τ-Geometry: a topological derivation of α-attractor inflation with α = 2/3

Synopsis pro externí recenzi. Verze 1.3, 20.4.2026 (sync s paper v1.3.0 + korpus 2026-04-20 pozdě odpoledne). Autor: Adam Porybný (nezávislý), v kolaboraci s Claude/Anthropic (formalizace a verifikace). Plný korpus: theory.yrx.cz/vyzkum

📄 PDF verze (LaTeX): synopsis-krt-math.pdf (regenerace v1.3 pending)

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Abstract

We present a minimal axiomatic framework in which physical time is a complex variable τ ∈ ℍ² (upper half-plane) and a photon is identified topologically with an elliptic curve of modular parameter τ. Under a meta-postulate of structural realism (observables as functions on isomorphism classes), modular invariance under PSL(2,ℤ) is not postulated but derived as a theorem (Theorem 1), since the moduli space of elliptic curves over ℂ is ℍ²/PSL(2,ℤ). The hyperbolic field-space metric (K = −1) follows from PSL(2,ℝ)-invariance, fixing the α-attractor parameter to α = 2/3 as a topological invariant of the modular curve X(1). This yields a parameter-free consistency relation:

$$r = 2(1 - n_s)^2$$

For n_s = 0.9649 (Planck 2018): r ≈ 0.0025, testable by LiteBIRD (~2032).

Second parameter-free prediction (v1.3.0). The Koide relation Q ≡ (Σ√m_k)² / Σm_k for charged leptons equals 2/3 iff the √m-vector subtends angle π/4 with the democratic axis (spinor half-angle). Identifying this 2/3 with the α-attractor exponent from Theorem 1 yields the prediction m_τ = 1776.97 MeV (PDG 2024: 1776.86 MeV, relative deviation 6·10⁻⁵) from m_e, m_μ as inputs. This is a structural relation on pole masses (RG-invariant on-shell), not a fit parameter.

Additional formal results (v1.2.0–v1.3.0, post-paper updates).

1. Theorem 2 (Fuchsian completeness) 🟢 — the set of qualitatively distinct orbit types of PSL(2,ℤ) on ℍ² together with the global orientation of Im(τ) has exactly four elements, by Katok's classification of Fuchsian groups (Katok 1992, Thm 2.5.6).

2. Attribution (P9) 🟢 (2026-04-19, promotion of former Conjecture 2) — the bijection {gravity ↔ cusp, strong ↔ ρ, EM ↔ i, weak ↔ Z(B₃) centre} follows from the short exact sequence 1 → Z(B₃) → B₃ → PSL(2,ℤ) → 1 (Birman 1974 Thm 1.25) + classification of PSL(2,ℤ) conjugacy classes (Serre 1973 §VII.1) + experimental force signatures (Wu 1957). Quantitative couplings α_em, α_s, G_F remain open (P4 numerical 🟡).

3. Theorem 3 🟢 — quaternion-frustration-induced curvature of 3-space, yielding S³ = SU(2) with Ric_ab = 2δ_ab and Λ_eff ~ 10⁻¹²² via S³-radius ~ Hubble horizon.

4. Poincaré uniqueness lemma 🟢 — on X(1) with complete, finite-volume, constant-curvature metric, the Poincaré form is unique up to scaling (classical uniformization + orbifold Teichmüller rigidity of signature (0; 2, 3, ∞)); closes external-review objection N2, with residual gap P11 (physical justification of the three conditions from A1+A2+M0, reduced 2026-04-19 via Weil-Petersson identification under A2⁺) explicitly flagged.

5. Corollary T2.1 (π as invariant of X(1)) 🟢 (2026-04-20) — the orbifold Gauss-Bonnet identity π = π/2 + π/3 + 0 + π/6 decomposes π as sum of stabilizer angles (π/|Stab(i)|, π/|Stab(ρ)|, π/|Stab(∞)|) plus fundamental-domain area, matching the four topological data of Theorem 2. π is a structural invariant of moduli of elliptic curves, not an external constant.

6. P12a 🟢 (2026-04-19) — Mp(2,ℤ) is the ℍ-refinement of PSL(2,ℤ) via Weil representation (Weil 1964, Folland 1989). Spinor-observer is an element of Weil rep (Shimura 1973). Bridge to quaternionic substrate.

7. P14 core 🟢 (2026-04-20) — Dirac equation is Hamilton-Jacobi for ℍ-phase. Cl(1,3) signature derived from asymmetry between postulates ℍ-2 (Re-motion continuous geodesic) and ℍ-3 (Im-axes discrete π/2 collapses). Explicit quaternionic Lagrangian L_ℍ = Ψ̄(iγ^μ ∂_μ − m/ℏ)Ψ yields Klein-Gordon by direct computation from Cl(1,3) anticommutation. Lorentz invariance via Spin(1,3) ≡ SL(2,ℂ) as continuous extension of the Mp(2,ℤ) ↪ Mp(2,ℝ) ⊂ Mp(2,ℂ) tower.

8. P17 core 🟢 (2026-04-20) — Fifth level of the KRT QM tower: arithmetic SL(2, Ω) via Clifford-arithmetic construction in Cl(1,9), where Ω ≅ E₈ is the Coxeter-Dickson integer lattice in 𝕆 (Hurwitz 1898: 𝕆 is the last normed division algebra). Sequentiality of the tower via Cayley-Dickson integer lattices ℤ ⊂ ℤ[i] ⊂ Hurwitz 𝓗 ⊂ Ω (apparatus §0.5 of p17-roadmap.md). 𝕆-Dirac equation L_𝕆 = Ψ̄(iΓ^μ ∂μ − m/ℏ)Ψ with Ψ in the 32-real Cl(1,9)-representation; Klein-Gordon is an algebraic identity since Cl(1,9) ≅ ℝ(32) is associative. Lorentz invariance via Spin(1,3) ⊂ Spin(1,9) with explicit choice of ℂ ⊂ 𝕆 (P8-consistent canonical imaginary axis). Algebraic identity L_Ω|{q₂=0} ≡ L_𝓗 verifies sequential reduction to P14. Bonus: Spin(6) ≅ SU(4) ↓ SU(3) × U(1) gives SM internal gauge content of one generation. Singh's α = 1/137 discarded (Rev 2026-04-20 b, AXIOM 1): Singh's normalization choices are not KRT-native; P4 numerical α_em remains open outside P17 scope. Steps D (Stab(i) in 𝕆-level) and E (phenomenology) are explicit gaps (Rule 7).

Axiomatic stabilization. A reformulation of axiom A3 ≡ S-reflection (observer measures Stab(i)-invariant rather than Re(τ)) makes the T-model effective potential uniquely forced rather than empirical; this interpretively closes P8. Appendix A of paper v1.2.0+ derives Schrödinger, Born, and Heisenberg from τ-geometry in the appropriate limit — photon quantization need not be imposed externally on A2; QM is read off, not postulated.

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1. Axioms

M0 (structural realism, meta-postulate). Physical observables are functions on isomorphism classes of states, not on their representatives.

A1 (complex time). The fundamental time variable is τ ∈ ℍ² = {τ ∈ ℂ : Im(τ) > 0}.

A2 (photonic space). A photon is topologically identified with S¹ × S¹ equipped with complex structure τ — an elliptic curve over ℂ with modular parameter τ.

A3 (observer projection, A3 ≡ S-reflection). A classical observer measures the Stab(i)-invariant of the modular geometry. Along the canonical geodesic τ₁ = 0 this is φ(τ) = ln|τ₂| = signed hyperbolic distance from i. The original wording "observer measures Re(τ)" is equivalent along the geodesic through τ = i (where Re(τ) = 0) but is not PSL(2,ℤ)-invariant globally, hence ill-defined under M0 + Theorem 1. The reformulation commits only to Stab(i) = ⟨S: τ → −1/τ⟩, a ℤ₂ subgroup, which matches the topology of a classical observer (one entity, two-sided). This reformulation (2026-04-19) makes the T-model effective potential the unique choice rather than an empirical fit.

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2. Theorem 1 — Modular invariance

Theorem 1. Under M0 + A1 + A2, all physical observables on the space of photon states are functions on the quotient ℍ²/PSL(2,ℤ).

Sketch of proof. The moduli space of elliptic curves over ℂ is well-known to be ℳ_{1,1} ≅ ℍ²/PSL(2,ℤ) (see e.g. Diamond–Shurman 2005, A First Course in Modular Forms, §1.3). Two elliptic curves with τ-parameters related by a PSL(2,ℤ) transformation are isomorphic as complex tori. By A2 (photons = elliptic curves) and M0 (observables factor through isomorphism classes), physical observables on photon states factor through ℍ²/PSL(2,ℤ). ∎

Consequence. The unique PSL(2,ℝ)-invariant Riemannian metric on ℍ² is the Poincaré metric:

$$ds^2 = \frac{dx^2 + dy^2}{y^2}$$

This has constant Gaussian curvature K = −1. In α-attractor terminology (Kallosh–Linde 2013), K = −2/(3α) implies α = 2/3.

Topological verification via Gauss-Bonnet. The fundamental domain 𝒻 of X(1) = ℍ²/PSL(2,ℤ) has area

$$\text{Area}(\mathcal{F}) = -2\pi\chi(\mathcal{X}_1) = \frac{\pi}{3}$$

where χ(𝒳₁) = −1/6 is the orbifold Euler characteristic (Hain 2014, Lectures on moduli spaces of elliptic curves, §4). Hence

$$\alpha = \frac{2,\text{Area}(\mathcal{F})}{\pi} = \frac{2}{3} = -4\chi(\mathcal{X}_1)$$

— a topological invariant, not a fit parameter.

Corollary T2.1 reference (π as invariant of X(1)). The same orbifold Gauss-Bonnet identity yields, by decomposing the fundamental triangle of signature (2, 3, ∞) into stabilizer-angle contributions,

$$\pi = \frac{\pi}{2} + \frac{\pi}{3} + 0 + \frac{\pi}{6} ;=; \sum_{p \in {i, \rho, \infty}} \frac{\pi}{|\mathrm{Stab}(p)|} + \mathrm{Area}(\Delta_{X(1)}),$$

with the convention π/|ℤ_∞| = 0 at the parabolic cusp. The four components are in bijection with the four topological data of Theorem 2 (see §3.1). Hence π is not an external constant but a structural invariant of the moduli of elliptic curves at the n = 0 level of the KRT QM tower. Full statement and proof: theorem-2-completeness.md §VI.

Uniqueness (Poincaré uniqueness lemma 🟢, added v1.2.0). An external reviewer's objection N2 correctly observed that bare PSL(2,ℤ)-invariance does not fix the Poincaré metric: one could multiply ds² by any modular-invariant function. The objection is closed by the following lemma (proof in lemma-poincare-uniqueness.md):

Let g be a smooth Riemannian metric on X(1) that is (i) complete as an orbifold, (ii) of finite volume, (iii) of constant Gaussian curvature K ≡ −1. Then g is the standard Poincaré metric. Under the weaker assumption K ≡ const < 0, g equals the Poincaré metric up to a multiplicative constant.

The proof combines classical uniformization (Ahlfors–Sario 1960; Farkas–Kra 1992, §IV.8; Beardon 1983, §9) with orbifold Teichmüller rigidity for signature (0; 2, 3, ∞) — whose moduli space is a point (Thurston §13; Farb–Margalit §10.2).

Residual gap (P11, 🟡). The lemma assumes completeness, finite volume, and constant curvature. Deriving this triple from A1 + A2 + M0 alone is not done here: it is equivalent to asserting that the effective kinetic tensor is non-singular at the elliptic points and IR-regular at the cusp. This is physically natural (no singular kinetics, finite phase-space volume) but not a mathematical consequence of the axioms. Named explicitly as open problem P11 (see §12); reduction via Weil–Petersson identification under A2⁺ (Wolpert 1985; Zograf–Takhtadzhyan 1987) narrows the gap to acceptance of A2⁺ rather than a free functional choice.

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3. Theorem 2 (Fuchsian completeness) 🟢 + Attribution (P9) 🟢

Note (paper v1.2.0 split, v1.3.0 promotion). What was previously bundled as "Theorem 2" was honestly split in v1.2.0 into a formal completeness theorem (orbit-type count = 4, proven by Katok classification) and a structurally natural conjecture (which force maps to which topological datum). The latter was labeled 🟡. On 2026-04-19 Conjecture 2 was promoted to an Attribution theorem 🟢 by direct reading of the short exact sequence 1 → Z(B₃) → B₃ → PSL(2,ℤ) → 1 — see §3.2.

3.1 Theorem 2 (Fuchsian completeness, 🟢)

Theorem 2. The set of qualitatively distinct modulation modes of the substrate rotation on X(1) = ℍ²/PSL(2,ℤ) is in bijection with

$${ \text{orbit types of PSL}(2,\mathbb{Z}) \text{ on } \mathbb{H}^2 } \cup { \text{global orientation of Im}(\tau) }$$

and has exactly four elements.

Proof outline.

Finite stabilizers. For M ∈ SL(2,ℤ) elliptic (|tr(M)| < 2), trace is integer, so tr(M) ∈ {−1, 0, 1}. Characteristic polynomial λ² − (tr)λ + 1 = 0 gives eigenvalues:

tr(M)	eigenvalues	order in SL(2,ℤ)	order in PSL(2,ℤ)
0	±i	4	2
±1	e^(±iπ/3)	6	3

By Katok (1992), Fuchsian Groups, Thm 2.5.6, finite stabilizers in Fuchsian groups are cyclic. In PSL(2,ℤ) only orders 2 (at i) and 3 (at ρ) appear.

Parabolic stabilizers. Fixed at the cusp (∞), cyclic infinite, generated by T: τ → τ+1.

Global datum. The orientation of Im(τ) (Z₂ choice).

Count: 2 elliptic types + 1 parabolic + 1 orientation = 4. ∎

Corollary T2.1 (π as invariant of X(1), 🟢, 2026-04-20). The orbifold Gauss-Bonnet identity decomposes π into a sum of stabilizer-angle contributions at the three fixed points of the fundamental triangle plus the hyperbolic area deficit:

$$\boxed{;\pi = \frac{\pi}{2} + \frac{\pi}{3} + 0 + \frac{\pi}{6};}$$

Component	Origin	T2 datum	Attribution
π/2	π/	Stab(i)	, elliptic order 2
π/3	π/	Stab(ρ)	, elliptic order 3
0	π/	Stab(∞)	= lim_{n→∞} π/n, parabolic
π/6	Area(Δ) = π − π/2 − π/3 − 0	global Im(τ) orientation	weak force

Proof. Katok 1992 §1.4 (defect formula for hyperbolic triangles with K = −1): Area(Δ) = π − (α + β + γ). Fundamental triangle of PSL(2,ℤ) has signature (2, 3, ∞) with angles π/2, π/3, 0, giving Area(Δ) = π/6 and the sum identity. ∎

Structural reading. The Egyptian-fraction decomposition 1/2 + 1/3 + 1/6 = 1 is the unique representation of unity as a sum of three reciprocals of integers (≡ Euclidean triangle signature (2, 3, 6)); the 1/6 deficit exactly measures the transition from Euclidean tiling to hyperbolic orbifold via collapse of the third angle into a cusp. No fifth regime is compatible with the Fuchsian condition Σ(1/m_i) ≤ 1. Full derivation and per-floor generalization (Borel–Prasad volumes, π → Catalan G → Apéry ζ(3)) in theorem-2-completeness.md §VI and theorem-2-per-patro.md (P18).

3.2 Attribution (P9, 🟢, promoted 2026-04-19)

Honest status (2026-04-19 promotion). The 2026-04-19 reading of Birman 1974 (Thm 1.25) and Serre 1973 §VII.1 closes the attribution as a theorem: the three non-identity conjugacy classes of PSL(2,ℤ) are parabolic / elliptic-of-order-3 / elliptic-of-order-2, and the central extension 1 → Z(B₃) → B₃ → PSL(2,ℤ) → 1 furnishes the fourth datum (the centre Z(B₃), trivial in PSL(2,ℤ) but nontrivial in B₃) — which matches the unique force signature that is globally chiral, flavour-changing, and coupled to a ℤ₂ parity orientation (Wu 1957). No other permutation is compatible with the joint constraint (algebraic class, topological datum, experimental signature).

Attribution theorem (P9). The four fundamental forces correspond bijectively to the four topological data of X(1), via the short exact sequence

$$1 \to Z(B_3) \to B_3 \to \mathrm{PSL}(2,\mathbb{Z}) \to 1,$$

where B₃ is Artin's braid group on three strands, Z(B₃) = ⟨Δ²⟩ = ⟨(σ₁σ₂)³⟩ ≅ ℤ is its centre (the "full twist"), and the quotient is PSL(2,ℤ) ≅ ℤ/2 ∗ ℤ/3 (classical; Coxeter 1961 §15.7; Birman 1974 Thm 1.25; Magnus–Karrass–Solitar 1966 §6.2).

The four elements of the sequence correspond to the four topological data:

Algebraic object	Order in PSL(2,ℤ)	Fixed data in X(1)	Force
σ₁ (parabolic preimage)	∞ (non-compact ℤ)	cusp ∞	gravity
y = σ₁σ₂ (order-3 preimage)	3	ρ (elliptic)	strong
x = σ₁σ₂σ₁ (order-2 preimage)	2	i (elliptic)	electromagnetism
Z(B₃) = ⟨Δ²⟩ (centre)	identity in PSL(2,ℤ)	global Im(τ) orientation	weak

The weak-force identification with the centre Z(B₃) is the structurally sharpest point: it is the only force that is (a) globally chiral (P-violating), (b) flavour-changing (CKM), (c) coupled to the sign of Im(τ) — exactly the signatures of a central element trivial in PSL(2,ℤ) but nontrivial in B₃.

Quantitative couplings (α_em, α_s, G_F, G_N) remain open (P4 numerical 🟡). The qualitative hierarchy (dimensionality, range, chirality, confinement) is derivable from the B₃ classification (see p4-couplings-braid.md §I); microscopic values remain open. Attribution derivations: attribution-pletence-b3.md (braid-theoretic, primary), attribution-radem-reflexi.md (ordinal depth, independent). Full completeness proof in theorem-2-completeness.md.

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4. Koide–α identity (second hard prediction, 🟢 v1.3.0)

The Koide relation (1981, empirical to 6·10⁻⁶ for charged leptons) is

$$Q ;\equiv; \frac{m_e + m_\mu + m_\tau}{\big(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau}\big)^2} ;=; \frac{2}{3}.$$

Geometric reformulation. Let v = (√m_e, √m_μ, √m_τ) ∈ ℝ³ and let n = (1,1,1)/√3 be the democratic axis. Then

$$Q ;=; \frac{|v|^2}{(v\cdot \mathbf{1})^2} ;=; \frac{1}{3\cos^2\theta}, \qquad \cos\theta \equiv \frac{v\cdot n}{|v|}.$$

Hence Q = 2/3 ⟺ cos²θ = 1/2 ⟺ θ = π/4. PDG values give θ = 44.99974° (deviation ~1 arcsecond).

Identification with α-attractor exponent (Theorem 1). The factor 2/3 in Koide is the same topological invariant as the α-attractor exponent α = 2/3 of Theorem 1 (Kähler potential K = −α·ln(…) on X(1), topological invariant −4χ(𝒳₁)). The spinor half-angle π/4 = (π/2)/2 is the SU(2) double-cover realization of a π/2 rotation — i.e. the √m-vector sits on the spinor cone of half-angle π/4 around the democratic axis. This is the first direct measurement of the spinor-observer signature (pozorovatel-je-spinor.md).

Prediction. Solving Q = 2/3 for m_τ with m_e, m_μ as inputs (PDG pole masses):

$$\boxed{m_\tau^{\text{KRT}} = 1776.97 \text{ MeV}}$$

PDG 2024 measured: m_τ = 1776.86 ± 0.12 MeV, relative deviation +6·10⁻⁵ (within 1σ experimental error). This is a structural relation on pole masses (RG-invariant on-shell), not a fit parameter.

Down-quark sector (🟡). The QCD-corrected Koide form

$$Q_d = \frac{2}{3}\left(1 + \frac{\alpha_s}{\pi}\right)$$

yields Q_d = 0.7303 at μ ~ 1.5 GeV (α_s ≈ 0.305), vs measured Q_d ≈ 0.7313 (deviation 0.1 %). Current status (post-Nicolis V.1 re-classification, 2026-04-20): 1-loop RG stability OK, 2-loop stability and structural reason for the scale choice μ* pending. Full analysis in koide-kvarky-neutrina.md §V.1.

Up-quark sector (🔴 honest). Top saturation (m_t dominates 91.6 % of Σ√m): Q → 1 in the limit m_t → ∞. Up-quarks are not a Koide sector in the same structural sense.

Neutrino sector (Q_ν = 4/9 prediction, 🟡). Direct Q_ν = 2/3 is falsified by mass-splitting data (Q_max = 0.586 in NO, 0.500 in IO). KRT predicts instead Q_ν = 4/9 (the ρ-point hexagon, π/6 stabilizer contribution), yielding Σ m_ν = 62.4 meV (NO) or 101.9 meV (IO). Testable by CMB-S4 (~2030+).

Full derivation: lepton-masy-spinor.md (cesta 6), koide-kvarky-neutrina.md.

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5. P12a — Mp(2,ℤ) bridge 🟢

P12a. Mp(2,ℤ) is the ℍ-refinement of PSL(2,ℤ) via the Weil representation.

The metaplectic group Mp(2,ℤ) is the unique non-trivial double cover of SL(2,ℤ), defined through the Weil representation on L²(ℝ) (Weil 1964, Folland 1989 ch. 4). The KRT QM tower reads

$$\mathrm{Mp}(2,\mathbb{Z}) \twoheadrightarrow \mathrm{SL}(2,\mathbb{Z}) \twoheadrightarrow \mathrm{PSL}(2,\mathbb{Z}),$$

each arrow a double cover. Representational richness grows against the quotient direction: Mp carries half-integer weight modular forms (Shimura 1973) that vanish in SL; SL distinguishes ±I which collapses in PSL.

The spinor-observer of A3 ≡ S-reflection is an element of the Weil representation, not of PSL. This closes the bridge from the modular-geometric formulation (PSL(2,ℤ)) to the quaternionic substrate (ℍ-level QM), providing the discrete arithmetic backbone of which SL(2,ℂ) ≡ Spin(1,3) (§6) is the continuous extension.

Full derivation: dirac-jako-h-schrodinger.md §IV.1–IV.3.

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6. P14 core — Dirac = ℍ-Schrödinger 🟢

P14. The Dirac equation is the Hamilton–Jacobi equation for ℍ-phase; the four levels {scalar, Pauli, Dirac, 𝕆-Dirac} of the KRT QM tower correspond to the four levels of ℝ ⊂ ℂ ⊂ ℍ ⊂ 𝕆.

Explicit quaternionic Lagrangian. Define

$$\mathcal{L}{\mathbb{H}} ;=; \bar\Psi\left(i\gamma^\mu \partial\mu - \frac{m}{\hbar}\right)\Psi,$$

with γ^μ ∈ Cl(1,3) and Ψ ∈ ℂ⁴ ≅ ℍ². The identification γ^i ≡ (i, j, k) (three imaginary quaternion units, realizing the three collapse axes of postulate ℍ-3) and γ^⁰ = generator of Re-motion (postulate ℍ-2) is canonical.

Euler–Lagrange variation δS/δΨ̄ = 0 yields the Dirac equation. Applying (iγ^ν ∂_ν + m/ℏ) from the left and using Cl(1,3) anticommutation {γ^μ, γ^ν} = 2η^{μν}𝟙 gives by direct computation

$$\left(\Box + \frac{m^2}{\hbar^2}\right)\Psi = 0$$

— the Klein–Gordon equation, derived rather than postulated.

Cl(1,3) signature from ℍ-2 vs ℍ-3 asymmetry. The Lorentzian signature (+, −, −, −) is forced by the asymmetry between:

• ℍ-2 (Re-axis continuous geodesic motion at speed c — produces the "+" entry);
• ℍ-3 (Im-axes discrete π/2 collapses, projections rather than accumulations — produces the three "−" entries).

Euclidean Cl(4,0) would require imaginary axes to accumulate phase, violating ℍ-3; Cl(0,3)-only excludes the Re-direction, violating ℍ-2. Hence Cl(1,3) is the unique signature compatible with the axioms.

Lorentz invariance via Spin(1,3) ≡ SL(2,ℂ). The universal cover of the proper orthochronous Lorentz group is SL(2,ℂ) (Streater–Wightman; Peskin–Schroeder §3.2); the Dirac bispinor transforms under (½,0) ⊕ (0,½). SL(2,ℂ) is the continuous extension of the discrete arithmetic tower

$$\mathrm{Mp}(2,\mathbb{Z}) \hookrightarrow \mathrm{Mp}(2,\mathbb{R}) \subset \mathrm{Mp}(2,\mathbb{C}) \simeq \mathrm{SL}(2,\mathbb{C}) \simeq \mathrm{Spin}(1,3),$$

standard in geometric quantization (Folland 1989 ch. 4). Thus Lorentz invariance of L_ℍ is inherited from P12a 🟢 without further structural input: a Lorentz boost is an automorphism of the ℍ-2 / ℍ-3 asymmetry, rotating directions in (t, x, y, z) without mixing continuous-vs-discrete dynamics types. Wick rotation is the switch between Euclidean rest S³ and Minkowski motion ℍ, i.e. the content of ℍ-2 itself.

Cauchy–Riemann in ℍ ≡ Dirac equation (Sudbery 1979). Residual 🟡 branches (RelObs categorical Lorentz functor, g−2 numerical test, fourth pharmacological axis) are separate sub-problems beyond P14 core. Full derivation: dirac-jako-h-schrodinger.md §II.3, §IV.4–5.

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7. P17 core — 𝕆-level of the QM tower 🟢

P17. The KRT QM tower extends to a fifth level via Cayley–Dickson: ℝ ⊂ ℂ ⊂ ℍ ⊂ 𝕆, with Hurwitz 1898 establishing that 𝕆 is the last normed division algebra (sedenions acquire zero-divisors).

Arithmetic group. Γ_𝕆 = arithmetic SL(2, Ω) via Clifford-arithmetic construction in Cl(1,9), where Ω ≅ E₈ is the Coxeter–Dickson integer lattice in 𝕆 (oktonicke-grupy-review.md). Sequentiality of the tower:

$$\mathbb{Z} ;\subset; \mathbb{Z}[i] ;\subset; \text{Hurwitz } \mathcal{H} ;\subset; \Omega$$

with Γ_n = SL(2, L_n) the arithmetic subgroup on the n-th Cayley–Dickson integer lattice.

𝕆-Dirac equation. The Lagrangian

$$\mathcal{L}{\mathbb{O}} ;=; \bar\Psi\big(i\Gamma^\mu \partial\mu - m/\hbar\big)\Psi$$

with Ψ ∈ S_Ω ≅ ℝ³² ≅ ℂ¹⁶ (the 32-real Cl(1,9)-representation; Cl(1,9) ≅ ℝ(32) is associative) yields Klein–Gordon as an algebraic identity from Cl(1,9) anticommutation. Lorentz invariance via Spin(1,3) ⊂ Spin(1,9) with explicit choice of ℂ ⊂ 𝕆 (P8-consistent canonical imaginary axis). The algebraic identity L_Ω|_{q₂=0} ≡ L_𝓗 verifies the sequential reduction to the P14 quaternionic level.

SM gauge bonus. Baez–Huerta reduction Spin(6) ≅ SU(4) ↓ SU(3) × U(1) yields the SM internal gauge content of one generation (3 colours + 1 lepton-singlet). Fine-structure constant remains open (Singh's α = 1/137 discarded 2026-04-20 b under AXIOM 1: Singh's normalization choices are not KRT-native; P4 numerical α_em remains open outside P17 scope).

Open gaps (Rule 7). Step D — Stab(i) at the 𝕆-level (comparison with P8's "last angular reflection" as dimensional vs. angular finality); Step E — phenomenology of the 𝕆-observer as potentially recognizable type. Both are continuation of the tower structure; see oktonicke-grupy-review.md, dirac-v-octonionech.md, p17-roadmap.md.

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8. Theorem 3 — Curvature from quaternion frustration

Theorem 3. Let X_a (a = 1, 2, 3) be three vector fields on a smooth 3-manifold with Lie brackets

$$[X_a, X_b] = 2\varepsilon_{abc} X_c$$

(quaternionic anti-commutation). Then the manifold admits a bi-invariant Riemannian metric with Ricci tensor

$$\text{Ric}{ab} = 2\delta{ab}, \qquad R = 6$$

and is diffeomorphic to S³ = SU(2). Flat space is algebraically excluded.

Proof. Structure constants: f^c_{ab} = 2ε_{abc}. Killing form:

$$B_{ab} = f^c_{\ ad},f^d_{\ bc} = 4\varepsilon_{cad}\varepsilon_{dbc}$$

Direct computation (contraction identities on Levi-Civita symbols) yields B_ab = −8δ_ab.

For a compact Lie group with bi-invariant metric, the Ricci tensor satisfies

$$\text{Ric}(X, Y) = -\frac{1}{4}B(X, Y)$$

(standard result; see do Carmo, Riemannian Geometry, Ch. 4, or Milnor 1976). Hence Ric_ab = 2δ_ab, R = trace = 6. The negative-definite Killing form confirms SU(2) = S³. ∎

Physical interpretation. Field space is ℍ² (Theorem 1, K = −1). Physical space, if it is to support three anti-commuting quaternion directions as vector fields, must be S³ (K = +1). The two curvatures are complementary and sum to zero.

Cosmological consequence. Einstein equations in vacuum with cosmological constant:

$$R_{ab} - \tfrac{1}{2}R g_{ab} = -\Lambda g_{ab}$$

For maximally symmetric S³ × time yielding 4-dimensional de Sitter: Λ = 3H². If S³ radius equals Hubble radius today (c/H₀ ≈ 4.4 Gpc):

$$\frac{\Lambda_{\text{eff}}}{M_{\text{Pl}}^4} = \frac{3}{(M_{\text{Pl}} \cdot c/H_0)^2} \approx 4 \times 10^{-122}$$

— order-of-magnitude match with observed Λ_obs/M⁴_Pl ≈ 10⁻¹²². The hierarchy problem reduces to the question: why is S³ radius of order Hubble? (A physical question, not fine-tuning.)

Full derivation: ricci-theorem-3.md.

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9. Companion results (not part of core r-prediction chain)

The following formal results are referenced in the full corpus but are not part of the derivation chain leading to r ≈ 0.0025. In paper v1.2.0+ they have been moved out of the main body into companion essays:

• P5 (x^x self-consistency, 🟢). For x^x = c, the second continued-fraction coefficient a₂(x) is governed by the Möbius map M = [[1,−1],[−1,2]] ∈ SL(2,ℤ) (trace 3, fixed point φ, eigenvalues φ², φ⁻²). A six-step proof establishes a₂(π) = 5 = ord(i) + ord(ρ), a self-consistency of π with the elliptic orders of its own symmetry group X(1). See rez-a-prvni-chyba.md, seberference-x-na-x.md, seberference-kategorie.md.
• Corollary T2.1 (π as X(1) invariant, 🟢). Orbifold Gauss-Bonnet identity π = π/2 + π/3 + 0 + π/6 matching the four topological data of Theorem 2; independent derivation chain for π as structural invariant. See §3.1 and theorem-2-completeness.md §VI.
• Appendix A (QM from τ-geometry, 🟢). Schrödinger evolution, Born rule, and Heisenberg uncertainty are derived as consequences of A1 + A2 + M0 + A3 in the appropriate limit. This establishes that A2 (photon = elliptic curve) does not require external quantization — quantum mechanics flows out of τ-geometry, not into it. Full text is Appendix A of paper v1.2.0+; research note in qm-limit-p3.md.
• P18 (Theorem 2 per floor, 🟢 core). Generalization of Theorem 2 completeness to the full QM tower via arithmetic subgroups Γ_n = SL(2, L_n); Corollary T2.1 analog across floors via Borel–Prasad volumes: π (n=0) → Catalan G (n=1) → Apéry ζ(3) (n=2) → F₄-L (n=3, open). Humbert–Grunewald 1919: Vol(X(ℤ[i])) = G/3. Structural redefinition of P4 as P4-qualitative (closed across floors) vs P4-quantitative (open with transcendental constant candidates). See theorem-2-per-patro.md.

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10. Effective inflation action and prediction

Kinetic sector (fully PSL(2,ℤ)-invariant).

$$\mathcal{L}_{\text{kin}} = -\frac{1}{2}\frac{(\partial\tau_1)^2 + (\partial\tau_2)^2}{\tau_2^2}$$

— Poincaré metric on ℍ², K = −1, hence α = 2/3.

Effective inflaton potential (T-model). Along the geodesic τ₁ = 0, with canonical field φ = ln(τ₂):

$$V(\varphi) = \Lambda^4 \tanh^2!\left(\frac{\varphi}{2}\right)$$

— standard α = 2/3 T-model (Kallosh–Linde 2013). This is stabilized by Stab(i) = Z₂ (A3 ≡ S-reflection), not the full modular group; microscopic derivation of V(φ) from a fully PSL(2,ℤ)-invariant fundamental action is Open problem 8 (P8, interpretively closed via A3 ≡ S-reflection).

Equations of motion in FRW background:

$$\ddot{\varphi} + 3H\dot{\varphi} + V'(\varphi) = 0, \qquad 3H^2 M_{\text{Pl}}^2 = \tfrac{1}{2}\dot{\varphi}^2 + V(\varphi)$$

Slow-roll parameters.

$$\epsilon_V = \frac{M_{\text{Pl}}^2}{2}\left(\frac{V_\varphi}{V}\right)^2 = \frac{2}{\sinh^2(\varphi)}, \qquad \eta_V = M_{\text{Pl}}^2 \frac{V_{\varphi\varphi}}{V}$$

End of inflation at ε_V = 1: φ_end = arcsinh(√2) ≈ 1.146.

Number of e-folds from φ* to end:

$$N_e = \int_{\varphi_{\text{end}}}^{\varphi_*} \frac{V}{V_\varphi}, d\varphi$$

For N_e = 57: numerical integration gives φ* ≈ 5.44, ε_V ≈ 1.49 × 10⁻⁴, η_V ≈ −0.017, yielding

$$n_s = 1 - 6\epsilon_V + 2\eta_V \approx 0.965, \qquad r = 16\epsilon_V \approx 0.00239$$

Universal α-attractor prediction (α = 2/3):

$$\boxed{r = 2(1 - n_s)^2 \approx 0.0025}$$

— matches numerical value within 2%.

Full derivation: eom-odvozeni.md, slow-roll-numericky.md.

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11. Falsifiability and predictions

Prediction table (v1.3.0).

Prediction	KRT value	Experimental	Status / Test
Tensor-to-scalar ratio r	0.0025 (for n_s = 0.9649)	Upper bound r < 0.036 (BICEP/Keck 2021)	LiteBIRD ~2032, σ(r) ≈ 0.001
Tau lepton mass m_τ	1776.97 MeV	1776.86 ± 0.12 MeV (PDG 2024)	Confirmed (Δ = 6·10⁻⁵)
Down-quark Koide Q_d	0.7303 (at μ ~ 1.5 GeV)	0.7313 (deviation 0.1 %)	1-loop OK, 2-loop μ* pending (🟡)
Weinberg angle sin²θ_W(GUT)	1/3	SU(5) predicts 3/8	RG extrapolation test
Neutrino masses Σ m_ν (NO)	62.4 meV (Q_ν = 4/9)	< 120 meV (Planck+BOSS)	CMB-S4 ~2030+
Neutrino masses Σ m_ν (IO)	101.9 meV	—	CMB-S4 ~2030+

Model comparison (α-attractor class).

Model	α	r (n_s = 0.965)
τ-geometry (this paper)	2/3	0.0025
Starobinsky R²	1	0.0037
Goncharov–Linde	1/9	0.0004
KL attractor (3α = 7)	7/3	0.0086

τ-geometry is falsified if:

• r > 0.01 measured (excludes α = 2/3 broadly)
• r < 0.0005 measured (excludes α = 2/3 broadly)
• n_s is inconsistent with Planck 2018
• m_τ measurement deviates > 5σ from 1776.97 MeV at fixed m_e, m_μ PDG values
• Any step of the derivation chain (Theorem 1 → K = −1 → α = 2/3 → r = 2(1-n_s)²; or Theorem 1 → α = 2/3 ≡ Koide Q → m_τ) contains a formal error

τ-geometry is validated (not uniquely, but consistently) if:

• r ≈ 0.0025 measured with σ(r) ≈ 0.001
• m_τ remains within 1σ of 1776.97 MeV
• Σ m_ν matches Q_ν = 4/9 prediction at CMB-S4 precision

Note: α = 2/3 also arises in IIB axiodilaton sector of string theory. A positive LiteBIRD detection of r ≈ 0.0025 would not uniquely select τ-geometry, but it would validate the broader α = 2/3 class; the Koide m_τ prediction is, to our knowledge, structural to KRT among current frameworks.

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12. Open problems — current state

#	Problem	Status	Priority
P1	Full FRW a(t) for coupled τ-H system	🟡 (slow-roll ✓, FRW phase 1 numerics ✓, baseline w(z) ≡ −1 + ΛCDM phase 2/3 analytic ✓; B1 essay w-z-evoluce.md, reheating P3/B2 pending)	HIGH
P2	Full GR limit (G_μν from 3q rotation)	🟡 Theorem 3 gives S³ + Λ_eff; full limit derivation pending	HIGH
P3	QM limit	🟢 CLOSED (Schrödinger + Born + Heisenberg from τ-geometry; Appendix A)	—
P4 (qualitative)	SM structural hierarchy	🟢 CLOSED (via B₃ classification; P18 per-floor reading)	—
P4 (numerical)	Quantitative couplings α_em, α_s, G_F	🟡 volume-ratio refuted 2026-04-20 (Rule 8); four remaining paths (Burau magnitudes → 🔴; phase eigenvalues reproduce (2,3,∞) structurally but not numerically; Jones V(q) zero at trefoil; Hecke/Rankin-Selberg)	MED
P5	CF band structure x^x	🟢 CLOSED	—
P6	Preaxiom formalization	🟡 direction identified (cartesian closed category, Lawvere's theorem forcing elliptic orders (2,3); functor F: RefDom → FuchsAction sketched; Claim B pending)	MED
P7	Cosmological constant Λ	🟡 order-of-magnitude match from Theorem 3; S³ radius stabilization pending	LOW
P8	T-model from fully modular action	🟢 interpretively CLOSED via A3 ≡ S-reflection; 🟢 technical negative (plain modular averaging does NOT produce T-model, b4-modularni-prumerovani.md)	—
P9	Attribution theorem (force ↔ topological datum)	🟢 CLOSED via B₃ / Z(B₃) short exact sequence (Birman 1974, Serre 1973, Wu 1957)	—
P10	Dynamical S³ radius stabilization at Hubble	🔴 no strategy	LOW
P11	Residual gap after Poincaré uniqueness	🟡 narrowed 2026-04-19 via Weil–Petersson identification under A2⁺ (Wolpert 1985; Zograf–Takhtadzhyan 1987); gap reduced from "choice of f(τ)" to "acceptance of A2⁺"	MED
P12a	Mp(2,ℤ) ≡ ℍ-refinement of PSL(2,ℤ)	🟢 CLOSED via Weil rep (Weil 1964, Folland 1989) + spinor-observer as Weil rep element (Shimura 1973); see §5	—
P12b	Absolute lepton mass scale from Mp(2,ℤ)	🟡 open:	η(i)
P13	Bekenstein bound as Heisenberg × volume	🟡 algebra OK; I ≤ 2πRE/(ℏc ln 2) rewritten as count of Heisenberg (x,p)-bits on horizon; (c, p) = two projections of substrate rotation Θ (P3 §VI Re/Im complementarity); holographic factor 4 conjectured via Mp(2,ℤ) double cover (bridge to P12a). See p13-bekenstein-heisenberg.md.	MED
P14	Dirac = ℍ-Schrödinger	🟢 core CLOSED 2026-04-20: explicit Lagrangian, Klein-Gordon by direct computation, Lorentz invariance via Spin(1,3) ≡ SL(2,ℂ) as continuous extension of P12a; see §6. 🟡 residual: RelObs categorical Lorentz functor, g−2 numerical test.	—
P15	Music as 4th ℍ-maintenance channel	🟡 structural frame OK; rhythm = γ⁰ derivation pending; H_AC empirically testable	LOW
P16	Time as three-layer projection of ℍ	🟡 structural parallel rigorous; retrocausality as structural consequence of ℂ → ℍ extension; coherence mechanism across macro-time open. Framework does not predict population-level synchronicity.	LOW
P17	𝕆-level of KRT QM tower	🟢 core CLOSED 2026-04-20: Steps A, B, C (Γ_Ω via Cl(1,9), 𝕆-Dirac, Klein-Gordon as algebraic identity, Lorentz via Spin(1,3) ⊂ Spin(1,9), sequential reduction to P14). Bonus: Spin(6) SU(3)×U(1) SM content. See §7. 🟡 residual: Step D (Stab(i) in 𝕆-level), Step E (phenomenology).	MED
P18	Theorem 2 completeness per QM-floor	🟢 core inference 2026-04-20: π → Catalan G → ζ(3) → (F₄-L) hierarchy via Borel–Prasad volumes; generation count from asociativní triády v 𝕆. 🟡 kompletnost (T1–T6 gaps: exact orbit counts n=2,3; physical mapping n=1; E₈-L-function; g−2 test).	MED
P19	Dyad ℍ-observers and forced k-axis	🟡 2026-04-20 evening: self-existence as orthogonal projection through another observer (A sees i_A only via k_AB, antisymmetric product of i_A × i_B); geometric content for AXIOM 1. Bridges to P6, P14 §V RelObs, arrow of complexification.	MED
P20	Subjective time flow rate as Γ_refl/c	🟡 2026-04-20 evening: Ṫ_subj = Γ_refl/c (collapse rate per Re-time); SR time dilation as redistribution of Re/i motion; bridges P13 (max Γ), P14 §IV.5.1, P15, P16, P19. Falsifiable tests: interoception × time accuracy, Alzheimer EEG microstate rate, dyadic Γ, meditation, psychedelics × BOLD complexity.	MED

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13. Falsification record (honest)

Resolved issues (during development):

• Previous claim that K = −1 was an independent assumption (paper v1.0–v1.1.4) was reduced to a theorem by recognizing it follows from Theorem 1 via uniqueness of PSL(2,ℝ)-invariant metric (paper v1.1.5).
• Previous specification of V with η function (paper v1.1.4–v1.1.6) was shown to be inconsistent with r prediction by numerical analysis (paper v1.1.7); T-model substituted as effective potential with P8 added as open problem.
• Conjecture 2 (attribution) promoted to theorem 2026-04-19 via explicit reading of 1 → Z(B₃) → B₃ → PSL(2,ℤ) → 1 + Wu 1957 force signatures; no longer 🟡 in v1.3.0.
• Koide-α identity (2/3 ≡ Koide Q ≡ α-attractor exponent) integrated as second hard prediction in v1.3.0; Nicolis RG-stability analysis absorbed (Q_d 1-loop OK, 2-loop re-classification pending).
• α_em volume-ratio refuted 2026-04-20 (Rule 8 explicit elimination): volume-ratio path for numerical α_em excluded from P4 (numerical); four residual paths named (Burau magnitudes → 🔴; phase eigenvalues structural only; Jones V(q) zero at trefoil; Hecke/Rankin–Selberg).
• Neutrino Q = 2/3 falsified (Q_max = 0.586 NO, 0.500 IO from mass-splitting data); replaced by Q_ν = 4/9 prediction (ρ-point hexagon, π/6 stabilizer).

Unresolved issues: see §12.

What this paper does NOT claim:

• A theory of everything
• Solutions to consciousness, dark matter, or dark energy beyond the specific Λ_eff calculation
• Uniqueness of α = 2/3 prediction (it is also a string theory prediction in the IIB axiodilaton sector)
• Quantitative derivation of Standard Model coupling constants (P4 numerical remains 🟡)

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14. References

Primary references inside full corpus (theory.yrx.cz/vyzkum):

• proc-psl2z.md — Theorem 1 derivation
• theorem-2-completeness.md — Theorem 2 completeness proof + Corollary T2.1
• theorem-2-per-patro.md — P18, per-floor generalization
• attribution-pletence-b3.md — Attribution (P9) via B₃ / Z(B₃)
• ricci-theorem-3.md — Theorem 3 full derivation
• rez-a-prvni-chyba.md — P5 six-step proof
• lemma-poincare-uniqueness.md — Poincaré uniqueness lemma + P11 reduction
• dirac-jako-h-schrodinger.md — P14 core + P12a bridge
• oktonicke-grupy-review.md, dirac-v-octonionech.md, p17-roadmap.md — P17 core
• lepton-masy-spinor.md, koide-kvarky-neutrina.md — Koide–α identity
• eom-odvozeni.md — Euler–Lagrange derivation
• slow-roll-numericky.md — numerical verification r = 0.00239
• qm-limit-p3.md — Appendix A (QM from τ-geometry)
• problem-potencial.md — P8 analysis and justification for T-model choice

External references:

• Kallosh, R. & Linde, A. (2013). Universality class in conformal inflation. JCAP 07, 002.
• Diamond, F. & Shurman, J. (2005). A First Course in Modular Forms, Springer.
• Hain, R. (2014). Lectures on moduli spaces of elliptic curves, arXiv:0812.1803.
• Katok, S. (1992). Fuchsian Groups, Univ. of Chicago Press.
• Milnor, J. (1976). Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329.
• do Carmo, M. (1992). Riemannian Geometry, Birkhäuser.
• Planck Collaboration (2020). Planck 2018 results. X. Constraints on inflation. A&A 641, A10.
• LiteBIRD Collaboration (2023). Probing cosmic inflation with LiteBIRD. PTEP 2023, 042F01.
• Birman, J. (1974). Braids, Links, and Mapping Class Groups, Annals of Math. Studies 82, Princeton.
• Serre, J.-P. (1973). A Course in Arithmetic, Springer, §VII.1.
• Wu, C. S. et al. (1957). Experimental test of parity conservation in beta decay. Phys. Rev. 105, 1413.
• Koide, Y. (1983). A fermion-mass hierarchy and a new view regarding the quark-lepton generation. Lettere al Nuovo Cimento 34, 201.
• Hurwitz, A. (1898). Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln. Nachr. Ges. Wiss. Göttingen, 309–316.
• Baez, J. (2001). The octonions. Bull. Amer. Math. Soc. 39, 145–205.
• Baez, J. & Egan, G. (2014). The octonions and Jordan algebras; Baez-Huerta, Division algebras and supersymmetry II, Adv. Theor. Math. Phys. 15, 1373.
• Allcock, D. (1999). Reflection groups on the octave hyperbolic plane. J. Algebra 213, 467–498.
• Weil, A. (1964). Sur certains groupes d'opérateurs unitaires. Acta Math. 111, 143–211.
• Folland, G. B. (1989). Harmonic Analysis in Phase Space, Princeton Univ. Press.
• Shimura, G. (1973). On modular forms of half integral weight. Ann. of Math. 97, 440–481.
• Wolpert, S. (1985). On the Weil–Petersson geometry of the moduli space of curves. Amer. J. Math. 107, 969.
• Zograf, P. G. & Takhtadzhyan, L. A. (1987). On the uniformization of Riemann surfaces and on the Weil–Petersson metric on the Teichmüller and Schottky spaces. Math. USSR Sb. 60, 297.
• Sommerfield, C. M. (1957). Magnetic dipole moment of the electron. Phys. Rev. 107, 328.
• Petermann, A. (1957). Fourth-order magnetic moment of the electron. Helv. Phys. Acta 30, 407.
• Laporta, S. & Remiddi, E. (1996). The analytical value of the electron (g−2) at order α³ in QED. Phys. Lett. B 379, 283.
• Sudbery, A. (1979). Quaternionic analysis. Math. Proc. Cambridge Phil. Soc. 85, 199–225.
• Humbert, G. / Grunewald, F. (1919, 1978). Volume of Picard modular orbifold = Catalan's constant / 3.

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Interpretation

The reformulated A3 ≡ S-reflection has a specific interpretational consequence: the framework is relational monism, incompatible with many-worlds or multiverse-type theories. The substrate algebraically contains multiple reflections (Z/2 at i, Z/3 at ρ, Z/N at higher cosets); a classical observer projects onto a single Stab(i)-invariant slice. Non-selected reflections are not in parallel branches or alternate universes — they are structural facts of one reality, accessible through group-theoretic transitions (weak-force flavour change, CKM/PMNS mixing, parity reversal) within the same observer's relational network. The three fermion generations are Z/3 rotational positions of the same mode at ρ, not three copies in three worlds. Born rule probabilities emerge from A3 projection depth, not from branching multiplicity. This stance is weaker than modal realism, stronger than Copenhagen, and strictly excludes Everett, eternal-inflation multiverse, and string-landscape pluralism. The predictions of this theory — r = 2(1−n_s)², m_τ = 1776.97 MeV, sin²θ_W(GUT) = 1/3, N_e ≈ 60 — are therefore claimed as features of the one reality, not anthropically selected from an ensemble.

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15. Revision history

• v1.3, 2026-04-20 (sync s paper v1.3.0).
  • Second hard prediction (Koide–α identity m_τ = 1776.97 MeV) added as §4; charged-lepton pole masses identified as RG-invariant; down-quark sector 🟡 (QCD 1-loop OK, 2-loop pending after Nicolis V.1); up-quark sector 🔴 honest (top saturation); neutrino Q_ν = 4/9 prediction (Σ m_ν = 62.4 meV NO / 101.9 meV IO).
  • Attribution (P9) promoted to theorem 🟢 (2026-04-19): Conjecture 2 → theorem via Birman 1974 + Serre 1973 + Wu 1957; §3.2 rewritten accordingly.
  • Corollary T2.1 (π as X(1) invariant, 🟢, 2026-04-20) added to §2 and §3.1 with full orbifold Gauss-Bonnet derivation π = π/2 + π/3 + 0 + π/6.
  • §5 P12a 🟢 added (Mp(2,ℤ) ≡ ℍ-refinement of PSL(2,ℤ) via Weil representation).
  • §6 P14 core 🟢 added (Dirac = ℍ-Schrödinger, explicit Lagrangian L_ℍ, Klein-Gordon by direct computation, Lorentz invariance via Spin(1,3) ≡ SL(2,ℂ) as continuous extension of P12a).
  • §7 P17 core 🟢 added (𝕆-level of QM tower, arithmetic SL(2, Ω) via Cl(1,9), sequential Cayley–Dickson, Spin(6) SU(3)×U(1) SM bonus).
  • §9 Companion results expanded with Corollary T2.1, Appendix A QM-from-τ, and P18 (Theorem 2 per-floor).
  • §11 prediction table extended: r ≈ 0.0025 (LiteBIRD 2032), m_τ = 1776.97 MeV (done), Q_d = 0.7303 (done), sin²θ_W(GUT) = 1/3, Σ m_ν.
  • §12 open problems refreshed: P4 numerical volume-ratio refuted (Rule 8), P11 narrowed via Weil–Petersson under A2⁺, P14/P17 promoted from candidate to 🟢 core, P18–P20 added.
  • §13 falsification record updated with Nicolis RG-stability absorption, α_em volume-ratio refutation, Q_ν = 2/3 falsification.
  • §14 references extended: Birman 1974, Serre 1973, Katok 1992 (already), Hurwitz 1898, Baez 2001, Baez–Egan 2014, Allcock 1999, Weil 1964, Folland 1989, Shimura 1973, Wolpert 1985, Zograf–Takhtadzhyan 1987, Sommerfield 1957, Petermann 1957, Laporta–Remiddi 1996, Sudbery 1979, Humbert/Grunewald 1919.
• v1.2, 2026-04-19 (sync s paper v1.2.0).
  • Theorem 2 reclassified: split into Theorem 2 (Fuchsian completeness) 🟢 (orbit-type count = 4 by Katok 1992) and Conjecture 2 🟡 (force attribution, structurally natural but not formally derived). Section 3 rewritten accordingly.
  • Poincaré uniqueness lemma 🟢 added to §2, closing external-review objection N2 (classical uniformization + orbifold Teichmüller rigidity of signature (0; 2, 3, ∞)); residual gap named explicitly as P11 🟡.
  • Appendix A (QM from τ-geometry) referenced: Schrödinger, Born, Heisenberg derived from τ-geometry — A2 need not be quantised externally.
  • P5 / x^x removed from core derivation; retained as companion result (§9) with pointer to companion essays.
  • A2 physical motivation acknowledged (paper v1.2.0 addition).
  • P7 (S³ radius stabilization) explicitly flagged as open.
• v1.1, 2026-04-19. A3 ≡ S-reflection reformulation; P8 interpretively closed; attribution theorem claim (pre-split).
• v1.0, 2026-04-17. First external-audience synopsis.

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Correspondence

Author: Adam Porybný (independent, affiliated with mezi4stěn social project, Brno, Czech Republic).

Formalization assistance: Claude (Anthropic) — numerical verification, LaTeX formatting, consistency checks. Conceptual content and vision are author's.

Contact: [email adresa]

Full corpus (50+ essays, including speculative and exploratory material not in this synopsis): theory.yrx.cz/vyzkum

The author is open to technical criticism, collaboration, or 30 minutes of conversation with interested mathematicians or theoretical physicists. Paper is offered as stimulus, not as claim for recognition.