeomorphism parameters and the gauge transformation they generate on X. — #6 §2; #11 §7.1 — The program’s version
π_μ — the four would-be Goldstone modes of the broken dual diffeomorphisms, which X absorbs. #6 §3 says X eats the four π_μ; #8 §3 says it eats “the vector π_μ and the longitudinal scalar of the λ-gradient sector”; no document reconciles the two wordings. — #6 §3; #8 §3 — Step 4; What X absorbs: two unreconciled wordings; The modes X absorbs
A_μ — in the textbook Stückelberg construction, the vector field added beside h_μν that restores the broken gauge symmetry. — [T] — Stückelberg fields
M_X — the mass of X, “the polarization-breaking scale” (#11 §7.1). #11 §7.5 writes M_X ∝ 2(β₁ + β₂), from #8 §3’s “m_FP² ∝ β₁ + 2β₂ + β₃ = 2(β₁+β₂), i.e. M_X”; flag 5 notes that the β-combination fixes the squared mass. #11 §12’s candidate table writes m_X. No document gives M_X a value. — #6 §3; #8 §3; #11 §7.1, §7.5 — Step 5: the mass scale; No value for M_X
m_FP² — the Fierz–Pauli mass combination, m_FP² ∝ β₁ + 2β₂ + β₃ = 2(β₁+β₂) on the palindromic branch. — #8 §3 — What remains; The Fierz–Pauli weight

The tuning test#

J₀ — the vacuum polarization [[0, 1], [1, 0]] in d × d blocks, used in the tuning test. — #7 §2 — The polarization J and the vacuum polarization J₀
{J, I[Z]} = [[S, A], [A, S]] — the anticommutator that measures misalignment between the polarization and Z, with blocks S = Y + XY⁻¹X − Y⁻¹ and A = [X, Y⁻¹]. #7 reads its zero set as X = 0 with Y = Y⁻¹; flag 3 finds it larger, X and Y commuting with X² + Y² = 1. — #7 §2; #11 §7.2 — The computation
h; x; ε — the metric tier’s expansion variables, Y = 1 + h and X = x in #7 §2, and the small parameter of battery test T4, which writes Y = 1 + εh and X = εx and finds Tr(S²)/ε² → 4Tr(h²). #7 §2 writes S = 2h + x² + O(3); flag 4 notes a missing −h². — #7 §2; #11 §2; verify/battery.py T4 — The expansion
E; E₀[J] — the symplectic vielbein, with H = EᵀE, and the polarization frame. — #7 §3; #11 §7.2 — The symplectic vielbein E; The polarization frame E₀[J]
e, f — the two frames of the program’s bimetric sector, with f = E₀[J]. — #8 §1 — The symplectic vielbein and the polarization frame
S = S_H + S_τ + S_x + S_p + S_mix; M_B — the lab notebook’s action and its five terms, with S_H = ½M_B² ∫√|H| R_B[H] over the section and M_B the Born mass scale, the g-block giving the Einstein–Hilbert action “with M_P fixed by M_B and the section volume”. Superseded. — #1 §3 — The five terms

The palindromic branch#

βₙ = β₄₋ₙ — the palindrome: β₀ = β₄, β₁ = β₃, β₂ free. — #8 §1; #11 §7.3 — Statement
T; V(1) — the tadpole combination T = β₁ + 3β₂ + 3β₃ + β₄, and the potential at the aligned point 𝕏 = 1. Under the palindrome V(1) = 2T; the flatness condition is V(1) = T. — #8 §2; #11 §7.3 — The value V(1) and the tadpole T
δᵢ — the small shifts of the eigenvalues near the aligned point, 1 + δᵢ. — [T] — The polynomials at and near the unit matrix
Λ_wedge — the vacuum energy of the wedge sector; #9 §11 calls the identity “an unsolicited theorem (Λ_wedge = 0)”. — HANDOFF.md §3; #9 §11 — Names and history; Cosmological constants in the bimetric potential

Dark matter#

Conjugate matter and the gate#

χ — conjugate (p-polarized) matter, a complex scalar or Dirac fermion. — #1 §3; #11 §8 — The field χ; Conjugate matter and the velocity gate
M(k, q) — the nonrelativistic self-interaction amplitude, M(k, q) = 4πα̃m_χ/(q² + m_φ²)·F(k²/μ²). — #11 §8 — The amplitude
m_χ, m_φ, α̃ — the χ mass, the mediator mass and the gate coupling. #11 §8 uses them without defining them; the wiki takes them from #1 §3 and #9 §6. — #1 §3 — The amplitude
k; q; v_rel — the incoming relative momentum k = m_χv_rel/2, which is the gate variable; the momentum transfer; and the relative velocity. — #1 §3 — The amplitude
F — the gate form factor F(k²/μ²), whose shape, Gaussian or rational, is open (OP4). — #11 §8 — The question
μ; p₁, p₂ — the gate width in momentum, since #1’s v0.4 addendum the χ quantum’s intrinsic momentum-fiber width, and the momenta of the two quanta in #6’s overlap form. The sources write the gate as e^{−k²/μ²} (#1 §3) and as e^{−(p₁−p₂)²/2μ²} (#6 §4), which differ by a factor √2 in μ if #6’s form is read as a gate on the amplitude, as #1’s is (flag 17, which the audit found to depend on that reading; see Collisions). #13 §4 reads μ ~ ℏ/λ_χ. — #1 §3; #6 §4; #11 §8; #13 §4 — What μ is; Two conventions, not reconciled
v* — the knee velocity, the gate scale in relative velocity, defined in #1 §5.1 as v* ≡ √2μ/m_χ and repeated in #13 §4; #11 does not restate the relation. With #6’s fiber width, read as the width of an amplitude gate, the knee would sit at v* = μ/m_χ (flag 17, convention-dependent in the audit). — #1 §5.1; #11 §8; #13 §4 — What μ is; Row 1: the knee in σ_T(v)
σ_T(v); σ₀ — the momentum-transfer cross-section as a function of relative velocity, σ_T(v) = σ₀F(v²/v*²), and its low-velocity value; σ₀/m is the cross-section per mass that dwarf-galaxy cores fix. — #11 §8 — The law; The transfer cross-section σ_T; The scattering benchmark
T̃ — the dual stress, built from momentum derivatives of the χ bilinear; X couples to it as the metric couples to the stress-energy tensor. — #5 §E — Definition

Halos and wave dark matter#

σ_T = ∫ dΩ (1 − cos θ) dσ/dΩ; σ_tot; σ_V — the transfer cross-section, which weights each scattering angle θ by the forward momentum the collision removes; the total cross-section; and the viscosity cross-section, weighted by sin²θ. — [T] — The definition
Γ = nσ_T v — the rate at which one particle scatters, with n the number density and v the speed. — [T] — Per unit mass
ρ_d, σ_d — the density and velocity dispersion of a classical dwarf core, Scenario A’s normalization point. — #3 — The velocity dispersion σ; The environment table
N_occ; λ_dB — the occupation number N_occ = nλ_dB³ = ρ/(m⁴σ³), the number of quanta per de Broglie volume, and the de Broglie wavelength. Debt 3b also writes a peak occupation N_occ ≈ (2π)^{3/2}ρ/(m⁴σ³) and does not reconcile the two. — #2; #3 — Definition; Two normalizations in Debt 3b
Γ_gr; Λ — the gravitational condensation rate Γ_gr ≈ 8πG²ρ²Λ/(m³σ⁶) of Levkov, Panin and Tkachev, which always runs; Debt 3b leaves Λ undefined. — #3 — The gravitational rate
M_crit; M_P — the critical mass of an attractive condensate, M_crit ≈ 3.6 M_P/(m√g), above which it collapses (the bosenova). The Debt-3 notes do not define M_P; the wiki reads it as G^{−1/2} (Collisions). — #3; #4 — Why there is a maximum mass; What is unchecked
φ_i; χ_i — the initial value of a light field in the textbook misalignment mechanism, and the initial amplitude of wave χ that sets its relic abundance on the wave island. — [T]; #3 — The mechanism; Relic abundance by misalignment on the wave island

Momentum clumping (Debt 3)#

f(x, p); f₀; m — the phase-space distribution of the dark-matter particles, which evolves by the Vlasov equation; the homogeneous Maxwellian f₀(p) of dispersion σ about which Debt 3a linearizes; and the particle mass, written m in the Debt-3 notes. — #2 — What the model is
σ — the velocity dispersion of a halo, the spread of its particles’ velocities about their mean along one direction. — #2; #3 — The velocity dispersion σ; What the model is
g; G — the gated contact coupling g ≡ 4πα̃/m_φ² = √(πσ₀/m²), in GeV⁻², and the Gaussian gate G, with G(0) = 1. — #2 — What the model is
ε(x, p); Φ_grav — a particle’s energy, ε(x, p) = p²/2m + mΦ_grav(x) − g ∫d³p′ G((p − p′)/μ) f(x, p′), with Φ_grav the gravitational potential. — #2 — What the model is
ε(p; f); U; f̄ — the lab notebook’s earlier form of the same energy, ε(p; f) = p²/2m_χ + ∫d³p′ U(p − p′) f̄(p′), with U the interaction in momentum space and f̄ the averaged distribution. — #1 §5.2 — Obituary: velocity clumping, the founding image
ω, k; δf — a frequency and a wavevector, in perturbations δf ∝ e^{i(k·x − ωt)}. — #2 — Linear response about a homogeneous equilibrium
k_J — the Jeans wavenumber k_J = √(4πGρ)/σ; perturbations of longer wavelength grow. — [T] — The static threshold and the Jeans instability
γ — the growth rate of the stream instability, which #1 left to be computed; the claim it served is dead. — #1 §5.2 — Obituary: velocity clumping, the founding image
ε_int, ε_gate; n_gate — the interaction energies compared with kinetic energy in the stability criterion, and the density of co-movers inside the gate, n_gate ≈ n(v*/σ)³/(2π)^{3/2}. The criterion g·n_gate ≥ m·min(σ, v*)² fails by 10²⁸–10³¹ at the scattering benchmark. — #2 — Statement
S ≡ σ₀/m; m_crit — the cross-section per mass held fixed in the classicality argument, and the classicality threshold m_crit ≈ 10⁻¹³ GeV ~ 0.1 meV for Milky Way conditions; flag 19 concerns the regime of that estimate. — #2 — Step 1; Step 2: the critical mass
Γ_self; Γ_cond; c — the gated wave-kinetic condensation rate, Γ_self ≈ c·g²ρ²min(σ, v*)⁴/(m³σ⁶), named Γ_cond in Debt 3a, with an uncalibrated coefficient c = O(5–15). — #3; #11 §9 — The rate; Open: calibration of the coefficient c
σ_scat — a scattering cross-section in the generic wave-kinetic rate Γ ~ nσ_scat v × N_occ. — #3 — Background: wave kinetics with Bose enhancement
Ω; t_dyn, t_age — the orbital stirring rate Ω = 1/t_dyn, the dynamical time and the age of a system. — #3 — The race against orbital stirring
R_X; X; κ — the stream-to-stirring ratio R_X ≡ Γ_self(X)/Ω_X in an environment X, with ρ_X, σ_X and t_dyn,X its values there; and Scenario A’s normalization Γ_self(dwarf)·t_age = κ ∼ 1. Scenario A is dead. — #3 — The environment table; Obituary: Scenario A and the ultra-faint-dwarf endpoint
X (mean-field parameter) — X ≡ gn/(mσ²), Debt 3a’s mean-field parameter, which has to stay below 1; under Scenario B the stream condition becomes X ≥ √(Ωσ²/(c m v*⁴)). — #2; #3 — The mean-field parameter X
M_core; τ_cond; R — an ultra-faint dwarf’s core mass, the condensation time τ_cond = t_dyn/R, and Segue 1’s rate ratio R. — #3; #4; #11 §9 — Segue 1 and the condensation time

Candidates and relic abundance#

Ω_χ, Ω_b, Ω_DM, Ω_x^(0) — the density fractions of conjugate matter, baryons, dark matter, and x-polarized matter before baryogenesis; the abundance question is Ω_χ : Ω_b = 5.4 : 1. — #1 §5.3, §8; #11 §14 — The question
η_B — the baryon asymmetry, the template for that ratio: “an input awaiting a mechanism, exactly as η_B awaited baryogenesis”. — #1 §5.3 — The template: baryogenesis
Ω h²; h — a density fraction times the square of h, the present Hubble rate in a conventional unit; the round-3 record takes the dark-matter density from Planck 2018’s Ω_c h² = 0.120. — [T] — What is being set; The dark-matter density
n, n_eq; ⟨σv⟩; x_f, T_f; σ₀ — the number density and its equilibrium value in the Boltzmann equation dn/dt + 3Hn = −⟨σv⟩(n² − n_eq²); the thermally averaged annihilation cross-section times velocity; the freeze-out point x_f = m/T_f; and, for s-wave annihilation, the constant σv, written σ₀. — [T] — The Boltzmann equation; The s-wave formula as the program codes it
m_X — the mass of X as #11 §12’s candidate table writes it, in the X-condensate’s stability entry “m_X-protected”; M_X in #11 §7. — #11 §12 — The three handles

The retired soliton candidate#

These symbols belong to #14, withdrawn after the round-3 review. The candidate is dead (Obituary: superheavy conjugate solitons), and #14’s body numbers are superseded by the round-3 rulings.

SU(2)_d → U(1)_d; Φ, v — the hidden gauge group, broken by a real adjoint scalar Φ with vacuum value v. — #14 §2 — Obituary: the hidden SU(2) → U(1) proposal
π₂ — the second homotopy group of the vacuum manifold, which classifies monopoles; the handoff called the candidate “π₂ monopole-class” solitons. — HANDOFF.md §2 — Why it is stable: winding
W′±, m_W′; h′ — the charged hidden vector bosons, of mass m_W′ = g_d v, which verify/soliton_obituary.py writes as α_W′ M/f; and the hidden Higgs boson, in the round-3 referee’s final states h′h′ and g′h′. — #14 §2, §3; review/round3/referee/audit-notes.md Step 2 — The masses at the benchmark
ψ, ψ̄; γ′ — the hidden Dirac doublet, with U(1)_d charges ±g_d/2, and its antiparticle, whose mass the round-3 audit writes m_psi; and the hidden photon, gamma' in the same notes. — #14 §2; review/round3/referee/audit-notes.md — Why ψ existed; The window
g_d; g_m — the hidden gauge coupling and the monopoles’ magnetic charge, g_m = 4π/g_d. — #14 §2 — The relation in #14
α_W′; α_m — α_W′ = g_d²/4π, and the monopoles’ mutual coupling α_m = g_m²/4π = 1/α_W′. — #14 §2 — The relation in #14
α_d; α_e,d; e_d — the dark coupling, “α_d ≈ 34” in HANDOFF.md §2, dead as derived from Dirac quantization against our photon; the v3 worklog’s mirror electric coupling; and v2’s dark electric charge, in M ~ (4π/e_d²)Λ_c. — HANDOFF.md §2, §6; review/v3-worklog.md F2 — Obituary: the coupling α_d ≈ 34
H (hypothesis H) — #14’s hypothesis that the smallest hidden charge equals ours, (g_d/2)²/4π = α. — #14 §2 — Attempt 3: hypothesis H
M = 4πf v/g_d; f; r — the monopole mass, with f = 1 at the Bogomolny–Prasad–Sommerfield (BPS) limit and about 1.79 at large scalar self-coupling (1.787 in verify/soliton_obituary.py), and the ratio r = M/v = 4πf/g_d. — #14 §3 — The mass; The relation
n/s = p(T_c/M_Pl)³; p; T_c; Λ_c — the Kibble yield, its normalization and the transition temperature; Λ_c is the breaking scale in #14 v2 and the v3 worklog. — #14 §3; #11 §12 — Obituary: the Kibble normalization; One convention per formula
M* — the monopole mass at which the Kibble abundance matches the observed dark-matter density. — #14 §3; review/round3-adjudication.md ruling 2 — The relation
M_* — the scale suppressing the dimension-five operator Φ^a F^a_μν B^μν/M_*. — #14 §2(b) — Obituary: Planck-suppressed adjoint mixing
ε; C; C_A, C_X — kinetic mixing between the field strengths of our photon and the hidden one, which #11’s P4 writes as ε ≲ 10⁻¹³ from gravity alone, a bound the round-3 referee reads as a floor (unresolved); hidden charge conjugation, whose exactness, #11’s P4 says, makes gravity-induced mixing vanish; and the two separate charge conjugations of Gherghetta et al., as the round-3 referee quotes them. — #11 §4; #14 §2 — The term; The result; An unresolved reading: ceiling or floor
θ_d; q_vis — the hidden theta angle, and the visible electric charge it gives a monopole through the Witten effect when ε ≠ 0. — review/round3/referee/audit-notes.md 4(a) — The Witten effect
ξ; ξ₀ — the hidden-to-visible temperature ratio, and its starting value in the referee’s thermal integration. — #14 §4; review/round3/referee/audit-notes.md Step 2 — The ratio ξ; The referee’s numbers
σ_T/M; F; ln Λ — the monopoles’ transfer cross-section per mass, the flux they would have had as monopoles of our U(1), and the Coulomb logarithm, which #14 takes as 10–30. — #14 §4 — Benchmark-conditional arithmetic; The soliton candidate’s σ_T/M
λ_p — the Higgs-portal coupling in the round-3 audit’s list of omitted constraints. — review/round3/referee/audit-notes.md 4(c) — The Higgs portal

Cosmology#

Scale-factor duality#

a, ã — our scale factor and the conjugate one, with a·ã = 1, the cosmological constraint. — #1 §6; #11 §10 — Statement
H = ȧ/a; N; k — the Hubble rate; the lapse, which #1’s minisuperspace shares between the two frames; and the spatial curvature, k > 0 for a closed universe. — [T]; #1 §6 — The homogeneous universe; Minisuperspace
M_P; M̃; k̃ — the Planck mass in #1 §6’s 3H²(M_P² + M̃²a⁻⁶) = ρ, which the notebook leaves unstated and the wiki reads as the reduced mass; the dual Planck mass; and the dual curvature parameter, the optional geometric parameters of #1’s count. — #1 §6, §10 — The program’s extension: a second scale factor; Where a document states no convention; Row 10: an early step in G_eff
G_eff — the effective gravitational coupling, which the dual Einstein term changes by an a⁻⁶ step at early times. — #1 §6; #11 §10 — The a⁻⁶ step in G_eff
ρ; ρ̃, ρ_eff — the energy density in the Friedmann equation, the energy density of a fluid of the conjugate universe, and the effective density it contributes to ours. — #1 §6; #11 §10 — The derivation
w, w̃, w_eff — the equation of state of a fluid in our frame and of a fluid in the dual frame, and the effective value the dual fluid contributes to ours, w_eff = −w̃. The earlier rule w_eff = −2 − w̃ is superseded, and the claim it supported is dead (Obituary: conjugate collapse powers our expansion). — #1 §6; #11 §10 — Statement; Superseded: the earlier heuristic rule
a₀, a_e; z — the scale factor today and at emission, and the redshift, 1 + z = a₀/a_e. — [T] — Redshift
D_x, D_p, N — the universe’s spatial and momentum diameters and its photon count, in the chat-only relation D_x · D_p = h · N^(1/3) per dimension. — HANDOFF.md §4 — The phase-space diameter relation (chat-only)

Dark energy#

w; w₀, wₐ — the equation of state of dark energy, and the Chevallier–Polarski–Linder (CPL) parameters of w(a) = w₀ + wₐ(1 − a); the thawing corridor is wₐ ≈ −1.5(1 + w₀). — #1 §7; #11 §10 — The CPL parametrization; Thawing models in the w₀–wₐ plane; Thawing and freezing quintessence
w_p, a_p; a_c; ρ_DE — the pivot value of w and the pivot scale factor of the decorrelated CPL form, w(a) = w_p + wₐ(a_p − a); the scale factor at which a CPL w(a) crosses −1; and the dark-energy density. — [T] — The pivot form; A phantom crossing in the w₀–wₐ plane; The density it implies
Λ; ρ_Λ = M_P²Λ; ΛCDM — the cosmological constant, the energy density of empty space it amounts to, with M_P the reduced Planck mass and w = −1 exactly, and the standard cosmological model of a cosmological constant plus cold dark matter. — [T] — The cosmological constant Λ; An energy density of empty space
Λ = 3H²; a ∝ e^{Ht} — the constant Hubble rate of de Sitter space, set by Λ alone, and its exponential expansion. — [T] — A universe with only a cosmological constant
φ; V(φ) — on the quintessence page, a canonical scalar field and its potential. — [T] — Energy density and pressure
Ω_DE; t_turn — the dark-energy density fraction, which fixes V₀, and the time of the future turnaround. — #1 §7 — The count, step by step
T_μν k^μ k^ν ≥ 0; u^μ — the null energy condition, for every light-like vector k, and the four-velocity of a perfect fluid, T_μν = (ρ + p)u_μu_ν + p g_μν. — [T] — The null energy condition; For a perfect fluid
z — redshift. The Dark Energy Spectroscopic Instrument’s Data Release 2 (DESI DR2) favors a phantom crossing above z ≃ 0.4 in #11’s quotation, z ≈ 0.4 in the root CLAUDE.md. — #11 §10 — What DESI reported
r_d; D_M, D_H; θ — the sound horizon, the ruler of baryon acoustic oscillations; the transverse comoving distance and the distance D_H = c/H(z) along the line of sight; and the angle θ = r_d/D_M(z) that the ruler subtends across it. — [T] — From the ruler to distances

The early universe#

H = 1.66√g* T²/M_Pl; g* — the radiation-era expansion rate, with g* the number of relativistic degrees of freedom and M_Pl the unreduced Planck mass; the coefficient is the first anchor of verify/soliton_obituary.py. — [T] — The expansion rate
M_Pl; M̄_Pl — the Planck mass G^{−1/2}, 1.22089×10¹⁹ GeV in verify/soliton_obituary.py, and the reduced Planck mass M_Pl/√(8π), 2.43536×10¹⁸ GeV there. — [T] — Two conventions for the Planck mass; Two Planck masses
s, s₀; g*s; Y = n/s — the entropy density and its value today, the effective number of degrees of freedom for entropy, in g*s T³ a³ = constant, and the yield, the number per unit entropy that a relic keeps as the universe expands. — [T] — The relic yield Y = n/s and entropy dilution
N_eff; ΔN_eff; T_ν, T_γ — the effective number of neutrino species, its excess from extra light species, and the neutrino and photon temperatures, T_ν = (4/11)^{1/3} T_γ. — #1 §6; #14 §4 — N_eff: counting light species; What the abundances measure; N_eff
Y_p; n/p; Δm; G_F — the primordial helium mass fraction, Y_p ≈ 2(n/p)/(1 + n/p); the neutron-to-proton ratio, frozen at exp(−Δm/T_f); the neutron–proton mass difference; and the Fermi constant of the weak interaction, in Γ ∝ G_F² T⁵. — [T] — The epoch
C_ℓ, ℓ; r*, θ*, z*; r_D; c_s — the angular power spectrum of the microwave-background temperature and its multipole; the sound horizon at last scattering, the angle it subtends today and the redshift of last scattering; the diffusion length behind the damping tail; and the sound speed of the photon–baryon fluid. — [T] — The anisotropies
ξ; T_h, T_x; g_h, g_vis — the hidden-to-visible temperature ratio, the temperatures of a hidden sector and of an extra species, and the degrees of freedom on each side of T_h/T_ν = ξ · [(g_h(high)/g_h(late)) / (g_vis(high)/g_ν,dec)]^{1/3}. — #14 §4 — The ratio at late times
T_RH; N (e-folds) — the reheating temperature, and the number of e-folds of inflation, N = ln(a_end/a_start). — [T] — What it is; Reheating
T*; β/H*; f*, f₀; a*; Ω_gw — for a first-order phase transition: its temperature; its inverse duration in units of the Hubble rate; the peak frequency of the gravitational waves at emission and today; the scale factor at the transition; and the waves’ energy density per logarithmic frequency interval as a fraction of the critical density. — [T] — The parameters; Redshifting the peak to today; The amplitude

Electric–magnetic duality#

e, g; q = (e, g) — a particle’s electric and magnetic charge, and its charge as a point of the charge lattice. — #11 §11.3 — The charge lattice
⟨q, q′⟩ = e₁g₂ − e₂g₁ ∈ 2πℤ — the Dirac–Schwinger–Zwanziger (DSZ) pairing, an antisymmetric integer form, so that the charge lattice is a symplectic space. — #11 §11.3 — Dyons and the DSZ pairing; The DSZ pairing
e·g ~ 2π; g_D — Dirac’s condition, which #11 §11.2 says “pins the magnetic coupling near 4π/e² ≈ 34” (flag 28), and the Dirac charge g_D = 2π/e, the smallest magnetic charge. — #11 §11.2 — Dirac’s argument; The convention behind 34
α; α₃; α_em — the fine-structure constant α = e²/4π, the strong-force coupling, and the running electromagnetic coupling, as the v3 worklog’s convention item R2 uses them. — [T]; review/v3-worklog.md R2 — The convention behind 34
τ_c; θ — the complexified gauge coupling τ_c = θ/2π + 4πi/g², unrelated to the dark-energy field τ, and the vacuum angle θ. — #11 §11.3 — The complexified coupling; The θ term
S, T; SL(2, ℤ) — the generators of the duality group: S = [[0, 1], [−1, 0]], τ_c → −1/τ_c (electric ↔ magnetic), the same matrix as ρ, and T = [[1, 1], [0, 1]], τ_c → τ_c + 1 (θ shifted by 2π). — #11 §11.3 — The modular group SL(2, ℤ); One matrix, three appearances; The dyon lattice and the shift T
q = e(n + mθ/2π); (n, m) — the electric charge of the dyon with labels (n, m) in a theory with vacuum angle θ: the Witten effect. — [T] — Witten’s result
N = 4 — the number of supersymmetries of N = 4 super Yang–Mills theory (SYM), in which Montonen–Olive duality is exact. — [T] — Montonen–Olive duality
π₀, π₁, π₂, π₃ — the homotopy groups of the vacuum manifold that classify a transition’s defects: domain walls, strings, monopoles and textures. — [T] — The classification; Defects from a cooling field
ξ; ξ̂ — the correlation length of the field at a transition, with of the order of one monopole per correlation volume, n ~ ξ⁻³, and its frozen value in Kibble–Zurek dynamics. — [T] — Defects from a cooling field; Continuous transitions: Kibble–Zurek
n/s = p (T_c/M_Pl)³ — the Kibble yield of monopoles per unit entropy; one defect per Hubble volume gives p = 104.31 with M_Pl and 0.828 with the reduced Planck mass. #14 v3’s p ≈ 0.023 dropped a factor and is dead. — #14 §3; #11 §12; verify/soliton_obituary.py — The causal bound; Obituary: the Kibble normalization; One convention per formula
m_P; n_monop — Gangui’s symbols for the Planck mass and the monopole number density, in “n_monop/s ~ 100 (T_c/m_P)^3”, as the round-3 referee quotes it. — review/round3/referee/audit-notes.md Step 1 — The causal bound; Symbols in the documents
ρ_M — the mass density of relic monopoles today, ρ_M = M · (n/s) · s₀. — [T] — Too many monopoles
M_W; ξ (radius); H(ξ), K(ξ); μ ∝ λ/g² — the mass of the W boson, and, in the round-3 referee’s monopole profile, the dimensionless radius ξ = g v r, the scalar and gauge profile functions, and a parameter proportional to the scalar self-coupling λ over g². — [T]; review/round3/referee/step2b.py — The energy functional

Conjugate special relativity#

λ; ε — the Born length and its conjugate momentum scale ε = ℏ/λ; at the Planck value λ² = ℏG/c³. — #13 §1; #1 §2 — The Born length λ; The Born length at the Planck length
c̃ — the invariant speed of the momentum polarization, a maximum force: c̃ = ℏc/λ² = c⁴/G. — #13 §1 — The result
ds̃²; F; γ̃; φ — the dual interval ds̃² = c̃²dt² − dp², preserved by the dual Lorentz group; the dual boost parameter F, a force; the dual Lorentz factor γ̃ = (1 − F²/c̃²)^{−1/2}; and the dual rapidity, with F = c̃ tanh φ. — #13 §2 — The dual interval and the dual Lorentz group; The force boost
F₁ ⊕ F₂ — force composition, (F₁ + F₂)/(1 + F₁F₂/c̃²). — #13 §2 — Force composition
τ̃ — the dual proper time, dτ̃ = dt√(1 − F²/c̃²). — #13 §3 — The dual proper time
λ_χ — the invariant length of a conjugate quantum, on the shell x_μx^μ = λ_χ², the counterpart of the mass shell. — #13 §4 — The shell
l_P; E_Planck — the Planck length and the Planck energy; the dual rest-energy relation E = λc̃ gives E_Planck at λ = l_P. — #13 §4 — At the Planck length; Units from ℏ, c and G
γ; v; x′, t′; τ — in the template, the Lorentz factor, the boost velocity, the boosted coordinates x′ = γ(x − vt) and t′ = γ(t − vx/c²), and proper time, dτ = ds/c. — [T] — Boosts and the Lorentz factor; Simultaneity, proper time and length contraction
r_s — the Schwarzschild radius r_s = 2GM/c², the template of the chat-only momentum-space black holes. — [T]; HANDOFF.md §4 — Background

Predictions#

E_QG; ξ₁, ξ₂ — the energy scale of quantum-gravity corrections to photon propagation, near the Planck energy, and the coefficients of the linear and quadratic terms in E² = p²c²[1 + ξ₁(E/E_QG) + ξ₂(E/E_QG)² + …], the wiki’s reading of the standard parametrization. — [T] — Row 9: no linear photon dispersion; The scale of quantum gravity
ξ (dispersion) — the coefficient of the lab notebook’s quadratic correction, E² = p² + m² + ξp⁴/M_B². — #1 §4 — Row 9: no linear photon dispersion

Labels and marks#

P1–P5; P2; P2′ — the postulates; P2 is the superseded original, P2′ its repair. — #11 §4 — Postulates P1–P5; Original P2 (superseded)
#1–#15 — the artifact numbers of HANDOFF.md §3. — Documents
T1–T7 — the seven tests of the verification battery. — HANDOFF.md §5; verify/battery.py — The verification battery T1–T7
S0–S5 — the toy models’ success criteria, fixed before construction. — worklog — Success criteria S0–S5
OP1–OP11 — the wiki’s names for the open problems of #11 §14, items 1–11. — #11 §14 — Open problems and the queue
Rows 1–11; old rows 3 and 4 — the rows of #11 §13’s predictions table. #1 §9 and the Debt-3 notes number an older table, whose rows 3 and 4 were withdrawn and are dead. — #11 §13 — Numbering; Obituary: velocity-space granularity in halos (old row 3); Obituary: Fornax friction suppression (old row 4)
R1–R7; F1, F2, L1, O1–O3 — the round-2 pending items, and the v3 worklog’s own findings, literature point and open items. — review/v3-worklog.md — Review rounds 1 and 2
F1–F6; C1–C3 — round 4’s numbered findings and the referee’s conceptual points. — review/round4-adjudication.md; review/round4/referee/audit-notes.md — The rulings
Rulings 1–13; Decisions 1–4; Lessons 1–2 — round 3’s rulings, the decisions taken after it and its two lessons. — review/round3-adjudication.md — The rulings; Round-3 decisions and lessons
A, B, C — the referees’ verdict scale: (A) coherent speculative physics, (B) flawed but containing salvageable ideas, (C) confabulation. — review/skeptic-review-console.html — The verdict scale
Tier 1–3 — #11 §2’s claim tiers: explicit algebra, literature, novel structural physics. — #11 §2 — Claim tiers and selection bias
Flags 1–44 — the items of review/wiki-build-flags.md, with executed checks in review/wiki-build-checks/. — What this page does not list
C20, C21, C27, C28, C29, C54 — the inventory numbers of conflicts between sources that survive in the repository. — Conflicts between sources
[D] [T] [P] [O] [V]; † [S] — the provenance tags (derived here, established literature, postulate, open, machine-verified) and the citation marks (cited from model memory, retrieved by live search). In #1, [T] meant “theorem — established in the literature, cited”. — #11 header; #1 §0 — The tags and the marks
✓; conjecture-tier; [NOTE-tier] — the handoff’s mark for a machine check; #13’s label for a conjecture; and the v3 worklog’s label on its item O2, whose numbers its script printed as NOTE lines, “adjudication inputs not yet quoted by the manuscript”. — HANDOFF.md §2; #13 §5; review/v3-worklog.md header, O2 — Variants in other documents; The worklog’s table and the corrected bound

Collisions#

Each entry below is a letter or mark with more than one meaning, or one quantity written in more than one convention. The first group are conventions that differ between documents for the same quantity, the second the letters with several meanings, the third the marks and labels. Where a suspected source error bears on an entry, the flag is named, and an arrow (→) points to the pages that discuss the point. Words with several meanings (the gate, soliton, condensate, island, verdict C) are on the Glossary.

One quantity, two conventions#

J and K: which operator is “the real structure”. #11 §5 defines both, J = η⁻¹H and K = η⁻¹ω. #11’s P2′ writes η = HJ, “with the real structure J chosen spontaneously” (§4), as do #5 §B and the notebook’s v0.3 addendum. The toy note writes “η = ωK (K a real structure, K² = 1)”, with Kᵀ ω K = −ω (§1), and #11 §14 item 11 follows it: “η = ωK is the order parameter”. The two formulas give one η: on the compatible family H = diag(g, g⁻¹), the check review/wiki-build-checks/jk_check.py prints that HJ, ωK and the standard η agree, so the clash is one of naming; the documents differ in which operator they call the structure chosen spontaneously. The letter K also names different matrices: at d = 1 and H = 1 the same check prints #11’s K as [[−1, 0], [0, 1]], the negative of the toy note’s reference vacuum K₀ = diag(1, −1), and #11’s J as [[0, 1], [1, 0]], the toy’s K(φ) at φ = π/2 (review/wiki-build-checks/jk_check.out). Pages on #11’s postulates write J, pages on the toy models K. → Notation: which operator is “the real structure”; Which operator is “the real structure” (unreconciled); conflicts between sources
The Planck mass, reduced or not. Two conventions are in common use [T]: M_Pl = G^{−1/2} and the reduced M̄_Pl = M_Pl/√(8π); verify/soliton_obituary.py takes 1.22089×10¹⁹ GeV and 2.43536×10¹⁸ GeV. Any formula that combines the expansion rate with a power of the Planck mass has to use one of them throughout. The documents use them as follows.
- The soliton record names both: one defect per Hubble volume gives p = 104.31 with M_Pl and 0.828 with the reduced mass, and #14 v3’s “factor of 3” between conventions was an artefact of mixing them (round-3 ruling 2). → The reduced-Planck “factor of 3” was an artefact
- The Debt-3 notes write M_crit ≈ 3.6 M_P/(m√g) without defining M_P; the wiki reads M_P there as G^{−1/2}. → What is unchecked
- #1 §6 writes 3H²(M_P² + M̃²a⁻⁶) = ρ without stating its convention; the equation is the ordinary Friedmann equation when M_P is the reduced mass, which is how the wiki reads it, and the quintessence page writes H² = ρ/(3M_P²) with the reduced mass too. → The program’s extension: a second scale factor; Expansion and the Friedmann equation
- The round-3 referee quotes Gangui’s estimate with m_P. The scripts write the two masses as M_PL and M_PL_RED, and the referee’s output writes the reduced one as MPl_red.
- Related masses: M_g and M_f, the two Planck masses of bimetric gravity; M̃, #1 §6’s dual Planck mass; M_B, #1 §3’s Born mass scale, with “M_P fixed by M_B and the section volume”.
→ Two conventions for the Planck mass; Two Planck masses
μ and the gate’s exponent (flag 17). #1 §3 postulates the gate as e^{−k²/μ²}, with k = m_χv_rel/2 the relative momentum, and defines v* ≡ √2μ/m_χ (§5.1). Calc note #6 §4 derives the overlap e^{−(p₁−p₂)²/2μ²}, with μ the χ quantum’s momentum width, and calls it “the postulated gate”. #11 §8 writes F(k²/μ²) and fixes neither form, and #13 §4 combines #6’s fiber width with #1’s v*. “With p₁ − p₂ = 2k these differ by √2 in μ”, so with #6’s width the knee sits at v* = μ/m_χ (flag 17; review/wiki-build-checks/gate_conventions.py). No stated number changes. The flag’s audit found the √2 present only if #6’s form is read as the gate on the amplitude; read as the factor on the rate, it equals #1 §5.1’s e^{−v²/v*²} with #1’s μ, and the documents that identify that μ with the fiber width are consistent with #13 §4. → Two conventions, not reconciled; What μ is; conflicts between sources
The DSZ pairing, 2πℤ or 4πℤ. #11 and #14 write ⟨q, q′⟩ ∈ 2πℤ, Dirac’s normalization; Zwanziger’s abstract states 4πℤ (round-3 ruling 12, accepted as minor). With eg = 2πn the magnetic coupling is 1/(4α) = 34.26, with eg = 4πn it is 1/α = 137.04 (v3 worklog R2). #11 §11.2 writes the 34 as 4π/e² (flag 28). → Which charge counts: 2π or 4π; The convention behind 34
The square root in the bimetric potential (flag 14). The potential uses the eigenvalues of √(g⁻¹f); #7 §2’s “Tr(g²) = p₄ of the square-root eigenvalues” holds for √(f⁻¹g). The conclusion, that the metric tier lies outside the eₙ span (battery T7), stands. → Power sums and Newton’s identities
M_X or M_X² (flag 5). #11 §7.5 writes M_X ∝ 2(β₁+β₂); the β-combination fixes the squared Fierz–Pauli mass. → Step 5: the mass scale
The complex structure, ω⁻¹H or H⁻¹ω. #11 §5 defines I = ω⁻¹H; the round-3 referee writes Freidel, Leigh and Minic’s statement with I = H⁻¹ω. The wiki follows #11. → conflicts between sources
The sign of Δ. Δ(β) = ⟨x²⟩_vacuum − ⟨x²⟩_dual with Δ(0) > 0, the β = 0 vacuum broader in x; the worklog first had the sign reversed, and the round-4 referee confirmed the corrected one. → Superseded: the reversed sign of the polarization
The occupation number. Debt 3b writes N_occ = ρ/(m⁴σ³), as 3a does, and a peak occupation N_occ ≈ (2π)^{3/2}ρ/(m⁴σ³), and reconciles neither with the other. → Two normalizations in Debt 3b
The tadpole’s growth, ln N or 2 ln N (flag 11). → The coefficient: ln N or 2 ln N
Row numbers. #1 §9 and the Debt-3 notes number the predictions table one way, #9 and #11 another; the wiki uses #11’s numbers and calls the withdrawn rows “old row 3” and “old row 4”. → Numbering

Letters with several meanings#

Greek letters come first, in alphabetical order, then Latin letters.

α — the gate coupling α̃ (#1 §3); the fine-structure constant (#14 §2; Dirac quantization); the dead dark coupling α_d (HANDOFF.md §2; obituary); α_W′ and α_m (#14 §2); α_e,d (v3 worklog F2); the strong coupling α₃ and the running α_em (v3 worklog R2); the partially massless parameter (#11 §7.4); the phase in e^{−iαN} (toy note §3); the strength α of a fifth force relative to gravity (Yukawa forces).
β — the bimetric coefficients βₙ (#8; #11 §7.3); the relative phase of model II’s complex vacua (toy note §3); the β-function (toy note §1); the rate β of a first-order phase transition, the inverse of its duration, quoted as β/H* and entering the round-3 referee’s bubble count p ~ 104 (β/H)³ (gravitational waves from phase transitions).
γ, Γ — γ: the stream-instability growth rate (#1 §5.2); the Lorentz factor of the template and γ̃, the dual one (#13 §2); γ′, the hidden photon (round-3 audit); Euler’s constant (verify/toy_gw.py). Γ: the gated condensation rate Γ_self, named Γ_cond in #2 (#3); the gravitational rate Γ_gr (#3); a scattering or annihilation rate per particle (the transfer cross-section; thermal freeze-out); the Christoffel symbols Γ^λ_μν.
Δ — model II’s polarization Δ(β) (toy note §3); the excess ΔN_eff (#1 §6; #14 §4).
ε — the momentum scale ε = ℏ/λ (#1 §2; #13 §1); kinetic mixing (#11 §4; #14 §2); the small parameter of battery test T4 (verify/battery.py); a particle’s energy, ε(p; f) (#1 §5.2) and ε(x, p) (#2), with ε_int and ε_gate (#2); the antisymmetric symbols ε_abcd and εₐᵢⱼ.
η — the neutral metric, the order parameter under P2′; η_B, the baryon asymmetry (#1 §5.3); η_μν, the Minkowski metric, and η_ab, the frame metric (general relativity); η^{AB}, the neutral metric of double field theory’s doubled space, and the program’s η written with indices in the toy note’s section condition (§5).
θ — the Moyal parameter (toy note §3); a gauge theory’s vacuum angle (#11 §11.3; the Witten effect) and θ_d, the hidden one (round-3 audit); a scattering angle (the transfer cross-section); θ*, the angle the sound horizon at last scattering subtends today (the cosmic microwave background); θ_pol, the polarization tilt (#1 §8); the angles θᵢ of the XY model; the weak mixing angle θ_W (kinetic mixing); the angle a sound horizon subtends (baryon acoustic oscillations).
κ — Scenario A’s normalization κ ∼ 1 (#3); the annihilation coefficient, κ = 1 and 10, in the W′ relic estimate (v3 worklog O1; verify/soliton_obituary.py), which the round-3 audit writes c (W′ overclosure); the coupling κ ∝ 1/M̄_Pl of a spin-2 field to matter (Planck units).
λ — the Born length (#13 §1) and λ_χ, a conjugate quantum’s invariant length (#13 §4); the dual-diffeomorphism parameters λ_μ (#6 §2); model I’s compatibility coupling and model II’s quartic coupling (toy note §2, §3); the scalar self-coupling of the monopole calculation and of the Kibble–Zurek estimate (round-3 audit); λ_p, the Higgs portal (round-3 audit); λ_dB (#2); the eigenvalues λᵢ of √(g⁻¹f) (elementary symmetric polynomials); the range λ of a fifth force (Yukawa forces).
Λ — the Lagrangian Grassmannian Λ(d) (#1 §2; #11 §6); the cosmological constant (#11 §7.4, §10; the cosmological constant Λ; the cosmological-constant problem), whose page lists the letter’s other uses (the letter Λ); Λ_wedge (HANDOFF.md §3); a factor Debt 3b leaves undefined in Γ_gr (#3); Λ_c, #14 v2’s transition scale; ln Λ, the Coulomb logarithm (#14 §4).
μ — the gate width in momentum, in two conventions that may differ by √2 (#1 §3; #6 §4; flag 17, above); the coupling of model I’s ρ-preserving potential (toy note §4); μ², model II’s mass parameter (toy note §3); a parameter proportional to λ/g² in the round-3 referee’s monopole profile (review/round3/referee/step2b.py); a spacetime index (X_μν, λ_μ).
ξ — the hidden-to-visible temperature ratio (#14 §4); the coefficient of #1 §4’s p⁴ term, and ξ₁, ξ₂ in the standard photon-dispersion series; a correlation length and its frozen value ξ̂ (Kibble mechanism); the dimensionless radius of the round-3 referee’s monopole profile; the gauge vector ξ^μ of a coordinate change.
π, Π — π_μ, the would-be Goldstones X absorbs (#6 §3); π, a Goldstone field such as a membrane’s displacement (inverse-Higgs constraints); π₀, π₁, π₂, homotopy groups, with π₁ = ℤ the Maslov class of Λ(d); Π, parity on model II’s matrix base (toy note §3).
ρ — the Born rotation (#11 §3); an energy or mass density: in the Friedmann equation (#1 §6), in Γ_self and N_occ (#2; #3; #11 §9), ρ_DM (#14 §3), ρ_d and ρ_X (#3), ρ̃ and ρ_eff (the duality dictionary), ρ_vac (the cosmological-constant problem) and ρ_M (the monopole problem); the electric and magnetic charge densities ρ_e, ρ_m (Dirac quantization).
σ — the velocity dispersion of a halo (#2; #3), with σ_d and σ_X; cross-sections: σ_T(v) and σ₀ (#11 §8), σ_scat (#3), σ_T/M (#14 §4), and the annihilation σv and ⟨σv⟩, with σ₀ the constant σv of s-wave annihilation (v3 worklog O1; round-3 ruling 4; thermal freeze-out); σ_tot and σ_V (the transfer cross-section); the σ-modes of Λ(d) (#5 §D); standard deviations, as in “non-phantom compatible at ~2σ” (#11 §10).
τ — the dark-energy field (#1 §2; #11 §6), with τᵢ its initial value; τ_c, the complexified gauge coupling (#11 §11.3; S-duality), on which S-duality acts and which is unrelated to τ; τ_cond, the condensation time (#4); τ̃, the dual proper time (#13 §3); the proper time τ of special relativity.
φ, Φ — φ: model I’s polarization angle and the lattice angles φ_i (toy note §2, §4); model II’s field and its components φ_mn (toy note §3); the dual rapidity (#13 §2) and the template’s rapidity; the quintessence field; m_φ, the gate mediator’s mass (#1 §3); φ_i, a light field’s initial value in the misalignment mechanism. Φ: the hidden adjoint scalar (#14 §2); a generic vacuum Φ₀ (vacua as a coset); Φ_grav, the gravitational potential (#2).
χ — conjugate matter (#11 §8); χ_i, an initial field amplitude (#3); χ = v/M, a kinetic-mixing parameter in Brümmer and Jaeckel’s notation (v3 worklog R4; Planck-suppressed adjoint mixing).
ψ — the hidden Dirac doublet (#14 §2); a wavefunction (toy note §5); the alternating form ψ = Πφ (toy note §3); matter fields in the action S_matter[ψ; g].
ω — the symplectic form; a frequency in the Vlasov response, δf ∝ e^{i(k·x − ωt)} (#2), and in the dispersion relation ω² = k² + M² of massive X waves (chat-only results); ω_p, the monopoles’ magnetic plasma frequency (round-3 audit 4(c)).
Ω — model II’s oscillator frequency (toy note §1); the stirring rate Ω = 1/t_dyn and Ω_X (#3); the density fractions Ω_χ, Ω_b, Ω_DE, Ω_DM, Ω_x^(0), Ω_c and the product Ω h²; Ω_gw, the energy fraction in gravitational waves.
A, B, C — A, B, C, the proof blocks (#5 §A); A = [X, Y⁻¹] (#7 §2); A_μ, the Stückelberg vector; B, the constant symmetric matrix in the shift Z → Z + B (#6 §1), and the antisymmetric Kalb–Ramond B-field (#6 §2); C = KI + IK in model I’s potential (toy note §2); C, hidden charge conjugation (#14 §2), with C_A and C_X in Gherghetta et al.; C, the lowest referee grade.
c — the speed of light; Γ_self’s coefficient (#3); the W′ annihilation coefficient, “c=1..10” in the round-3 audit’s table and κ in the v3 worklog (W′ overclosure); a nonzero constant in the toy check of the section condition.
D — the Langmann–Szabo transform (toy note §3); the base dimension of a sigma model; D_x and D_p, the diameters of the chat-only relation; D_M and D_H, the distances of baryon acoustic oscillations.
E — the symplectic vielbein (#7 §3); an energy (#13 §4); the lattice energy (toy note §4) and that of the XY model; E1 and E2, the round-3 referee’s residual matrices; E_QG and E_Planck.
e — the electric charge (#11 §11.3) and e_d, v2’s dark charge; the elementary symmetric polynomials eₙ (#7; #8); the frame e and the vielbein e^a_μ (#8 §1).
F — the gate form factor (#11 §8); a force and the dual-boost parameter (#13 §2); a monopole flux (#14 §4); the free energies F(±) (toy note §3); the Fourier transform (toy note §5); field strengths F_μν. The labels F1–F6 (round 4) and F1, F2 (v3 worklog) name different findings.
f — τ’s period scale (#1 §2); the second frame or metric (#8 §1); the monopole mass factor (#14 §3); a distribution function (#2), with f₀ the Maxwellian and f̄ the average (#1 §5.2); model I’s couplings f_H, f_K and a sigma model’s coupling f; model II’s matrix base f_mn (toy note §3); a test function in the section condition (toy note §5); frequencies f* and f₀ of phase-transition gravitational waves.
G — Newton’s constant (#3; #13 §1); the Gaussian gate (#2); G_eff (#1 §6); the propagator G_ij and its entries G₀ₖ, Gₖ₀ (round-4 referee’s notes); a symmetry group G and its stabilizer G₀; the Einstein tensor G_μν.
g — the spacetime metric, g = Im Z (#11 §6), and g_μν; the first metric of bimetric gravity; the gated contact coupling, in GeV⁻² (#2; #3), also in M_crit ≈ 3.6 M_P/(m√g); a magnetic charge (#11 §11.3), with g_D the Dirac charge and g_m the hidden monopoles’ charge; a gauge coupling, as in τ_c = θ/2π + 4πi/g², with g_d the hidden coupling (#14 §2) and g′ the hypercharge coupling; degrees of freedom, g* and g*s, g_h and g_vis; a test function (toy note §5); an element of a symmetry group; a sigma model’s target metric g_ab. → the gated Vlasov model’s symbol collisions
H — the generalized metric (#11 §4); the Hubble rate (#1 §6; #14 §1), with H₀, H* and H(z); the oscillator ½x² of model II (toy note §3); hypothesis H (#14 §2); the Higgs doublet H of the Higgs portal (round-3 rulings); the scalar profile H(ξ) of the round-3 referee’s monopole; the H-theorem of kinetic theory (the collisional route); a subgroup, in the coset G/H of the sigma-model page’s summary, which its text writes G/G₀.
h — the metric perturbation (#7 §2) and h_μν; Planck’s constant h in the chat-only diameter relation (HANDOFF.md §4); the potential strength of the lattice model (toy note §4); h in Ω h²; h = 2π/e, the Dirac magnetic charge in the round-3 referee’s note on Preskill 1979; h′, the hidden Higgs boson.
I — I = ω⁻¹H (#11 §5), written H⁻¹ω where the round-3 audit restates Freidel–Leigh–Minic; I[Z] (#7 §2); the matter coupling I[W] (toy note §5).
J — J = η⁻¹H (#11 §5), the real structure of #11’s P2′; J₀, the vacuum polarization (#7 §2); the lattice stiffness (toy note §4) and the XY coupling.
K — K = η⁻¹ω (#11 §5); the toy note’s dynamical real structure, with K₀, K_ref and K(φ) (toy note §2, §4), which at d = 1 is a different matrix from #11’s K (above); the gauge profile K(ξ) of the round-3 referee’s monopole (review/round3/referee/step2b.py).
k — the gate variable k = m_χv_rel/2 (#1 §3); a wavevector (#2), with k_J the Jeans wavenumber; a light-like vector in the null energy condition; the spatial curvature, and k̃, the dual one (#1 §6, §10).
L — a Lagrangian subspace, the eigenlines L± and the worklog’s L and L̃; the lattice size (toy note §4); model I’s Lagrangian (toy note §2); a rest length L₀; the label L1 (v3 worklog).
M, m — M_X, the mass of X (m_X in #11 §12); M_B; M_P, M_Pl, M̄_Pl and m_P (above; Planck units); M, the mass of a Yukawa mediator (Yukawa forces); m_inert and m_grav (the equivalence principle); M̃; M_f and M_g; M* and M_*; M_core; M_crit, the critical mass of a condensate, distinct from m_crit, the classicality threshold; the amplitude M(k, q) (#11 §8); the monopole mass M (#14 §3); M_W, the W mass; the Monte Carlo’s U(1) order parameter M and its ρ-odd m (toy note §4); m and M in the Hassan–Rosen prefactor 2m²M²; m_χ, m_φ, m_FP², m_g, m_W′ and the round-3 audit’s m_psi; m, the particle mass of the Debt-3 notes; m and n as matrix indices (toy note §3).
N — the occupation number N_occ (#2); the lapse (#1 §6); the photon count of the chat-only relation (HANDOFF.md §4); model II’s truncation and its level-number operator (toy note §3); N_eff; the number of e-folds; the N = 4 of super Yang–Mills theory (#14 §2).
p, P — momentum, and p₁, p₂ in #6’s gate overlap; the Kibble normalization p (#14 §3; #11 §12); the power sums pₖ (#7 §2); pressure, in the perfect fluid; the postulates P1–P5.
q — the momentum transfer (#1 §3); a charge q = (e, g) (#11 §11.3); the occupations q₀, q₁, q_m (toy note §3); q_vis (round-3 audit).
R — #1’s rotation matrix (#1 §2); R_B[H] (#1 §2); the curvature scalar R; R_X (#3); Segue 1’s ratio R (#4); the labels R1–R7.
r — a radius, as in r_s = 2GM/c²; r = M/v = 4πf/g_d (#14 §3); r_d, the sound horizon at the baryons’ release, and r*, the sound horizon at last scattering, with r_D the photons’ diffusion length (the cosmic microwave background).
S, s — the action and its parts S_H, S_τ, S_x, S_p, S_mix (#1 §3), S_EH and S_matter; the block S (#7 §2); S ≡ σ₀/m (#2); a symplectic matrix S, and the referee’s path S(t) (toy note §2; round 4); the terms Sₙ (#8 §1); the criteria S0–S5; the S-duality generator; S_min, model II’s least action, and S_r, the round-4 referee’s real-model action; s, the entropy density, with s₀ today; s₁, s₂, s₃ in verify/toy_coset.py’s symplectic matrix.
T — the tadpole T = β₁ + 3β₂ + 3β₃ + β₄ (#8 §2); temperatures: T, T_c, T_RH, T_h, T_x, T_ν, T_γ, T_f, T* and the Monte Carlo’s T (toy note §4); the battery tests T1–T7; T̃, the dual stress (#5 §E); T_μν, the stress-energy tensor; T-duality and the SL(2, ℤ) generator T (#11 §11.3); Lakatos’s sequence of theories T₁, T₂, T₃ (Lakatos’s methodology); the round-4 referee’s tadpole T₀(N).
U — the stabilizer U(1, d−1) (#6 §3) and the unitary groups of U(d)/O(d); the Binder cumulant (round-4 referee’s notes); U-duality (#11 §11.3); the momentum-space interaction U(p − p′) (#1 §5.2).
V, v — potentials: V(τ) (#1 §3), the bimetric V (#8 §1), model I’s V_b, V_μ, V_λ and V′ (toy note §4; round 4), the quintessence V(φ); v: a velocity, v*, v_rel; the hidden Higgs vacuum value v (#14 §2).
W, w — the Wigner function (toy note §5); the hidden vectors W′± (#14 §2) and the W boson; w, the equation of state, with w̃, w_eff, w₀ and wₐ.
X — the momentum graviton (#5 §C), with X_μν; the mean-field parameter X ≡ gn/(mσ²) (#2; #3) and the environment label in R_X, Ω_X, ρ_X (#3); X as a heavy particle charged under both groups in the Chen–Cline–Frey wording quoted in round-3 ruling 7; 𝕏 = √(g⁻¹f), the frame ratio (#8 §1); x, the position, and X’s expansion variable in the metric tier (#7 §2). → Scenario B: the mean-field parameter X
Y — Im Z (#5 §C); the yield n/s (#14 §3), with Y_eq and the freeze-in yield Y_FI in the round-3 audit; Y_p, the primordial helium mass fraction (Big Bang nucleosynthesis).
Z, ℤ — the modular field Z (#5 §C); the integers ℤ, with ℤ₂ and ℤ₄ (#1 §2; toy note §2, §4).

Marks and labels#

GW — Grosse–Wulkenhaar on this wiki; gravitational waves in several documents (#1 §9, #11 §7.5, round-3 ruling 5), always written out here.
† — the citation mark, a reference cited from model memory; the Hermitian adjoint inside the toy note’s formulas.
“The gate” — the velocity gate of conjugate matter’s self-interaction, and the build check scripts/check.sh. → A name collision
Tier — #11 §2’s claim tiers (Tier 1 explicit algebra, Tier 2 literature, Tier 3 novel structural physics), and the two tiers of the toy models’ success criteria, “the letter” and “the spirit” (success criteria); also #7’s “metric tier” and “frame tier” of the tuning test.
Appendix A — in HANDOFF.md, the code of the verification battery; in #11, the audit of the eleven founding suppositions.
Rows 3 and 4 — granularity and Fornax friction in #1 §9 and the Debt-3 notes; the island discriminant and dark energy in #9 and #11.
Open problem 5 — #1 §11’s item 5, the relic abundance, closed on the wave island; #11 §14’s item 5 (OP5), the dual-stress enhancement.

Sources#

papers/broken-born-checkpoint-v2.md §3–§14 — the current symbols, from ρ and the three structures through the modular field, the bimetric potential, the gate, cosmology and the charge lattice; m_X in §12
papers/reciprocity-toy-models.md §1–§5 — the toy models’ symbols, including K as the real structure with η = ωK
papers/conjugate-special-relativity.md §1–§5 — c̃, λ, λ_χ and the dual kinematics
notes/broken-born-theory.md §0–§8 — the lab notebook’s definitions: R, ρ², J as order parameter, τ, the action, χ and the gate variables, the cosmology with its unstated Planck-mass convention
notes/debt1a-geometric-core.md, notes/debt1a-ii-X-classification.md, notes/debt1a-iii-tuning-test.md, notes/debt1a-iv-beta-coefficients.md — H_Z, the shift symmetry, π_μ, the misalignment blocks, E, 𝕏, βₙ, T and the gate overlap with its width μ
notes/debt3a-stream-stability.md, notes/debt3b-wave-island.md, notes/debt3c-ufd-endpoint.md — the momentum-clumping symbols: g, G, n_gate, N_occ, Γ_self, R_X, κ, M_crit with its undefined M_P
notes/knot-dark-matter-standalone.md §2–§4 — the retired candidate’s hidden-sector symbols
notes/toy-lagrangian.md — Π, L and L̃, dS₂, S0–S5
review/wiki-build-flags.md items 2, 3, 4, 5, 10, 11, 14, 17, 19, 23, 27, 28 — the suspected source errors named in the entries
review/wiki-build-checks/jk_check.out and review/wiki-build-checks/gate_conventions.out — the printed checks on the J/K naming and the two gate conventions
review/v3-worklog.md header, R2, R7, F2, O1, O2 — the NOTE lines, the 2π/4π convention, M_f and M_g, α_e,d, κ
review/round3/referee/audit-notes.md and review/round3/referee/step2b.py — the round-3 referee’s notation: γ′, θ_d, q_vis, Y, λ_p, E1 and E2, the monopole profile
review/round4/referee/audit-notes.md and review/round4-adjudication.md F1–F2 — V_μ, V_λ, V′, D₋, S(t), G_ij, T₀(N), U
review/round3-adjudication.md rulings 1, 2 and 12 — the Kibble normalization in both Planck-mass conventions, and Zwanziger’s 4πℤ
HANDOFF.md §2–§5 — the artifact numbers, α_d, Λ_wedge, the chat-only D_x, D_p and N, and the battery labels
verify/soliton_obituary.py, verify/toy_coset.py, verify/toy_gw.py, verify/battery.py — the two Planck masses, K₀ and K_ref, Π, the T4 expansion parameter, the value of f