TauLib · API Book II

TauLib.BookII.Transcendentals.IotaTauConfirmed

TauLib.BookII.Transcendentals.IotaTauConfirmed

Master constant iota_tau = 2/(pi + e) confirmed with earned pi and e.

Registry Cross-References

  • [II.T25] Master Constant Confirmed – iota_tau_confirmed_check

  • [II.D34] Archimedean Bridge – archimedean_bridge_check

  • [II.P07] Refinement Resolution Bound – refinement_resolution_check

Mathematical Content

The master constant iota_tau = 2/(pi + e) ~ 0.341304 is now confirmed using the pi and e earned in the previous modules. This closes the circle: iota_tau was introduced axiomatically in Book I; now it is derived from earned transcendentals.

The Archimedean bridge: iota_tau mediates between the Archimedean world (pi, e from limits/series) and the profinite world (tau^3 from primorial inverse system). The resolution crossover happens between stages k=1 and k=2: 1/P_1 = 0.5 > iota_tau > 1/P_2 ~ 0.167.

Refinement resolution bound: at stage k, the approximation error is bounded by 1/P_k, which decreases monotonically.

Tau.BookII.Transcendentals.iota_tau_computed

source def Tau.BookII.Transcendentals.iota_tau_computed (pi_terms e_terms scale : ℕ) :ℕ

Compute iota_tau * scale using earned pi and e approximations. iota_tau = 2 / (pi + e). In scaled arithmetic: iota_tau * scale = 2 * scale^2 / (pi_scaled + e_scaled).

pi_scaled and e_scaled are both in units of 10^6. Equations

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

Tau.BookII.Transcendentals.iota_tau_confirmed_check

source def Tau.BookII.Transcendentals.iota_tau_confirmed_check :Bool

[II.T25] Master constant confirmed: iota_tau 0.341304. iota_tau * 10^6 341304. With 2000 Leibniz terms and 12 factorial terms: pi 3141092, e 2718281, sum 5859373 iota 2 * 10^12 / 5859373 ~ 341381 (within 1% of 341304). Equations

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

Tau.BookII.Transcendentals.iota_tau_precision_check

source def Tau.BookII.Transcendentals.iota_tau_precision_check :Bool

Higher precision check: the computed iota_tau matches the known rational approximation 341304/10^6 to within 2%. Equations

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

Tau.BookII.Transcendentals.self_consistency_check

source def Tau.BookII.Transcendentals.self_consistency_check :Bool

Self-consistency: pi + e = 2 / iota_tau. In scaled arithmetic: (pi + e) * iota = 2 * scale. Check: pi_scaled + e_scaled ~ 2 * scale^2 / iota_scaled. Equations

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

Tau.BookII.Transcendentals.archimedean_bridge_check

source def Tau.BookII.Transcendentals.archimedean_bridge_check :Bool

[II.D34] Archimedean bridge: iota_tau mediates between the Archimedean world (pi, e from limits) and the profinite world (tau^3 from primorial inverse system).

The resolution at stage k is 1/P_k. The scale where resolution crosses iota_tau happens between k=1 and k=2: 1/P_1 = 1/2 = 0.5 > iota_tau 0.341 > 1/P_2 = 1/6 0.167.

In scaled integer arithmetic: scale/P_k vs iota_tau_scaled. Equations

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

Tau.BookII.Transcendentals.bridge_interpolation_check

source def Tau.BookII.Transcendentals.bridge_interpolation_check :Bool

Bridge characterization: iota_tau as the stage-1/stage-2 interpolation constant. The fact that iota_tau falls between 1/P_1 and 1/P_2 means it governs the first refinement step where the primorial ladder becomes finer than the master constant. Equations

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

Tau.BookII.Transcendentals.refinement_resolution_check

source def Tau.BookII.Transcendentals.refinement_resolution_check (stages : Denotation.TauIdx) :Bool

[II.P07] Refinement resolution bound: at stage k, the resolution 1/P_k decreases monotonically. P_{k+1} > P_k for all k >= 0. Equations

  • Tau.BookII.Transcendentals.refinement_resolution_check stages = Tau.BookII.Transcendentals.refinement_resolution_check.go stages 1 (stages + 1) Instances For

Tau.BookII.Transcendentals.refinement_resolution_check.go

source@[irreducible]

**def Tau.BookII.Transcendentals.refinement_resolution_check.go (stages : Denotation.TauIdx)

(k fuel : ℕ) :Bool**

Equations

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

Tau.BookII.Transcendentals.resolution_halving_check

source def Tau.BookII.Transcendentals.resolution_halving_check (stages : Denotation.TauIdx) :Bool

Resolution ratio: P_{k+1}/P_k = p_{k+1} >= 2. Each refinement step at least halves the resolution. Equations

  • Tau.BookII.Transcendentals.resolution_halving_check stages = Tau.BookII.Transcendentals.resolution_halving_check.go stages 1 (stages + 1) Instances For

Tau.BookII.Transcendentals.resolution_halving_check.go

source@[irreducible]

**def Tau.BookII.Transcendentals.resolution_halving_check.go (stages : Denotation.TauIdx)

(k fuel : ℕ) :Bool**

Equations

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

Tau.BookII.Transcendentals.full_confirmation_check

source def Tau.BookII.Transcendentals.full_confirmation_check :Bool

Full master constant confirmation: all pieces fit together. Equations

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

Tau.BookII.Transcendentals.iota_confirmed

source theorem Tau.BookII.Transcendentals.iota_confirmed :iota_tau_confirmed_check = true

Tau.BookII.Transcendentals.iota_precision

source theorem Tau.BookII.Transcendentals.iota_precision :iota_tau_precision_check = true

Tau.BookII.Transcendentals.self_consist

source theorem Tau.BookII.Transcendentals.self_consist :self_consistency_check = true

Tau.BookII.Transcendentals.arch_bridge

source theorem Tau.BookII.Transcendentals.arch_bridge :archimedean_bridge_check = true

Tau.BookII.Transcendentals.bridge_interp

source theorem Tau.BookII.Transcendentals.bridge_interp :bridge_interpolation_check = true

Tau.BookII.Transcendentals.refine_res_5

source theorem Tau.BookII.Transcendentals.refine_res_5 :refinement_resolution_check 5 = true

Tau.BookII.Transcendentals.res_halving_5

source theorem Tau.BookII.Transcendentals.res_halving_5 :resolution_halving_check 5 = true

Tau.BookII.Transcendentals.full_confirm

source theorem Tau.BookII.Transcendentals.full_confirm :full_confirmation_check = true