Gravity Master Theorem Bmv Positive Unconditional Proven

The entropy guarantee

In quantum information, the von Neumann entropy of a mixed state measures how much information is hidden when a system is observed as part of a larger whole. For a pure two-qubit state, the entropy of one qubit's reduced density matrix is never negative. The declaration bmv_positive_unconditional_proven establishes a precise, unconditional version of this fact: for any 2 by 2 complex matrix A whose squared entries sum to one, if the concurrence (a standard measure of entanglement) is positive, then the reduced density matrix von Neumann entropy is strictly greater than zero.

This is not a new physical law. It is a formal theorem in the framework's machine-checked library of formal theorems, proved with no unproved assumptions (zero sorry) and no framework-specific axioms. The theorem is named for the ledger, the framework's discrete record of recognition events, but its content is a standard result about quantum states. The proof relies on the library's existing theorem pure_two_qubit_entropy_positive_unconditional, which carries the same statement for pure two-qubit systems.

In the framework's larger project, this theorem serves as one of twelve clauses in a proposed master statement for quantum gravity. Eight of those twelve clauses are already closed, meaning they are proved. The entropy clause is one of them. The remaining five clauses, including the continuum limit of Regge calculus and the derivation of the Page curve, are still open hypotheses. The master theorem rs_quantum_gravity_master_conditional assembles all twelve clauses, but it is conditional: it holds only if the five open hypotheses are eventually discharged.

What the declaration does not claim is as important as what it proves. It does not claim that the full quantum gravity discovery has been made. The master statement is conditional, not unconditional. The entropy theorem alone does not imply anything about gravity, spacetime, or the framework's other constants. It is a single, isolated result about quantum information, and its role in the master theorem is structural: it is one brick in a wall that is still under construction.

theorem bmv_positive_unconditional_proven : bmv_positive_unconditional :=
  Quantum.PureTwoQubit.EntropyConcurrence.pure_two_qubit_entropy_positive_unconditional

theorem bmv_positive_unconditional_proven : bmv_positive_unconditional :=
  Quantum.PureTwoQubit.EntropyConcurrence.pure_two_qubit_entropy_positive_unconditional

/-- **MASTER THEOREM (conditional form, Track 7.A statement-authoring
closure).** Under the five hypothesis inputs corresponding to the still-
open tracks (1.B/1.C, 2.C/2.D unconditional, 3.C, 6.B, 6.C), the master
statement holds with the eight CLOSED clauses discharged from existing
Lean theorems (Sessions 89–96 anchors). -/
theorem rs_quantum_gravity_master_conditional
    (H_d2 : RegEHContinuumAndBianchi)
    (H_amp : AmplitudeLinearForcedUnconditional)
    (H_page : PageCurveDerived)
    (H_pta : PTAStochasticGWDistinctFromInflation)
    (H_strong : StrongFieldTestsDistinctFromGR) :
    RSQuantumGravityMaster H_d2 H_amp H_page H_pta H_strong := by
  refine ⟨?d1, ?d2, ?d3, ?d4, ?d5, ?d6⟩
  case d1 =>
    exact ⟨T0_T8_holds_proven, CostUniqueness_proven, Lorentzian_1_3_proven⟩
  case d2 =>
    exact ⟨H_d2.regge_holds, H_d2.bianchi_holds⟩
  case d3 =>
    exact ⟨H_amp.holds, bmv_positive_unconditional_proven⟩
  case d4 =>
    exact ⟨hawking_temperature_SI_proven,
            c_RS_observable_distinct_proven,
            H_page.holds,
            omega_lambda_from_phi_proven⟩
  case d5 =>
    exact ⟨rs_qnm_distinct_LQG_string_proven,
            H_pta.holds,
            H_strong.holds⟩
  case d6 =>
    exact gravity_sector_zero_free_parameters_proven

What this page does not claim

The declaration does not prove the full quantum gravity master statement. The theorem does not imply anything about gravity, spacetime, or the framework's other constants. The entropy result is not a new physical law; it is a formal theorem about standard quantum information.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/MasterTheorem.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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