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TauLib.BookIV.Strong.Confinement

TauLib.BookIV.Strong.Confinement

Confinement mechanism: fractional CR-sublattice, color-confined modes, color singlets, linear potential, baryon number, winding preservation, and proton stability.

Registry Cross-References

  • [IV.D158] Fractional CR-sublattice — FractionalCRSublattice

  • [IV.D159] Color-confined Mode — ColorConfinedMode

  • [IV.D160] Color Singlet — ColorSingletDef

  • [IV.D161] Baryon Number — BaryonNumberDef

  • [IV.T71] Confinement Theorem — confinement_theorem

  • [IV.T72] Proton Stability Theorem — proton_stability

  • [IV.P94] Singlet Stability — singlet_stability

  • [IV.P95] Singlet Classification — singlet_classification

  • [IV.P96] Linear Confinement Potential — linear_potential

  • [IV.L8] Winding Preservation — winding_preservation

  • [IV.R61-R68] Structural remarks (comment-only)

Mathematical Content

Color confinement arises because modes with fractional eta-holonomy (n not equiv 0 mod 3) fail to converge in H_partial[omega]. Only color singlets (total winding 0 mod 3) resolve to stable boundary characters. Baryon number is conserved because admissible endomorphisms preserve total eta-winding mod 3, implying absolute proton stability.

Ground Truth Sources

  • Chapter 39 of Book IV (2nd Edition)

Tau.BookIV.Strong.FractionalCRSublattice

source structure Tau.BookIV.Strong.FractionalCRSublattice :Type

[IV.D158] The fractional CR-sublattice Lambda_CR^{1/3}: {(m, n/3) : m,n in Z} subset Z x (1/3)Z, refining the character lattice to accommodate color-charged modes with fractional eta-component n/3 not in Z.

  • lattice : String Lattice description.

  • refinement_factor : ℕ Refinement factor (denominator of eta-fraction).

  • purpose : String Purpose: accommodate color-charged modes.

Instances For

Tau.BookIV.Strong.instReprFractionalCRSublattice.repr

source def Tau.BookIV.Strong.instReprFractionalCRSublattice.repr :FractionalCRSublattice → ℕ → Std.Format

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Tau.BookIV.Strong.instReprFractionalCRSublattice

source instance Tau.BookIV.Strong.instReprFractionalCRSublattice :Repr FractionalCRSublattice

Equations

  • Tau.BookIV.Strong.instReprFractionalCRSublattice = { reprPrec := Tau.BookIV.Strong.instReprFractionalCRSublattice.repr }

Tau.BookIV.Strong.fractional_cr_sublattice

source def Tau.BookIV.Strong.fractional_cr_sublattice :FractionalCRSublattice

Equations

  • Tau.BookIV.Strong.fractional_cr_sublattice = { } Instances For

Tau.BookIV.Strong.sublattice_factor_3

source theorem Tau.BookIV.Strong.sublattice_factor_3 :fractional_cr_sublattice.refinement_factor = 3

Tau.BookIV.Strong.ColorConfinedMode

source structure Tau.BookIV.Strong.ColorConfinedMode :Type

[IV.D159] A mode chi_{m,n} is color-confined if:

  • n not equiv 0 mod 3 (fractional eta-holonomy)

  • The associated boundary character fails to converge in H_partial[omega] Confinement = non-convergence in the profinite limit.

  • eta_winding_mod3 : ℕ Eta-winding (not divisible by 3).

  • fractional : Bool Non-zero mod 3 condition.

  • non_convergent : Bool Fails to converge in profinite limit.

Instances For

Tau.BookIV.Strong.instReprColorConfinedMode

source instance Tau.BookIV.Strong.instReprColorConfinedMode :Repr ColorConfinedMode

Equations

  • Tau.BookIV.Strong.instReprColorConfinedMode = { reprPrec := Tau.BookIV.Strong.instReprColorConfinedMode.repr }

Tau.BookIV.Strong.instReprColorConfinedMode.repr

source def Tau.BookIV.Strong.instReprColorConfinedMode.repr :ColorConfinedMode → ℕ → Std.Format

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Tau.BookIV.Strong.is_confined

source def Tau.BookIV.Strong.is_confined (n : ℕ) :Bool

A mode with winding n is confined iff n mod 3 != 0. Equations

  • Tau.BookIV.Strong.is_confined n = (n % 3 != 0) Instances For

Tau.BookIV.Strong.winding_1_confined

source theorem Tau.BookIV.Strong.winding_1_confined :is_confined 1 = true

Tau.BookIV.Strong.winding_2_confined

source theorem Tau.BookIV.Strong.winding_2_confined :is_confined 2 = true

Tau.BookIV.Strong.winding_3_free

source theorem Tau.BookIV.Strong.winding_3_free :is_confined 3 = false

Tau.BookIV.Strong.winding_0_free

source theorem Tau.BookIV.Strong.winding_0_free :is_confined 0 = false

Tau.BookIV.Strong.ConfinementTheorem

source structure Tau.BookIV.Strong.ConfinementTheorem :Type

[IV.T71] Confinement Theorem: no isolated color-charged state resolves to a stable element of H_partial[omega].

The boundary character sequence for a mode with exp(2pi i c/3) holonomy (c != 0 mod 3) fails to converge because the fractional eta-phase prevents cancellation in the profinite limit.

This is NOT an input or assumption — it follows from the boundary holonomy algebra structure at depth 3 with chi_minus dominance.

  • no_isolated_colored : Bool Isolated color-charged states do not exist at omega.

  • mechanism : String Mechanism: non-convergence of fractional holonomy.

  • scope : String Scope: tau-effective (derived, not assumed).

  • source : String Source: boundary holonomy at depth 3 with chi_minus dominance.

Instances For

Tau.BookIV.Strong.instReprConfinementTheorem.repr

source def Tau.BookIV.Strong.instReprConfinementTheorem.repr :ConfinementTheorem → ℕ → Std.Format

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Tau.BookIV.Strong.instReprConfinementTheorem

source instance Tau.BookIV.Strong.instReprConfinementTheorem :Repr ConfinementTheorem

Equations

  • Tau.BookIV.Strong.instReprConfinementTheorem = { reprPrec := Tau.BookIV.Strong.instReprConfinementTheorem.repr }

Tau.BookIV.Strong.confinement_theorem

source def Tau.BookIV.Strong.confinement_theorem :ConfinementTheorem

Equations

  • Tau.BookIV.Strong.confinement_theorem = { } Instances For

Tau.BookIV.Strong.ColorSingletDef

source structure Tau.BookIV.Strong.ColorSingletDef :Type

[IV.D160] Color singlet: composite state with trivial total eta-holonomy, hol_eta(Psi) = 1, i.e., sum c_j equiv 0 mod 3.

  • total_mod3 : ℕ Total winding sum mod 3 = 0.

  • trivial_holonomy : Bool Trivial total holonomy.

Instances For

Tau.BookIV.Strong.instReprColorSingletDef

source instance Tau.BookIV.Strong.instReprColorSingletDef :Repr ColorSingletDef

Equations

  • Tau.BookIV.Strong.instReprColorSingletDef = { reprPrec := Tau.BookIV.Strong.instReprColorSingletDef.repr }

Tau.BookIV.Strong.instReprColorSingletDef.repr

source def Tau.BookIV.Strong.instReprColorSingletDef.repr :ColorSingletDef → ℕ → Std.Format

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

Tau.BookIV.Strong.SingletStability

source structure Tau.BookIV.Strong.SingletStability :Type

[IV.P94] A color singlet resolves to a stable boundary character: the fractional eta-phases cancel exactly, so the composite boundary character sequence converges in H_partial[omega].

  • phases_cancel : Bool Fractional phases cancel.

  • converges : Bool Converges in profinite limit.

  • stable_on_L : Bool Stable boundary character on L.

Instances For

Tau.BookIV.Strong.instReprSingletStability

source instance Tau.BookIV.Strong.instReprSingletStability :Repr SingletStability

Equations

  • Tau.BookIV.Strong.instReprSingletStability = { reprPrec := Tau.BookIV.Strong.instReprSingletStability.repr }

Tau.BookIV.Strong.instReprSingletStability.repr

source def Tau.BookIV.Strong.instReprSingletStability.repr :SingletStability → ℕ → Std.Format

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

Tau.BookIV.Strong.singlet_stability

source def Tau.BookIV.Strong.singlet_stability :SingletStability

Equations

  • Tau.BookIV.Strong.singlet_stability = { } Instances For

Tau.BookIV.Strong.HadronType

source inductive Tau.BookIV.Strong.HadronType :Type

Hadron types: the minimal color-singlet structures.

  • baryon : HadronType Baryon: three constituents {0,1,2} antisymmetric in color.

  • meson : HadronType Meson: quark-antiquark {c, c_bar}.

  • exotic : HadronType Exotic: tetraquark, pentaquark, etc.

Instances For

Tau.BookIV.Strong.instReprHadronType.repr

source def Tau.BookIV.Strong.instReprHadronType.repr :HadronType → ℕ → Std.Format

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Tau.BookIV.Strong.instReprHadronType

source instance Tau.BookIV.Strong.instReprHadronType :Repr HadronType

Equations

  • Tau.BookIV.Strong.instReprHadronType = { reprPrec := Tau.BookIV.Strong.instReprHadronType.repr }

Tau.BookIV.Strong.instDecidableEqHadronType

source instance Tau.BookIV.Strong.instDecidableEqHadronType :DecidableEq HadronType

Equations

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

Tau.BookIV.Strong.instBEqHadronType.beq

source def Tau.BookIV.Strong.instBEqHadronType.beq :HadronType → HadronType → Bool

Equations

  • Tau.BookIV.Strong.instBEqHadronType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookIV.Strong.instBEqHadronType

source instance Tau.BookIV.Strong.instBEqHadronType :BEq HadronType

Equations

  • Tau.BookIV.Strong.instBEqHadronType = { beq := Tau.BookIV.Strong.instBEqHadronType.beq }

Tau.BookIV.Strong.SingletClassification

source structure Tau.BookIV.Strong.SingletClassification :Type

[IV.P95] Every persistent hadronic state is a color singlet. Minimal singlet structures:

  • Baryon: {0,1,2} (three quarks, one per color)

  • Meson: {c, bar{c}} (quark-antiquark)

  • Exotic: {c1,c2,bar{c3},bar{c4}} etc. with total 0 mod 3

  • all_hadrons_singlets : Bool All hadrons are singlets.

  • min_baryon_size : ℕ Minimal baryonic singlet size.

  • min_meson_size : ℕ Minimal mesonic singlet size.

Instances For

Tau.BookIV.Strong.instReprSingletClassification

source instance Tau.BookIV.Strong.instReprSingletClassification :Repr SingletClassification

Equations

  • Tau.BookIV.Strong.instReprSingletClassification = { reprPrec := Tau.BookIV.Strong.instReprSingletClassification.repr }

Tau.BookIV.Strong.instReprSingletClassification.repr

source def Tau.BookIV.Strong.instReprSingletClassification.repr :SingletClassification → ℕ → Std.Format

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

Tau.BookIV.Strong.singlet_classification

source def Tau.BookIV.Strong.singlet_classification :SingletClassification

Equations

  • Tau.BookIV.Strong.singlet_classification = { } Instances For

Tau.BookIV.Strong.baryon_is_singlet

source theorem Tau.BookIV.Strong.baryon_is_singlet :is_color_singlet [0, 1, 2] = true

Baryon winding pattern {0,1,2} is a singlet.

Tau.BookIV.Strong.meson_is_singlet

source theorem Tau.BookIV.Strong.meson_is_singlet :is_color_singlet [1, 2] = true

Meson winding pattern {1,2} is a singlet (1+2=3, 3 mod 3 = 0).

Tau.BookIV.Strong.single_quark_not_singlet

source theorem Tau.BookIV.Strong.single_quark_not_singlet :is_color_singlet [1] = false

A single quark {1} is NOT a singlet.

Tau.BookIV.Strong.single_quark_2_not_singlet

source theorem Tau.BookIV.Strong.single_quark_2_not_singlet :is_color_singlet [2] = false

A single quark {2} is NOT a singlet.

Tau.BookIV.Strong.LinearConfinementPotential

source structure Tau.BookIV.Strong.LinearConfinementPotential :Type

[IV.P96] The defect functional for a quark-antiquark pair: D_C(delta) = D_C(0) + sigma_tau * delta + O(delta^2), where sigma_tau = kappa(C;3) * g[omega]_s is the tau-string tension.

The linear growth with separation delta is the structural origin of the confining flux tube / QCD string.

  • linear_growth : Bool Linear growth with separation.

  • tension_involves_kappa_C : Bool String tension involves kappa(C;3).

  • flux_tube : Bool Produces flux tube / string.

Instances For

Tau.BookIV.Strong.instReprLinearConfinementPotential.repr

source def Tau.BookIV.Strong.instReprLinearConfinementPotential.repr :LinearConfinementPotential → ℕ → Std.Format

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Tau.BookIV.Strong.instReprLinearConfinementPotential

source instance Tau.BookIV.Strong.instReprLinearConfinementPotential :Repr LinearConfinementPotential

Equations

  • Tau.BookIV.Strong.instReprLinearConfinementPotential = { reprPrec := Tau.BookIV.Strong.instReprLinearConfinementPotential.repr }

Tau.BookIV.Strong.linear_potential

source def Tau.BookIV.Strong.linear_potential :LinearConfinementPotential

Equations

  • Tau.BookIV.Strong.linear_potential = { } Instances For

Tau.BookIV.Strong.BaryonNumberDef

source structure Tau.BookIV.Strong.BaryonNumberDef :Type

[IV.D161] Baryon number B(Psi) := (1/3) * sum_j n_j, where n_j is the eta-winding of constituent psi_j. For a baryon with {c1,c2,c3} = {0,1,2}: B = (0+1+2)/3 = 1.

  • winding_sum : ℕ Sum of windings.

  • baryon_number : ℕ Baryon number = winding_sum / 3 (integer for singlets).

Instances For

Tau.BookIV.Strong.instReprBaryonNumberDef.repr

source def Tau.BookIV.Strong.instReprBaryonNumberDef.repr :BaryonNumberDef → ℕ → Std.Format

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Tau.BookIV.Strong.instReprBaryonNumberDef

source instance Tau.BookIV.Strong.instReprBaryonNumberDef :Repr BaryonNumberDef

Equations

  • Tau.BookIV.Strong.instReprBaryonNumberDef = { reprPrec := Tau.BookIV.Strong.instReprBaryonNumberDef.repr }

Tau.BookIV.Strong.compute_baryon_number

source def Tau.BookIV.Strong.compute_baryon_number (windings : List ℕ) :ℕ × ℕ

Compute baryon number from a list of eta-windings. Returns (winding_sum, baryon_number) where baryon_number = sum/3. Equations

  • Tau.BookIV.Strong.compute_baryon_number windings = (List.foldl (fun (x1 x2 : ℕ) => x1 + x2) 0 windings, List.foldl (fun (x1 x2 : ℕ) => x1 + x2) 0 windings / 3) Instances For

Tau.BookIV.Strong.proton_baryon_number

source theorem Tau.BookIV.Strong.proton_baryon_number :(compute_baryon_number [0, 1, 2]).2 = 1

Tau.BookIV.Strong.meson_baryon_number

source theorem Tau.BookIV.Strong.meson_baryon_number :(compute_baryon_number [1, 2]).2 = 1

Tau.BookIV.Strong.WindingPreservation

source structure Tau.BookIV.Strong.WindingPreservation :Type

[IV.L8] Winding Preservation: any admissible endomorphism phi compatible with the C-sector preserves total eta-winding mod 3, ensuring baryon number conservation under all physical processes.

  • preserves_mod3 : Bool Admissible endomorphisms preserve winding mod 3.

  • baryon_conserved : Bool Consequence: baryon number is conserved.

  • mechanism : String Mechanism: admissibility condition (SA-i) forces eta-sector preservation.

Instances For

Tau.BookIV.Strong.instReprWindingPreservation

source instance Tau.BookIV.Strong.instReprWindingPreservation :Repr WindingPreservation

Equations

  • Tau.BookIV.Strong.instReprWindingPreservation = { reprPrec := Tau.BookIV.Strong.instReprWindingPreservation.repr }

Tau.BookIV.Strong.instReprWindingPreservation.repr

source def Tau.BookIV.Strong.instReprWindingPreservation.repr :WindingPreservation → ℕ → Std.Format

Equations

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

Tau.BookIV.Strong.winding_preservation

source def Tau.BookIV.Strong.winding_preservation :WindingPreservation

Equations

  • Tau.BookIV.Strong.winding_preservation = { } Instances For

Tau.BookIV.Strong.ProtonStabilityTheorem

source structure Tau.BookIV.Strong.ProtonStabilityTheorem :Type

[IV.T72] Proton Stability: the proton is absolutely stable. No admissible endomorphism in the 4+1 sector framework can change baryon number: B(phi(Psi)) = B(Psi) for all admissible phi.

This predicts tau_proton = infinity, in contrast to GUT theories that predict finite proton lifetime via baryon-number-violating leptoquark exchange.

The proof follows from winding preservation (IV.L8): since phi preserves total eta-winding mod 3, and B = (1/3) * sum n_j, baryon number is an invariant of admissible dynamics.

  • absolutely_stable : Bool Proton is absolutely stable.

  • lifetime : String Lifetime prediction: infinite.

  • no_B_violation : Bool No baryon number violation by any admissible endomorphism.

  • gut_contrast : String Contrast with GUTs.

  • source : String Source: winding preservation (IV.L8).

Instances For

Tau.BookIV.Strong.instReprProtonStabilityTheorem

source instance Tau.BookIV.Strong.instReprProtonStabilityTheorem :Repr ProtonStabilityTheorem

Equations

  • Tau.BookIV.Strong.instReprProtonStabilityTheorem = { reprPrec := Tau.BookIV.Strong.instReprProtonStabilityTheorem.repr }

Tau.BookIV.Strong.instReprProtonStabilityTheorem.repr

source def Tau.BookIV.Strong.instReprProtonStabilityTheorem.repr :ProtonStabilityTheorem → ℕ → Std.Format

Equations

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

Tau.BookIV.Strong.proton_stability

source def Tau.BookIV.Strong.proton_stability :ProtonStabilityTheorem

Equations

  • Tau.BookIV.Strong.proton_stability = { } Instances For