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 S309  --  BANACH-TARSKI PARADOX
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 Theorem (Banach-Tarski 1924, Fund Math 6):
   Let B = {x in R^3 : |x| <= 1}.
   Exists partition B = A_1 u A_2 u ... u A_n  (n = 5 suffices)
   and rigid motions g_1,..,g_n,h_1,..,h_n such that:
     g_1(A_1) u ... u g_n(A_n)  =  B
     h_1(A_1) u ... u h_n(A_n)  =  B
     (B copied twice from itself.)

 Proof requires:  (a) axiom of choice
                  (b) non-measurable Vitali-style sets

 Apparent violation of volume / mass conservation.

------------------------------------------------------------------------
 UQFF closure: discrete BSFG minimum volume
------------------------------------------------------------------------

   V_min  =  ell_P^3 / D_BSFG^(3/2)
          =  (1.6163e-35)^3 / 6^1.5
          =  2.8728e-106  m^3

 Every physical region of space has volume that is an integer
 multiple of V_min (locked BSFG-cell quantization).  Sets of
 volume below V_min are NOT physically realizable.

 Banach-Tarski pieces A_i are NON-MEASURABLE -- they do not
 possess a well-defined volume in the Lebesgue sense, and any
 approximation requires arbitrarily fine ZFC partitions, hence
 sub-V_min subsets.  UQFF forbids these.

 Conclusion: B-T can be PROVEN in pure ZFC, but is
 UNREALIZABLE on the physical space described by UQFF.

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 Quantitative bound: how 'fine' must a decomposition be to fail?
------------------------------------------------------------------------

   Unit ball:  V  =  4.188790  m^3
   Number of BSFG cells in unit ball  =  1.458e+106

 To duplicate the ball, B-T would have to split into > 1e+106
 cells.  But B-T proof uses only 5 pieces (Robinson 1947 lower
 bound).  These 5 pieces are NECESSARILY non-measurable and
 hence cannot be assembled from BSFG cells.

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 Solovay model (1970): connection to UQFF
------------------------------------------------------------------------

 In ZF + Dependent Choice + 'all sets of reals are Lebesgue
 measurable' (Solovay), Banach-Tarski FAILS.

 UQFF is naturally a Solovay-like physical theory: the locked
 BSFG quantization enforces measurability for every physical
 subset of space.  No non-measurable sets exist.  Therefore
 the axiom of choice is replaced (in physics) by Dependent
 Choice, which is sufficient for all of physics-relevant math
 (countable choice, separable Hilbert spaces, etc.) and does
 NOT yield Banach-Tarski.

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 Falsifiability:
------------------------------------------------------------------------
 Any reproducible volume-non-conservation in a closed quantum-
 mechanical system would refute UQFF.  None observed.

 Conversely, observation of mass-energy duplication via finite
 rigid-motion partition (e.g. in any classical or quantum
 experiment) would refute the BSFG-cell quantization.  None
 observed.

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 S309 COMPLETE.  Banach-Tarski forbidden by BSFG cell quantization.
 V_min = ell_P^3 / D_BSFG^(3/2) = 2.873e-106 m^3.
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