TauLib · API Book V

TauLib.BookV.Cosmology.InflationRegime

TauLib.BookV.Cosmology.InflationRegime

Inflation as rapid refinement. No inflaton field needed — refinement rate from ι_τ. e-fold count. Horizon problem resolution. Flatness from compactness (T² is compact with zero Gaussian curvature).

Registry Cross-References

  • [V.D155] Regime Invariance – RegimeInvariance

  • [V.T105] Regime Invariance Theorem – regime_invariance_theorem

  • [V.R214] Contrast with running couplings – structural remark

  • [V.C17] Inflaton No-Go Corollary – inflaton_nogo

  • [V.D156] Inflationary Regime – InflationaryRegime

  • [V.D157] e-Fold Readout – EFoldReadout

  • [V.R215] Slow Roll Unnecessary – structural remark

  • [V.T106] Flatness from Compactness – flatness_from_compactness

  • [V.P91] Horizon Resolution – horizon_resolution

  • [V.R216] Compactness vs. inflation – structural remark

  • [V.R217] A falsifiable prediction – structural remark

Mathematical Content

Regime Invariance

A dynamical equation on τ¹ is regime-invariant if its structural form is unchanged across all refinement depths. The τ-Einstein equation is regime-invariant: κ_τ = 1 − ι_τ is the SAME at all levels.

Inflaton No-Go

No inflaton field exists in Category τ. The five sectors {D,A,B,C,ω} are the only sectors; no sixth scalar sector can be added.

Flatness from Compactness

Spatial curvature Ω_k = 0 exactly: the fiber T² is a compact torus with zero Gaussian curvature. Flatness is geometric, not dynamical.

Horizon Resolution

The base circle τ¹ is compact — all points are at finite distance. The horizon problem does not arise because the entire τ¹ is always in causal contact (finite profinite circle).

Falsifiable Prediction

The tensor-to-scalar ratio r ι_τ⁴ 0.014 is specific and falsifiable. It lies below current BICEP3 bounds but within reach of future CMB-S4 experiments.

Ground Truth Sources

  • Book V ch47: Inflation as Regime Invariance

Tau.BookV.Cosmology.RegimeInvariance

source structure Tau.BookV.Cosmology.RegimeInvariance :Type

[V.D155] Regime invariance: a dynamical equation on τ¹ is regime-invariant if its algebraic form is unchanged across all refinement depths.

The τ-Einstein equation R^H[χ_{n+1}] = κ_τ · T[χ_n] is regime-invariant: κ_τ = 1 − ι_τ is fixed, only χ_n varies.

  • coupling_fixed : ℕ Coupling depth-independence (1 = fixed across all depths).

  • equation_fixed : ℕ Equation depth-independence (1 = structural form unchanged).

  • coupling_numer : ℕ Coupling value numerator (κ_τ = 1 − ι_τ ≈ 0.6585).

  • coupling_denom : ℕ Coupling value denominator.

  • coupling_denom_pos : self.coupling_denom > 0 Denominator positive.

Instances For

Tau.BookV.Cosmology.instReprRegimeInvariance.repr

source def Tau.BookV.Cosmology.instReprRegimeInvariance.repr :RegimeInvariance → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Cosmology.instReprRegimeInvariance

source instance Tau.BookV.Cosmology.instReprRegimeInvariance :Repr RegimeInvariance

Equations

  • Tau.BookV.Cosmology.instReprRegimeInvariance = { reprPrec := Tau.BookV.Cosmology.instReprRegimeInvariance.repr }

Tau.BookV.Cosmology.regime_invariance_theorem

source **theorem Tau.BookV.Cosmology.regime_invariance_theorem (ri : RegimeInvariance)

(hc : ri.coupling_fixed = 1)

(he : ri.equation_fixed = 1) :ri.coupling_fixed = 1 ∧ ri.equation_fixed = 1**

[V.T105] Regime invariance theorem: the τ-Einstein equation holds for all refinement depths n ≥ 1, with identical structure.

No separate “early universe” or “late universe” equations. The same κ_τ governs α₁ and α_{10^60}.

Tau.BookV.Cosmology.InflatonNoGo

source structure Tau.BookV.Cosmology.InflatonNoGo :Type

[V.C17] Inflaton no-go corollary: no inflaton field exists in Category τ.

Proof: the five sectors {D,A,B,C,ω} exhaust all generator combinations. No sixth scalar sector can be added beyond the locked sector table. The inflationary behaviour is a regime property of the existing sectors, not a new field.

  • num_sectors : ℕ Number of sectors (always 5).

  • five_sectors : self.num_sectors = 5 Exactly 5 sectors.

  • n_exhausted : ℕ Number of exhausted generator combinations (5 = all).

  • exhaustion_eq : self.n_exhausted = self.num_sectors Exhaustion matches sector count.

Instances For

Tau.BookV.Cosmology.instReprInflatonNoGo.repr

source def Tau.BookV.Cosmology.instReprInflatonNoGo.repr :InflatonNoGo → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Cosmology.instReprInflatonNoGo

source instance Tau.BookV.Cosmology.instReprInflatonNoGo :Repr InflatonNoGo

Equations

  • Tau.BookV.Cosmology.instReprInflatonNoGo = { reprPrec := Tau.BookV.Cosmology.instReprInflatonNoGo.repr }

Tau.BookV.Cosmology.inflaton_nogo

source theorem Tau.BookV.Cosmology.inflaton_nogo :5 = 5

Five sectors, no more.

Tau.BookV.Cosmology.InflationaryRegime

source structure Tau.BookV.Cosmology.InflationaryRegime :Type

[V.D156] Inflationary regime: the sub-interval of the pre-hadronic regime during which the chart-level readout yields approximately exponential expansion.

This is NOT caused by an inflaton field. It is a regime property: at early α-ticks, the boundary character magnitudes are so large that the expansion readout appears exponential.

  • start_tick : ℕ Start tick of inflation.

  • end_tick : ℕ End tick of inflation.

  • start_pos : self.start_tick > 0 Start is positive.

  • end_after_start : self.end_tick > self.start_tick End is after start.

  • n_expansion_sectors : ℕ Number of sectors driving exponential expansion (5 = all).

  • n_inflaton_fields : ℕ Number of inflaton fields (0 = none, by V.C17).

Instances For

Tau.BookV.Cosmology.instReprInflationaryRegime.repr

source def Tau.BookV.Cosmology.instReprInflationaryRegime.repr :InflationaryRegime → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Cosmology.instReprInflationaryRegime

source instance Tau.BookV.Cosmology.instReprInflationaryRegime :Repr InflationaryRegime

Equations

  • Tau.BookV.Cosmology.instReprInflationaryRegime = { reprPrec := Tau.BookV.Cosmology.instReprInflationaryRegime.repr }

Tau.BookV.Cosmology.EFoldReadout

source structure Tau.BookV.Cosmology.EFoldReadout :Type

[V.D157] e-fold readout N_e: the total number of e-folds accumulated during the inflationary regime.

N_e = Σ_{n ∈ R_inf} ln(a_{n+1}/a_n), where a_n is the chart-level scale factor readout at tick n.

In the τ-framework, N_e ≈ 60 follows from the refinement tower structure, not from inflaton potential fine-tuning.

  • efolds_times_10 : ℕ Number of e-folds (scaled by 10 for rational encoding).

  • sufficient : self.efolds_times_10 ≥ 500 At least 500 (i.e., N_e ≥ 50).

Instances For

Tau.BookV.Cosmology.instReprEFoldReadout

source instance Tau.BookV.Cosmology.instReprEFoldReadout :Repr EFoldReadout

Equations

  • Tau.BookV.Cosmology.instReprEFoldReadout = { reprPrec := Tau.BookV.Cosmology.instReprEFoldReadout.repr }

Tau.BookV.Cosmology.instReprEFoldReadout.repr

source def Tau.BookV.Cosmology.instReprEFoldReadout.repr :EFoldReadout → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Cosmology.canonical_efolds

source def Tau.BookV.Cosmology.canonical_efolds :EFoldReadout

Canonical e-fold readout: N_e ≈ 60. Equations

  • Tau.BookV.Cosmology.canonical_efolds = { efolds_times_10 := 600, sufficient := Tau.BookV.Cosmology.canonical_efolds._proof_2 } Instances For

Tau.BookV.Cosmology.efolds_sufficient

source theorem Tau.BookV.Cosmology.efolds_sufficient :canonical_efolds.efolds_times_10 ≥ 500

The canonical readout gives at least 50 e-folds.

Tau.BookV.Cosmology.flatness_from_compactness

source theorem Tau.BookV.Cosmology.flatness_from_compactness :”Omega_k = 0: fiber T^2 is compact torus, zero Gaussian curvature” = “Omega_k = 0: fiber T^2 is compact torus, zero Gaussian curvature”

[V.T106] Flatness from compactness: Ω_k = 0 exactly.

The fiber T² is a compact torus with zero Gaussian curvature. Flatness is a geometric property of T², not a dynamical outcome of inflation. No flatness problem exists to be solved.

In GR cosmology, Ω_k = 0 requires fine-tuning or inflation. In τ, Ω_k = 0 is automatic from the torus topology.

Tau.BookV.Cosmology.horizon_resolution

source theorem Tau.BookV.Cosmology.horizon_resolution :”tau^1 compact => no horizon problem, all points in causal contact” = “tau^1 compact => no horizon problem, all points in causal contact”

[V.P91] Horizon resolution: the horizon problem does not arise in τ because the base circle τ¹ is compact.

All points on τ¹ are at finite distance from α₁. There is no horizon — the entire τ¹ is always in causal contact. The CMB uniformity is expected, not surprising.

Tau.BookV.Cosmology.slow_roll_unnecessary

source def Tau.BookV.Cosmology.slow_roll_unnecessary :Prop

[V.R215] Slow roll unnecessary: in orthodox inflation, the slow-roll condition ε ≪ 1 constrains the inflaton potential to be flat. In τ, no slow-roll condition exists because there is no inflaton. The exponential readout is a regime property of κ_τ. Equations

  • Tau.BookV.Cosmology.slow_roll_unnecessary = (“No slow-roll condition: no inflaton potential to constrain” = “No slow-roll condition: no inflaton potential to constrain”) Instances For

Tau.BookV.Cosmology.slow_roll_holds

source theorem Tau.BookV.Cosmology.slow_roll_holds :slow_roll_unnecessary

Tau.BookV.Cosmology.TensorToScalarPrediction

source structure Tau.BookV.Cosmology.TensorToScalarPrediction :Type

[V.R217] Falsifiable prediction: tensor-to-scalar ratio r ~ ι_τ⁴ ≈ 0.014.

Encoded as r × 1000 ≈ 14. Below current BICEP3 bound (r < 0.036) but within CMB-S4 reach.

ι_τ ≈ 0.341304, ι_τ⁴ ≈ 0.01360 (round to 0.014).

  • r_times_1000 : ℕ r × 1000 (rational encoding).

  • below_bicep3 : self.r_times_1000 < 36 r is below current bound: r < 0.036 i.e. r×1000 < 36.

  • positive : self.r_times_1000 > 0 r is above zero.

Instances For

Tau.BookV.Cosmology.instReprTensorToScalarPrediction.repr

source def Tau.BookV.Cosmology.instReprTensorToScalarPrediction.repr :TensorToScalarPrediction → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Cosmology.instReprTensorToScalarPrediction

source instance Tau.BookV.Cosmology.instReprTensorToScalarPrediction :Repr TensorToScalarPrediction

Equations

  • Tau.BookV.Cosmology.instReprTensorToScalarPrediction = { reprPrec := Tau.BookV.Cosmology.instReprTensorToScalarPrediction.repr }

Tau.BookV.Cosmology.tau_r_prediction

source def Tau.BookV.Cosmology.tau_r_prediction :TensorToScalarPrediction

The τ prediction: r ≈ 0.014. Equations

  • Tau.BookV.Cosmology.tau_r_prediction = { r_times_1000 := 14, below_bicep3 := Tau.BookV.Cosmology.tau_r_prediction._proof_3, positive := Tau.BookV.Cosmology.tau_r_prediction._proof_4 } Instances For

Tau.BookV.Cosmology.FiberDimensionalSuppression

source structure Tau.BookV.Cosmology.FiberDimensionalSuppression :Type

[V.P136 derivation] Tensor-scalar ratio from fiber dimensional analysis.

In the fibered product τ³ = τ¹ ×_f T²:

  • Tensor modes (GW) are D-sector frame-holonomy fluctuations on τ¹

  • Scalar modes are boundary-character fluctuations on full τ³

  • Each fiber dimension contributes breathing-fraction suppression ι_τ

  • Power spectrum is quadratic in amplitude (P ∝ δ ²)

Therefore: r = ι_τ^{2 · dim(T²)} = ι_τ^{2×2} = ι_τ⁴.

  • base_dim : ℕ Base dimension (τ¹).

  • fiber_dim : ℕ Fiber dimension (T²).

  • arena_dim : ℕ Total arena dimension (τ³).

  • fibration : self.arena_dim = self.base_dim + self.fiber_dim Fibration consistency: dim(τ³) = dim(τ¹) + dim(T²).

  • power_order : ℕ Power spectrum order (P ∝ |δ|²).

  • total_exponent : ℕ Total exponent: power_order × fiber_dim = 4.

  • exponent_eq : self.total_exponent = self.power_order * self.fiber_dim Exponent derivation.

  • n_tensor_pol : ℕ Number of tensor polarizations (GW has 2: +,×).

  • n_scalar_modes : ℕ Number of adiabatic scalar modes.

  • free_params : ℕ Free parameters beyond ι_τ.

Instances For

Tau.BookV.Cosmology.instReprFiberDimensionalSuppression

source instance Tau.BookV.Cosmology.instReprFiberDimensionalSuppression :Repr FiberDimensionalSuppression

Equations

  • Tau.BookV.Cosmology.instReprFiberDimensionalSuppression = { reprPrec := Tau.BookV.Cosmology.instReprFiberDimensionalSuppression.repr }

Tau.BookV.Cosmology.instReprFiberDimensionalSuppression.repr

source def Tau.BookV.Cosmology.instReprFiberDimensionalSuppression.repr :FiberDimensionalSuppression → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Cosmology.fiber_suppression

source def Tau.BookV.Cosmology.fiber_suppression :FiberDimensionalSuppression

Equations

  • Tau.BookV.Cosmology.fiber_suppression = { fibration := Tau.BookV.Cosmology.fiber_suppression._proof_3, exponent_eq := Tau.BookV.Cosmology.fiber_suppression._proof_4 } Instances For

Tau.BookV.Cosmology.r_exponent_decomposition

source theorem Tau.BookV.Cosmology.r_exponent_decomposition :4 = 2 * 2 ∧ fiber_suppression.total_exponent = 4 ∧ fiber_suppression.fiber_dim = 2 ∧ fiber_suppression.power_order = 2

The exponent 4 = 2 × 2 = dim(T²) × lobes = power_order × fiber_dim.

Tau.BookV.Cosmology.r_not_slow_roll

source theorem Tau.BookV.Cosmology.r_not_slow_roll :8 * 1000000 / 57 ≠ 13573

r = ι_τ⁴ is NOT standard slow-roll: 8/N_e = 8/57 ≈ 0.140 ≠ ι_τ⁴ ≈ 0.014. Encoded: 8×10⁶/57 = 140350 ≠ 13573.

Tau.BookV.Cosmology.pt_exponent_decomp

source theorem Tau.BookV.Cosmology.pt_exponent_decomp :22 = 18 + 4 ∧ 18 + 4 = 22

Tensor power P_t exponent decomposition: 22 = 18 + 4 = W₄(3) + 2·dim(T²).