A globally static quantum state contains ordinary time evolution for internal observers: the solver-found null state of a clock+qubit constraint is stationary to machine precision (⟨H²⟩ ≈ 6×10⁻³⁰, global qubit maximally mixed), yet conditioned on the clock the qubit Rabi-oscillates at full amplitude, matching Schrödinger evolution to 4×10⁻¹⁶ (Page–Wootters 1983, recomputed). Time as clock–system entanglement in a toy model — not a claim about nature's time.

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TEST 2: TIME FROM ENTANGLEMENT (Page-Wootters 1983) -- REPRODUCED
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Parameters:
  clock dimension D            = 64
  clock tick dt                = 1.000000
  clock quantum w=2pi/(D*dt)   = 0.098174770425
  qubit Omega = 4*w           = 0.392699081699
  commensurability Omega*D*dt/(2*pi) = 4.000000000000  (exact integer =>
    constraint exactly solvable; DISCLOSED as a deliberate choice)
Clock sanity checks:
  max |F^dag F - I|                        = 2.742420e-15
  max |e^(-i H_c dt) - cyclic shift|       = 4.671668e-15
Total constraint operator H_tot: shape (128, 128), Hermitian by construction
Diagonalization of H_tot (numpy.linalg.eigh):
  4 eigenvalues closest to 0: 3.296341e-17  3.625976e-16  9.817477e-02  -9.817477e-02
  null-space dimension (|E| < 1e-8)        = 2
  smallest NON-null |eigenvalue| (gap)     = 9.817477e-02
  (2-fold degeneracy = one timeless solution per qubit initial
   condition; a 1-D null space would force a stationary qubit)
  |P_null (|t_0> x |0>)| (gauge fixing)    = 1.250000e-01  (analytic 1/sqrt(D) = 1.250000e-01)
------------------------------------------------------------------------
(a) GLOBAL STATICITY of the found state |Psi>:
  <Psi| H_tot   |Psi>  = 2.681557e-17
  <Psi| H_tot^2 |Psi>  = 8.588121e-30
  => e^{-i H_tot T}|Psi> = |Psi| for all T: no global evolution.
  global reduced qubit state (clock traced out):
    <sigma_x> = -1.205501e-15  <sigma_y> = 6.021600e-15  <sigma_z> = 9.992007e-16
    |global Bloch vector| = 6.221841e-15  (maximally mixed: globally, 'nothing happens')
------------------------------------------------------------------------
(b) CONDITIONAL DYNAMICS vs exact Schrodinger evolution
    (project clock on |t_k>, normalize; compare to e^{-i H_s t_k}|psi_0>):
  max infidelity  1-|<cond|exact>|^2  over 64 readings = 2.220446e-16
  max trace distance                  over 64 readings = 1.490116e-08
  max |P(t_k) - 1/D| (clock-reading uniformity)        = 1.543904e-16
------------------------------------------------------------------------
(c) CONDITIONAL <sigma_z>(t_k) sweep (full table, 8 values/row):
  k= 0.. 7:  +1.000000  +0.923880  +0.707107  +0.382683  +0.000000  -0.382683  -0.707107  -0.923880
  k= 8..15:  -1.000000  -0.923880  -0.707107  -0.382683  -0.000000  +0.382683  +0.707107  +0.923880
  k=16..23:  +1.000000  +0.923880  +0.707107  +0.382683  +0.000000  -0.382683  -0.707107  -0.923880
  k=24..31:  -1.000000  -0.923880  -0.707107  -0.382683  +0.000000  +0.382683  +0.707107  +0.923880
  k=32..39:  +1.000000  +0.923880  +0.707107  +0.382683  -0.000000  -0.382683  -0.707107  -0.923880
  k=40..47:  -1.000000  -0.923880  -0.707107  -0.382683  +0.000000  +0.382683  +0.707107  +0.923880
  k=48..55:  +1.000000  +0.923880  +0.707107  +0.382683  -0.000000  -0.382683  -0.707107  -0.923880
  k=56..63:  -1.000000  -0.923880  -0.707107  -0.382683  +0.000000  +0.382683  +0.707107  +0.923880
  expected cos(Omega*t_k): max |z_k - cos(Omega t_k)| = 1.782650e-14
  Rabi oscillation amplitude (max-min)/2 = 1.000000e+00
  full periods across one clock cycle    = 4
========================================================================
HEADLINES:
  <H_tot>   of found state          = 2.681557e-17
  <H_tot^2> of found state          = 8.588121e-30
  max infidelity vs Schrodinger     = 2.220446e-16
  conditional Rabi amplitude        = 1.000000e+00
  global Bloch vector length        = 6.221841e-15
========================================================================
VERDICT: MECHANISM REPRODUCED. A single globally static state
(<H_tot^2> = 8.588121e-30) contains, via clock-system entanglement, an
internal qubit whose conditional state executes full-amplitude Rabi
oscillation matching ordinary Schrodinger evolution to machine
precision. Time as entanglement bookkeeping WORKS in this toy model.
This does NOT show that time in our universe arises this way.

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