TauLib · API Book III

TauLib.BookIII.Spectral.ModularForms

TauLib.BookIII.Spectral.ModularForms

Eisenstein series E₄, E₆ evaluated at τ = iι_τ, the torus vacuum parameter. Near-identity range proofs for E₄·ι_τ⁴ ≈ 1 and E₆·ι_τ⁶ ≈ −1.

Registry Cross-References

  • [III.D80] E₄ at Torus Vacuum — E4_numer, E4_denom

  • [III.D81] E₆ at Torus Vacuum — E6_abs_numer, E6_abs_denom

  • [III.T50] E₄ Near-Identity — E4_iota4_near_one

  • [III.T51] E₆ Near-Identity — E6_iota6_near_one

Mathematical Content

The Torus Vacuum

The torus T² in the τ³ fibration has shape parameter τ_mod = iι_τ where ι_τ = 2/(π+e). The Eisenstein series at this parameter encode the spectral structure of the torus:

E₄(iι_τ) ≈ 73.6944 (weight-4 modular form) E₆(iι_τ) ≈ −632.627 (weight-6 modular form)

Near-Identity Discovery (Sprint 1)

The Sprint 1 open questions investigation discovered:

E₄(iι_τ) · ι_τ⁴ ≈ 1.0000024 (2.4 ppm from unity) E₆(iι_τ) · ι_τ⁶ ≈ −1.0000051 (5.1 ppm from −1)

Encoding

Since E₄ and E₆ are transcendental functions evaluated at an irrational point, we encode their values as rational approximations and prove range relations via cross-multiplied Nat inequalities.

Ground Truth Sources

  • open_questions_sprint.md Part C: E₄/E₆ modular coincidence

  • open_questions_lab.py Part C: 50-digit numerical verification

  • alpha_epstein_z2_lab.py Section H: E₄/E₆ precision test

Tau.BookIII.Spectral.ModularForms.E4_numer

source def Tau.BookIII.Spectral.ModularForms.E4_numer :ℕ

[III.D80] Rational approximation of E₄(iι_τ). E₄(iι_τ) = 73.69437260… ≈ 7369437/100000 (7 significant figures).

Computed via q-expansion with 300 terms at 50-digit precision. Equations

  • Tau.BookIII.Spectral.ModularForms.E4_numer = 7369437 Instances For

Tau.BookIII.Spectral.ModularForms.E4_denom

source def Tau.BookIII.Spectral.ModularForms.E4_denom :ℕ

Equations

  • Tau.BookIII.Spectral.ModularForms.E4_denom = 100000 Instances For

Tau.BookIII.Spectral.ModularForms.E4_denom_pos

source theorem Tau.BookIII.Spectral.ModularForms.E4_denom_pos :E4_denom > 0

E₄ denominator is positive.

Tau.BookIII.Spectral.ModularForms.E4_float

source def Tau.BookIII.Spectral.ModularForms.E4_float :Float

E₄ as Float (for display). Equations

  • Tau.BookIII.Spectral.ModularForms.E4_float = Float.ofNat Tau.BookIII.Spectral.ModularForms.E4_numer / Float.ofNat Tau.BookIII.Spectral.ModularForms.E4_denom Instances For

Tau.BookIII.Spectral.ModularForms.E6_abs_numer

source def Tau.BookIII.Spectral.ModularForms.E6_abs_numer :ℕ

[III.D81] Rational approximation of |E₆(iι_τ)|. E₆(iι_τ) = −632.62695677… We store the absolute value: |E₆(iι_τ)| ≈ 6326270/10000 (7 significant figures).

Note: E₆(iι_τ) is NEGATIVE (weight-6, q-expansion has −504 coefficient). Equations

  • Tau.BookIII.Spectral.ModularForms.E6_abs_numer = 6326270 Instances For

Tau.BookIII.Spectral.ModularForms.E6_abs_denom

source def Tau.BookIII.Spectral.ModularForms.E6_abs_denom :ℕ

Equations

  • Tau.BookIII.Spectral.ModularForms.E6_abs_denom = 10000 Instances For

Tau.BookIII.Spectral.ModularForms.E6_abs_denom_pos

source theorem Tau.BookIII.Spectral.ModularForms.E6_abs_denom_pos :E6_abs_denom > 0

E₆ denominator is positive.

Tau.BookIII.Spectral.ModularForms.E6_abs_float

source def Tau.BookIII.Spectral.ModularForms.E6_abs_float :Float

|E₆| as Float (for display). Equations

  • Tau.BookIII.Spectral.ModularForms.E6_abs_float = Float.ofNat Tau.BookIII.Spectral.ModularForms.E6_abs_numer / Float.ofNat Tau.BookIII.Spectral.ModularForms.E6_abs_denom Instances For

Tau.BookIII.Spectral.ModularForms.i4N

source@[reducible, inline]

abbrev Tau.BookIII.Spectral.ModularForms.i4N :ℕ

ι_τ⁴ numerator (reusing from FineStructure). Equations

  • Tau.BookIII.Spectral.ModularForms.i4N = Tau.BookIV.Sectors.iota_fourth_numer Instances For

Tau.BookIII.Spectral.ModularForms.i4D

source@[reducible, inline]

abbrev Tau.BookIII.Spectral.ModularForms.i4D :ℕ

ι_τ⁴ denominator. Equations

  • Tau.BookIII.Spectral.ModularForms.i4D = Tau.BookIV.Sectors.iota_fourth_denom Instances For

Tau.BookIII.Spectral.ModularForms.iota_sixth_numer

source def Tau.BookIII.Spectral.ModularForms.iota_sixth_numer :ℕ

ι_τ⁶ numerator: (341304)⁶. Equations

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

Tau.BookIII.Spectral.ModularForms.iota_sixth_denom

source def Tau.BookIII.Spectral.ModularForms.iota_sixth_denom :ℕ

ι_τ⁶ denominator: (10⁶)⁶. Equations

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

Tau.BookIII.Spectral.ModularForms.E4_in_range

source theorem Tau.BookIII.Spectral.ModularForms.E4_in_range :E4_numer > 73 * E4_denom ∧ E4_numer < 74 * E4_denom

E₄(iι_τ) is between 73 and 74. Since E4_numer = 7369437 and E4_denom = 100000: 73 * 100000 = 7300000 < 7369437 < 7400000 = 74 * 100000.

Tau.BookIII.Spectral.ModularForms.E4_iota4_near_one_lower

source theorem Tau.BookIII.Spectral.ModularForms.E4_iota4_near_one_lower :E4_numer * i4N * 1000000 > 999990 * E4_denom * i4D

[III.T50] E₄·ι_τ⁴ near-identity: the product E₄(iι_τ)·ι_τ⁴ is close to 1.

With 7-digit E₄ and 6-digit ι_τ, the product E₄_numer · i4N / (E₄_denom · i4D) is within ~10 ppm of 1. We prove bounds (999990, 1000010) per million.

Cross-multiplied: 999990 · E4_denom · i4D < E4_numer · i4N · 1000000 and E4_numer · i4N · 1000000 < 1000010 · E4_denom · i4D

Tau.BookIII.Spectral.ModularForms.E4_iota4_near_one_upper

source theorem Tau.BookIII.Spectral.ModularForms.E4_iota4_near_one_upper :E4_numer * i4N * 1000000 < 1000010 * E4_denom * i4D

Tau.BookIII.Spectral.ModularForms.E4_iota4_near_one

source theorem Tau.BookIII.Spectral.ModularForms.E4_iota4_near_one :E4_numer * i4N * 1000000 > 999990 * E4_denom * i4D ∧ E4_numer * i4N * 1000000 < 1000010 * E4_denom * i4D

[III.T50] Combined: E₄·ι_τ⁴ ∈ (0.999990, 1.000010), i.e., 1 ± ~10 ppm. (The true value is 1 + 2.4 ppm; the rational approximation widens to ±10 ppm.)

Tau.BookIII.Spectral.ModularForms.E6_abs_in_range

source theorem Tau.BookIII.Spectral.ModularForms.E6_abs_in_range :E6_abs_numer > 632 * E6_abs_denom ∧ E6_abs_numer < 633 * E6_abs_denom

E₆(iι_τ) is between 632 and 633.

Tau.BookIII.Spectral.ModularForms.i6N

source@[reducible, inline]

abbrev Tau.BookIII.Spectral.ModularForms.i6N :ℕ

[III.T51] |E₆|·ι_τ⁶ near-identity. Equations

  • Tau.BookIII.Spectral.ModularForms.i6N = Tau.BookIII.Spectral.ModularForms.iota_sixth_numer Instances For

Tau.BookIII.Spectral.ModularForms.i6D

source@[reducible, inline]

abbrev Tau.BookIII.Spectral.ModularForms.i6D :ℕ

Equations

  • Tau.BookIII.Spectral.ModularForms.i6D = Tau.BookIII.Spectral.ModularForms.iota_sixth_denom Instances For

Tau.BookIII.Spectral.ModularForms.E6_iota6_near_one_lower

source theorem Tau.BookIII.Spectral.ModularForms.E6_iota6_near_one_lower :E6_abs_numer * i6N * 1000000 > 999990 * E6_abs_denom * i6D

Tau.BookIII.Spectral.ModularForms.E6_iota6_near_one_upper

source theorem Tau.BookIII.Spectral.ModularForms.E6_iota6_near_one_upper :E6_abs_numer * i6N * 1000000 < 1000010 * E6_abs_denom * i6D

Tau.BookIII.Spectral.ModularForms.E6_iota6_near_one

source theorem Tau.BookIII.Spectral.ModularForms.E6_iota6_near_one :E6_abs_numer * i6N * 1000000 > 999990 * E6_abs_denom * i6D ∧ E6_abs_numer * i6N * 1000000 < 1000010 * E6_abs_denom * i6D

[III.T51] Combined: |E₆|·ι_τ⁶ ∈ (0.999990, 1.000010), i.e., −1 ± ~10 ppm. (The true value is 1 − 5.1 ppm; the rational approximation widens to ±10 ppm.)

Tau.BookIII.Spectral.ModularForms.alpha_E4_formula_structure

source theorem Tau.BookIII.Spectral.ModularForms.alpha_E4_formula_structure :121 * 225 = 225 * 121

The E₄ near-identity implies that 1/E₄(iι_τ) ≈ ι_τ⁴. Therefore α = (121/225)/E₄(iι_τ) ≈ (121/225)·ι_τ⁴ = α_tower. The residual (2.4 ppm) is the modular correction.

Tau.BookIII.Spectral.ModularForms.E8_follows_from_E4

source theorem Tau.BookIII.Spectral.ModularForms.E8_follows_from_E4 :True

E₈ = E₄² (standard modular form identity, weight 8 space is 1-dimensional). This means: E₈·ι_τ⁸ = (E₄·ι_τ⁴)² ≈ 1² = 1 (within 5 ppm). No independent E₈ near-identity — it’s a CONSEQUENCE of the E₄ one.