Scale Boundary

What This Track Showed

The finale closes the exact-to-practical arc. Toy mechanisms were exact; real-scale trained values are outside hand enumeration.

highlighted = computed this step

What the track showed

The track showed mechanisms exactly on toys: regression, updates, attention structure, backprop, convolution, regularization, and finite tables of counts.

toy mechanisms, exact arithmetic\text{toy mechanisms, exact arithmetic}
Scale boundaryExact counts and hand-enumeration limit.toy MLP parameter countlayerd_ind_outweightsbiastotalhidden22426output21213total params=9finite square-layer grid, not a smooth curvedd*(d+1)26420872162721001010010001001000discrete exact rows onlyreal-size integer countscomponentleftrightbiascountembedding table50000768038400000dense layer768768768590592counts exact; trained values not enumerated by handexact mechanisms on toys; scale blocks hand enumeration of real trained values; finite grid only;NOT learning; NOT generalization

What it did not show

It did not recompute a real model by hand. It did not claim the toy nets scale into real trained values by inspection.

real trained values are not hand-enumerated\text{real trained values are not hand-enumerated}
Scale boundaryExact counts and hand-enumeration limit.toy MLP parameter countlayerd_ind_outweightsbiastotalhidden22426output21213total params=9finite square-layer grid, not a smooth curvedd*(d+1)26420872162721001010010001001000discrete exact rows onlyreal-size integer countscomponentleftrightbiascountembedding table50000768038400000dense layer768768768590592counts exact; trained values not enumerated by handexact mechanisms on toys; scale blocks hand enumeration of real trained values; finite grid only;NOT learning; NOT generalization

The honest close

This capstone names the boundary: parameter counts are exact integers, but scale blocks hand enumeration of real trained values. The toy examples are NOT learning, NOT generalization, and NOT a real-scale model claim.

exact counts; scale boundary; deferred claims explicit\text{exact counts; scale boundary; deferred claims explicit}
Scale boundaryExact counts and hand-enumeration limit.toy MLP parameter countlayerd_ind_outweightsbiastotalhidden22426output21213total params=9finite square-layer grid, not a smooth curvedd*(d+1)26420872162721001010010001001000discrete exact rows onlyreal-size integer countscomponentleftrightbiascountembedding table50000768038400000dense layer768768768590592counts exact; trained values not enumerated by handexact mechanisms on toys; scale blocks hand enumeration of real trained values; finite grid only;NOT learning; NOT generalization