TauLib · API Book V

TauLib.BookV.Temporal.HighEnergy

TauLib.BookV.Temporal.HighEnergy

The opening (high-energy) epoch: maximal coupling, the opening regime, unique τ-Einstein solutions, refinement progression rate, and the inflationary interpretation.

Registry Cross-References

  • [V.P05] Full Spectrum at Ignition — full_spectrum_at_ignition

  • [V.D24] Maximal Coupling Condition — MaximalCouplingCondition

  • [V.D25] Opening Regime Interval — OpeningRegime

  • [V.T13] Unique Solution in Opening — opening_has_solution

  • [V.D26] Refinement Progression Rate — RefinementRate

  • [V.R33] Inflationary Epoch — inflation_remark

Mathematical Content

Full Spectrum at Ignition [V.P05]

At the ignition depth n_ign, all 5 sector spectral labels become present in the boundary holonomy algebra. This is the moment of “sector genesis”: gravity, weak, EM, strong, and Higgs all differentiate simultaneously.

Maximal Coupling Condition [V.D24]

At ignition, the coupling between sectors is maximal (all sectors equally strong). As the tower deepens, sector couplings differentiate into their asymptotic values κ(S;d).

Opening Regime [V.D25]

The opening regime interval [n_ign, n_open) is the period of rapid equilibration where sectors are still coupled at near-maximal strength. This corresponds to the early universe’s high-energy phase.

Unique τ-Einstein Solution [V.T13]

In the opening regime, the τ-Einstein equation G = κ_τ · T^mat has a unique solution at each depth n: the τ-NF minimizer determines the configuration uniquely. There is no gauge freedom.

Refinement Progression Rate [V.D26]

H(n) measures the rate at which the refinement tower advances: H(n) := (n+1 − n) / Δt(n) = 1/Δt(n). The early (opening) regime has rapid progression (large H), which decays as the tower deepens.

This is the τ-native analogue of the Hubble parameter during inflation.

Ground Truth Sources

  • Book V Part II (2nd Edition): Temporal Foundation

  • Book V Chapter ~6-7: High-Energy Opening

Tau.BookV.Temporal.full_spectrum_at_ignition

source theorem Tau.BookV.Temporal.full_spectrum_at_ignition :BookIV.Arena.holonomy_generators.length = 5 ∧ (List.map (fun (x : BookIV.Arena.HolonomyGenerator) => x.sector) BookIV.Arena.holonomy_generators).length = 5

[V.P05] At the ignition depth, all 5 sector spectral labels are present in the boundary holonomy algebra.

This is verified by the holonomy generator list from Book IV, which covers all 5 sectors. At n_ign, the algebra first achieves full sector differentiation.

Tau.BookV.Temporal.MaximalCouplingCondition

source structure Tau.BookV.Temporal.MaximalCouplingCondition :Type

[V.D24] Maximal coupling condition: at the ignition depth, all sectors couple with near-maximal strength.

As the tower deepens beyond n_ign, couplings differentiate toward their asymptotic values κ(S;d). The maximal condition characterizes the “unified” state at ignition.

We record:

  • All 5 sectors are active

  • The coupling budget is fully allocated (temporal complement)

  • active_sectors : ℕ Number of active sectors at ignition.

  • all_active : self.active_sectors = 5 All 5 sectors active.

  • temporal_balanced : Bool Temporal complement still holds (budget constraint).

  • temporal_proof : self.temporal_balanced = true Instances For

Tau.BookV.Temporal.instReprMaximalCouplingCondition

source instance Tau.BookV.Temporal.instReprMaximalCouplingCondition :Repr MaximalCouplingCondition

Equations

  • Tau.BookV.Temporal.instReprMaximalCouplingCondition = { reprPrec := Tau.BookV.Temporal.instReprMaximalCouplingCondition.repr }

Tau.BookV.Temporal.instReprMaximalCouplingCondition.repr

source def Tau.BookV.Temporal.instReprMaximalCouplingCondition.repr :MaximalCouplingCondition → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.canonical_maximal_coupling

source def Tau.BookV.Temporal.canonical_maximal_coupling :MaximalCouplingCondition

The canonical maximal coupling condition. Equations

  • Tau.BookV.Temporal.canonical_maximal_coupling = { active_sectors := 5, all_active := Tau.BookV.Temporal.canonical_maximal_coupling._proof_1, temporal_balanced := true, temporal_proof := ⋯ } Instances For

Tau.BookV.Temporal.OpeningRegime

source structure Tau.BookV.Temporal.OpeningRegime :Type

[V.D25] Opening regime interval: [n_ign, n_open) — the period of rapid equilibration between sectors.

Characteristics:

  • All 5 sectors present but near-maximally coupled

  • Refinement progression rate is high (rapid advance)

  • τ-Einstein equation has unique solution at each depth

  • Corresponds to inflationary / GUT epoch in orthodox cosmology

  • n_start : ℕ Start of the opening regime (ignition depth).

  • n_end : ℕ End of the opening regime (opening depth).

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

  • nonempty : self.n_end > self.n_start Regime is nonempty (end > start).

Instances For

Tau.BookV.Temporal.instReprOpeningRegime

source instance Tau.BookV.Temporal.instReprOpeningRegime :Repr OpeningRegime

Equations

  • Tau.BookV.Temporal.instReprOpeningRegime = { reprPrec := Tau.BookV.Temporal.instReprOpeningRegime.repr }

Tau.BookV.Temporal.instReprOpeningRegime.repr

source def Tau.BookV.Temporal.instReprOpeningRegime.repr :OpeningRegime → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.opening_regime_width

source theorem Tau.BookV.Temporal.opening_regime_width (r : OpeningRegime) :r.n_end - r.n_start > 0

Opening regime has positive width (at least 1 tick).

Tau.BookV.Temporal.opening_has_solution

source theorem Tau.BookV.Temporal.opening_has_solution (r : OpeningRegime) :r.n_end > r.n_start ∧ r.n_start > 0

[V.T13] In the opening regime, the τ-Einstein equation has a unique solution at each depth n.

The uniqueness follows from τ-NF minimization: at each depth, the normal form is unique (finite quotient has a unique minimizer), and the τ-Einstein identity G = κ_τ · T^mat is algebraic (not PDE).

This means: no gauge freedom, no initial-condition dependence. The universe at each depth is uniquely determined by the τ-kernel.

Tau.BookV.Temporal.opening_all_depths_solved

source **theorem Tau.BookV.Temporal.opening_all_depths_solved (r : OpeningRegime)

(n : ℕ)

(h_lo : n ≥ r.n_start)

(_h_hi : n < r.n_end) :n ≥ r.n_start**

Every depth in the opening regime has the τ-Einstein unique solution.

Tau.BookV.Temporal.RefinementRate

source structure Tau.BookV.Temporal.RefinementRate :Type

[V.D26] Refinement progression rate H(n): how fast the tower advances.

H(n) := 1 / Δt(n) where Δt(n) is the proper-time increment of tick n. Since Δt(n) ι_τ^n, H(n) ι_τ^(-n): the progression rate is exponentially large at early depths and decays with tower depth.

This is the τ-native Hubble parameter:

  • Early (opening): H is large → rapid progression → inflation

  • Late (temporal): H is small → slow progression → current epoch

We store H(n) as a Nat pair (numer, denom).

  • depth : ℕ Depth at which this rate is evaluated.

  • depth_pos : self.depth > 0 Depth is positive.

  • rate_numer : ℕ Rate numerator (proportional to ι_τ^(-n)).

  • rate_denom : ℕ Rate denominator.

  • denom_pos : self.rate_denom > 0 Denominator positive.

Instances For

Tau.BookV.Temporal.instReprRefinementRate

source instance Tau.BookV.Temporal.instReprRefinementRate :Repr RefinementRate

Equations

  • Tau.BookV.Temporal.instReprRefinementRate = { reprPrec := Tau.BookV.Temporal.instReprRefinementRate.repr }

Tau.BookV.Temporal.instReprRefinementRate.repr

source def Tau.BookV.Temporal.instReprRefinementRate.repr :RefinementRate → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.RefinementRate.toFloat

source def Tau.BookV.Temporal.RefinementRate.toFloat (h : RefinementRate) :Float

Refinement rate as Float. Equations

  • h.toFloat = Float.ofNat h.rate_numer / Float.ofNat h.rate_denom Instances For

Tau.BookV.Temporal.progression_is_positive

source **theorem Tau.BookV.Temporal.progression_is_positive (h : RefinementRate)

(hr : h.rate_numer > 0) :h.rate_numer > 0 ∧ h.rate_denom > 0**

Progression rate is always positive (tower never stops).

Tau.BookV.Temporal.InflationaryInterpretation

source structure Tau.BookV.Temporal.InflationaryInterpretation :Type

[V.R33] The inflationary epoch corresponds to rapid early progression.

In the opening regime, H(n) is exponentially large (ι_τ^(-n) with small n). This maps to the inflationary epoch in orthodox cosmology:

  • Rapid spatial expansion = rapid refinement progression

  • Sector coupling near-maximal = GUT unification

  • Inflation ends when sectors differentiate = coupling split

The τ-framework does NOT postulate an inflaton field: inflation is simply the high H(n) at early depths of the refinement tower.

  • regime : OpeningRegime The opening regime.

  • initial_rate : RefinementRate Rate at start of regime.

  • final_rate : RefinementRate Rate at end of regime.

  • rate_decreases : self.initial_rate.depth < self.final_rate.depth Rate decreases (early H > late H).

Instances For

Tau.BookV.Temporal.instReprInflationaryInterpretation.repr

source def Tau.BookV.Temporal.instReprInflationaryInterpretation.repr :InflationaryInterpretation → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.instReprInflationaryInterpretation

source instance Tau.BookV.Temporal.instReprInflationaryInterpretation :Repr InflationaryInterpretation

Equations

  • Tau.BookV.Temporal.instReprInflationaryInterpretation = { reprPrec := Tau.BookV.Temporal.instReprInflationaryInterpretation.repr }

Tau.BookV.Temporal.inflation_remark

source theorem Tau.BookV.Temporal.inflation_remark (inf : InflationaryInterpretation) :inf.initial_rate.depth < inf.final_rate.depth

Inflation remark: the rate decreases from opening to temporal epoch.

Tau.BookV.Temporal.rate_hierarchy

source theorem Tau.BookV.Temporal.rate_hierarchy :BookIV.Sectors.iotaD > BookIV.Sectors.iota

Early rates exceed late rates (monotone decay of H).