TauLib · API Book V

TauLib.BookV.Temporal.MacroReadout

TauLib.BookV.Temporal.MacroReadout

Macroscopic readouts: null intertwiners, operational distance, redshift, expansion, and the Hubble parameter.

Registry Cross-References

  • [V.D27] Null Intertwiner — NullIntertwiner

  • [V.T14] Boundary Supports Null — boundary_supports_null

  • [V.P06] Null Selects EM — null_selects_em

  • [V.D28] Operational Distance — OperationalDistance

  • [V.T15] Distance-Time Duality — distance_time_dual

  • [V.D29] Refinement Drift — RefinementDrift

  • [V.T16] Refinement Drift Formula — drift_formula_positive

  • [V.D30] Readout Expansion — ReadoutExpansion

  • [V.D31] Hubble Readout Parameter — HubbleReadout

Ground Truth Sources

  • Book V Part II (2nd Edition): Temporal Foundation

Tau.BookV.Temporal.NullIntertwiner

source structure Tau.BookV.Temporal.NullIntertwiner :Type

[V.D27] Null intertwiner: massless morphism in boundary holonomy. Null transport moves along base τ¹ at c_τ. Sector B (EM) is uniquely selected by the null (zero fiber stiffness) condition.

  • sector : BookIII.Sectors.Sector The sector supporting this null intertwiner.

  • null_is_em : self.sector = BookIII.Sectors.Sector.B Null selects EM.

  • carrier : BookIV.Physics.CarrierType The carrier type (always Base for null transport).

  • carrier_is_base : self.carrier = BookIV.Physics.CarrierType.Base Null transport is base-only.

  • massless : Bool Massless flag.

  • massless_true : self.massless = true Must be massless.

Instances For

Tau.BookV.Temporal.instReprNullIntertwiner.repr

source def Tau.BookV.Temporal.instReprNullIntertwiner.repr :NullIntertwiner → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.instReprNullIntertwiner

source instance Tau.BookV.Temporal.instReprNullIntertwiner :Repr NullIntertwiner

Equations

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

Tau.BookV.Temporal.photon_null

source def Tau.BookV.Temporal.photon_null :NullIntertwiner

The photon as canonical null intertwiner. Equations

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

Tau.BookV.Temporal.boundary_supports_null

source theorem Tau.BookV.Temporal.boundary_supports_null :BookIV.Sectors.em_sector.generator = Kernel.Generator.gamma ∧ BookIV.Sectors.em_sector.depth = 2 ∧ photon_null.sector = BookIII.Sectors.Sector.B

[V.T14] The boundary holonomy algebra supports null intertwiners. EM generator gamma at depth 2 allows null transport on τ¹.

Tau.BookV.Temporal.null_selects_em

source theorem Tau.BookV.Temporal.null_selects_em (n : NullIntertwiner) :n.sector = BookIII.Sectors.Sector.B

[V.P06] The null condition uniquely selects B-sector (EM). Only B supports null transport: D/A are depth 1 temporal, C is confined (χ₋), Omega is massive (crossing).

Tau.BookV.Temporal.em_unique_null

source theorem Tau.BookV.Temporal.em_unique_null :BookIV.Sectors.em_sector.polarity = BookIV.Sectors.PolaritySign.ChiPlus ∧ BookIV.Sectors.em_sector.depth = 2 ∧ BookIV.Sectors.strong_sector.polarity = BookIV.Sectors.PolaritySign.ChiMinus ∧ BookIV.Sectors.higgs_sector.polarity = BookIV.Sectors.PolaritySign.Crossing

EM is the unique null carrier among the 5 sectors.

Tau.BookV.Temporal.OperationalDistance

source structure Tau.BookV.Temporal.OperationalDistance :Type

[V.D28] Operational distance: tick count of the null intertwiner connecting two events at depth n₀. The τ-native spatial distance is NOT a primitive metric but a counting readout of null transport.

  • ref_depth : ℕ Reference depth for the measurement.

  • ref_depth_pos : self.ref_depth > 0

  • dist_numer : ℕ Distance numerator (tick count * scale).

  • dist_denom : ℕ
  • denom_pos : self.dist_denom > 0 Instances For

Tau.BookV.Temporal.instReprOperationalDistance

source instance Tau.BookV.Temporal.instReprOperationalDistance :Repr OperationalDistance

Equations

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

Tau.BookV.Temporal.instReprOperationalDistance.repr

source def Tau.BookV.Temporal.instReprOperationalDistance.repr :OperationalDistance → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.OperationalDistance.toFloat

source def Tau.BookV.Temporal.OperationalDistance.toFloat (d : OperationalDistance) :Float

Distance as Float. Equations

  • d.toFloat = Float.ofNat d.dist_numer / Float.ofNat d.dist_denom Instances For

Tau.BookV.Temporal.distance_time_dual

source theorem Tau.BookV.Temporal.distance_time_dual :canonical_base_circle.seed = Kernel.Generator.alpha ∧ photon_null.sector = BookIII.Sectors.Sector.B ∧ canonical_base_circle.profinite.seed = Kernel.Generator.alpha

[V.T15] Distance and proper time are dual readouts. Time counts α-ticks along base; distance counts null-intertwiner ticks between events. Both derived from the refinement tower.

Tau.BookV.Temporal.RefinementDrift

source structure Tau.BookV.Temporal.RefinementDrift :Type

[V.D29] Refinement drift (cosmological redshift): z(n_s, n_r) := Δt(n_s)/Δt(n_r) − 1. Since Δt(n) ~ ι_τ^n and ι_τ < 1, source earlier (n_s < n_r) gives z > 0 (redshift). The τ-framework predicts redshift WITHOUT metric expansion.

  • n_source : ℕ
  • n_receiver : ℕ
  • source_pos : self.n_source > 0
  • receiver_pos : self.n_receiver > 0
  • source_earlier : self.n_source < self.n_receiver Source precedes receiver (cosmological redshift).

Instances For

Tau.BookV.Temporal.instReprRefinementDrift

source instance Tau.BookV.Temporal.instReprRefinementDrift :Repr RefinementDrift

Equations

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

Tau.BookV.Temporal.instReprRefinementDrift.repr

source def Tau.BookV.Temporal.instReprRefinementDrift.repr :RefinementDrift → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.RefinementDrift.depth_diff

source def Tau.BookV.Temporal.RefinementDrift.depth_diff (d : RefinementDrift) :ℕ

Depth difference (positive for redshift). Equations

  • d.depth_diff = d.n_receiver - d.n_source Instances For

Tau.BookV.Temporal.drift_formula_positive

source theorem Tau.BookV.Temporal.drift_formula_positive (d : RefinementDrift) :d.depth_diff > 0

[V.T16] Drift depth difference is positive for cosmological observations (source earlier than receiver).

Tau.BookV.Temporal.ReadoutExpansion

source structure Tau.BookV.Temporal.ReadoutExpansion :Type

[V.D30] Readout expansion a(n) ~ ι_τ^(-n): cumulative proper-time scaling. The universe “expands” because the tower deepens and proper-time increments shrink — not because space stretches.

  • depth : ℕ
  • depth_pos : self.depth > 0
  • expansion_numer : ℕ
  • expansion_denom : ℕ
  • denom_pos : self.expansion_denom > 0 Instances For

Tau.BookV.Temporal.instReprReadoutExpansion.repr

source def Tau.BookV.Temporal.instReprReadoutExpansion.repr :ReadoutExpansion → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.instReprReadoutExpansion

source instance Tau.BookV.Temporal.instReprReadoutExpansion :Repr ReadoutExpansion

Equations

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

Tau.BookV.Temporal.ReadoutExpansion.toFloat

source def Tau.BookV.Temporal.ReadoutExpansion.toFloat (a : ReadoutExpansion) :Float

Expansion as Float. Equations

  • a.toFloat = Float.ofNat a.expansion_numer / Float.ofNat a.expansion_denom Instances For

Tau.BookV.Temporal.HubbleReadout

source structure Tau.BookV.Temporal.HubbleReadout :Type

[V.D31] Hubble readout H(n) := Δa/a per tick. NOT constant: decays with depth. Early (opening) H is large (inflation), late H is small. H(n) ~ 1 − ι_τ to leading order.

  • depth : ℕ
  • depth_pos : self.depth > 0
  • hubble_numer : ℕ
  • hubble_denom : ℕ
  • denom_pos : self.hubble_denom > 0 Instances For

Tau.BookV.Temporal.instReprHubbleReadout.repr

source def Tau.BookV.Temporal.instReprHubbleReadout.repr :HubbleReadout → ℕ → Std.Format

Equations

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

Tau.BookV.Temporal.instReprHubbleReadout

source instance Tau.BookV.Temporal.instReprHubbleReadout :Repr HubbleReadout

Equations

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

Tau.BookV.Temporal.HubbleReadout.toFloat

source def Tau.BookV.Temporal.HubbleReadout.toFloat (h : HubbleReadout) :Float

Hubble readout as Float. Equations

  • h.toFloat = Float.ofNat h.hubble_numer / Float.ofNat h.hubble_denom Instances For

Tau.BookV.Temporal.null_structural

source theorem Tau.BookV.Temporal.null_structural (n : NullIntertwiner) :n.massless = true ∧ n.carrier = BookIV.Physics.CarrierType.Base

Null intertwiner is always massless and base-carried.

Tau.BookV.Temporal.null_transport_scale

source theorem Tau.BookV.Temporal.null_transport_scale :BookIV.Sectors.em_sector.coupling_numer = BookIV.Sectors.iota_sq_numer ∧ BookIV.Sectors.em_sector.coupling_denom = BookIV.Sectors.iota_sq_denom

The EM sector coupling ι_τ² is the null-transport scale.

Tau.BookV.Temporal.redshift_requires_earlier

source theorem Tau.BookV.Temporal.redshift_requires_earlier (d : RefinementDrift) :d.n_source < d.n_receiver

Redshift requires source < receiver.