TauLib · API Book IV

TauLib.BookIV.ManyBody.CondensedMatter

TauLib.BookIV.ManyBody.CondensedMatter

Condensed matter from the defect lattice: monotone melting sequence, topological branch of the defect spectrum, fiber-base factorization of the universal defect functional, and fiber-level physics completeness.

Registry Cross-References

  • [IV.P143] Melting Sequence Monotone Mobility — MeltingSequenceMobility

  • [IV.D240] Topological Branch — TopologicalBranch

  • [IV.T94] Fiber-Base Factorization — FiberBaseFactorization

  • [IV.R178] Why the separation is clean — comment-only

  • [IV.P144] Fiber-Level Physics is Complete — FiberLevelComplete

  • [IV.R179] The base is not the rest — comment-only

Mathematical Content

This module synthesizes the many-body physics of Book IV Part VII by establishing three structural results:

  • Melting sequence: the six non-topological regimes are ordered by monotonically increasing macroscopic mobility, from crystal (arrested) to plasma (fully mobile).

  • Topological branch: the superfluid and superconductor regimes form a separate branch characterized by nonzero maximal mobility, zero bulk vorticity, and quantized topological charge.

  • Fiber-base factorization: the universal defect functional on tau^3 = tau^1 x_f T^2 factorizes exactly as delta[omega]{tau^3} = delta[omega]{T^2} tensor delta[omega]_{tau^1}, separating fiber (T^2, spatial) from base (tau^1, temporal) physics.

This factorization closes the fiber-level physics of Book IV and exports a clean interface to Book V (base tau^1 = macrocosm).

Ground Truth Sources

  • Chapter 54 of Book IV (2nd Edition)

  • fluid-condensed-matter.json: spectrum-complete-exports

Tau.BookIV.ManyBody.MeltingSequenceMobility

source structure Tau.BookIV.ManyBody.MeltingSequenceMobility :Type

[IV.P143] The six non-topological regimes are ordered by monotonically increasing macroscopic mobility:

mu_crystal <= mu_quasi <= mu_glass < mu_Euler <= mu_NS < mu_MHD <= mu_plasma

This defines the melting sequence: each step increases the degrees of freedom available to the many-body system.

The two topological regimes (superfluid, superconductor) lie on a separate branch with maximal mobility but constrained theta.

  • num_regimes : ℕ Number of non-topological regimes.

  • monotone : Bool Ordering is monotone in mobility.

  • sequence : List String Regime labels in order.

  • topological_separate : Bool Topological branch separate.

Instances For

Tau.BookIV.ManyBody.instReprMeltingSequenceMobility

source instance Tau.BookIV.ManyBody.instReprMeltingSequenceMobility :Repr MeltingSequenceMobility

Equations

  • Tau.BookIV.ManyBody.instReprMeltingSequenceMobility = { reprPrec := Tau.BookIV.ManyBody.instReprMeltingSequenceMobility.repr }

Tau.BookIV.ManyBody.instReprMeltingSequenceMobility.repr

source def Tau.BookIV.ManyBody.instReprMeltingSequenceMobility.repr :MeltingSequenceMobility → ℕ → Std.Format

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Tau.BookIV.ManyBody.melting_sequence

source def Tau.BookIV.ManyBody.melting_sequence :MeltingSequenceMobility

Equations

  • Tau.BookIV.ManyBody.melting_sequence = { } Instances For

Tau.BookIV.ManyBody.six_nontopo_regimes

source theorem Tau.BookIV.ManyBody.six_nontopo_regimes :melting_sequence.num_regimes = 6

Tau.BookIV.ManyBody.melting_monotone

source theorem Tau.BookIV.ManyBody.melting_monotone :melting_sequence.monotone = true

Tau.BookIV.ManyBody.melting_sequence_count

source theorem Tau.BookIV.ManyBody.melting_sequence_count :melting_sequence.sequence.length = 6

Tau.BookIV.ManyBody.TopologicalBranch

source structure Tau.BookIV.ManyBody.TopologicalBranch :Type

[IV.D240] A topological branch of the defect spectrum is a regime with:

  • Nonzero maximal mobility (free base-direction translation)

  • Zero bulk vorticity nu_bulk = 0 (except at quantized cores)

  • Quantized topological charge theta in Z

The two topological regimes are:

  • Superfluid: theta quantized on fluid vortex cores

  • Superconductor: theta quantized on magnetic flux tubes

They are distinguished by whether the B-sector (EM) is coupled.

  • num_regimes : ℕ Number of topological regimes.

  • maximal_mobility : Bool Common: maximal mobility.

  • zero_bulk_vorticity : Bool Common: zero bulk vorticity.

  • quantized_theta : Bool Common: quantized theta.

  • superfluid_no_em : Bool Superfluid: no EM coupling.

  • superconductor_em : Bool Superconductor: EM coupled.

  • distinguished_by_em : Bool Distinguished by B-sector coupling.

Instances For

Tau.BookIV.ManyBody.instReprTopologicalBranch

source instance Tau.BookIV.ManyBody.instReprTopologicalBranch :Repr TopologicalBranch

Equations

  • Tau.BookIV.ManyBody.instReprTopologicalBranch = { reprPrec := Tau.BookIV.ManyBody.instReprTopologicalBranch.repr }

Tau.BookIV.ManyBody.instReprTopologicalBranch.repr

source def Tau.BookIV.ManyBody.instReprTopologicalBranch.repr :TopologicalBranch → ℕ → Std.Format

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Tau.BookIV.ManyBody.topological_branch

source def Tau.BookIV.ManyBody.topological_branch :TopologicalBranch

Equations

  • Tau.BookIV.ManyBody.topological_branch = { } Instances For

Tau.BookIV.ManyBody.two_topological_regimes

source theorem Tau.BookIV.ManyBody.two_topological_regimes :topological_branch.num_regimes = 2

Tau.BookIV.ManyBody.topological_distinguished_by_em

source theorem Tau.BookIV.ManyBody.topological_distinguished_by_em :topological_branch.distinguished_by_em = true

Tau.BookIV.ManyBody.FiberBaseFactorization

source structure Tau.BookIV.ManyBody.FiberBaseFactorization :Type

[IV.T94] The universal defect functional on tau^3 = tau^1 x_f T^2 factorizes exactly:

delta[omega]{tau^3} = delta[omega]{T^2} tensor delta[omega]_{tau^1}

The fiber component delta[omega]_{T^2} contains all of:

  • Quantum mechanics (Part III)

  • Particle spectrum (Part VI)

  • Electroweak and strong forces (Parts IV-V)

  • Many-body and condensed matter (Part VII)

The base component delta[omega]_{tau^1} contains:

  • Gravity (D-sector)

  • Temporal structure

  • Cosmological evolution

The factorization is exact by axiom K5 (fibered-product structure) and the lemniscate L. The only fiber-base coupling passes through the omega-sector (Kirchhoff junction).

  • tensor_product : Bool Factorizes as tensor product.

  • fiber : String Fiber: T^2 (spatial physics).

  • base : String Base: tau^1 (temporal physics).

  • exact : Bool Exact by K5 + lemniscate.

  • coupling_omega_only : Bool Only coupling: omega-sector.

  • fiber_parts : ℕ Number of fiber parts covered.

Instances For

Tau.BookIV.ManyBody.instReprFiberBaseFactorization.repr

source def Tau.BookIV.ManyBody.instReprFiberBaseFactorization.repr :FiberBaseFactorization → ℕ → Std.Format

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Tau.BookIV.ManyBody.instReprFiberBaseFactorization

source instance Tau.BookIV.ManyBody.instReprFiberBaseFactorization :Repr FiberBaseFactorization

Equations

  • Tau.BookIV.ManyBody.instReprFiberBaseFactorization = { reprPrec := Tau.BookIV.ManyBody.instReprFiberBaseFactorization.repr }

Tau.BookIV.ManyBody.fiber_base_factorization

source def Tau.BookIV.ManyBody.fiber_base_factorization :FiberBaseFactorization

Equations

  • Tau.BookIV.ManyBody.fiber_base_factorization = { } Instances For

Tau.BookIV.ManyBody.factorization_exact

source theorem Tau.BookIV.ManyBody.factorization_exact :fiber_base_factorization.exact = true

Tau.BookIV.ManyBody.coupling_through_omega

source theorem Tau.BookIV.ManyBody.coupling_through_omega :fiber_base_factorization.coupling_omega_only = true

Tau.BookIV.ManyBody.FiberLevelComplete

source structure Tau.BookIV.ManyBody.FiberLevelComplete :Type

[IV.P144] At the close of Part VII, every tau-admissible phenomenon on the fiber T^2 is classified:

  • Single particles (Part VI): quark/lepton spectrum, generations

  • Quantum mechanics (Part III): uncertainty, measurement, energy/entropy

  • Mass derivation (Part III): R = m_n/m_e, 10-link chain

  • Electroweak forces (Part IV): EM, weak, Weinberg mixing, Higgs

  • Strong force (Part V): confinement, mass gap, quarks/gluons

  • Many-body (Part VII): 10 regimes, phase transitions, magnetism, NFL theorem

Nothing on the fiber T^2 remains unclassified. Book V addresses the base tau^1 (gravity, cosmology, temporal structure).

  • all_classified : Bool All fiber phenomena classified.

  • num_parts : ℕ Number of Parts covering fiber.

  • parts : List String Parts: III (QM), IV (EW), V (Strong), VI (Particles), VII (Many-body).

  • book_v_handles_base : Bool Book V handles the base.

  • export_to_book_v : Bool Export contract to Book V.

Instances For

Tau.BookIV.ManyBody.instReprFiberLevelComplete.repr

source def Tau.BookIV.ManyBody.instReprFiberLevelComplete.repr :FiberLevelComplete → ℕ → Std.Format

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Tau.BookIV.ManyBody.instReprFiberLevelComplete

source instance Tau.BookIV.ManyBody.instReprFiberLevelComplete :Repr FiberLevelComplete

Equations

  • Tau.BookIV.ManyBody.instReprFiberLevelComplete = { reprPrec := Tau.BookIV.ManyBody.instReprFiberLevelComplete.repr }

Tau.BookIV.ManyBody.fiber_level_complete

source def Tau.BookIV.ManyBody.fiber_level_complete :FiberLevelComplete

Equations

  • Tau.BookIV.ManyBody.fiber_level_complete = { } Instances For

Tau.BookIV.ManyBody.fiber_all_classified

source theorem Tau.BookIV.ManyBody.fiber_all_classified :fiber_level_complete.all_classified = true

Tau.BookIV.ManyBody.fiber_five_parts

source theorem Tau.BookIV.ManyBody.fiber_five_parts :fiber_level_complete.num_parts = 5

Tau.BookIV.ManyBody.fiber_parts_count

source theorem Tau.BookIV.ManyBody.fiber_parts_count :fiber_level_complete.parts.length = 5

Tau.BookIV.ManyBody.RegimeSummary

source structure Tau.BookIV.ManyBody.RegimeSummary :Type

Summary of the complete regime classification.

  • name : String Regime name.

  • branch : String Branch: non-topological or topological.

  • discriminant : String Key discriminant in defect tuple.

Instances For

Tau.BookIV.ManyBody.instReprRegimeSummary

source instance Tau.BookIV.ManyBody.instReprRegimeSummary :Repr RegimeSummary

Equations

  • Tau.BookIV.ManyBody.instReprRegimeSummary = { reprPrec := Tau.BookIV.ManyBody.instReprRegimeSummary.repr }

Tau.BookIV.ManyBody.instReprRegimeSummary.repr

source def Tau.BookIV.ManyBody.instReprRegimeSummary.repr :RegimeSummary → ℕ → Std.Format

Equations

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

Tau.BookIV.ManyBody.regime_summary_table

source def Tau.BookIV.ManyBody.regime_summary_table :List RegimeSummary

The complete 10-regime classification. Equations

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

Tau.BookIV.ManyBody.ten_regimes_total

source theorem Tau.BookIV.ManyBody.ten_regimes_total :regime_summary_table.length = 10