A region of empty space has no state of its own: an 8-site block of vacuum has nonzero entropy and is mixed (1 − Tr ρ² = 0.988) while the whole ring is pure, and its mutual information with the complement equals exactly twice its entropy — 100% of an empty region's entropy is correlation with the rest. Proving correlations never vanish at maximum separation required 120-digit arithmetic (3.68×10⁻⁵⁷).

MENT does reveal A -- and still sends
      nothing.  Homodyne x on all 504 sites of B (ideal projective
      measurement of the complement of A in a pure global state).
      m=1e-06:  cond(X_B) = 1.968e+06
               S(A) unconditioned            = 4.922063078 nats
               S(A | outcome of x_B)         = 0.000000000 nats
               max|nu_cond - 1/2|            = 1.311e-12  (1/2 = pure)
               law of total covariance residual
                 max|Cov_cond + Cov_of_means - Cov_A| = 1.071e-08
                 same, divided by max|X_A| = 9.774e+02   -> 1.096e-11
      m=5e-01:  cond(X_B) = 4.123e+00
               S(A) unconditioned            = 0.270931445 nats
               S(A | outcome of x_B)         = 0.000000000 nats
               max|nu_cond - 1/2|            = 4.441e-16  (1/2 = pure)
               law of total covariance residual
                 max|Cov_cond + Cov_of_means - Cov_A| = 5.551e-17
                 same, divided by max|X_A| = 4.370e-01   -> 1.270e-16
      S(A|x_B)=0 is a theorem, not a fit: projectively measuring the
      complement of A in a globally pure state must leave A pure.  The
      residuals above are float64 witnesses of the implementation, and
      the homodyne is the idealised infinite-squeezing limit.
      Reading ALL of B collapses A to a PURE state (S -> 0): the missing
      information really was in the correlations, not lost.  But the
      ENSEMBLE over outcomes reconstructs sigma_A identically, so every
      statistic available inside A alone is unchanged.  Encoded: yes.
      Readable from inside A: no.  Signalling: none.

==============================================================================
HEADLINE NUMBERS
  (every '+/-' anywhere above is an OLS fit standard error = fit quality
   in a disclosed window.  It is NOT an evidence sigma.  Window drift,
   printed next to each fit, is the honest systematic and is larger.)
==============================================================================
  [H1] massless (m=1e-06) FIELD correlator does NOT decay:
       X(255)/X(1) = 0.999176302 ; log-law coefficient of ln sin(pi r/N) =
       -0.159126 vs CFT -1/(2pi) = -0.159155  (ratio 0.99982).
       NEGATIVE RESULT: it is a LOGARITHM, not a power law -- 1+1D is the
       special case.  Slower than any power law, so 'never zero' holds.
  [H2] massless POWER-LAW SLOPE of the zero-mode-invariant correlator
       G(r) = <grad phi grad phi> = -Delta^2 X = P - m^2 X, fitted over the
       disclosed window r in [4,32]:  -1.999030   [continuum: -2 exactly]
       slope movement across 5 windows = 0.121709 (finite-size ring images)
  [H3] massive (m=0.5) EXPONENTIAL RATE, window r in [8,40], prefactor-
       corrected: kappa_fit = 0.494312  vs LATTICE 2 asinh(m/2) = 0.494933
       (uncorrected log-linear fit gives 0.517658 -- the 1/sqrt(r) prefactor)
       correlation length 1/kappa_fit = 2.0230 sites vs 1/m = 2.0
  [H4] FLOAT64 NOISE ONSET, massive X: relative error exceeds 100% at
       r = 72.  Last r with >=2 correct digits: r = 56.  Every value
       printed at r > 56 in float64 is ROUNDOFF, proven by (a) two
       independent float64 routes diverging, (b) an mpmath reference,
       (c) the value tracking the working precision until converged.
       Converged extended-precision X(255) = 3.676757e-57 -- nonzero, and
       ~39 orders BELOW float64's ~1e-17 floor.
       The massless correlator needs none of this: at r=255 it is 5.9924e-06,
       ~10^11 above the float64 floor, certified against mpmath to 1e-8.
  [H5] S(L=8) for a block of EMPTY SPACE (no particles anywhere):
       m=1e-06: 4.922063078 nats    m=0.5: 0.270931445 nats    -- both > 0
       (the m=1e-6 value carries a non-universal ln(1/m) zero-mode piece;
        the sign and the mixedness are what is load-bearing, not the size)
  [H6] MIXEDNESS of that empty region, 1 - Tr rho_A^2:
       m=1e-06: 0.988230179      m=0.5: 0.109659861
       rho_A is MIXED: the region has no state of its own.
  [H7] S(WHOLE RING) = 3.881e-08 (m=1e-06), 9.949e-12 (m=0.5).
       Exact zero BY CONSTRUCTION (4XP=I); residual is a float64 witness
       of correct implementation, NOT evidence strength.
       Mutual information I(A:A^c) = 9.844126155 = 2.0000 x S(A) (m=1e-06):
       100% of the region's entropy is correlation with everything else.
  [H8] NO READOUT: random local symplectic on all of B changes sigma_A by
       0.000e+00 (m=1e-06) / 0.000e+00 (m=0.5) -- exact zero, structurally.
       Homodyning ALL of B drives S(A|x_B) to 0 (the information IS
       there) while the outcome ensemble restores sigma_A exactly.
==============================================================================
INTERPRETATION (honesty rules apply)
==============================================================================
  'Empty' is not 'independent'.

  (a) Vacuum correlations between distant EMPTY regions never vanish.
      Massless: the field correlator does not decay at all (it is a
      logarithm) and its local, zero-mode-invariant part falls as a
      power law r^-2.00, still 10^11 above the numerical floor at the
      ring's maximum separation.  Massive: exponential with a genuine
      correlation length, and by r~70 the float64 answer is pure
      roundoff -- yet extended precision shows the true value is ~1e-57,
      not zero.  BOTH HALVES MATTER: never exactly zero, and past a few
      correlation lengths never remotely measurable either.

  (b) Every region of the vacuum is a MIXED half of a PURE whole.
      S(L=8) > 0 for a patch containing no particles, while S of the
      entire ring is zero by construction.  A part is mixed, the whole
      is pure, and the whole of the part's entropy is mutual information
      with the complement.  The region therefore has no state of its own:
      its 'missing' information is in correlations with everything else.
      That is the precise, checkable content of 'the vacuum is one
      entangled object'.

  (d) None of this reads out or signals.  Anything done inside B leaves
      rho_A bit-identical; measuring the entire complement purifies A
      conditionally but reconstructs rho_A on the ensemble.  Information
      being ENCODED in vacuum correlations is not the same as its being
      ACCESSIBLE, and the numbers above separate the two cleanly.

  WHAT THIS IS NOT.  Nothing here is new.  It is Reeh-Schlieder /
  Summers-Werner / Srednicki / no-communication, recomputed on a 512-site
  free scalar ring to check that this project's machinery reproduces them
  and to find out where its arithmetic quits.  Two honest failures are on
  the record: the m=1e-6 zero mode costs ~12 of 16 digits on the absolute
  correlator (only its r-dependence survives), and the massless field
  correlator is NOT the power law the framing expected.  No statement
  above extends beyond a free scalar field on a lattice.
==============================================================================

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