TauLib · API Book IV

TauLib.BookIV.Sectors.CouplingFormulas

TauLib.BookIV.Sectors.CouplingFormulas

The 10-entry coupling ledger: all inter-sector couplings as rational functions of ι_τ = 2/(π+e), with structural theorems.

Registry Cross-References

  • [IV.D07] Coupling Formula Map — coupling_formula, all_coupling_formulas

  • [IV.T01] Temporal Complement — temporal_complement

  • [IV.T02] Temporal Multiplicative Closure — temporal_multiplicative

  • [IV.P01] All Couplings Positive — all_formulas_positive

  • [IV.P03] Power Hierarchy — em_is_weak_squared, weak_strong_is_multiplicative

Mathematical Content

The No Knobs Principle (III.T08) determines all 10 inter-sector couplings from ι_τ alone. This module gives the explicit rational function formulas:

Self-couplings (4)

Sector Formula Physical meaning

D (Gravity) 1 − ι_τ Temporal flow magnitude

A (Weak) ι_τ Temporal arrow (= master constant)

B (EM) ι_τ² Spatial distance scale

C (Strong) ι_τ³/(1−ι_τ) Confinement coupling

Cross-couplings (6)

Pair Formula Physical meaning

(A,B) ι_τ³ Electroweak (multiplicative closure κ(A)·κ(B))

(A,C) ι_τ⁴/(1−ι_τ) Weak-Strong (multiplicative closure κ(A)·κ(C))

(A,D) ι_τ(1−ι_τ) Weak-Gravity (temporal consistency)

(B,C) ι_τ³/(1+ι_τ) EM-Strong = Higgs/mass crossing

(B,D) ι_τ²(1−ι_τ) EM-Gravity (lensing)

(C,D) ι_τ³(1−ι_τ) Strong-Gravity (mass curves time)

Key structural relations

  • κ(A;1) + κ(D;1) = 1 (temporal complement)

  • κ(A,D) = κ(A;1) · κ(D;1) (temporal multiplicative closure)

  • κ(B;2) = κ(A;1)² (EM = Weak squared)

  • κ(A,C) = κ(A;1)·κ(C;3) (multiplicative closure)

Ground Truth Sources

  • temporal_spatial_decomposition.md §5: complete coupling reinterpretation

  • Book III ch63 No Knobs Ledger: 10-entry inventory

Tau.BookIV.Sectors.CouplingFormula

source structure Tau.BookIV.Sectors.CouplingFormula :Type

[IV.D07] A coupling formula: rational expression of ι_τ between two sectors, evaluated at the rational approximation.

  • sector_i : BookIII.Sectors.Sector First sector (ordered by Sector.toNat).

  • sector_j : BookIII.Sectors.Sector Second sector.

  • numer : ℕ Numerator of coupling (scaled).

  • denom : ℕ Denominator of coupling (scaled).

  • denom_pos : self.denom > 0 Denominator is positive.

Instances For

Tau.BookIV.Sectors.instReprCouplingFormula.repr

source def Tau.BookIV.Sectors.instReprCouplingFormula.repr :CouplingFormula → ℕ → Std.Format

Equations

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

Tau.BookIV.Sectors.instReprCouplingFormula

source instance Tau.BookIV.Sectors.instReprCouplingFormula :Repr CouplingFormula

Equations

  • Tau.BookIV.Sectors.instReprCouplingFormula = { reprPrec := Tau.BookIV.Sectors.instReprCouplingFormula.repr }

Tau.BookIV.Sectors.CouplingFormula.toFloat

source def Tau.BookIV.Sectors.CouplingFormula.toFloat (c : CouplingFormula) :Float

Coupling formula as Float. Equations

  • c.toFloat = Float.ofNat c.numer / Float.ofNat c.denom Instances For

Tau.BookIV.Sectors.kappa_DD

source def Tau.BookIV.Sectors.kappa_DD :CouplingFormula

κ(D,D) = 1 − ι_τ: Gravity self-coupling. Equations

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Tau.BookIV.Sectors.kappa_AA

source def Tau.BookIV.Sectors.kappa_AA :CouplingFormula

κ(A,A) = ι_τ: Weak self-coupling. Equations

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

Tau.BookIV.Sectors.kappa_BB

source def Tau.BookIV.Sectors.kappa_BB :CouplingFormula

κ(B,B) = ι_τ²: EM self-coupling. Equations

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

Tau.BookIV.Sectors.kappa_CC

source def Tau.BookIV.Sectors.kappa_CC :CouplingFormula

κ(C,C) = ι_τ³/(1−ι_τ): Strong self-coupling (confinement). Equations

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

Tau.BookIV.Sectors.kappa_AB

source def Tau.BookIV.Sectors.kappa_AB :CouplingFormula

κ(A,B) = ι_τ³: Electroweak coupling (multiplicative closure κ(A)·κ(B)). Equations

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

Tau.BookIV.Sectors.kappa_AC

source def Tau.BookIV.Sectors.kappa_AC :CouplingFormula

κ(A,C) = ι_τ⁴/(1−ι_τ): Weak-Strong coupling (multiplicative closure κ(A)·κ(C)). Equations

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

Tau.BookIV.Sectors.kappa_AD

source def Tau.BookIV.Sectors.kappa_AD :CouplingFormula

κ(A,D) = ι_τ(1−ι_τ): Weak-Gravity coupling (temporal consistency). Equations

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Tau.BookIV.Sectors.kappa_BC

source def Tau.BookIV.Sectors.kappa_BC :CouplingFormula

κ(B,C) = ι_τ³/(1+ι_τ): EM-Strong = Higgs/mass crossing. Equations

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

Tau.BookIV.Sectors.kappa_BD

source def Tau.BookIV.Sectors.kappa_BD :CouplingFormula

κ(B,D) = ι_τ²(1−ι_τ): EM-Gravity (gravitational lensing). Equations

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

Tau.BookIV.Sectors.kappa_CD

source def Tau.BookIV.Sectors.kappa_CD :CouplingFormula

κ(C,D) = ι_τ³(1−ι_τ): Strong-Gravity (mass curves time). Equations

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

Tau.BookIV.Sectors.all_coupling_formulas

source def Tau.BookIV.Sectors.all_coupling_formulas :List CouplingFormula

The complete 10-entry coupling ledger. Equations

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

Tau.BookIV.Sectors.temporal_complement

source theorem Tau.BookIV.Sectors.temporal_complement :kappa_AA.numer + kappa_DD.numer = kappa_AA.denom

[IV.T01] κ(A;1) + κ(D;1) = 1: the temporal pair exhausts the depth-1 coupling budget. Gravity and Weak are complements.

Proof: ι + (D − ι) = D, so ι/D + (D−ι)/D = 1.

Tau.BookIV.Sectors.temporal_complement_denom

source theorem Tau.BookIV.Sectors.temporal_complement_denom :kappa_AA.denom = kappa_DD.denom

The shared denominator confirms they sum to exactly 1.

Tau.BookIV.Sectors.temporal_multiplicative

source theorem Tau.BookIV.Sectors.temporal_multiplicative :kappa_AD.numer * (kappa_AA.denom * kappa_DD.denom) = kappa_AA.numer * kappa_DD.numer * kappa_AD.denom

[IV.T02] κ(A,D) = κ(A;1) · κ(D;1): the temporal cross-coupling is exactly the product of the two temporal self-couplings. This means the temporal pair is “multiplicatively closed.”

Proof: ι(D−ι)/D² = (ι/D)·((D−ι)/D).

Tau.BookIV.Sectors.em_is_weak_squared

source theorem Tau.BookIV.Sectors.em_is_weak_squared :kappa_BB.numer * (kappa_AA.denom * kappa_AA.denom) = kappa_AA.numer * kappa_AA.numer * kappa_BB.denom

[IV.P03a] κ(B;2) = κ(A;1)²: EM self-coupling equals Weak squared. Proof: ι²/D² = (ι/D)².

Tau.BookIV.Sectors.weak_strong_is_multiplicative

source theorem Tau.BookIV.Sectors.weak_strong_is_multiplicative :kappa_AC.numer * (kappa_AA.denom * kappa_CC.denom) = kappa_AA.numer * kappa_CC.numer * kappa_AC.denom

[IV.P03b] κ(A,C) = κ(A;1)·κ(C;3): Weak-Strong = Weak × Strong (multiplicative closure). Proof: (ι⁴·D)/(D⁴·(D−ι)) = (ι/D) · (ι³·D/(D³·(D−ι))).

Tau.BookIV.Sectors.all_formulas_positive

source theorem Tau.BookIV.Sectors.all_formulas_positive :kappa_DD.numer > 0 ∧ kappa_AA.numer > 0 ∧ kappa_BB.numer > 0 ∧ kappa_CC.numer > 0 ∧ kappa_AB.numer > 0 ∧ kappa_AC.numer > 0 ∧ kappa_AD.numer > 0 ∧ kappa_BC.numer > 0 ∧ kappa_BD.numer > 0 ∧ kappa_CD.numer > 0

[IV.P01] All 10 coupling numerators are positive. Since all denominators are positive by construction, all coupling values are strictly positive.

Tau.BookIV.Sectors.coupling_ledger_count

source theorem Tau.BookIV.Sectors.coupling_ledger_count :all_coupling_formulas.length = 10

The ledger has exactly 10 entries.

Tau.BookIV.Sectors.coupling_formula

source def Tau.BookIV.Sectors.coupling_formula (si sj : BookIII.Sectors.Sector) :CouplingFormula

Retrieve the coupling formula for a sector pair. Symmetric: coupling(i,j) = coupling(j,i). Equations

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

Tau.BookIV.Sectors.self_coupling_order

source theorem Tau.BookIV.Sectors.self_coupling_order :kappa_CC.toFloat < kappa_BB.toFloat ∧ kappa_BB.toFloat < kappa_AA.toFloat ∧ kappa_AA.toFloat < kappa_DD.toFloat

Self-couplings are ordered: κ(C) < κ(B) < κ(A) < κ(D). Strong < EM < Weak < Gravity in coupling strength.