The Framework

Abstract

We propose a minimal axiomatic framework in which physical time is a complex variable τ ∈ ℍ² and the photon is topologically an elliptic curve with modular parameter τ. Modular invariance under PSL(2,ℤ) is then not a postulate but a classical consequence of the theory of modular forms — the moduli space of elliptic curves over ℂ is precisely ℍ²/PSL(2,ℤ). PSL(2,ℝ) invariance fixes the field-space metric only up to an overall positive scale; the canonical normalization (hyperbolic metric, Gaussian curvature K = −1 in Planck units) is stated as an explicit normalization postulate N — the framework's only continuous input. Given N, the α-attractor inflation parameter is fixed at α = 2/3 (3α = 2, one of the discrete benchmark values of the α-attractor literature). The result is a consistency relation between two CMB observables, r = 2(1 − nₛ)². For Planck 2018 (nₛ = 0.9649) this gives r ≈ 0.0025 ± 0.0006; for combinations including ACT DR6 (nₛ ≈ 0.971–0.974), r ≈ 0.0013–0.0017. Testable by LiteBIRD (σ(r) ≈ 0.001, ~2032); fully separating the T-model branch from Starobinsky R² (rᴋᴿᴛ/rStar = 2/3 at fixed nₛ) requires CMB-S4-class sensitivity.

In the particle sector, the empirical Koide relation Q = 2/3 is equivalent to a cone condition with half-angle θ = π/4; under that assumption m_τ = 1776.97 MeV (agreement at the 0.9 σ level). Deriving θ = π/4 from observer projection (A3) remains an open problem (P12a). The absolute electron mass (P12b), the relation of the S³ scale to the cosmological constant (P15), RG running as a geodesic shift (P16), and the projective reading of the fourth topological datum (P17) are stated as candidate, hypothesis, and program — not results.

Meta-postulate M0 (structural realism)

Physical observables are functions on isomorphism classes of physical states, not on their representatives. Equivalently: if two configurations differ only by an isomorphism of their internal structure, no physical measurement distinguishes them. This is the meta-level assumption under which all three axioms below operate — and it is also the assumption from which Theorem 1 derives PSL(2,ℤ)-invariance (see the paper).

M0 is weaker and more general than full PSL(2,ℤ)-invariance: it does not say which symmetry holds, only that physics cares about structure rather than labels on representatives. The same assumption is used implicitly throughout gauge theory.

The Three Axioms

A1. Complex time. The fundamental time variable is $\tau \in \mathbb{H}^2 = {\tau \in \mathbb{C} : \text{Im}(\tau) > 0}$.

A2. Photonic topology. The photon is topologically identified with S¹ × S¹ equipped with complex structure τ — i.e., with an elliptic curve $E_\tau \cong \mathbb{C}/(\mathbb{Z} + \mathbb{Z}\tau)$. One S¹ factor corresponds to the photon's U(1) gauge symmetry, the other encodes the internal phase cycle parametrized by Im(τ). The whole chain M0 → moduli space → PSL(2,ℤ) is available only over ℂ: quaternions are excluded (they are noncommutative, carry no theory of holomorphic functions and no analog of the modular curve ℍ/PSL(2,ℤ)). (Change relative to versions through v4, where space was introduced as a "quaternionic extension of τ"; from v5 the entire derivation chain is built exclusively over ℂ — see Remark 1 in the paper.)

A3. Observer projection. A classical observer measures Re(τ). Measurement is a projection from the full complex τ onto the real axis.

Modular invariance under PSL(2,ℤ) is not an axiom. It is a theorem (Theorem 1) derived from A1 + A2 + M0 — the moduli space of elliptic curves over ℂ is ℍ²/PSL(2,ℤ), so observables factor through this quotient. See the paper and proc-psl2z.md for the full derivation.

Postulate N (kinetic-metric normalization), narrowed to A2⁺. PSL(2,ℝ) invariance fixes the shape of the metric on ℍ², but not its scale — invariance alone. The scale, however, is not a free numerical fiat: the Weil–Petersson metric on the moduli space of elliptic curves M₁,₁ (whose quotient is X(1) — A2) equals the Poincaré metric with no arbitrary modular-invariant conformal factor, by a classical theorem (Wolpert 1985; Zograf–Takhtadzhyan 1987) — see lemma-poincare-uniqueness.md §O. The residual input is a single one-line geometric postulate A2⁺ (the field's kinetic tensor = the WP metric on M₁,₁), plus the Planck-unit convention for the overall scale — not a free choice of K itself. (Through v1.4, K = −1 was presented as a consequence of Theorem 1; from v1.5.0 as postulate N; from 2026-07 narrowed to A2⁺, see below.)

Derivation Chain

Step 1: Hyperbolic geometry from Theorem 1

Up to an overall positive factor, the only Riemannian metric on ℍ² invariant under PSL(2,ℝ) ⊃ PSL(2,ℤ) is the Poincaré metric: $$ds^2 = \lambda,\frac{dx^2 + dy^2}{y^2}$$ The shape of the metric is thus forced by invariance; the normalization choice λ = 1, i.e. constant Gaussian curvature K = −1 in Planck units, is the content of postulate N (above) — the framework's only continuous input.

Step 2: The fundamental domain

PSL(2,ℤ) identifies equivalent points in ℍ². The fundamental domain 𝒻 has hyperbolic area: $$\text{Area}(\mathcal{F}) = \frac{\pi}{3}$$ (from Gauss-Bonnet for the modular orbifold with Euler characteristic χ = −1/6).

Step 3: α = 2/3

The α-attractor class of inflation models (Kallosh & Linde 2013) places the inflaton on a negatively curved field space with curvature $K_\text{fields} = -2/(3\alpha)$. If cosmological evolution is dynamics on the τ-plane, then $K_\text{fields} = K = -\lambda$ (under postulate N: λ = 1), giving: $$\alpha = \frac{2\lambda}{3} ;\xrightarrow{;N:;\lambda=1;}; \frac{2}{3} = -4\chi(\mathcal{X}_1) = \frac{2,\text{Area}(\mathcal{F})}{\pi}$$ The ratio 3α = 2 (one of the discrete benchmark values in the α-attractor literature) is topological — the shape is forced by invariance — but the absolute value of α is fixed only by the continuous normalization N (λ = 1). So α is neither a free fit nor a pure topological invariant: it is a topological ratio × postulate N.

Step 4: The prediction

From universal α-attractor formulae, eliminating the e-folding number Nₑ: $$\boxed{r = 2(1 - n_s)^2}$$

• Planck 2018 ($n_s = 0.9649 \pm 0.0042$): r ≈ 0.0025 ± 0.0006.
• P-ACT (Planck + ACT DR6) ($n_s ≈ 0.971$–$0.974$): r ≈ 0.0013–0.0017. The higher nₛ implies a tension in the e-folding number (Nₑ ≈ 69–78, above the standard window).

Model Comparison

At fixed nₛ, rᴋᴿᴛ/rStar = 2/3. LiteBIRD (σ(r) ≈ 0.001) discriminates the T-model branch from the strongly suppressed (fully PSL(2,ℤ)-invariant) branch where r ~ 10⁻⁵; fully separating it from Starobinsky R² requires CMB-S4-class sensitivity.

What Would Falsify This

• LiteBIRD measures r > 0.01 or r < 0.0005 → α = 2/3 ruled out
• LiteBIRD measures r ≈ 0.004 → Starobinsky preferred, τ-geometry needs revision
• A formal error found in the chain Theorem 1 → shape of the metric → (postulate N) → α = 2/3
• Effective action for the inflaton fails to reproduce α-attractor dynamics

What would NOT falsify this: speculative addenda independent of the main r prediction — the cosmological φ(z) branch (LRDs/JWST) and, from v5, also the hadron mass defect from observer projection (IR addendum: the projection P onto the Stab(i)-invariant yields ~98% of hadron mass as the gluonic contribution / QCD trace anomaly). The collapse of either leaves the core (the r-relation) unchanged.

The single continuous input (status, v4)

The kinetic term of the effective action is the Poincaré metric on ℍ²: the Euler–Lagrange equations exist explicitly, the canonical inflaton φ = ln(τ₂) gives the standard slow-roll α-attractor, and the prediction r = 2(1−nₛ)² follows directly from the equations of motion. What this forces is the shape of the metric.

Status sharpening (v1.5.0 → v4/v1.6.0). Earlier the curvature K = −1 was presented as a direct consequence of Theorem 1. But PSL(2,ℝ) invariance fixes the metric only up to a positive scale, so the normalization K = −1 in Planck units is now carried explicitly as postulate N: the single place a continuous choice enters. Either N is derived from first principles, or it is accepted as the framework's one input. Status of the core (the r-relation conditional on N): formal.

Further sharpening (2026-07). The question above is partly answered: lemma-poincare-uniqueness.md §O shows that, of five candidate routes to forcing f(τ)≡1, one (the Weil–Petersson metric on M₁,₁) closes rigorously, modulo a single one-line postulate A2⁺. N therefore does not dissolve entirely, but reduces from a bare numerical choice to one named geometric postulate plus a unit convention. Status: 🟡 (P11 in 00-status.md, conservatively — A2⁺ itself is not derived from A1+A2+M0).

Theorem 3 proven (2026-04-18)

The algebra of rotation generators [Xₐ, X_b] = 2ε_{abc}X_c (su(2), i.e. the imaginary unit quaternions as generators of SU(2)) forces a physical space with Ricci tensor Ric_ab = 2δ_ab and scalar curvature R = 6. This manifold is S³ = SU(2) with a bi-invariant metric. Flat space is algebraically excluded. (Note: this su(2) structure of physical space is separate from the holomorphic chain on the time plane τ, which lives exclusively over ℂ — see A2; here SU(2) appears as the rotation group, not as a quaternionic extension of τ.) Field space ℍ² (K = −1, Theorem 1) and physical space S³ (K = +1, Theorem 3) have complementary curvatures, summing to 0. See ricci-theorem-3.md. Status: 🟢 formally proven.

Λ_eff ≈ 10⁻¹²² · M⁴_Pl: From Theorem 3 via the vacuum Einstein equations with the Hubble radius as a_S³, order-of-magnitude agreement with the observed Λ.

Theorem 2 completeness (2026-04-18)

Exactly 4 modulation regimes on X(1) = ℍ²/PSL(2,ℤ): the cusp (∞), i (Z₂), ρ (Z₃), orientation. Completeness follows from the classification of stabilizers of Fuchsian groups (Katok 1992): in PSL(2,ℤ) only orders {2, 3} exist, by integrality of the trace. The number of forces = 4 is a topological consequence. See theorem-2-completeness.md. Status: 🟢 formally proven (the count). The attribution hypothesis (force ↔ point) remains a candidate.

P3 (QM limit) solved (2026-04-18)

Quantum mechanics is a derived effective description of τ-geometry:

• State ψ = e^(iS/ℏ) = accumulated Im(τ) phase
• Schrödinger equation = Hamilton-Jacobi in this variable
• Born rule = the A3 projection gives |ψ|² = intensity
• Heisenberg = Fourier duality of the Re↔Im projections of e^(iΘ)

See qm-limit-p3.md. Status: 🟢 formally proven.

The potential problem and its resolution (2026-04-18)

Numerical analysis shows that the fully PSL(2,ℤ)-invariant potential V = Λ⁴tanh²(H(τ)/2) with H = −ln(τ₂|η|⁴) does not give r = 0.0025, but r ~ 10⁻⁵. The standard α=2/3 T-model V = Λ⁴tanh²(φ/2) gives r = 0.00239 for N=57 ✓. The paper needs to specify the T-model explicitly as an effective potential. See problem-potencial.md and slow-roll-numericky.md. Status: 🟢 documented, requires a paper revision.

Theorem 4: Nₑ = 60 = |A₅|, topologically forced (closed 2026-04-21)

The number of e-folds Nₑ = 60 is not a fitted parameter but the subgroup index [PSL(2,ℤ):Γ(5)] = |PSL(2,5)| = |A₅| = 60. The two remaining gaps were closed on the same day:

• Configuration A (i a saddle, ρ the global minimum, ∞ the maximum) follows for the canonical modular-invariant potential V_A = Λ⁴·|j(τ)| directly from complex analysis — the maximum-modulus principle (Ahlfors) plus the minimum-on-zeros principle (Conway) — with no Hessian computation.
• 60 = 10 × 6: A₅ has 20 elements of order 3, giving 10 order-3 axes after quotienting by 2 (SymPy-verified); Klein's (1884) spherical picture identifies these axes with the 10 order-3 vertices of the chiral (2,3,5) Schwarz tessellation of S², where 6 triangles surround each vertex.

Consequence: for the canonical potential V_A, r = 2/900 ≈ 0.00222 is fully parameter-free — no slow-roll fit, no open numerical computation. This is the first fully parameter-free TOE-level cosmological prediction in this framework. What remains is only the question of the canonical choice of V among modular-invariant potentials (an alternative choice could give Configuration B) — a question of choice, not of validity for V_A. See theorem-4-m1-dyn-uzavreni.md and m1-dyn-b-closure.md. Status: 🟢 formally proven for V_A.

Particle sector: the Koide relation (v1.6.0)

The empirical Koide relation for the three charged-lepton masses $$Q = \frac{m_e + m_\mu + m_\tau}{(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau})^2} = \frac{2}{3}$$ is equivalent to a cone condition with half-angle θ = π/4. Under that condition, $$m_\tau = 1776.97\ \text{MeV},$$ a deviation of 0.11 MeV ≈ 0.9 σ from experiment (1776.86 ± 0.12 MeV). This is a postdiction (the Koide relation predates the precise mτ measurement); the specific KRT claim is not the number itself but the derivation of Q = 2/3 from observer projection A3 — the open problem P12a.

Open problems and program (status index, v1.6.0)

Version 1.6.0 introduces a binding status index. The core (the relation r = 2(1−nₛ)², conditional on postulate N) is formal; the following are explicitly flagged as candidate / hypothesis / program, not results:

• P12a — θ = π/4 from A3. Derive the Koide cone condition from observer projection. Status: OPEN.
• P12b — absolute electron mass. One-loop QED matching protocol: the geometric value is read as a running mass m(μ); a 0.31 % deviation from the pole mass fixes μ/mₑ ≈ 1.25; candidate scales are pre-registered ahead of full evaluation. Status: CANDIDATE.
• P12c — strange-quark generation balanced in the neutrino subsystem. Status: CANDIDATE / program.
• P15 — hierarchy a/ℓ_Pl ≈ 10⁶¹. The relation of the S³ spatial-curvature scale to the cosmological constant is so far a consistency observation, not a derivation.
• P16 — RG running as a geodesic along Re(τ) = 0 (A3 as the carrier of renormalization running); candidate point τ = iπ/2. Status: HYPOTHESIS.
• P17 — projective structure of the four data: three local data as the canvas, the orientation choice as the center of projection. Status: PROGRAM.

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