TauLib · API Book II

TauLib.BookII.Transcendentals.JReplacesI

TauLib.BookII.Transcendentals.JReplacesI

j replaces i: the split-complex unit j (j^2 = +1) replaces the imaginary unit i (i^2 = -1) as the fundamental algebraic unit.

Registry Cross-References

  • [II.D32] Interior j-Unit – j_unit, j_squared_check

  • [II.D33] Bipolar Idempotents Interior – idempotent_check

  • [II.T24] j Replaces i – j_vs_i_check

Mathematical Content

The split-complex unit j with j^2 = +1 forces hyperbolic (wave) geometry rather than elliptic (Laplacian) geometry. The idempotents e_+ = (1+j)/2 and e_- = (1-j)/2 provide the bipolar decomposition.

Key contrast:

  • i^2 = -1 (Gaussian): rotation, elliptic PDE, NO nontrivial idempotents over Z

  • j^2 = +1 (split-complex): polarity, wave PDE, idempotents e_+, e_-

The split-complex structure is FORCED by the bipolar prime partition (I.T10). This module verifies the algebraic properties at the interior level.

Tau.BookII.Transcendentals.j_unit

source def Tau.BookII.Transcendentals.j_unit :Polarity.SplitComplex

[II.D32] The interior j-unit: j = (0, 1) in SplitComplex. Equations

  • Tau.BookII.Transcendentals.j_unit = { re := 0, im := 1 } Instances For

Tau.BookII.Transcendentals.j_squared_check

source def Tau.BookII.Transcendentals.j_squared_check :Bool

j^2 = 1: the defining property. (0,1)(0,1) = (00+11, 01+1*0) = (1,0). Equations

  • Tau.BookII.Transcendentals.j_squared_check = (Tau.BookII.Transcendentals.j_unit.mul Tau.BookII.Transcendentals.j_unit == { re := 1, im := 0 }) Instances For

Tau.BookII.Transcendentals.j_nontrivial_check

source def Tau.BookII.Transcendentals.j_nontrivial_check :Bool

j is not trivial: j != 0 and j != 1. Equations

  • Tau.BookII.Transcendentals.j_nontrivial_check = (Tau.BookII.Transcendentals.j_unit != Tau.Polarity.SplitComplex.zero && Tau.BookII.Transcendentals.j_unit != Tau.Polarity.SplitComplex.one) Instances For

Tau.BookII.Transcendentals.j_involution_check

source def Tau.BookII.Transcendentals.j_involution_check :Bool

Polarity involution: sigma(j) = -j. Equations

  • Tau.BookII.Transcendentals.j_involution_check = (Tau.Polarity.polarity_inv Tau.BookII.Transcendentals.j_unit == Tau.BookII.Transcendentals.j_unit.neg) Instances For

Tau.BookII.Transcendentals.idempotent_check

source def Tau.BookII.Transcendentals.idempotent_check :Bool

[II.D33] Bipolar idempotents in sector coordinates. e_+ = (1, 0) and e_- = (0, 1) in SectorPair.

Algebraic properties:

  • e_+^2 = e_+ (idempotent)

  • e_-^2 = e_- (idempotent)

  • e_+ * e_- = 0 (orthogonal)

  • e_+ + e_- = (1,1) = unity

Equations

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

Tau.BookII.Transcendentals.interior_projection_check

source def Tau.BookII.Transcendentals.interior_projection_check :Bool

Interior projection: idempotent action on interior bipolar pairs. e_+ * (b, c) = (b, 0): keeps B-sector, kills C-sector. e_- * (b, c) = (0, c): kills B-sector, keeps C-sector. Equations

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

Tau.BookII.Transcendentals.j_vs_i_check

source def Tau.BookII.Transcendentals.j_vs_i_check :Bool

[II.T24] j replaces i: comprehensive comparison.

  • j^2 = +1 (split-complex, hyperbolic)

  • i^2 = -1 (Gaussian, elliptic)

  • j admits nontrivial idempotents; i does not (I.T10)

  • Zero divisors in H_tau witness polarity structure

The wave equation u_tt = u_xx comes from j^2 = +1. The Laplace equation u_tt + u_xx = 0 would come from i^2 = -1. Category tau chooses j (waves, polarity) over i (rotation). Equations

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

Tau.BookII.Transcendentals.wave_vs_laplace_check

source def Tau.BookII.Transcendentals.wave_vs_laplace_check :Bool

Wave vs Laplace: the two signatures in sector coordinates. j^2 = +1: sector product is componentwise (wave eq characteristic coords) i^2 = -1: Gaussian product mixes components (elliptic) Equations

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

Tau.BookII.Transcendentals.interior_bipolar_via_j

source def Tau.BookII.Transcendentals.interior_bipolar_via_j (bound : Denotation.TauIdx) :Bool

Every tau-admissible point inherits bipolar decomposition via j. The B-coordinate maps to the e_+-sector, C to e_-. Equations

  • Tau.BookII.Transcendentals.interior_bipolar_via_j bound = Tau.BookII.Transcendentals.interior_bipolar_via_j.go bound 2 (bound + 1) Instances For

Tau.BookII.Transcendentals.interior_bipolar_via_j.go

source@[irreducible]

**def Tau.BookII.Transcendentals.interior_bipolar_via_j.go (bound : Denotation.TauIdx)

(x fuel : ℕ) :Bool**

Equations

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

Tau.BookII.Transcendentals.j_sq

source theorem Tau.BookII.Transcendentals.j_sq :j_squared_check = true

Tau.BookII.Transcendentals.j_nontriv

source theorem Tau.BookII.Transcendentals.j_nontriv :j_nontrivial_check = true

Tau.BookII.Transcendentals.j_invol

source theorem Tau.BookII.Transcendentals.j_invol :j_involution_check = true

Tau.BookII.Transcendentals.idemp

source theorem Tau.BookII.Transcendentals.idemp :idempotent_check = true

Tau.BookII.Transcendentals.int_proj

source theorem Tau.BookII.Transcendentals.int_proj :interior_projection_check = true

Tau.BookII.Transcendentals.j_vs_i

source theorem Tau.BookII.Transcendentals.j_vs_i :j_vs_i_check = true

Tau.BookII.Transcendentals.wave_laplace

source theorem Tau.BookII.Transcendentals.wave_laplace :wave_vs_laplace_check = true

Tau.BookII.Transcendentals.int_bipolar_30

source theorem Tau.BookII.Transcendentals.int_bipolar_30 :interior_bipolar_via_j 30 = true