Constants Hbar Eq Phi Inv Fifth

/-- Native action quantum in RS-native units: `hbar = E_coh · tau0 = φ⁻⁵ · 1`.

## ATTACKER BREADCRUMB (read before declaring ℏ = φ⁻⁵ "true by definition")

The lemma `hbar_eq_phi_inv_fifth` below closes by `unfold; simp`. That is
intentional: this file *defines* the RS-native action unit. Two separate
questions then arise, and they have different answers; do not collapse them.

* Is the EXPONENT `5` forced, or a free choice? Forced, modulo one modeling
  step. The coherence energy carries one factor of `φ⁻¹` per configuration
  degree of freedom of a recognition event, and a recognition event has
  `D + 2` such degrees: `D` spatial (lattice, T8), `1` temporal (tick advance,
  T2), `1` balance (ledger neutrality `J(x)=J(x⁻¹)`, T3). With `D = 3` forced
  by T8 this gives `configDim = 5`, hence `E_coh = φ^(-(D+2)) = φ⁻⁵`. This is
  boundary item B-22, proved in `Foundation/GapDerivation.lean`
  (`configDim_at_D3`, `E_coh_gap_eq`, `Gap45Cert.ecoh`). The link back to THIS
  constant is machine-checked there:
  `GapDerivation.Constants_E_coh_eq_configDim` and
  `GapDerivation.hbar_exponent_eq_configDim` prove
  `E_coh = hbar = φ^(-(configDim D))`. The forced content is the count
  `D + 2`; the only modeling input is the `φ⁻¹`-per-dof rule. So the honest tag
  for the exponent is derived-modulo-one-modeling-step, NOT pure unit choice.
* Is the SI VALUE of `ℏ` (in J·s) predicted? No. A pure-number theory cannot
  output an absolute dimensionful SI constant without a dimensional anchor:
  see `Constants/NativeDimensionalBoundary.no_nontrivial_dimensionless_monomial`.

So "true by definition" is correct only at the level of native units (one tick
= the time unit ⟹ the native action quantum is `φ⁻⁵` as a pure number). The
substantive, non-definitional content is that the exponent equals the forced
configuration dimension `D + 2 = 5`.

## What the SI calibration looks like

Mapping `hbar_RS = φ⁻⁵` to SI units requires a dimensional anchor.  The
conversion is uniquely determined once the anchor is supplied
(`Foundation/SIBridgeClosure.lean`, `Verification/FirstPrinciplesToSI.lean`,
`Measurement/RSNative/Calibration/SingleAnchor.lean`); the boundary theorem
explaining why an anchor is required lives in
`Constants/NativeDimensionalBoundary.lean`. -/
noncomputable def hbar : ℝ := cLagLock * tau0

/-- Canonical locked C_lag constant: C_lock = φ^{−5}. -/
@[simp] noncomputable def cLagLock : ℝ := phi ^ (-(5 : ℝ))

/-- The fundamental time unit τ₀ (duration of one tick) in RS-native units. -/
@[simp] noncomputable def tau0 : ℝ := tick

/-- **THEOREM C-004.5**: Bounds on ℏ from φ bounds.

    With φ ∈ (1.61, 1.62), we get ℏ ∈ (0.088, 0.093). -/
theorem hbar_bounds : (0.088 : ℝ) < hbar ∧ hbar < (0.093 : ℝ) := by
  rw [hbar_eq_phi_inv_fifth]
  have h1 : (1.61 : ℝ) < phi := phi_gt_onePointSixOne
  have h2 : phi < (1.62 : ℝ) := phi_lt_onePointSixTwo
  -- We want 0.088 < φ^(-5) < 0.093
  -- Since hbar = 1/φ^5, we need bounds on φ^5
  -- Lower bound: φ < 1.62, so φ^5 < 1.62^5, so 1/φ^5 > 1/1.62^5
  -- Upper bound: φ > 1.61, so φ^5 > 1.61^5, so 1/φ^5 < 1/1.61^5
  have h_phi5_lower : phi ^ (5 : ℝ) > (1.61 : ℝ) ^ (5 : ℝ) := by
    apply Real.rpow_lt_rpow
    · linarith
    · linarith
    · norm_num
  have h_phi5_upper : phi ^ (5 : ℝ) < (1.62 : ℝ) ^ (5 : ℝ) := by
    apply Real.rpow_lt_rpow
    · linarith
    · linarith
    · norm_num
  -- Convert to hbar = φ^(-5) bounds
  have hbar_lower : phi ^ (-(5 : ℝ)) > (0.088 : ℝ) := by
    have h_inv : phi ^ (-(5 : ℝ)) = 1 / (phi ^ (5 : ℝ)) := by
      rw [show (-(5 : ℝ)) = - (5 : ℝ) by norm_num]
      rw [Real.rpow_neg]
      · ring
      · exact le_of_lt phi_pos
    rw [h_inv]
    -- Since φ^5 < 1.62^5, we have 1/φ^5 > 1/1.62^5
    -- Compute 1.62^5 = 11.158... and 1/11.158 ≈ 0.0896 > 0.088
    have h_div : 1 / (phi ^ (5 : ℝ)) > 1 / ((1.62 : ℝ) ^ (5 : ℝ)) := by
      apply (one_div_lt_one_div (by positivity) (by positivity)).mpr
      linarith [h_phi5_upper]
    have h_numeric : 1 / ((1.62 : ℝ) ^ (5 : ℝ)) > (0.088 : ℝ) := by
      rw [show (5 : ℝ) = (5 : ℕ) by norm_num, Real.rpow_natCast]
      norm_num
    linarith
  have hbar_upper : phi ^ (-(5 : ℝ)) < (0.093 : ℝ) := by
    have h_inv : phi ^ (-(5 : ℝ)) = 1 / (phi ^ (5 : ℝ)) := by
      rw [show (-(5 : ℝ)) = - (5 : ℝ) by norm_num]
      rw [Real.rpow_neg]
      · ring
      · exact le_of_lt phi_pos
    rw [h_inv]
    -- Since φ^5 > 1.61^5, we have 1/φ^5 < 1/1.61^5
    -- Compute 1.61^5 = 10.817... and 1/10.817 ≈ 0.0924 < 0.093
    have h_div : 1 / (phi ^ (5 : ℝ)) < 1 / ((1.61 : ℝ) ^ (5 : ℝ)) := by
      apply (div_lt_div_iff₀ (by positivity) (by positivity)).mpr
      linarith [h_phi5_lower]
    have h_numeric : 1 / ((1.61 : ℝ) ^ (5 : ℝ)) < (0.093 : ℝ) := by
      rw [show (5 : ℝ) = (5 : ℕ) by norm_num, Real.rpow_natCast]
      norm_num
    linarith
  exact ⟨hbar_lower, hbar_upper⟩

/-- **THEOREM C-004.2**: ℏ is positive (required for quantum dynamics). -/
theorem hbar_positive : hbar > 0 := hbar_pos

/-- **THEOREM C-004.3**: ℏ < 1 (the action quantum is small compared to natural units).

    Proof: φ > 1 ⟹ φ⁵ > 1 ⟹ φ⁻⁵ < 1. -/
theorem hbar_lt_one : hbar < 1 := by
  rw [hbar_eq_phi_inv_fifth]
  have h1 : phi ^ (5 : ℝ) > 1 := by
    have hphi : phi > 1 := one_lt_phi
    have hexp : (5 : ℝ) > 0 := by norm_num
    have h1_lt : (1 : ℝ) < phi ^ (5 : ℝ) := by
      rw [← Real.one_rpow (5 : ℝ)]
      apply Real.rpow_lt_rpow
      · norm_num
      · linarith
      · norm_num
    linarith
  have h2 : phi ^ (-(5 : ℝ)) = 1 / (phi ^ (5 : ℝ)) := by
    rw [show (-(5 : ℝ)) = - (5 : ℝ) by norm_num]
    rw [Real.rpow_neg]
    · ring
    · exact le_of_lt phi_pos
  rw [h2]
  have h3 : phi ^ (5 : ℝ) > 0 := by positivity
  apply (div_lt_iff₀ h3).mpr
  linarith