TauLib · API Book IV

TauLib.BookIV.Particles.SpectrumComplete

TauLib.BookIV.Particles.SpectrumComplete

Particle spectrum completeness: the ontic entity definition, complete ontic register, non-ontic entities list, the ontological line, dictionary limits, the parameter count summary, and temperature as derived readout.

Registry Cross-References

  • [IV.D209] Ontic Entity — OnticEntity, OnticCriterion

  • [IV.R149] Parameter Count — parameter_count

  • [IV.R150] Ontic Entities List — comment-only (not_applicable)

  • [IV.R151] Non-ontic Entities List — non_ontic_entities

  • [IV.R152] Where the Ontological Line Falls — ontological_line

  • [IV.R153] Dictionary Limits — dictionary_limits

  • [IV.R154] Temperature is not Fundamental — temperature_not_fundamental

Mathematical Content

Chapter 51 closes Part VI with the spectrum completeness theorem: every observed particle is accounted for by τ³ mode structure, no BSM particles are predicted, and the complete ontic/non-ontic classification is a mathematical consequence of the fibration τ³ = τ¹ ×_f T².

An entity is ontic iff it can be constructed as a mode, character, or finite composite on τ³. Wave functions as independent objects, virtual particles, path integral measures, renormalization group flow, and gravitons as particles are non-ontic (computational devices, not things).

The entire Part VI uses only two inputs: ι_τ = 2/(π+e) and m_n.

Ground Truth Sources

  • Chapter 51 of Book IV (2nd Edition)

Tau.BookIV.Particles.OnticCriterion

source inductive Tau.BookIV.Particles.OnticCriterion :Type

[IV.D209] An entity is ontic in Category τ if it satisfies at least one of four criteria.

  • fiberMode : OnticCriterion Well-defined mode on fiber T² (particle).

  • baseMode : OnticCriterion Well-defined mode on base τ¹ (temporal/gravitational).

  • crossingMode : OnticCriterion Well-defined crossing mode at ω = γ ∩ η (Higgs-type).

  • finiteComposite : OnticCriterion Finite composite of ontic modes (hadrons, nuclei, atoms).

Instances For

Tau.BookIV.Particles.instReprOnticCriterion

source instance Tau.BookIV.Particles.instReprOnticCriterion :Repr OnticCriterion

Equations

  • Tau.BookIV.Particles.instReprOnticCriterion = { reprPrec := Tau.BookIV.Particles.instReprOnticCriterion.repr }

Tau.BookIV.Particles.instReprOnticCriterion.repr

source def Tau.BookIV.Particles.instReprOnticCriterion.repr :OnticCriterion → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.instDecidableEqOnticCriterion

source instance Tau.BookIV.Particles.instDecidableEqOnticCriterion :DecidableEq OnticCriterion

Equations

  • Tau.BookIV.Particles.instDecidableEqOnticCriterion x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookIV.Particles.instBEqOnticCriterion

source instance Tau.BookIV.Particles.instBEqOnticCriterion :BEq OnticCriterion

Equations

  • Tau.BookIV.Particles.instBEqOnticCriterion = { beq := Tau.BookIV.Particles.instBEqOnticCriterion.beq }

Tau.BookIV.Particles.instBEqOnticCriterion.beq

source def Tau.BookIV.Particles.instBEqOnticCriterion.beq :OnticCriterion → OnticCriterion → Bool

Equations

  • Tau.BookIV.Particles.instBEqOnticCriterion.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookIV.Particles.OnticEntity

source structure Tau.BookIV.Particles.OnticEntity :Type

[IV.D209] An ontic entity in Category τ.

  • name : String Entity name.

  • criterion : OnticCriterion Primary ontic criterion.

  • sectors : List BookIII.Sectors.Sector Sector(s).

  • stable : Bool Is stable?

Instances For

Tau.BookIV.Particles.instReprOnticEntity.repr

source def Tau.BookIV.Particles.instReprOnticEntity.repr :OnticEntity → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.instReprOnticEntity

source instance Tau.BookIV.Particles.instReprOnticEntity :Repr OnticEntity

Equations

  • Tau.BookIV.Particles.instReprOnticEntity = { reprPrec := Tau.BookIV.Particles.instReprOnticEntity.repr }

Tau.BookIV.Particles.ontic_register

source def Tau.BookIV.Particles.ontic_register :List OnticEntity

The complete list of fundamental ontic entities constructed in Book IV. Equations

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

Tau.BookIV.Particles.ontic_register_count

source theorem Tau.BookIV.Particles.ontic_register_count :ontic_register.length = 15

Tau.BookIV.Particles.ParameterCount

source structure Tau.BookIV.Particles.ParameterCount :Type

[IV.R149] Across all 25+ results of Part VI:

  • 1 dimensionless constant: ι_τ = 2/(π+e), derived from K0-K6

  • 1 dimensional anchor: m_n = 939.565421 MeV

  • 0 fitting parameters, effective couplings, or ad hoc mass ratios

  • dimensionless : ℕ Dimensionless constants.

  • anchors : ℕ Dimensional anchors.

  • fitting : ℕ Fitting parameters.

  • effective : ℕ Effective couplings.

  • ad_hoc : ℕ Ad hoc mass ratios.

Instances For

Tau.BookIV.Particles.instReprParameterCount.repr

source def Tau.BookIV.Particles.instReprParameterCount.repr :ParameterCount → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.instReprParameterCount

source instance Tau.BookIV.Particles.instReprParameterCount :Repr ParameterCount

Equations

  • Tau.BookIV.Particles.instReprParameterCount = { reprPrec := Tau.BookIV.Particles.instReprParameterCount.repr }

Tau.BookIV.Particles.parameter_count

source def Tau.BookIV.Particles.parameter_count :ParameterCount

Equations

  • Tau.BookIV.Particles.parameter_count = { } Instances For

Tau.BookIV.Particles.zero_fitting

source theorem Tau.BookIV.Particles.zero_fitting :parameter_count.fitting = 0

Tau.BookIV.Particles.zero_effective

source theorem Tau.BookIV.Particles.zero_effective :parameter_count.effective = 0

Tau.BookIV.Particles.zero_ad_hoc

source theorem Tau.BookIV.Particles.zero_ad_hoc :parameter_count.ad_hoc = 0

Tau.BookIV.Particles.total_inputs

source theorem Tau.BookIV.Particles.total_inputs :parameter_count.dimensionless + parameter_count.anchors = 2

Tau.BookIV.Particles.NonOnticEntity

source structure Tau.BookIV.Particles.NonOnticEntity :Type

[IV.R151] Non-ontic entities: computational devices useful in orthodox calculations but NOT representing τ³ objects.

  • name : String Entity name.

  • reason : String Why non-ontic.

Instances For

Tau.BookIV.Particles.instReprNonOnticEntity

source instance Tau.BookIV.Particles.instReprNonOnticEntity :Repr NonOnticEntity

Equations

  • Tau.BookIV.Particles.instReprNonOnticEntity = { reprPrec := Tau.BookIV.Particles.instReprNonOnticEntity.repr }

Tau.BookIV.Particles.instReprNonOnticEntity.repr

source def Tau.BookIV.Particles.instReprNonOnticEntity.repr :NonOnticEntity → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.non_ontic_entities

source def Tau.BookIV.Particles.non_ontic_entities :List NonOnticEntity

Equations

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

Tau.BookIV.Particles.five_non_ontic

source theorem Tau.BookIV.Particles.five_non_ontic :non_ontic_entities.length = 5

Tau.BookIV.Particles.OntologicalLine

source structure Tau.BookIV.Particles.OntologicalLine :Type

[IV.R152] The ontic/non-ontic distinction is not philosophical preference but a mathematical consequence of the τ³ fibration. An entity is ontic iff it can be constructed as a mode, character, or finite composite on τ³ = τ¹ ×_f T².

  • mathematical : Bool Mathematical, not philosophical.

  • criterion : String Criterion: constructible on τ³.

  • resolves_wf : Bool Resolves wave function reality.

  • resolves_virtual : Bool Resolves virtual particle reality.

  • resolves_spacetime : Bool Resolves spacetime reality.

Instances For

Tau.BookIV.Particles.instReprOntologicalLine

source instance Tau.BookIV.Particles.instReprOntologicalLine :Repr OntologicalLine

Equations

  • Tau.BookIV.Particles.instReprOntologicalLine = { reprPrec := Tau.BookIV.Particles.instReprOntologicalLine.repr }

Tau.BookIV.Particles.instReprOntologicalLine.repr

source def Tau.BookIV.Particles.instReprOntologicalLine.repr :OntologicalLine → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.ontological_line

source def Tau.BookIV.Particles.ontological_line :OntologicalLine

Equations

  • Tau.BookIV.Particles.ontological_line = { } Instances For

Tau.BookIV.Particles.line_is_mathematical

source theorem Tau.BookIV.Particles.line_is_mathematical :ontological_line.mathematical = true

Tau.BookIV.Particles.DictionaryLimits

source structure Tau.BookIV.Particles.DictionaryLimits :Type

[IV.R153] The τ-to-SM translation dictionary covers sector decomposition, coupling ledger, QM infrastructure, and particle content but has limits:

No SM counterpart for: H_∂[ω], breathing operator, defect functional. No τ counterpart for: virtual particles, path integral, RG flow, gravitons.

  • tau_only : List String Tau concepts without SM counterpart.

  • sm_only : List String SM concepts without tau counterpart.

Instances For

Tau.BookIV.Particles.instReprDictionaryLimits.repr

source def Tau.BookIV.Particles.instReprDictionaryLimits.repr :DictionaryLimits → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.instReprDictionaryLimits

source instance Tau.BookIV.Particles.instReprDictionaryLimits :Repr DictionaryLimits

Equations

  • Tau.BookIV.Particles.instReprDictionaryLimits = { reprPrec := Tau.BookIV.Particles.instReprDictionaryLimits.repr }

Tau.BookIV.Particles.dictionary_limits

source def Tau.BookIV.Particles.dictionary_limits :DictionaryLimits

Equations

  • Tau.BookIV.Particles.dictionary_limits = { } Instances For

Tau.BookIV.Particles.three_tau_only

source theorem Tau.BookIV.Particles.three_tau_only :dictionary_limits.tau_only.length = 3

Tau.BookIV.Particles.four_sm_only

source theorem Tau.BookIV.Particles.four_sm_only :dictionary_limits.sm_only.length = 4

Tau.BookIV.Particles.TemperatureNotFundamental

source structure Tau.BookIV.Particles.TemperatureNotFundamental :Type

[IV.R154] Temperature is not fundamental in Category τ but a readout: the gradient of the defect functional with respect to the entropy component. Part VII will use the defect functional as organizing variable with temperature as derived quantity.

  • derived : Bool Temperature is derived.

  • derivation : String Derivation: gradient of defect functional.

  • fundamental : String Fundamental variable: defect functional.

  • used_in_part_vii : Bool Part VII uses this.

Instances For

Tau.BookIV.Particles.instReprTemperatureNotFundamental.repr

source def Tau.BookIV.Particles.instReprTemperatureNotFundamental.repr :TemperatureNotFundamental → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.instReprTemperatureNotFundamental

source instance Tau.BookIV.Particles.instReprTemperatureNotFundamental :Repr TemperatureNotFundamental

Equations

  • Tau.BookIV.Particles.instReprTemperatureNotFundamental = { reprPrec := Tau.BookIV.Particles.instReprTemperatureNotFundamental.repr }

Tau.BookIV.Particles.temperature_not_fundamental

source def Tau.BookIV.Particles.temperature_not_fundamental :TemperatureNotFundamental

Equations

  • Tau.BookIV.Particles.temperature_not_fundamental = { } Instances For

Tau.BookIV.Particles.temp_is_derived

source theorem Tau.BookIV.Particles.temp_is_derived :temperature_not_fundamental.derived = true

Tau.BookIV.Particles.SpectrumSummary

source structure Tau.BookIV.Particles.SpectrumSummary :Type

Summary of the complete particle spectrum:

  • All observed SM particles accounted for

  • No BSM particles predicted

  • Two inputs only (ι_τ, m_n)

  • Ontic/non-ontic line is mathematical

  • sm_complete : Bool All SM particles accounted for.

  • no_bsm : Bool No BSM predicted.

  • num_inputs : ℕ Number of inputs.

  • ontic_count : ℕ Ontic entities in register.

  • non_ontic_count : ℕ Non-ontic entities identified.

Instances For

Tau.BookIV.Particles.instReprSpectrumSummary.repr

source def Tau.BookIV.Particles.instReprSpectrumSummary.repr :SpectrumSummary → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.instReprSpectrumSummary

source instance Tau.BookIV.Particles.instReprSpectrumSummary :Repr SpectrumSummary

Equations

  • Tau.BookIV.Particles.instReprSpectrumSummary = { reprPrec := Tau.BookIV.Particles.instReprSpectrumSummary.repr }

Tau.BookIV.Particles.spectrum_summary

source def Tau.BookIV.Particles.spectrum_summary :SpectrumSummary

Equations

  • Tau.BookIV.Particles.spectrum_summary = { } Instances For

Tau.BookIV.Particles.spectrum_complete

source theorem Tau.BookIV.Particles.spectrum_complete :spectrum_summary.sm_complete = true

Tau.BookIV.Particles.spectrum_no_bsm

source theorem Tau.BookIV.Particles.spectrum_no_bsm :spectrum_summary.no_bsm = true

Tau.BookIV.Particles.spectrum_two_inputs

source theorem Tau.BookIV.Particles.spectrum_two_inputs :spectrum_summary.num_inputs = 2

Tau.BookIV.Particles.total_entities

source theorem Tau.BookIV.Particles.total_entities :spectrum_summary.ontic_count + spectrum_summary.non_ontic_count = 20

Ontic and non-ontic together account for all discussed entities.