# tau-geometry.cz — Technical Overview & Epistemic Map (v1.6.0)

> Independent theoretical physics working repository exploring modular geometry (H^2 / PSL(2,Z)) in relation to cosmological alpha-attractors. No peer review. Co-developed using human-AI workflows with public errata tracking.

## 1. Core Mathematical & Physical Architecture

### 1.1 Conventions & Units
- Reduced Planck units are used throughout: M_Pl = (8 * pi * G)^(-1/2) = 1.
- Complex modular parameter: tau = tau_1 + i * tau_2 in H^2 (upper half-plane).
- Kinetic Lagrangian for the modulus: L_kin = - (3 * alpha / 4) * (d_mu tau_2)^2 / tau_2^2.
- Canonical inflaton field definition (Axiom A4): phi = sqrt(3 * alpha / 2) * ln(tau_2). For alpha = 2/3, this simplifies to phi = ln(tau_2) with standard canonical kinetic term -(1/2)*(d_mu phi)^2.

### 1.2 Epistemic Stratification (Strict Separation of Claims)
To prevent conflating textbook mathematics with speculative physics, claims on this website are categorized into four distinct layers:

1. **[⚪ Classical Math / Literature]:**
   - Orbifold Euler characteristic chi(H^2 / PSL(2,Z)) = -1/6 and Poincaré area A = pi/3 (Gauss-Bonnet theorem).
   - Bi-invariant metric on compact Lie group SU(2) is isometric to S^3.
   - Standard Kallosh-Linde alpha-attractor slow-roll relations: n_s = 1 - 2/N_e, r = 12 * alpha / N_e^2.
   - Elementary numerical bounds (e.g., rational approximations of pi).

2. **[🟢 Conditionally Derived (Core Model - Branch A)]:**
   - *Condition 1 (Postulate N):* Setting Gaussian curvature K = -1 on the fundamental domain fixes alpha = -4 * chi = 2/3. (Note: Choosing K = -1 is equivalent to selecting alpha = 2/3; it is a single normalization postulate, not a parameter-free derivation).
   - *Condition 2 (T-Model Potential Ansatz):* Assuming V(phi) = V_0 * tanh^2(phi / sqrt(6*alpha)) yields the leading-order observable relation r = 2 * (1 - n_s)^2.

3. **[🟡 Open Problems & Heuristic Ansätze (Active Sandbox)]:**
   - *Modular Symmetry Breaking & P8:* Deriving the effective T-model potential from a fully modular-invariant action is open (P8). A strictly PSL(2,Z)-invariant potential (Branch B) yields r ~ 10^-5.
   - *Quaternionic Spatial Frame Ansatz (Ansatz 3):* Assuming physical 3-space carries an orthonormal su(2) frame yields Ric_ab = 2*delta_ab and local S^3 geometry (R = 6), but parallelizing physical space with this frame is an ansatz, not an unconditioned proof that flat space R^3 is excluded.
   - *E-foldings Coincidence (Ansatz 4):* The identification N_e = 60 = |A_5| = [PSL(2,Z) : Gamma(5)] is treated as a group-theoretic ansatz, not a thermodynamic reheating derivation. Note: N_e = 60 gives n_s = 0.9667, which exhibits ~2 sigma tension with recent P-ACT data (n_s ~ 0.971–0.974, requiring N_e ~ 70).
   - *Kinematic Photon Hypothesis (Axiom A2):* Identifying internal photon states with elliptic curves E_tau lacks an explicit U(1) gauge action and Maxwell dynamics.
   - *WKB / Hamilton-Jacobi Analogy (P3):* Mapping Re(tau) to classical action S provides a structural analogy to quantum phase, not a formal derivation of the Schrödinger equation or the Born rule.
   - *Modified Koide Scale (P12b):* The scale factor mu / m_e ~ 1.25 absorbs the 0.31% deviation as a 1-parameter fit, not an ab initio prediction.

4. **[🟣 Philosophical Essays & Numerology (/tales — Quarantined)]:**
   - Speculative essays on observer ontology, consciousness, pre-axiom x^x, and geometric metaphors. Strictly isolated from the formal derivation chain in /theory.

## 2. Falsifiability & Experimental Thresholds

- **Primary Testable Target (Branch A):** r = 2 * (1 - n_s)^2 => r ~ 0.0022 (for N_e = 60) or r = 0.0025 +/- 0.0006 (propagated from Planck/BICEP n_s).
- **Degeneracy Warning (String Theory & Supergravity):** This prediction is identical at leading order in 1/N_e to any standard Kallosh-Linde T-model with alpha = 2/3 (e.g., string theory IIB axiodilaton compactifications, supergravity ubiquity models). LiteBIRD detection confirms the alpha = 2/3 class, but does not uniquely select tau-geometry over string theory without leptonic sector data (Koide Q = 2/3).
- **Experimental Timeline & Sensitivity:**
  - *LiteBIRD (JAXA schedule ~2032, sigma(r) ~ 0.001):* Can rule out r > 0.01 (falsifying both alpha = 2/3 and Starobinsky alpha = 1). However, for r ~ 0.0022, a null result provides only a ~2 sigma tension.
  - *CMB-S4 / Next-Gen (sigma(r) ~ 0.0003–0.0005):* Required to decisively falsify Branch A (threshold: upper bound r < 0.001 at 95% CL) and to distinguish alpha = 2/3 from Starobinsky inflation (alpha = 1, r ~ 0.0033).
- **Methodological Note on Branch B:** Retreating to the fully PSL(2,Z)-invariant Branch B (r ~ 10^-5) if Branch A is falsified is explicitly recognized as an untestable immunizing strategy under current technological limits.

## 3. Site Map & Frozen Snapshots
- `/theory` — Formal definitions, Lagrangian conventions, and derivation chain.
- `/falsifikace` — Experimental bounds, error propagation, and degeneracy analysis.
- `/errata` — Public log of retracted claims and corrected LLM-assisted hallucinations (e.g., 2026-07-24 Weil-Petersson citation correction).
- `/archive` — Immutable timestamped PDF snapshots of working papers (v8.1, v12.0).
- `/tales` — Non-scientific philosophical essays and metaphors.