TauLib · API Book IV

TauLib.BookIV.Physics.MassEnergy

TauLib.BookIV.Physics.MassEnergy

Mass, energy, and the mass-energy structural relation in the τ-framework.

Registry Cross-References

  • [IV.D20] Mass Index — MassIndex

  • [IV.D21] Energy Index — EnergyIndex

  • [IV.D22] Speed Constant — SpeedConstant

  • [IV.D23] Mass-Energy Relation — MassEnergyRelation

  • [IV.R04] Neutron as first ontic particle — structural remark

Mathematical Content

Mass in the τ-Framework

Mass is the inertial invariant of a persistent localized defect bundle with stable 2-torus T² fiber. It is NOT a postulated scalar but a boundary fixed-point of the defect bundle’s coherence functional:

M_n(x) := MassIdx(NF_ω(x))

= α-Idx readout from normal-form stabilized configuration.

The neutron is the first ontic particle (minimal mass-bearing defect bundle in τ). Beta decay is the shedding of subsidiary defect modes.

Energy in the τ-Framework

Energy is the coherence cost of maintaining a localized defect bundle structure. It is proportional to the defect-tuple magnitude and stability degree.

Mass-Energy Relation

E = m_τ · c²_τ

where c_τ is the τ-derived speed-of-light constant. This relation holds as a structural identity between the mass and energy indices.

Particle Ontology

  • Ontic particles: Persistent defect bundles with stable T² fiber (neutron, proton, …)

  • Radiation: Null transport with degenerate S¹ fiber (photon)

  • E = ℏω_τ: Photon energy from transport frequency

Ground Truth Sources

  • particle-physics-defects.json: particle-ontology, mass definition

  • quantum-mechanics.json: mass-energy equivalence

  • gravity-einstein.json: BH mass index

Tau.BookIV.Physics.MassIndex

source structure Tau.BookIV.Physics.MassIndex :Type

[IV.D20] Mass index: the inertial invariant of a persistent localized defect bundle.

Mass = boundary fixed-point of the defect bundle’s coherence functional = α-Idx readout from NF-stabilized configuration.

Properties:

  • Not a primitive scalar but a resistance/scale index

  • Comes with minimal carrier that can host it

  • Monotone under admissible evolution (No-Shrink for BH)

  • numer : ℕ Mass value numerator (scaled).

  • denom : ℕ Mass value denominator.

  • denom_pos : self.denom > 0 Denominator is positive.

  • carrier : CarrierType Carrier type (must be Fiber for ontic particles).

  • is_persistent : Bool Whether the defect bundle is persistent (stable T² fiber).

Instances For

Tau.BookIV.Physics.instReprMassIndex.repr

source def Tau.BookIV.Physics.instReprMassIndex.repr :MassIndex → ℕ → Std.Format

Equations

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

Tau.BookIV.Physics.instReprMassIndex

source instance Tau.BookIV.Physics.instReprMassIndex :Repr MassIndex

Equations

  • Tau.BookIV.Physics.instReprMassIndex = { reprPrec := Tau.BookIV.Physics.instReprMassIndex.repr }

Tau.BookIV.Physics.MassIndex.toFloat

source def Tau.BookIV.Physics.MassIndex.toFloat (m : MassIndex) :Float

Float display for mass index. Equations

  • m.toFloat = Float.ofNat m.numer / Float.ofNat m.denom Instances For

Tau.BookIV.Physics.EnergyIndex

source structure Tau.BookIV.Physics.EnergyIndex :Type

[IV.D21] Energy index: the coherence cost of maintaining a localized defect bundle structure.

Energy ∝ defect-tuple magnitude × stability degree. Each sector provides its own excitation cost scale.

  • numer : ℕ Energy value numerator (scaled).

  • denom : ℕ Energy value denominator.

  • denom_pos : self.denom > 0 Denominator is positive.

  • sector : BookIII.Sectors.Sector Which sector provides the excitation.

Instances For

Tau.BookIV.Physics.instReprEnergyIndex.repr

source def Tau.BookIV.Physics.instReprEnergyIndex.repr :EnergyIndex → ℕ → Std.Format

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

Tau.BookIV.Physics.instReprEnergyIndex

source instance Tau.BookIV.Physics.instReprEnergyIndex :Repr EnergyIndex

Equations

  • Tau.BookIV.Physics.instReprEnergyIndex = { reprPrec := Tau.BookIV.Physics.instReprEnergyIndex.repr }

Tau.BookIV.Physics.EnergyIndex.toFloat

source def Tau.BookIV.Physics.EnergyIndex.toFloat (e : EnergyIndex) :Float

Float display for energy index. Equations

  • e.toFloat = Float.ofNat e.numer / Float.ofNat e.denom Instances For

Tau.BookIV.Physics.SpeedConstant

source structure Tau.BookIV.Physics.SpeedConstant :Type

[IV.D22] Speed-of-light constant c²_τ: the τ-derived structural constant relating mass and energy indices.

c²_τ is not postulated but earned from the τ-kernel as the canonical conversion factor between defect-bundle mass indices and coherence cost indices.

  • c_sq_numer : ℕ c² numerator (scaled).

  • c_sq_denom : ℕ c² denominator.

  • denom_pos : self.c_sq_denom > 0 Denominator is positive.

Instances For

Tau.BookIV.Physics.instReprSpeedConstant.repr

source def Tau.BookIV.Physics.instReprSpeedConstant.repr :SpeedConstant → ℕ → Std.Format

Equations

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

Tau.BookIV.Physics.instReprSpeedConstant

source instance Tau.BookIV.Physics.instReprSpeedConstant :Repr SpeedConstant

Equations

  • Tau.BookIV.Physics.instReprSpeedConstant = { reprPrec := Tau.BookIV.Physics.instReprSpeedConstant.repr }

Tau.BookIV.Physics.SpeedConstant.toFloat

source def Tau.BookIV.Physics.SpeedConstant.toFloat (s : SpeedConstant) :Float

Float display for speed constant. Equations

  • s.toFloat = Float.ofNat s.c_sq_numer / Float.ofNat s.c_sq_denom Instances For

Tau.BookIV.Physics.MassEnergyRelation

source structure Tau.BookIV.Physics.MassEnergyRelation :Type

[IV.D23] Mass-energy relation template: E = m · c²_τ.

This is a structural identity relating the mass index to the energy index via the τ-derived speed constant. The relation holds as a cross-multiplication equality on scaled rationals:

energy/1 = mass × c² means: E_numer · m_denom · c_denom = m_numer · E_denom · c_numer

  • mass : MassIndex Mass index.

  • energy : EnergyIndex Energy index.

  • speed : SpeedConstant Speed constant c²_τ.

  • relation : self.energy.numer * self.mass.denom * self.speed.c_sq_denom = self.mass.numer * self.energy.denom * self.speed.c_sq_numer The structural identity: E = m · c².

Instances For

Tau.BookIV.Physics.instReprMassEnergyRelation.repr

source def Tau.BookIV.Physics.instReprMassEnergyRelation.repr :MassEnergyRelation → ℕ → Std.Format

Equations

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

Tau.BookIV.Physics.instReprMassEnergyRelation

source instance Tau.BookIV.Physics.instReprMassEnergyRelation :Repr MassEnergyRelation

Equations

  • Tau.BookIV.Physics.instReprMassEnergyRelation = { reprPrec := Tau.BookIV.Physics.instReprMassEnergyRelation.repr }

Tau.BookIV.Physics.OnticParticle

source structure Tau.BookIV.Physics.OnticParticle :Type

Ontic particles have fiber carrier (stable T² topology).

  • mass : MassIndex Mass of the particle.

  • persistent_proof : self.mass.is_persistent = true Ontic particles are persistent.

  • fiber_proof : self.mass.carrier = CarrierType.Fiber Ontic particles live on the fiber T².

Instances For

Tau.BookIV.Physics.instReprOnticParticle.repr

source def Tau.BookIV.Physics.instReprOnticParticle.repr :OnticParticle → ℕ → Std.Format

Equations

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

Tau.BookIV.Physics.instReprOnticParticle

source instance Tau.BookIV.Physics.instReprOnticParticle :Repr OnticParticle

Equations

  • Tau.BookIV.Physics.instReprOnticParticle = { reprPrec := Tau.BookIV.Physics.instReprOnticParticle.repr }

Tau.BookIV.Physics.RadiationCarrier

source structure Tau.BookIV.Physics.RadiationCarrier :Type

Radiation has degenerate fiber (S¹, not T²).

  • energy : EnergyIndex Energy of the radiation.

  • sector_proof : self.energy.sector = BookIII.Sectors.Sector.B ∨ self.energy.sector = BookIII.Sectors.Sector.D Which sector (typically EM for photon).

Instances For

Tau.BookIV.Physics.instReprRadiationCarrier.repr

source def Tau.BookIV.Physics.instReprRadiationCarrier.repr :RadiationCarrier → ℕ → Std.Format

Equations

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

Tau.BookIV.Physics.instReprRadiationCarrier

source instance Tau.BookIV.Physics.instReprRadiationCarrier :Repr RadiationCarrier

Equations

  • Tau.BookIV.Physics.instReprRadiationCarrier = { reprPrec := Tau.BookIV.Physics.instReprRadiationCarrier.repr }

Tau.BookIV.Physics.NeutronRole

source structure Tau.BookIV.Physics.NeutronRole :Type

[IV.R04] The neutron is the first ontic particle: the minimal mass-bearing defect bundle in τ.

Properties:

  • Toroidal defect bundle on τ¹ (the “micro-donut”)

  • Beta decay: neutron → proton + electron + ν_e (shedding subsidiary defect modes)

  • Stable in bound states; β-decay only when free

This structure records the neutron’s structural role. The numerical mass comes from the calibration cascade (Part VII).

  • mass : MassIndex Neutron mass index (first/minimal).

  • is_ontic : self.mass.is_persistent = true ∧ self.mass.carrier = CarrierType.Fiber The neutron is ontic (persistent T² fiber).

  • mass_positive : self.mass.numer > 0 The neutron mass is positive.

Instances For

Tau.BookIV.Physics.instReprNeutronRole

source instance Tau.BookIV.Physics.instReprNeutronRole :Repr NeutronRole

Equations

  • Tau.BookIV.Physics.instReprNeutronRole = { reprPrec := Tau.BookIV.Physics.instReprNeutronRole.repr }

Tau.BookIV.Physics.instReprNeutronRole.repr

source def Tau.BookIV.Physics.instReprNeutronRole.repr :NeutronRole → ℕ → Std.Format

Equations

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

Tau.BookIV.Physics.ontic_has_fiber

source theorem Tau.BookIV.Physics.ontic_has_fiber (p : OnticParticle) :p.mass.carrier = CarrierType.Fiber

Ontic particles have Fiber carrier type.

Tau.BookIV.Physics.ontic_is_persistent

source theorem Tau.BookIV.Physics.ontic_is_persistent (p : OnticParticle) :p.mass.is_persistent = true

Ontic particles are persistent.

Tau.BookIV.Physics.mass_energy_positive

source **theorem Tau.BookIV.Physics.mass_energy_positive (r : MassEnergyRelation)

(hm : r.mass.numer > 0)

(hc : r.speed.c_sq_numer > 0) :r.energy.numer * r.mass.denom * r.speed.c_sq_denom > 0**

The mass-energy relation implies energy > 0 when mass > 0 and speed > 0 (structural).