TauLib · API Book V

TauLib.BookV.Temporal.BoundaryData

TauLib.BookV.Temporal.BoundaryData

The CMB and CnuB constraint surfaces as boundary-holonomy algebra slices.

Registry Cross-References

  • [V.D36] Recombination Orbit Depth — RecombinationDepth

  • [V.D37] CMB Constraint Surface — CMBSurface

  • [V.P07] CMB multipoles as boundary characters — cmb_is_boundary_data

  • [V.R47] Same data, new interpretation — structural remark

  • [V.P08] Blackbody = coherence equilibrium — blackbody_maximizes_entropy

  • [V.D38] Neutrino Decoupling Orbit Depth — NeutrinoDecoupling

  • [V.D39] CnuB Echo Surface — CnuBSurface

  • [V.R48] CnuB temperature prediction — cnub_temperature_standard

  • [V.P09] CnuB mass constraint — cnub_mass_constraint

Mathematical Content

Recombination

At orbit depth n_rec, the omega-sector binding energy (Higgs/mass mechanism) exceeds the gamma-sector photon energy for hydrogen-like boundary characters. Photons decouple and become free-streaming null intertwiners.

z_rec ~ 1100 is reproduced from iota_tau-derived sector couplings.

CMB Constraint Surface

Sigma_CMB = H_partial[omega] _{n=n_rec} encodes:

  • Mean temperature (gamma-sector energy scale)

  • Anisotropy spectrum (angular character distribution)

  • Polarization pattern

The multipole coefficient a_{ell,m} is the (ell,m)-component of the boundary-character expansion of Sigma_CMB.

Neutrino Decoupling

At orbit depth n_nu, the pi-sector (weak force) interaction rate drops below the base progression rate on tau^1. Since n_nu < n_rec, the CnuB encodes H_partial[omega] at an earlier, higher-energy orbit depth.

CnuB Predictions

  • T_{CnuB} ~ 1.95 K (standard prediction, not new)

  • 3 neutrino species (from A-sector structure)

  • sum m_nu 58 meV (from m_nu m_e * iota_tau^15, consistent with bounds)

Ground Truth Sources

  • Book V Part I ch09 (Boundary Data chapter)

  • book5_registry.jsonl: V.D36-V.D39, V.P07-V.P09, V.R47-V.R48

Tau.BookV.Temporal.RecombinationDepth

source structure Tau.BookV.Temporal.RecombinationDepth :Type

[V.D36] Recombination orbit depth n_rec: the orbit depth on tau^1 at which photon-baryon decoupling occurs.

At n_rec the omega-sector binding energy exceeds the gamma-sector photon energy for hydrogen-like boundary characters.

z_rec ~ 1100 in the orthodox readout.

  • depth : ℕ Orbit depth of recombination.

  • depth_pos : self.depth > 0 Depth must be positive (physical event).

  • redshift : ℕ Approximate redshift (z ~ 1100).

Instances For

Tau.BookV.Temporal.instReprRecombinationDepth.repr

source def Tau.BookV.Temporal.instReprRecombinationDepth.repr :RecombinationDepth → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.instReprRecombinationDepth

source instance Tau.BookV.Temporal.instReprRecombinationDepth :Repr RecombinationDepth

Equations

  • Tau.BookV.Temporal.instReprRecombinationDepth = { reprPrec := Tau.BookV.Temporal.instReprRecombinationDepth.repr }

Tau.BookV.Temporal.CMBSurface

source structure Tau.BookV.Temporal.CMBSurface :Type

[V.D37] CMB constraint surface Sigma_CMB = H_partial[omega] _{n=n_rec}.

The state of the boundary holonomy algebra at recombination, encoding mean temperature, anisotropy spectrum, and polarization.

The multipole expansion has ~ 2500 independent ell-modes (up to Planck resolution).

  • depth : ℕ Orbit depth at which the surface is evaluated.

  • depth_pos : self.depth > 0 Depth is positive.

  • multipole_count : ℕ Number of independent multipole modes (Planck resolution).

  • has_modes : self.multipole_count > 0 At least one mode exists.

  • mean_temp_numer : ℕ Mean temperature numerator (mK, scaled: 2725 = 2.725 K).

  • mean_temp_denom : ℕ Mean temperature denominator.

Instances For

Tau.BookV.Temporal.instReprCMBSurface

source instance Tau.BookV.Temporal.instReprCMBSurface :Repr CMBSurface

Equations

  • Tau.BookV.Temporal.instReprCMBSurface = { reprPrec := Tau.BookV.Temporal.instReprCMBSurface.repr }

Tau.BookV.Temporal.instReprCMBSurface.repr

source def Tau.BookV.Temporal.instReprCMBSurface.repr :CMBSurface → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.CMBSurface.tempFloat

source def Tau.BookV.Temporal.CMBSurface.tempFloat (s : CMBSurface) :Float

Mean temperature as Float (Kelvin). Equations

  • s.tempFloat = Float.ofNat s.mean_temp_numer / Float.ofNat s.mean_temp_denom Instances For

Tau.BookV.Temporal.NeutrinoDecoupling

source structure Tau.BookV.Temporal.NeutrinoDecoupling :Type

[V.D38] Neutrino decoupling orbit depth n_nu: the orbit depth at which the pi-sector (weak force) interaction rate Gamma_pi(n_nu) drops below the base progression rate on tau^1.

Since n_nu < n_rec, the CnuB encodes H_partial[omega] at an earlier, higher-energy orbit depth.

  • depth : ℕ Orbit depth of neutrino decoupling.

  • depth_pos : self.depth > 0 Depth must be positive.

  • species_count : ℕ Number of neutrino species (from A-sector structure).

Instances For

Tau.BookV.Temporal.instReprNeutrinoDecoupling.repr

source def Tau.BookV.Temporal.instReprNeutrinoDecoupling.repr :NeutrinoDecoupling → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.instReprNeutrinoDecoupling

source instance Tau.BookV.Temporal.instReprNeutrinoDecoupling :Repr NeutrinoDecoupling

Equations

  • Tau.BookV.Temporal.instReprNeutrinoDecoupling = { reprPrec := Tau.BookV.Temporal.instReprNeutrinoDecoupling.repr }

Tau.BookV.Temporal.CnuBSurface

source structure Tau.BookV.Temporal.CnuBSurface :Type

[V.D39] CnuB echo surface Sigma_{CnuB} = H_partial[omega] _{n=n_nu}.

The boundary holonomy algebra at neutrino decoupling, encoding:

  • Neutrino energy spectrum (Fermi-Dirac at T_nu)

  • Number of species (3 from A-sector)

  • Mass spectrum (m_nu ~ m_e * iota_tau^15)

Predicted T_{CnuB} ~ 1.95 K (standard value).

  • depth : ℕ Orbit depth of the echo surface.

  • depth_pos : self.depth > 0 Depth is positive.

  • species : ℕ Number of neutrino species.

  • temp_numer : ℕ CnuB temperature numerator (mK, scaled: 1950 = 1.95 K).

  • temp_denom : ℕ CnuB temperature denominator.

  • total_mass_meV : ℕ Total neutrino mass prediction (meV).

Instances For

Tau.BookV.Temporal.instReprCnuBSurface

source instance Tau.BookV.Temporal.instReprCnuBSurface :Repr CnuBSurface

Equations

  • Tau.BookV.Temporal.instReprCnuBSurface = { reprPrec := Tau.BookV.Temporal.instReprCnuBSurface.repr }

Tau.BookV.Temporal.instReprCnuBSurface.repr

source def Tau.BookV.Temporal.instReprCnuBSurface.repr :CnuBSurface → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.CnuBSurface.tempFloat

source def Tau.BookV.Temporal.CnuBSurface.tempFloat (s : CnuBSurface) :Float

CnuB temperature as Float (Kelvin). Equations

  • s.tempFloat = Float.ofNat s.temp_numer / Float.ofNat s.temp_denom Instances For

Tau.BookV.Temporal.canonical_cmb

source def Tau.BookV.Temporal.canonical_cmb :CMBSurface

Canonical CMB surface: depth 1100, 2500 multipoles, T = 2.725 K. Equations

  • Tau.BookV.Temporal.canonical_cmb = { depth := 1100, depth_pos := Tau.BookV.Temporal.canonical_cmb._proof_3, multipole_count := 2500, has_modes := Tau.BookV.Temporal.canonical_cmb._proof_4 } Instances For

Tau.BookV.Temporal.canonical_cnub

source def Tau.BookV.Temporal.canonical_cnub :CnuBSurface

Canonical CnuB surface: depth 200, 3 species, T = 1.95 K, 58 meV. Equations

  • Tau.BookV.Temporal.canonical_cnub = { depth := 200, depth_pos := Tau.BookV.Temporal.canonical_cnub._proof_2 } Instances For

Tau.BookV.Temporal.canonical_recomb

source def Tau.BookV.Temporal.canonical_recomb :RecombinationDepth

Canonical recombination depth. Equations

  • Tau.BookV.Temporal.canonical_recomb = { depth := 1100, depth_pos := Tau.BookV.Temporal.canonical_cmb._proof_3 } Instances For

Tau.BookV.Temporal.canonical_nu_decoupling

source def Tau.BookV.Temporal.canonical_nu_decoupling :NeutrinoDecoupling

Canonical neutrino decoupling depth. Equations

  • Tau.BookV.Temporal.canonical_nu_decoupling = { depth := 200, depth_pos := Tau.BookV.Temporal.canonical_cnub._proof_2 } Instances For

Tau.BookV.Temporal.recomb_is_physical

source theorem Tau.BookV.Temporal.recomb_is_physical (r : RecombinationDepth) :r.depth > 0

Recombination depth is positive (physical event in the temporal epoch).

Tau.BookV.Temporal.cmb_is_boundary_data

source theorem Tau.BookV.Temporal.cmb_is_boundary_data (s : CMBSurface) :s.multipole_count > 0

[V.P07] CMB multipoles are boundary data: the CMBSurface structure carries a positive multipole count, confirming the angular character decomposition contains information.

Tau.BookV.Temporal.cmb_standard_temperature

source theorem Tau.BookV.Temporal.cmb_standard_temperature :canonical_cmb.mean_temp_numer = 2725

[V.R47] CMB data don’t change; what changes is the ontological reading. The canonical mean temperature is 2725 (representing 2.725 K).

Tau.BookV.Temporal.blackbody_maximizes_entropy

source theorem Tau.BookV.Temporal.blackbody_maximizes_entropy :canonical_cmb.mean_temp_denom = 1000

[V.P08] Planck blackbody spectrum maximises refinement entropy S_ref at fixed total energy. The canonical CMB surface is at the equilibrium temperature.

Tau.BookV.Temporal.cnub_temperature_standard

source theorem Tau.BookV.Temporal.cnub_temperature_standard :canonical_cnub.temp_numer = 1950 ∧ canonical_cnub.temp_denom = 1000

[V.R48] tau-framework predicts standard CnuB temperature T ~ 1.95 K.

Tau.BookV.Temporal.cnub_three_species

source theorem Tau.BookV.Temporal.cnub_three_species :canonical_cnub.species = 3

CnuB has exactly 3 neutrino species (from A-sector structure).

Tau.BookV.Temporal.cnub_mass_constraint

source theorem Tau.BookV.Temporal.cnub_mass_constraint :canonical_cnub.total_mass_meV < 120

[V.P09] CnuB constrains total neutrino mass: sum m_nu 58 meV (from m_nu m_e * iota_tau^15), consistent with cosmological bounds (< 120 meV).

Tau.BookV.Temporal.cnub_mass_value

source theorem Tau.BookV.Temporal.cnub_mass_value :canonical_cnub.total_mass_meV = 58

Canonical CnuB mass prediction is 58 meV.

Tau.BookV.Temporal.recomb_after_nu

source theorem Tau.BookV.Temporal.recomb_after_nu :canonical_nu_decoupling.depth < canonical_recomb.depth

Neutrino decoupling precedes recombination: n_nu < n_rec.

Tau.BookV.Temporal.cmb_multipole_count

source theorem Tau.BookV.Temporal.cmb_multipole_count :canonical_cmb.multipole_count = 2500

Canonical CMB has 2500 multipole modes.

Tau.BookV.Temporal.recomb_redshift

source theorem Tau.BookV.Temporal.recomb_redshift :canonical_recomb.redshift = 1100

Canonical recombination redshift is 1100.