Geometric Entropy and the Multiplicative Collapse of Natural Numbers

This  paper  explains  a  two-phase  geometric  entropy  model  for  natural  numbers  in which  each  integer  is  characterised  by  the  tension  between  its  additive  stacking geometry  and  its  multiplicative  collapse  geometry.  In  the  additive  phase,  n  is represented by the most compact rectangular stacking of unit blocks, yielding a near-square  configuration  whose  deviation  from  perfect  symmetry  we  call  the  stacking asymmetry S ₁ (n). In the multiplicative phase, n collapses to the rectangle of smallest perimeter among all integer-sided rectangles of area n, with collapse perimeter P ₂ (n). The two-phase entropy is defined as E(n) = S ₁ (n) + |P ₁ (n) − P ₂ (n)| / n. 
We state that E(n) attains local maxima at prime numbers, that the collapse perimeter P ₂ (p) = 2(p+1) is maximal for primes (which admit only the trivial rectangle 1 × p), and that the stacking asymmetry vanishes for even n and equals 1 for odd n. Together these results give a physically motivated characterisation of primality: primes are the 
integers for which the tension between additive growth and multiplicative structure is greatest. 
We  further  introduce  a  hardware-native  parallel  divisor  scanner  exploiting  the geometric  mass-balance  condition  b  ·  h  −  (n−1)  =  1  for  integer  factorisation,  and demonstrate  factorisation  of  20-digit  numbers  in  approximately  83  seconds  on consumer hardware. An exact integer square root constructor without floating-point drift is also provided. Empirical data for n ≤ 50 000 and the Python source code are included.