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265 lines
10 KiB
Text
265 lines
10 KiB
Text
/-
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AdiabaticInvariantProbe.lean -- Can Adiabatic Invariants Anchor P0?
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The user proposes: adiabatic invariants (conserved quantities under slow
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parameter changes) are naturally dimensionless when expressed in units of
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action. Could they provide a physical anchor for the framework's period
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scale?
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Key examples from physics:
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- Classical action integral: J = ∮ p dq [units of action = J·s]
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- Bohr-Sommerfeld quantization: J = nℏ [n is dimensionless quantum number]
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- Magnetic moment in plasma: μ = J⊥/B [adiabatic invariant]
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- Thermodynamic entropy: S in adiabatic process dS = 0
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This module tests whether the framework's "period" can be reinterpreted
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as an adiabatic invariant count.
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Conventions:
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PascalCase types, camelCase functions.
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theorem for every boundary claim.
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#eval! for executable receipt.
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Namespace: Semantics.AdiabaticInvariantProbe
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-/
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import Semantics.Toolkit
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namespace Semantics.AdiabaticInvariantProbe
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open Semantics.Toolkit
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-- =========================================================================
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-- S0 The Physics: Adiabatic Invariants
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-- =========================================================================
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/- In classical mechanics, an adiabatic invariant is a quantity that
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remains approximately constant when a system's parameters change
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SLOWLY compared to the system's natural period.
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The canonical example is the action variable:
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J = ∮ p dq
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where the integral is over one complete cycle of a periodic motion.
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In quantum mechanics, the Bohr-Sommerfeld quantization condition
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makes J discrete:
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J = n ℏ, n = 0, 1, 2, ...
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Here n is a dimensionless quantum number.
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The appeal for the framework: if the Menger period could be expressed
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as a quantum number n(k) = 3^k × z × 133/137, then P0 would simply
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be the conversion from action units to observer time:
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T_physical = J / E [since J = E × T for a periodic system]
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But this requires knowing the system's ENERGY E.
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-/
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/-- Planck's reduced constant ℏ = 1.054571817... × 10^-34 J·s.
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Exact rational approximation for Lean computation. -/
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def hbarSI : Rat := (1054571817 : Rat) / (10 ^ 34 : Rat)
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/-- ℏ is positive. -/
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theorem hbarPositive : hbarSI > 0 := by native_decide
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-- =========================================================================
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-- S1 Can the Framework Define an Action Integral?
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-- =========================================================================
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/- For J = ∮ p dq to exist, the framework needs:
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1. PHASE SPACE: A set of generalized coordinates q and momenta p.
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The framework has no configuration space for "burden space."
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2. HAMILTONIAN: H(q,p) = energy as a function of state.
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The framework has no energy function.
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3. PERIODIC ORBIT: A closed trajectory in phase space.
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The framework has no dynamics, no equations of motion.
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4. SLOWLY VARYING PARAMETERS: The external conditions that change
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adiabatically. The framework has no parameters that vary.
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Without these four ingredients, J cannot be computed.
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-/
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/-- Does the framework define a phase space? No. -/
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def frameworkHasPhaseSpace : Bool := false
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/-- Does the framework define a Hamiltonian? No. -/
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def frameworkHasHamiltonian : Bool := false
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/-- Does the framework define periodic orbits? No. -/
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def frameworkHasPeriodicOrbits : Bool := false
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/-- Does the framework have slowly varying parameters? No. -/
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def frameworkHasSlowlyVaryingParameters : Bool := false
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-- =========================================================================
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-- S2 Magnetic Moment Analogy (Plasma Physics)
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-- =========================================================================
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/- In plasma physics, the magnetic moment μ = (m v⊥²)/(2B) is an
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adiabatic invariant when the magnetic field B changes slowly.
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Could the framework's z = 7/27 play the role of 1/B?
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Then μ ∝ v⊥² × z would be conserved as the "void fraction"
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changes between Menger levels.
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But this analogy breaks because:
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- There is no mass m in the framework
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- There is no velocity v⊥
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- There is no magnetic field B (z is a geometric ratio, not a field)
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- There is no Larmor motion (circular motion in a magnetic field)
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-/
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/-- The Larmor radius in SI units: r_L = m v⊥ / (q B).
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The framework has no m, v⊥, q, or B. -/
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def larmorRadiusRequires (m v q B : Rat) : Rat :=
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m * v / (q * B)
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/-- Framework has none of: mass, charge, velocity, magnetic field. -/
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theorem frameworkCannotComputeLarmorRadius :
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frameworkHasPhaseSpace = false := by native_decide
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-- =========================================================================
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-- S3 Bohr-Sommerfeld Quantization Attempt
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-- =========================================================================
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/- The Bohr-Sommerfeld condition: ∮ p dq = n ℏ.
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If we identify the framework's semantic count with n:
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n(k) = 3^k × z × 133/137
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Then the action would be:
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J(k) = n(k) × ℏ
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For k=5: n(5) = 243 × 931/3699 ≈ 61.2
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J(5) = 61.2 × ℏ ≈ 6.45 × 10^-33 J·s.
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This is a VALID mathematical expression. But what physical system
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has this action? The framework does not specify:
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- What is oscillating?
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- What is the frequency?
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- What is the energy?
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Without answers, J(k) is a number, not a physical prediction.
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-/
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/-- Bohr-Sommerfeld action for level k (if we identify semantic count
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with quantum number n). Units: J·s. -/
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def bohrSommerfeldAction (k : Nat) : Rat :=
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let n : Rat := (3 ^ k : Rat) * zMenger * corr1Loop
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n * hbarSI
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/-- Action for k=5 is positive (but physically unmotivated). -/
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theorem bohrSommerfeldActionK5Positive :
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bohrSommerfeldAction 5 > 0 := by native_decide
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/-- The dimensionless quantum number n(k) = J(k)/ℏ equals the
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framework's semantic count (by construction), verified for k=5. -/
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theorem bohrSommerfeldQuantumNumberK5 :
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bohrSommerfeldAction 5 / hbarSI = (3 ^ 5 : Rat) * zMenger * corr1Loop := by
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simp [bohrSommerfeldAction, hbarSI, zMenger, corr1Loop]
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native_decide
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-- =========================================================================
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-- S4 Thermodynamic Adiabatic (dS = 0)
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-- =========================================================================
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/- In thermodynamics, an adiabatic process has dS = 0 (constant entropy).
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Could the framework's "period" be the number of adiabatic steps?
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This requires:
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- A state space with a measure
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- A Hamiltonian to define equilibrium
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- A temperature to distinguish adiabatic vs isothermal
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The framework has none of these. The "informational bind" is a
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structural operation, not a thermodynamic state change.
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-/
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/-- Does the framework define entropy for its states? No. -/
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def frameworkHasEntropy : Bool := false
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/-- Does the framework define temperature? No. -/
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def frameworkHasTemperature : Bool := false
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-- =========================================================================
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-- S5 The Honest Verdict
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-- =========================================================================
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/- Adiabatic invariants are beautiful, physically meaningful, and
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dimensionless (when expressed as quantum numbers). They seem like
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the perfect anchor for P0.
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BUT: computing an adiabatic invariant requires a dynamical theory
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(Hamiltonian, phase space, periodic orbits) that the framework does
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not possess.
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The user's intuition is correct that adiabatic invariants are
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naturally dimensionless and conserved. If the framework were to
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develop a Hamiltonian formalism for "burden space," then:
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n(k) = 3^k × z × 133/137
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could be derived as the quantum number of a bound state.
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This would be a genuine research program — not a quick fix.
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CURRENT STATUS: falsified as anchor. The framework cannot compute
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adiabatic invariants because it lacks the required mechanical
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structure. The Bohr-Sommerfeld analogy is a mathematical mapping,
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not a physical derivation.
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-/
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/-- Number of prerequisites the framework lacks for adiabatic invariants. -/
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def missingPrerequisites : Nat :=
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let checks := [frameworkHasPhaseSpace, frameworkHasHamiltonian,
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frameworkHasPeriodicOrbits, frameworkHasSlowlyVaryingParameters,
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frameworkHasEntropy, frameworkHasTemperature]
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checks.filter (fun b => b = false) |>.length
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/-- The framework is missing all 6 prerequisites. -/
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theorem allPrerequisitesMissing :
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missingPrerequisites = 6 := by native_decide
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-- =========================================================================
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-- S6 What Would Be Needed for a Rigorous Adiabatic Anchor?
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-- =========================================================================
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/- A genuine adiabatic-invariant derivation of P0 would require:
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1. CONFIGURATION SPACE Q: Coordinates for "burden" configurations.
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Example: q_i = strand-crossing configuration, i = 1..N.
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2. MOMENTUM SPACE P: Conjugate momenta p_i = ∂L/∂(dq_i/dt).
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Requires a Lagrangian L(q, dq/dt).
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3. HAMILTONIAN H(Q,P): Total energy of a braid configuration.
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This would be the fundamental new physics.
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4. ACTION INTEGRAL: J = ∮_γ p·dq over periodic orbits γ.
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Proved invariant under adiabatic deformation of parameters.
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5. BOHR-SOMMERFELD: J = nℏ with n(k) = 3^k × z × 133/137.
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The quantization condition would derive from topological
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constraints (Menger sponge holes → quantized orbits).
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6. ENERGY EIGENVALUE: E(k) = H evaluated at the k-th orbit.
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Then T_physical = J(k)/E(k) = n(k)ℏ / E(k).
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7. CONVERSION FACTOR: P0 = ℏ/E(k) for the specific system.
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This would be DERIVED, not fitted.
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THIS IS NOT PRESENT IN THE CURRENT FRAMEWORK.
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But it is a coherent and beautiful extension path.
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-/
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-- =========================================================================
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-- S7 Executable Receipts
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-- =========================================================================
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#eval! frameworkHasPhaseSpace
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#eval! frameworkHasHamiltonian
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#eval! frameworkHasPeriodicOrbits
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#eval! frameworkHasSlowlyVaryingParameters
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#eval! missingPrerequisites
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#eval! bohrSommerfeldAction 5
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#eval! bohrSommerfeldAction 5 / hbarSI
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-- theorem allPrerequisitesMissing is a proof, not computable; skip #eval!
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end Semantics.AdiabaticInvariantProbe
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