NEGATIVE RESULT bounding a popular claim: in a finite spin chain, operations confined to k sites reach exactly 4^k dimensions — the full local-algebra bound, versus only 2^k on an unentangled product state — but that is merely 0.1–1.6% of the full 2^N space. A small patch does NOT reach the whole space in finite dimensions; full cyclicity needs k = N/2, where float64 fails outright (3264 of 4096).

 The limit is the
     INPUT: |psi> is known only to 6.9e-13, which is 17x
     LARGER than the singular value in question.  Extra
     precision downstream cannot rescue k=6; only a more
     accurate eigenvector could.  Honest verdict: unresolved.

(5c) THE REACH IS A RANK, SO IT IS DISCONTINUOUS -- AND THAT IS
     WHERE A TOLERANCE CAN MANUFACTURE AN ANSWER.
     State: sqrt(1-eps^2)|up..up> + eps|generic>, k = 2.
     Exact arithmetic gives reach = 4^2 = 16 for EVERY eps > 0,
     and reach = 4 = 2^k only at eps = 0 exactly.

       eps     S(A) nats   M s_min    reach@1e-9  reach@1e-13
--------------------------------------------------------------------------
     1e-02   8.752e-04   9.701e-03       16          16
     1e-04   1.571e-07   9.701e-05       16          16
     1e-06   2.266e-11   9.701e-07       16          16
     1e-08   3.108e-15   9.701e-09       16          16
     1e-10   3.582e-19   9.701e-11        4          16
     1e-12   4.278e-23   9.701e-13        4          16
     1e-14   4.973e-27   9.624e-15        4           4
--------------------------------------------------------------------------
     Independent sighting of the floor: s_min tracks 0.9701*eps
     exactly from 1e-2 down to 1e-12, then deviates (9.585e-15
     where the pattern predicts 9.701e-15, ~1% off).  That
     deviation is arithmetic entering the singular value itself,
     before any rank decision is made.

     The collapse 16 -> 4 happens at eps ~ tolerance, and it MOVES
     when the tolerance moves: that collapse is arithmetic, not a
     change in the state.  Two consequences, both honest:
     (i) any reach number is only as good as its disclosed
     tolerance and the margin over the numerical floor -- which is
     why Sections 1-4 print both;  (ii) a state 1e-8 away from a
     product state has the SAME reach (16) as the critical vacuum,
     with entanglement entropy ~1e-15 nats.  'A local patch
     reaches far' is therefore a statement about full Schmidt
     rank, which is generic, and NOT a strong statement about
     the vacuum being specially entangled.

==========================================================================
SECTION 6 -- CODA: THE REACH CARRIES NO SIGNAL (piece (d))
==========================================================================
  For all 4^3 = 64 Pauli strings P on the k=3 patch, compare the
  reduced state of the 9-site COMPLEMENT before and after:
    max trace distance || rho_comp(P psi) - rho_comp(psi) ||_1/2
      = 4.420e-16   (worst string: XXI)
  That is an exact algebraic identity (P unitary, traced out), so
  the number is a machine-precision code check -- NOT evidence.
  Meanwhile the GLOBAL state moved as far as it possibly could:
  the smallest overlap |<psi|P psi>| over non-identity strings is
  4.502e-18 (string YZI) -- P|psi> is ORTHOGONAL to |psi>.
  So an operation on 3 sites can send the global state to a state
  orthogonal to the vacuum, moving it through a 64-dimensional
  subspace, while leaving every observable on the other 9 sites
  EXACTLY unchanged.  Reach is not readout.  (Reproduction of the
  no-communication theorem, not a new result.)
  Caveat in the same breath: a general element of A(A) is not
  unitary, so realising an arbitrary vector in the reach means
  local measurement plus post-selection, whose success
  probability can be arbitrarily small.  The reach is the RANGE
  of an algebra, not a set of states preparable on demand.

==========================================================================
HEADLINE NUMBERS
==========================================================================
  tolerance (disclosed, primary)      = 1e-09
  rigorous bound on ||delta psi||     = 6.859e-13 (residual/gap)
  k=1: reach(critical) = 4 = 4^k    (4^k=4, 2^N=4096, reach/2^N=0.00098)
       reach(product)  = 2 = 2^k;  S(A) crit = 0.472573 nats, prod = 0.0e+00
       crit s_min = 6.0130e-01 (4.4e+11 above the M floor 1.4e-12; nothing discarded)
       prod s_min = 1.4142e+00 with 2 discarded singular values all exactly 0.0
  k=2: reach(critical) = 16 = 4^k    (4^k=16, 2^N=4096, reach/2^N=0.00391)
       reach(product)  = 4 = 2^k;  S(A) crit = 0.586180 nats, prod = 0.0e+00
       crit s_min = 9.6245e-02 (3.5e+10 above the M floor 2.7e-12; nothing discarded)
       prod s_min = 2.0000e+00 with 12 discarded singular values all exactly 0.0
  k=3: reach(critical) = 64 = 4^k    (4^k=64, 2^N=4096, reach/2^N=0.01562)
       reach(product)  = 8 = 2^k;  S(A) crit = 0.644338 nats, prod = 0.0e+00
       crit s_min = 3.3758e-03 (6.2e+08 above the M floor 5.5e-12; nothing discarded)
       prod s_min = 2.8284e+00 with 56 discarded singular values all exactly 0.0
  reach ratio to 4^k: critical = 1.000 for k=1,2,3 (saturates the
       trivial bound);  product = 2^-k = 0.500, 0.250, 0.125
  GHZ control reach   = 2*2^k = 4, 8, 16 (rank, not entropy)
  k=6 (=N/2) cyclicity: exact 4096, measured 3264 -> UNRESOLVED
  no-signaling on the complement: max trace distance = 4.420e-16
==========================================================================
WHAT THIS IS
  * A finite-dimensional demonstration of the FLAVOUR of
    Reeh-Schlieder: on an entangled vacuum-like state, operations
    confined to k sites move the global state through 4^k
    independent directions -- 2^k times more than the patch's own
    2^k dimensions -- and the extra directions live in the
    complement.  On an unentangled state the reach collapses to
    exactly 2^k: the patch is stuck inside itself.
  * Reproduced textbook structure (Reeh-Schlieder 1961; the
    cyclic/separating criterion for matrix algebras; the
    no-communication theorem).  Machinery check.  Not discovery.
WHAT THIS IS NOT
  * NOT a proof of the Reeh-Schlieder theorem.  That theorem is
    about QFT: infinite-dimensional local algebras on arbitrarily
    small open regions, giving a subspace DENSE in the whole
    Hilbert space.  Here the local algebra is finite (4^k
    elements) and the bound dim R_k <= 4^k is HARD.
  * So a small patch of this chain provably CANNOT reach the whole
    space: k=1 reaches 0.10% of 2^N, k=3 reaches 1.56%, and
    reaching everything needs k >= N/2 -- half the universe, not a
    small patch.  The 'arbitrarily small region' part of
    Reeh-Schlieder is exactly the part a finite chain cannot show.
  * NOT a statement that the vacuum is special: Section 5c shows a
    state 1e-8 from a product state has the same reach.  Full
    Schmidt rank is generic; that is all the reach measures.
  * NOT any claim about signalling or readout: Section 6 shows the
    complement's state is exactly invariant.  Encoded is not
    accessible.
  * The k=6 full-space test is UNRESOLVED in float64 and is
    reported as a failure, with the cause localised (Section 5b).
STATUS: reproduced-not-discovered; validates machinery and
        structure within a 12-spin model, not nature.
==========================================================================

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