TauLib · API Book IV

TauLib.BookIV.Particles.PeriodicTable

TauLib.BookIV.Particles.PeriodicTable

Periodic table from sector couplings: atom definition, electron quantum numbers, Madelung rule from T² geometry, period length sequence, chemical bonding (covalent, ionic, metallic), molecular geometry from mode repulsion, hybrid modes, and the five-rung donut ladder.

Registry Cross-References

  • [IV.D203] Atom as Dressed Nuclear Mode — AtomDef

  • [IV.D204] Electron Quantum Numbers — ElectronQuantumNumbers

  • [IV.D205] Covalent Bond — CovalentBond

  • [IV.D206] Ionic Bond — IonicBond

  • [IV.D207] Metallic Bond — MetallicBond

  • [IV.D208] Hybrid Modes — HybridMode

  • [IV.T88] Period Length Sequence — period_length_sequence

  • [IV.R140] Madelung Rule from T² Geometry — madelung_rule

  • [IV.R141] Topological not Accidental — topological_not_accidental

  • [IV.R142] Molecules as Mode-Sharing Graphs — comment-only (not_applicable)

  • [IV.R143] No New Parameters for Chemistry — comment-only (not_applicable)

  • [IV.R144] Mode-Repulsion Geometry — mode_repulsion_geometry

  • [IV.R145] Homochirality and Parity Violation — comment-only (not_applicable)

  • [IV.R146] Structural vs Quantitative Chemistry — comment-only (not_applicable)

  • [IV.R147] The Donut Ladder — comment-only (not_applicable)

  • [IV.R148] Comparison with Orthodox Physics — orthodox_comparison

Mathematical Content

The periodic table is a topological invariant of T²: shell capacities 2n² follow from winding mode counting, and period pairing from the Madelung energy ordering E_{n,l} ≈ −1/(n + l·ι_τ)² determined by the fiber shape ratio ι_τ. The sequence 2, 8, 8, 18, 18, 32, 32,… is fixed by geometry.

Chemical bonds (covalent, ionic, metallic) are different patterns of B-sector winding modes on T². Molecular geometry follows from mode repulsion: k mode pairs maximize angular separation.

Ground Truth Sources

  • Chapter 49 of Book IV (2nd Edition)

Tau.BookIV.Particles.AtomDef

source structure Tau.BookIV.Particles.AtomDef :Type

[IV.D203] An atom of atomic number Z is a stable composite of:

  • Nuclear core: Z protons + N neutrons (bound by C-sector)

  • Electron shell: Z electrons on T² (bound by B-sector EM)

A neutral atom has vanishing net B-sector winding: electromagnetically closed.

  • z : ℕ Atomic number Z.

  • n : ℕ Neutron number N.

  • electrons : ℕ Number of shell electrons in neutral atom.

  • neutral : self.electrons = self.z Neutral: electrons = Z.

  • em_closed : Bool Electromagnetically closed.

Instances For

Tau.BookIV.Particles.instReprAtomDef.repr

source def Tau.BookIV.Particles.instReprAtomDef.repr :AtomDef → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.instReprAtomDef

source instance Tau.BookIV.Particles.instReprAtomDef :Repr AtomDef

Equations

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

Tau.BookIV.Particles.hydrogen

source def Tau.BookIV.Particles.hydrogen :AtomDef

Hydrogen atom. Equations

  • Tau.BookIV.Particles.hydrogen = { z := 1, n := 0, electrons := 1, neutral := Tau.BookIV.Particles.hydrogen._proof_1 } Instances For

Tau.BookIV.Particles.helium

source def Tau.BookIV.Particles.helium :AtomDef

Helium atom. Equations

  • Tau.BookIV.Particles.helium = { z := 2, n := 2, electrons := 2, neutral := Tau.BookIV.Particles.helium._proof_1 } Instances For

Tau.BookIV.Particles.carbon

source def Tau.BookIV.Particles.carbon :AtomDef

Carbon atom. Equations

  • Tau.BookIV.Particles.carbon = { z := 6, n := 6, electrons := 6, neutral := Tau.BookIV.Particles.carbon._proof_1 } Instances For

Tau.BookIV.Particles.iron

source def Tau.BookIV.Particles.iron :AtomDef

Iron atom. Equations

  • Tau.BookIV.Particles.iron = { z := 26, n := 30, electrons := 26, neutral := Tau.BookIV.Particles.iron._proof_1 } Instances For

Tau.BookIV.Particles.ElectronQuantumNumbers

source structure Tau.BookIV.Particles.ElectronQuantumNumbers :Type

[IV.D204] An electron mode on T² bound to nuclear charge Z carries four quantum numbers:

  • n: principal (total winding depth)

  • l: angular (0 ≤ l ≤ n-1, lobe structure on L)

  • m_l: magnetic (-l ≤ m_l ≤ l, orientation on L)

  • m_s: spin (±1/2, chirality on T²)

Shell capacity: 2n² = 2 × sum_{l=0}^{n-1} (2l+1).

  • n_principal : ℕ Principal quantum number (n ≥ 1).

  • l_angular : ℕ Angular quantum number (0 ≤ l ≤ n-1).

  • m_l_magnetic : ℤ Magnetic quantum number (|m_l| ≤ l).

  • spin_up : Bool Spin (true = +1/2, false = -1/2).

  • n_pos : self.n_principal ≥ 1 n ≥ 1.

  • l_bound : self.l_angular < self.n_principal l < n.

Instances For

Tau.BookIV.Particles.instReprElectronQuantumNumbers

source instance Tau.BookIV.Particles.instReprElectronQuantumNumbers :Repr ElectronQuantumNumbers

Equations

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

Tau.BookIV.Particles.instReprElectronQuantumNumbers.repr

source def Tau.BookIV.Particles.instReprElectronQuantumNumbers.repr :ElectronQuantumNumbers → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.shell_capacity

source def Tau.BookIV.Particles.shell_capacity (n : ℕ) :ℕ

Shell capacity for principal quantum number n: 2n². Equations

  • Tau.BookIV.Particles.shell_capacity n = 2 * n * n Instances For

Tau.BookIV.Particles.shell_1

source theorem Tau.BookIV.Particles.shell_1 :shell_capacity 1 = 2

Tau.BookIV.Particles.shell_2

source theorem Tau.BookIV.Particles.shell_2 :shell_capacity 2 = 8

Tau.BookIV.Particles.shell_3

source theorem Tau.BookIV.Particles.shell_3 :shell_capacity 3 = 18

Tau.BookIV.Particles.shell_4

source theorem Tau.BookIV.Particles.shell_4 :shell_capacity 4 = 32

Tau.BookIV.Particles.MadelungRule

source structure Tau.BookIV.Particles.MadelungRule :Type

[IV.R140] The Madelung rule (subshells fill in order of increasing n+l) has no first-principles derivation in orthodox physics.

In Category τ, the breathing eigenvalue on T² with shape ratio ι_τ is E_{n,l} ≈ −1/(n + l·ι_τ)², and since ι_τ ≈ 0.34, the n+l ordering emerges naturally from the fiber geometry.

The subshell filling order is: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, …

  • ordering_param : String Ordering parameter: n + l.

  • origin : String Origin: breathing eigenvalue on T² with shape ι_τ.

  • no_orthodox_derivation : Bool No orthodox first-principles derivation.

  • tau_derived : Bool Tau-framework: emerges from geometry.

Instances For

Tau.BookIV.Particles.instReprMadelungRule.repr

source def Tau.BookIV.Particles.instReprMadelungRule.repr :MadelungRule → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.instReprMadelungRule

source instance Tau.BookIV.Particles.instReprMadelungRule :Repr MadelungRule

Equations

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

Tau.BookIV.Particles.madelung_rule

source def Tau.BookIV.Particles.madelung_rule :MadelungRule

Equations

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

Tau.BookIV.Particles.madelung_tau_derived

source theorem Tau.BookIV.Particles.madelung_tau_derived :madelung_rule.tau_derived = true

Tau.BookIV.Particles.PeriodLengthSequence

source structure Tau.BookIV.Particles.PeriodLengthSequence :Type

[IV.T88] The periodic table period lengths follow: 2, 8, 8, 18, 18, 32, 32,…

Each length = 2n² (twice a perfect square). Each value (except 2) appears twice.

This is a topological consequence of T² fiber geometry: shell capacity 2n² combined with Aufbau filling order determines noble gas closures.

  • lengths : List ℕ First 7 period lengths.

  • twice_perfect_square : Bool Each is twice a perfect square.

  • doubled : Bool Each (except 2) appears twice.

  • topological : Bool Topological origin.

Instances For

Tau.BookIV.Particles.instReprPeriodLengthSequence.repr

source def Tau.BookIV.Particles.instReprPeriodLengthSequence.repr :PeriodLengthSequence → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.instReprPeriodLengthSequence

source instance Tau.BookIV.Particles.instReprPeriodLengthSequence :Repr PeriodLengthSequence

Equations

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

Tau.BookIV.Particles.period_length_sequence

source def Tau.BookIV.Particles.period_length_sequence :PeriodLengthSequence

Equations

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

Tau.BookIV.Particles.seven_periods

source theorem Tau.BookIV.Particles.seven_periods :period_length_sequence.lengths.length = 7

Tau.BookIV.Particles.noble_gas_z

source def Tau.BookIV.Particles.noble_gas_z :List ℕ

Noble gas atomic numbers from cumulative period lengths. Equations

  • Tau.BookIV.Particles.noble_gas_z = [2, 10, 18, 36, 54, 86, 118] Instances For

Tau.BookIV.Particles.seven_noble_gases

source theorem Tau.BookIV.Particles.seven_noble_gases :noble_gas_z.length = 7

Tau.BookIV.Particles.first_noble_gas

source theorem Tau.BookIV.Particles.first_noble_gas :noble_gas_z.head? = some 2

Cumulative sum of period lengths gives noble gas Z values.

Tau.BookIV.Particles.period_2_is_2x1sq

source theorem Tau.BookIV.Particles.period_2_is_2x1sq :2 = 2 * 1 * 1

Period lengths are all twice perfect squares.

Tau.BookIV.Particles.period_8_is_2x2sq

source theorem Tau.BookIV.Particles.period_8_is_2x2sq :8 = 2 * 2 * 2

Tau.BookIV.Particles.period_18_is_2x3sq

source theorem Tau.BookIV.Particles.period_18_is_2x3sq :18 = 2 * 3 * 3

Tau.BookIV.Particles.period_32_is_2x4sq

source theorem Tau.BookIV.Particles.period_32_is_2x4sq :32 = 2 * 4 * 4

Tau.BookIV.Particles.topological_not_accidental

source def Tau.BookIV.Particles.topological_not_accidental :String

[IV.R141] The sequence 2, 8, 8, 18, 18, 32,… is a topological invariant of T²: the periodic table has its shape because the fiber torus has its shape. Equations

  • Tau.BookIV.Particles.topological_not_accidental = “Period sequence is topological invariant of T^2, not accidental” Instances For

Tau.BookIV.Particles.CovalentBond

source structure Tau.BookIV.Particles.CovalentBond :Type

[IV.D205] A covalent bond of order k: k electron winding modes on T² contribute simultaneously to shell closure of both atoms.

  • k=1: single bond (e.g., H₂)

  • k=2: double bond (e.g., O=O)

  • k=3: triple bond (e.g., N≡N)

Bond strength increases with k via mode-overlap integrals.

  • order : ℕ Bond order.

  • example_mol : String Example molecule.

  • order_valid : self.order ≥ 1 ∧ self.order ≤ 3 Order is 1, 2, or 3.

Instances For

Tau.BookIV.Particles.instReprCovalentBond.repr

source def Tau.BookIV.Particles.instReprCovalentBond.repr :CovalentBond → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.instReprCovalentBond

source instance Tau.BookIV.Particles.instReprCovalentBond :Repr CovalentBond

Equations

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

Tau.BookIV.Particles.single_bond

source def Tau.BookIV.Particles.single_bond :CovalentBond

Equations

  • Tau.BookIV.Particles.single_bond = { order := 1, example_mol := “H_2”, order_valid := Tau.BookIV.Particles.single_bond._proof_3 } Instances For

Tau.BookIV.Particles.double_bond

source def Tau.BookIV.Particles.double_bond :CovalentBond

Equations

  • Tau.BookIV.Particles.double_bond = { order := 2, example_mol := “O=O”, order_valid := Tau.BookIV.Particles.double_bond._proof_3 } Instances For

Tau.BookIV.Particles.triple_bond

source def Tau.BookIV.Particles.triple_bond :CovalentBond

Equations

  • Tau.BookIV.Particles.triple_bond = { order := 3, example_mol := “N_triple_N”, order_valid := Tau.BookIV.Particles.triple_bond._proof_3 } Instances For

Tau.BookIV.Particles.IonicBond

source structure Tau.BookIV.Particles.IonicBond :Type

[IV.D206] An ionic bond is a complete transfer of electron winding modes from atom A to atom B such that both ions achieve noble gas closure. Bond energy E_ionic ≈ −k²α/r_AB.

  • complete_transfer : Bool Complete electron transfer.

  • both_closed : Bool Both ions approach noble gas closure.

  • example_compound : String Example.

Instances For

Tau.BookIV.Particles.instReprIonicBond.repr

source def Tau.BookIV.Particles.instReprIonicBond.repr :IonicBond → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.instReprIonicBond

source instance Tau.BookIV.Particles.instReprIonicBond :Repr IonicBond

Equations

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

Tau.BookIV.Particles.ionic_bond

source def Tau.BookIV.Particles.ionic_bond :IonicBond

Equations

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

Tau.BookIV.Particles.MetallicBond

source structure Tau.BookIV.Particles.MetallicBond :Type

[IV.D207] A metallic bond is a collective binding mode in which outermost electron winding modes are delocalized across the lattice. Explains: conductivity, malleability, luster.

  • delocalized : Bool Delocalized modes.

  • properties : List String Properties explained.

  • few_outer : Bool Arises in elements with few outer-shell electrons.

Instances For

Tau.BookIV.Particles.instReprMetallicBond

source instance Tau.BookIV.Particles.instReprMetallicBond :Repr MetallicBond

Equations

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

Tau.BookIV.Particles.instReprMetallicBond.repr

source def Tau.BookIV.Particles.instReprMetallicBond.repr :MetallicBond → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.metallic_bond

source def Tau.BookIV.Particles.metallic_bond :MetallicBond

Equations

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

Tau.BookIV.Particles.three_metallic_properties

source theorem Tau.BookIV.Particles.three_metallic_properties :metallic_bond.properties.length = 3

Tau.BookIV.Particles.ModeRepulsionEntry

source structure Tau.BookIV.Particles.ModeRepulsionEntry :Type

[IV.R144] Molecular geometry from mode repulsion: k mode pairs maximize minimum angular separation.

k Geometry Angle

2 linear 180°

3 trigonal planar 120°

4 tetrahedral 109.5°

5 trigonal bipyramidal 90°/120°

6 octahedral 90°

Symmetry depends only on k, not on ι_τ.

  • k : ℕ Number of mode pairs.

  • geometry : String Geometry name.

  • angle_deg_x10 : ℕ Characteristic angle (degrees ×10).

Instances For

Tau.BookIV.Particles.instReprModeRepulsionEntry

source instance Tau.BookIV.Particles.instReprModeRepulsionEntry :Repr ModeRepulsionEntry

Equations

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

Tau.BookIV.Particles.instReprModeRepulsionEntry.repr

source def Tau.BookIV.Particles.instReprModeRepulsionEntry.repr :ModeRepulsionEntry → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.mode_repulsion_table

source def Tau.BookIV.Particles.mode_repulsion_table :List ModeRepulsionEntry

Equations

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

Tau.BookIV.Particles.mode_repulsion_geometry

source def Tau.BookIV.Particles.mode_repulsion_geometry :List ModeRepulsionEntry

Equations

  • Tau.BookIV.Particles.mode_repulsion_geometry = Tau.BookIV.Particles.mode_repulsion_table Instances For

Tau.BookIV.Particles.five_geometries

source theorem Tau.BookIV.Particles.five_geometries :mode_repulsion_table.length = 5

Tau.BookIV.Particles.HybridizationType

source inductive Tau.BookIV.Particles.HybridizationType :Type

Hybridization type.

  • sp : HybridizationType sp: 2 linear hybrids (180°).

  • sp2 : HybridizationType sp²: 3 planar hybrids (120°).

  • sp3 : HybridizationType sp³: 4 tetrahedral hybrids (109.5°).

Instances For

Tau.BookIV.Particles.instReprHybridizationType

source instance Tau.BookIV.Particles.instReprHybridizationType :Repr HybridizationType

Equations

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

Tau.BookIV.Particles.instReprHybridizationType.repr

source def Tau.BookIV.Particles.instReprHybridizationType.repr :HybridizationType → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.instDecidableEqHybridizationType

source instance Tau.BookIV.Particles.instDecidableEqHybridizationType :DecidableEq HybridizationType

Equations

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

Tau.BookIV.Particles.instBEqHybridizationType.beq

source def Tau.BookIV.Particles.instBEqHybridizationType.beq :HybridizationType → HybridizationType → Bool

Equations

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

Tau.BookIV.Particles.instBEqHybridizationType

source instance Tau.BookIV.Particles.instBEqHybridizationType :BEq HybridizationType

Equations

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

Tau.BookIV.Particles.HybridMode

source structure Tau.BookIV.Particles.HybridMode :Type

[IV.D208] A hybrid mode is a linear combination of s-type (l=0) and p-type (l=1) winding modes optimized for directional bonding.

  • hybridization : HybridizationType Hybridization type.

  • num_hybrids : ℕ Number of equivalent hybrids.

  • angle_deg_x10 : ℕ Bond angle (degrees ×10).

  • example_mol : String Example molecule.

Instances For

Tau.BookIV.Particles.instReprHybridMode

source instance Tau.BookIV.Particles.instReprHybridMode :Repr HybridMode

Equations

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

Tau.BookIV.Particles.instReprHybridMode.repr

source def Tau.BookIV.Particles.instReprHybridMode.repr :HybridMode → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.sp3_hybrid

source def Tau.BookIV.Particles.sp3_hybrid :HybridMode

Equations

  • Tau.BookIV.Particles.sp3_hybrid = { hybridization := Tau.BookIV.Particles.HybridizationType.sp3, num_hybrids := 4, angle_deg_x10 := 1095, example_mol := “CH_4” } Instances For

Tau.BookIV.Particles.sp2_hybrid

source def Tau.BookIV.Particles.sp2_hybrid :HybridMode

Equations

  • Tau.BookIV.Particles.sp2_hybrid = { hybridization := Tau.BookIV.Particles.HybridizationType.sp2, num_hybrids := 3, angle_deg_x10 := 1200, example_mol := “C_2H_4” } Instances For

Tau.BookIV.Particles.sp_hybrid

source def Tau.BookIV.Particles.sp_hybrid :HybridMode

Equations

  • Tau.BookIV.Particles.sp_hybrid = { hybridization := Tau.BookIV.Particles.HybridizationType.sp, num_hybrids := 2, angle_deg_x10 := 1800, example_mol := “C_2H_2” } Instances For

Tau.BookIV.Particles.OrthodoxComparison

source structure Tau.BookIV.Particles.OrthodoxComparison :Type

[IV.R148] In orthodox physics, the five rungs (QCD, nuclear, atomic, quantum chemistry, condensed matter) are separate disciplines with separate formalisms and effective parameters. In Category τ, all five are one continuous ascent on T² with derived scale separations κ(C) » κ(B) » κ(D) from a single master constant.

  • orthodox_disciplines : ℕ Orthodox: separate disciplines.

  • tau_unified : Bool Tau: one continuous ascent.

  • separations_derived : Bool Scale separations derived.

  • single_constant : Bool Single master constant.

Instances For

Tau.BookIV.Particles.instReprOrthodoxComparison

source instance Tau.BookIV.Particles.instReprOrthodoxComparison :Repr OrthodoxComparison

Equations

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

Tau.BookIV.Particles.instReprOrthodoxComparison.repr

source def Tau.BookIV.Particles.instReprOrthodoxComparison.repr :OrthodoxComparison → ℕ → Std.Format

Equations

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

Tau.BookIV.Particles.orthodox_comparison

source def Tau.BookIV.Particles.orthodox_comparison :OrthodoxComparison

Equations

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

Tau.BookIV.Particles.five_orthodox_disciplines

source theorem Tau.BookIV.Particles.five_orthodox_disciplines :orthodox_comparison.orthodox_disciplines = 5