A polymer with zero energy in every configuration exerts a real tension matching T·dS/dx from pure state counting (Monte Carlo vs exact enumeration, max deviation 4.5×10⁻⁴). Verlinde's holographic-screen composition (Unruh temperature + N = Ac³/Għ bits + equipartition) returns Newton's F = GMm/r² to 2×10⁻¹⁶ — an algebraic identity given its assumptions, recorded as a consistency check, not evidence that nature's gravity is entropic.

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TEST 4: ENTROPIC FORCE -- forces from pure counting
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scipy available: True

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PART A: 1D freely-jointed polymer, N=100, ALL microstates E = 0
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Entropic force F(x) = T dS/dx (finite difference vs analytic):
       x    S(x)=ln Omega         F_fd   F_analytic
     -60        47.730635     0.684055     0.693147
     -40        58.642096     0.418998     0.423649
     -20        64.790562     0.200692     0.202733
      20        64.790562    -0.200692    -0.202733
      40        58.642096    -0.418998    -0.423649
      60        47.730635    -0.684055    -0.693147
  max |F_fd - F_analytic| over |x|<=80: 2.1065e-02
  (force is restoring toward x=0, yet every microstate has E=0:
   the pull is 100% entropy bookkeeping)

Monte-Carlo (Metropolis, seed 42, 200000 sweeps, 20000 burn-in)
vs exact prediction x* = argmax[ S(x) + f*x/T ] (Stirling entropy;
saddle = exact canonical mean here, cross-checked 3 ways):
      f      x*_maxS  N*tanh(f/T)  <x>_discrete       <x>_MC    rel.dev
    0.1       9.9668       9.9668        9.9668       9.9655   1.35e-04
    0.3      29.1313      29.1313       29.1313      29.1198   3.93e-04
    0.5      46.2117      46.2117       46.2117      46.1908   4.52e-04
    1.0      76.1594      76.1594       76.1594      76.1290   4.00e-04
  cross-check: max rel diff, saddle vs exact discrete sum: 1.98e-07
  PART A max relative deviation MC vs exact: 4.5164e-04
  (residual is MC statistical noise; the correlated-sample
   standard error at these settings is of the same order --
   no sigma is quoted, per project honesty rules)
  Force balance at x*: lattice-FD entropic force vs applied -f:
    f=0.1:  T*dS/dx(x*) = -0.099014   (want -0.100000)
    f=0.3:  T*dS/dx(x*) = -0.296918   (want -0.300000)
    f=0.5:  T*dS/dx(x*) = -0.494297   (want -0.500000)
    f=1.0:  T*dS/dx(x*) = -0.982806   (want -1.000000)
  max |T*dS/dx + f| at equilibrium: 1.7194e-02
  (residual here is O(1/N) lattice discreteness of the FD route,
   consistent with the FD-vs-analytic deviation above)

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PART B: Verlinde composition -> Newton  (ALGEBRAIC IDENTITY,
        NOT EVIDENCE that gravity is entropic)
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Assumptions composed: (1) holographic screen with N = A c^3/(G hbar)
bits on A = 4 pi r^2; (2) equipartition E = (1/2) N kB T with
E = M c^2; (3) Verlinde displacement entropy dS/dx = 2 pi kB m c/hbar.
Then F = T dS/dx.  Ratio to Newton F_N = G M m / r^2:

        M [kg]       m [kg]        r [m]      F/(GMm/r^2)   |ratio-1|
     5.072e+37    1.038e-02    7.568e+18   1.000000000000   3.331e-16
     1.778e+36    2.334e-22    4.309e+20   1.000000000000   1.110e-16
     2.895e+37    6.392e+17    8.350e+07   1.000000000000   0.000e+00
     3.609e+31    1.366e-06    7.970e+19   1.000000000000   2.220e-16
     1.712e+35    7.896e+19    4.479e+12   1.000000000000   2.220e-16

  PART B max |F/(GMm/r^2) - 1| over 5 draws: 3.331e-16
  (exactly 1 to machine rounding, BY CONSTRUCTION: the three
   assumptions algebraically cancel to G M m / r^2. This is a
   consistency identity -- which is precisely why Verlinde's
   proposal is hard to test -- not evidence about nature.)

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HEADLINES
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partA_max_rel_dev_MC_vs_exact   : 4.5164e-04
partA_FD_force_vs_analytic      : 2.1065e-02
partA_force_balance_residual    : 1.7194e-02
partB_ratio_first_draw          : 1.000000000000
partB_max_abs_ratio_minus_1     : 3.331e-16

STATUS: reproduction of known results (polymer entropic elasticity;
Verlinde 2011 algebra). Part A: a real, measurable force can be
nothing but entropy bookkeeping -- established physics. Part B: the
Verlinde composition returns Newton EXACTLY given its assumptions;
it validates the machinery of the emergent picture, it does NOT
show that gravity in nature works this way.

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