# Abstract attenuation and layers

Run `python reproduce.py` with numpy and matplotlib. Standalone study:801 attenuation rows and1,681 layer rows (2,482 total), two original figures in PNG/SVG/PDF, recorded runtime/generator hash and validation. All parameters are dimensionless assigned examples; there are no measurements or uncertainty estimates.

`transmission(tau)` returns exp(-tau) for finite optical depth0..8. The attenuation CSV uses steps0.01. This is the solution of dT/dtau=-T with T(0)=1. Optical depth describes cumulative removal from an initially uncollided beam; the study supplies no material, physical thickness, incident energy, source strength or geometry. At depth ln2 the uncollided fraction halves; each additional equal depth multiplies it by the same factor. A logarithmic vertical axis makes this exponential curve straight.

`layers(a,b)` returns exp(-(a+b)), with each input nonnegative and total no more than8. Layer CSV uses a,b0..4 by0.1. For the assumed scalar attenuation model, Tcombined=Tfirst*Tsecond. A depth2 layer followed by depth1 has the same uncollided transmission as total depth3. This does not mean physical layers are interchangeable for all radiation fields: scattered or secondary components and energy changes are not modeled.

The complementary column1-T means removed from the uncollided component. It is not necessarily absorbed energy, destroyed radiation or avoided dose. Particles may scatter into other directions or produce secondary radiation. Total transmitted radiation, buildup, deposited energy, biological response and dose require additional models and measurements. Neither plot supplies a shielding design, material ranking, clinical recommendation or safe exposure threshold.

Independent checks verify multiplication against added depth across1,681 layer pairs, normalization of the uncollided/removed fractions, strict monotonicity, zero-depth and half-transmission limits. Explicit-midpoint integration of the differential equation at100/200/400steps shows second-order convergence to the analytic depth4 result. Five invalid guards pass; absolute errors are recorded. No statistical confidence bands are invented.