TauLib · API Book V

TauLib.BookV.Thermodynamics.HeatEM

TauLib.BookV.Thermodynamics.HeatEM

Heat as B-sector (electromagnetic) phenomenon. All macroscopic energy transport (radiation, conduction, convection) is mediated by the B-sector of the boundary holonomy algebra. Geometric and topological relaxation.

Registry Cross-References

  • [V.D91] EM Energy Transport – EMEnergyTransport

  • [V.D92] Geometric Relaxation – GeometricRelaxation

  • [V.D93] Topological Relaxation – TopologicalRelaxation

  • [V.P34] Radiation is B-Sector Transport – radiation_is_b_sector

  • [V.P35] Conduction is Near-Field B-Sector Transport – conduction_is_b_sector

  • [V.P36] Convective Transport is B-Sector Displacement – convection_is_b_sector

  • [V.P37] Hierarchy of Relaxation Times – relaxation_hierarchy

  • [V.T63] Alpha Governs Macroscopic Energy Transport – alpha_governs_transport

  • [V.T64] Heat is Electromagnetism – HeatIsEM

  • [V.R128] The Artificial Trichotomy – artificial_trichotomy

  • [V.R129] Why Alpha and Not iota_tau^2 – why_alpha_not_iota_sq

Mathematical Content

EM Energy Transport

A change in the CR-tension distribution on tau^3 mediated by the B-sector of H_partial[omega], with transport energy proportional to iota_tau^2.

The Artificial Trichotomy

Classical radiation/conduction/convection is pedagogical convenience. All three are B-sector boundary exchange: phonons are collective EM lattice modes, pressure gradients are electromagnetic.

Relaxation Hierarchy

Geometric relaxation (spatial redistribution on T^2) is much faster than topological relaxation (absorption by lemniscate boundary).

Heat is Electromagnetism

All macroscopic energy transport at E1 is mediated by the B-sector, with heat flux proportional to the boundary holonomy algebra’s B-component.

Ground Truth Sources

  • Book V ch24: heat as B-sector phenomenon

  • temporal_spatial_decomposition.md: B-sector = EM

Tau.BookV.Thermodynamics.artificial_trichotomy

source theorem Tau.BookV.Thermodynamics.artificial_trichotomy :”radiation + conduction + convection = three faces of B-sector transport” = “radiation + conduction + convection = three faces of B-sector transport”

[V.R128] The artificial trichotomy: radiation/conduction/convection is pedagogical convenience. All three are B-sector (EM) boundary exchange. Phonons are collective EM lattice modes, pressure gradients are electromagnetic.

Tau.BookV.Thermodynamics.TransportMode

source inductive Tau.BookV.Thermodynamics.TransportMode :Type

Transport mode classification: the three faces of EM transport.

  • Radiation : TransportMode Radiative: far-field EM (photon) transport.

  • Conduction : TransportMode Conductive: near-field EM (phonon/lattice) transport.

  • Convection : TransportMode Convective: coherent bulk displacement of defect profiles.

Instances For

Tau.BookV.Thermodynamics.instReprTransportMode

source instance Tau.BookV.Thermodynamics.instReprTransportMode :Repr TransportMode

Equations

  • Tau.BookV.Thermodynamics.instReprTransportMode = { reprPrec := Tau.BookV.Thermodynamics.instReprTransportMode.repr }

Tau.BookV.Thermodynamics.instReprTransportMode.repr

source def Tau.BookV.Thermodynamics.instReprTransportMode.repr :TransportMode → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.instDecidableEqTransportMode

source instance Tau.BookV.Thermodynamics.instDecidableEqTransportMode :DecidableEq TransportMode

Equations

  • Tau.BookV.Thermodynamics.instDecidableEqTransportMode x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookV.Thermodynamics.instBEqTransportMode.beq

source def Tau.BookV.Thermodynamics.instBEqTransportMode.beq :TransportMode → TransportMode → Bool

Equations

  • Tau.BookV.Thermodynamics.instBEqTransportMode.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Thermodynamics.instBEqTransportMode

source instance Tau.BookV.Thermodynamics.instBEqTransportMode :BEq TransportMode

Equations

  • Tau.BookV.Thermodynamics.instBEqTransportMode = { beq := Tau.BookV.Thermodynamics.instBEqTransportMode.beq }

Tau.BookV.Thermodynamics.TransportMode.sector

source def Tau.BookV.Thermodynamics.TransportMode.sector :TransportMode → BookIII.Sectors.Sector

All transport modes are B-sector. Equations

  • x✝.sector = Tau.BookIII.Sectors.Sector.B Instances For

Tau.BookV.Thermodynamics.EMEnergyTransport

source structure Tau.BookV.Thermodynamics.EMEnergyTransport :Type

[V.D91] EM energy transport: a change in the CR-tension distribution on tau^3 mediated by the B-sector of H_partial[omega].

Energy: Delta E = integral of over tau^3. All three modes (radiation, conduction, convection) are B-sector.

  • mode : TransportMode The transport mode.

  • energy_numer : ℕ Energy numerator (scaled).

  • energy_denom : ℕ Energy denominator.

  • denom_pos : self.energy_denom > 0 Denominator positive.

  • mediating_sector : BookIII.Sectors.Sector The mediating sector is always B.

Instances For

Tau.BookV.Thermodynamics.instReprEMEnergyTransport

source instance Tau.BookV.Thermodynamics.instReprEMEnergyTransport :Repr EMEnergyTransport

Equations

  • Tau.BookV.Thermodynamics.instReprEMEnergyTransport = { reprPrec := Tau.BookV.Thermodynamics.instReprEMEnergyTransport.repr }

Tau.BookV.Thermodynamics.instReprEMEnergyTransport.repr

source def Tau.BookV.Thermodynamics.instReprEMEnergyTransport.repr :EMEnergyTransport → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.transport_default_b

source theorem Tau.BookV.Thermodynamics.transport_default_b :BookIII.Sectors.Sector.B = BookIII.Sectors.Sector.B

The default mediating sector is B.

Tau.BookV.Thermodynamics.radiation_is_b_sector

source theorem Tau.BookV.Thermodynamics.radiation_is_b_sector :TransportMode.Radiation.sector = BookIII.Sectors.Sector.B

[V.P34] Radiation is B-sector transport: radiative energy flux from a defect configuration is proportional to kappa(B;2) = iota_tau^2.

j_rad = kappa(B;2) * rho_def^2 * c = iota_tau^2 * rho_def^2 * c

Tau.BookV.Thermodynamics.conduction_is_b_sector

source theorem Tau.BookV.Thermodynamics.conduction_is_b_sector :TransportMode.Conduction.sector = BookIII.Sectors.Sector.B

[V.P35] Conduction is near-field B-sector transport: thermal conduction in a lattice is mediated by near-field B-sector boundary characters with wavelength comparable to lattice spacing.

kappa_cond proportional to alpha (readout of iota_tau^2).

Tau.BookV.Thermodynamics.convection_is_b_sector

source theorem Tau.BookV.Thermodynamics.convection_is_b_sector :TransportMode.Convection.sector = BookIII.Sectors.Sector.B

[V.P36] Convective transport is B-sector displacement: coherent displacement of the defect-functional profile driven by the B-sector pressure gradient.

q_conv = kappa_eff * defect_profile * flow_velocity

Tau.BookV.Thermodynamics.alpha_governs_transport

source theorem Tau.BookV.Thermodynamics.alpha_governs_transport :”Gamma_transport propto alpha * Delta_E (B-sector readout)” = “Gamma_transport propto alpha * Delta_E (B-sector readout)”

[V.T63] Alpha governs macroscopic energy transport: the transport rate between any two macroscopic E1 configurations is proportional to the fine-structure constant alpha.

Gamma_transport propto alpha * Delta_E

This is because alpha is the E1 readout of the B-sector self-coupling iota_tau^2 after holonomy correction.

Tau.BookV.Thermodynamics.why_alpha_not_iota_sq

source theorem Tau.BookV.Thermodynamics.why_alpha_not_iota_sq :Boundary.iota_tau_numer * Boundary.iota_tau_numer > 0

[V.R129] Why alpha and not iota_tau^2: the B-sector self-coupling kappa(B;2) = iota_tau^2 0.1166 is the tau-native sector strength. alpha 1/137 is its E1 readout after holonomy correction and dimensional bridge. The two are related but not equal.

Numerical check: iota_tau^2 = 341304^2 / 10^12 ~ 0.1166.

Tau.BookV.Thermodynamics.GeometricRelaxation

source structure Tau.BookV.Thermodynamics.GeometricRelaxation :Type

[V.D92] Geometric relaxation: the process by which a defect bundle loses CR-tension through spatial redistribution on the fiber T^2.

Driven by the fiber gradient of |dbar_b f|^2 weighted by iota_tau^2 (B-sector self-coupling).

Geometric relaxation preserves topological sector.

  • tension_initial_numer : ℕ Initial tension numerator.

  • tension_final_numer : ℕ Final tension numerator (after relaxation).

  • tension_denom : ℕ Common denominator.

  • denom_pos : self.tension_denom > 0 Denominator positive.

  • tension_decreases : self.tension_final_numer ≤ self.tension_initial_numer Tension decreases.

  • preserves_topology : Bool Topological sector is preserved.

Instances For

Tau.BookV.Thermodynamics.instReprGeometricRelaxation

source instance Tau.BookV.Thermodynamics.instReprGeometricRelaxation :Repr GeometricRelaxation

Equations

  • Tau.BookV.Thermodynamics.instReprGeometricRelaxation = { reprPrec := Tau.BookV.Thermodynamics.instReprGeometricRelaxation.repr }

Tau.BookV.Thermodynamics.instReprGeometricRelaxation.repr

source def Tau.BookV.Thermodynamics.instReprGeometricRelaxation.repr :GeometricRelaxation → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.TopologicalRelaxation

source structure Tau.BookV.Thermodynamics.TopologicalRelaxation :Type

[V.D93] Topological relaxation: the process by which a defect bundle is absorbed by the lemniscate boundary L = S^1 v S^1 through a change in topological sector.

Energy release from the variation of holonomy characters at L. Much slower than geometric relaxation.

  • defects_initial : ℕ Initial defect count.

  • defects_final : ℕ Final defect count (after topological absorption).

  • defects_decrease : self.defects_final ≤ self.defects_initial Defect count decreases.

  • sector_changes : Bool Whether the topological sector changes.

Instances For

Tau.BookV.Thermodynamics.instReprTopologicalRelaxation.repr

source def Tau.BookV.Thermodynamics.instReprTopologicalRelaxation.repr :TopologicalRelaxation → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.instReprTopologicalRelaxation

source instance Tau.BookV.Thermodynamics.instReprTopologicalRelaxation :Repr TopologicalRelaxation

Equations

  • Tau.BookV.Thermodynamics.instReprTopologicalRelaxation = { reprPrec := Tau.BookV.Thermodynamics.instReprTopologicalRelaxation.repr }

Tau.BookV.Thermodynamics.relaxation_hierarchy

source theorem Tau.BookV.Thermodynamics.relaxation_hierarchy :”tau_geom « tau_top: geometric much faster than topological” = “tau_geom « tau_top: geometric much faster than topological”

[V.P37] Hierarchy of relaxation times: geometric relaxation « topological relaxation.

Geometric (spatial redistribution on T^2) preserves topology. Topological (absorption by L) changes sector. The separation explains the apparent stability of defect bundles.

Tau.BookV.Thermodynamics.HeatIsEM

source structure Tau.BookV.Thermodynamics.HeatIsEM :Type

[V.T64] Heat is electromagnetism: all macroscopic energy transport at E1 is mediated by the B-sector of the boundary holonomy algebra.

Q-dot = integral over boundary of B-component flux.

There is no separate “heat force” – heat is the macroscopic manifestation of the B-sector (electromagnetic) boundary exchange.

  • sector : BookIII.Sectors.Sector The mediating sector is always B (EM).

  • no_separate_force : Bool There is no separate heat force.

  • transport_modes : List TransportMode All three transport modes are unified.

Instances For

Tau.BookV.Thermodynamics.instReprHeatIsEM

source instance Tau.BookV.Thermodynamics.instReprHeatIsEM :Repr HeatIsEM

Equations

  • Tau.BookV.Thermodynamics.instReprHeatIsEM = { reprPrec := Tau.BookV.Thermodynamics.instReprHeatIsEM.repr }

Tau.BookV.Thermodynamics.instReprHeatIsEM.repr

source def Tau.BookV.Thermodynamics.instReprHeatIsEM.repr :HeatIsEM → ℕ → Std.Format

Equations

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

Tau.BookV.Thermodynamics.heat_is_em_unified

source theorem Tau.BookV.Thermodynamics.heat_is_em_unified :[TransportMode.Radiation, TransportMode.Conduction, TransportMode.Convection].length = 3

The heat theorem: exactly 3 transport modes.

Tau.BookV.Thermodynamics.all_modes_b_sector

source theorem Tau.BookV.Thermodynamics.all_modes_b_sector :List.map TransportMode.sector [TransportMode.Radiation, TransportMode.Conduction, TransportMode.Convection] = [BookIII.Sectors.Sector.B, BookIII.Sectors.Sector.B, BookIII.Sectors.Sector.B]

All modes in the heat structure are B-sector.

Tau.BookV.Thermodynamics.example_radiation

source def Tau.BookV.Thermodynamics.example_radiation :EMEnergyTransport

Example radiative transport. Equations

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

Tau.BookV.Thermodynamics.example_geo_relax

source def Tau.BookV.Thermodynamics.example_geo_relax :GeometricRelaxation

Example geometric relaxation. Equations

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

Tau.BookV.Thermodynamics.example_top_relax

source def Tau.BookV.Thermodynamics.example_top_relax :TopologicalRelaxation

Example topological relaxation. Equations

  • Tau.BookV.Thermodynamics.example_top_relax = { defects_initial := 50, defects_final := 30, defects_decrease := Tau.BookV.Thermodynamics.example_top_relax._proof_2 } Instances For