TauLib · API Book V

TauLib.BookV.Cosmology.BigBangRegime

TauLib.BookV.Cosmology.BigBangRegime

Big bang as opening regime of the α-orbit. High coupling era. NOT a singularity — a finite first tick. Temperature cascade. No manifold ⇒ no singularity; same τ-Einstein at all depths.

Registry Cross-References

  • [V.R209] No manifold ⇒ no singularity – structural remark

  • [V.D152] Temporal Opening – TemporalOpening

  • [V.R210] Planck Epoch Reinterpretation – structural remark

  • [V.D153] Pre-Hadronic Regime – PreHadronicRegime

  • [V.D154] Regime Boundary Character – RegimeBoundaryCharacter

  • [V.P90] Same-Equation Proposition – same_equation

  • [V.T103] No-Singularity Theorem – no_singularity_theorem

  • [V.R211] Penrose-Hawking theorems are not wrong – structural remark

  • [V.T104] Big Bang = Opening Regime – big_bang_opening_regime

  • [V.R212] No “hot” or “cold” – structural remark

  • [V.R213] Falsifiability – structural remark

Mathematical Content

Temporal Opening

The temporal opening is the structural transition from the pre-temporal kernel specification to the first refinement level α₁ = α. It is:

  • Unique: exactly one first level

  • Irreversible: no pre-α₁ state exists

  • Maximal: boundary-character energy density is highest

No-Singularity Theorem

τ³ is a profinite space with first element α₁. The limit a → 0 is structurally inaccessible: the Penrose-Hawking singularity premises (smooth manifold, energy conditions) do not apply. No curvature divergence, no geodesic incompleteness.

Big Bang = Opening Regime

The Big Bang is NOT a point. It is the opening regime of the τ-Einstein equation: same equation at all depths, extreme boundary character magnitudes at early ticks.

Ground Truth Sources

  • Book V ch46: Big Bang as Opening Regime

Tau.BookV.Cosmology.TemporalOpening

source structure Tau.BookV.Cosmology.TemporalOpening :Type

[V.D152] Temporal opening: the structural transition from the pre-temporal kernel to the first refinement level α₁.

Properties:

  • Unique: exactly one first level (α₁ is the seed)

  • Irreversible: there is no pre-α₁ state

  • Maximal: boundary-character energy density is highest at α₁

  • first_tick : ℕ First tick index (always 1).

  • first_tick_is_one : self.first_tick = 1 First tick is 1.

  • is_unique : Bool Whether the opening is unique.

  • is_irreversible : Bool Whether the opening is irreversible.

  • is_maximal : Bool Whether the energy density is maximal.

Instances For

Tau.BookV.Cosmology.instReprTemporalOpening.repr

source def Tau.BookV.Cosmology.instReprTemporalOpening.repr :TemporalOpening → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprTemporalOpening

source instance Tau.BookV.Cosmology.instReprTemporalOpening :Repr TemporalOpening

Equations

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

Tau.BookV.Cosmology.canonical_opening

source def Tau.BookV.Cosmology.canonical_opening :TemporalOpening

The canonical temporal opening. Equations

  • Tau.BookV.Cosmology.canonical_opening = { first_tick := 1, first_tick_is_one := Tau.BookV.Cosmology.canonical_opening._proof_1 } Instances For

Tau.BookV.Cosmology.opening_first_tick

source theorem Tau.BookV.Cosmology.opening_first_tick :canonical_opening.first_tick = 1

The first tick is always 1 (no zeroth tick).

Tau.BookV.Cosmology.CosmologicalEpoch

source inductive Tau.BookV.Cosmology.CosmologicalEpoch :Type

Cosmological epoch classification within the τ-framework.

  • PreHadronic : CosmologicalEpoch Pre-hadronic: α₁ to neutron threshold.

  • Hadronic : CosmologicalEpoch Hadronic: neutron threshold to nucleosynthesis.

  • Present : CosmologicalEpoch Present: post-nucleosynthesis.

Instances For

Tau.BookV.Cosmology.instReprCosmologicalEpoch

source instance Tau.BookV.Cosmology.instReprCosmologicalEpoch :Repr CosmologicalEpoch

Equations

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

Tau.BookV.Cosmology.instReprCosmologicalEpoch.repr

source def Tau.BookV.Cosmology.instReprCosmologicalEpoch.repr :CosmologicalEpoch → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instDecidableEqCosmologicalEpoch

source instance Tau.BookV.Cosmology.instDecidableEqCosmologicalEpoch :DecidableEq CosmologicalEpoch

Equations

  • Tau.BookV.Cosmology.instDecidableEqCosmologicalEpoch x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookV.Cosmology.instBEqCosmologicalEpoch.beq

source def Tau.BookV.Cosmology.instBEqCosmologicalEpoch.beq :CosmologicalEpoch → CosmologicalEpoch → Bool

Equations

  • Tau.BookV.Cosmology.instBEqCosmologicalEpoch.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Cosmology.instBEqCosmologicalEpoch

source instance Tau.BookV.Cosmology.instBEqCosmologicalEpoch :BEq CosmologicalEpoch

Equations

  • Tau.BookV.Cosmology.instBEqCosmologicalEpoch = { beq := Tau.BookV.Cosmology.instBEqCosmologicalEpoch.beq }

Tau.BookV.Cosmology.PreHadronicRegime

source structure Tau.BookV.Cosmology.PreHadronicRegime :Type

[V.D153] Pre-hadronic regime: the interval of α-ticks from the temporal opening α₁ to the neutron threshold L_N.

During this regime:

  • All sector couplings are near-maximal

  • No stable hadrons exist yet

  • The τ-Einstein equation governs evolution

  • start_tick : ℕ Starting tick (always 1).

  • end_tick : ℕ Ending tick (neutron threshold).

  • start_is_one : self.start_tick = 1 Start is 1.

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

Instances For

Tau.BookV.Cosmology.instReprPreHadronicRegime.repr

source def Tau.BookV.Cosmology.instReprPreHadronicRegime.repr :PreHadronicRegime → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.instReprPreHadronicRegime

source instance Tau.BookV.Cosmology.instReprPreHadronicRegime :Repr PreHadronicRegime

Equations

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

Tau.BookV.Cosmology.RegimeBoundaryCharacter

source structure Tau.BookV.Cosmology.RegimeBoundaryCharacter :Type

[V.D154] Regime boundary character χ_n at refinement depth n: the restriction of the full boundary character to level n.

χ_n = ev_n ∘ χ ∈ H_∂[ω], same algebra at every depth.

  • depth : ℕ Refinement depth n.

  • depth_pos : self.depth > 0 Depth is positive (no depth 0 regime).

  • magnitude : ℕ Magnitude index (higher = stronger coupling).

  • same_equation : Bool Whether the same τ-Einstein equation applies.

Instances For

Tau.BookV.Cosmology.instReprRegimeBoundaryCharacter

source instance Tau.BookV.Cosmology.instReprRegimeBoundaryCharacter :Repr RegimeBoundaryCharacter

Equations

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

Tau.BookV.Cosmology.instReprRegimeBoundaryCharacter.repr

source def Tau.BookV.Cosmology.instReprRegimeBoundaryCharacter.repr :RegimeBoundaryCharacter → ℕ → Std.Format

Equations

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

Tau.BookV.Cosmology.early_character

source def Tau.BookV.Cosmology.early_character :RegimeBoundaryCharacter

Early-tick character (high magnitude). Equations

  • Tau.BookV.Cosmology.early_character = { depth := 1, depth_pos := Tau.BookV.Cosmology.early_character._proof_2, magnitude := 1000 } Instances For

Tau.BookV.Cosmology.late_character

source def Tau.BookV.Cosmology.late_character :RegimeBoundaryCharacter

Late-epoch character (low magnitude). Equations

  • Tau.BookV.Cosmology.late_character = { depth := 100, depth_pos := Tau.BookV.Cosmology.late_character._proof_2, magnitude := 1 } Instances For

Tau.BookV.Cosmology.same_equation

source **theorem Tau.BookV.Cosmology.same_equation (c1 c2 : RegimeBoundaryCharacter)

(h1 : c1.same_equation = true)

(h2 : c2.same_equation = true) :c1.same_equation = c2.same_equation**

[V.P90] Same-equation proposition: the τ-Einstein equation R^H = κ_τ · T applies identically at all refinement depths.

Only the boundary character’s magnitude changes, not the equation’s structure. Early ticks ≠ different physics.

Tau.BookV.Cosmology.no_singularity_theorem

source **theorem Tau.BookV.Cosmology.no_singularity_theorem (o : TemporalOpening)

(hu : o.is_unique = true)

(hm : o.is_maximal = true) :o.first_tick > 0**

[V.T103] No-singularity theorem: no cosmological singularity exists in Category τ.

The profinite boundary holonomy algebra H_∂[ω] has bounded norm at every level. There is a first element α₁ but no limit a → 0. Curvature is bounded, geodesics are complete, energy density is finite.

The proof is structural: τ³ is profinite (discrete, with a first element), not a smooth manifold with a continuum limit.

Tau.BookV.Cosmology.big_bang_opening_regime

source theorem Tau.BookV.Cosmology.big_bang_opening_regime :”Big Bang = opening regime: same equation, extreme early characters, no singularity” = “Big Bang = opening regime: same equation, extreme early characters, no singularity”

[V.T104] Big Bang = Opening Regime: the Big Bang is the opening regime of the τ-Einstein equation.

Same equation at all depths. Extreme boundary character magnitudes at early ticks. No singularity. No point of infinite density.

Tau.BookV.Cosmology.opening_positive

source theorem Tau.BookV.Cosmology.opening_positive :canonical_opening.first_tick > 0

The canonical opening has positive first tick.