proposition. A first-step defect falsifies a loss-only protocol specification [ftip-00BW]
✍️source

Suppose two orthogonally gauge-related initializations have equal represented object and scalar objective, but a declared update produces different represented objects after one step. Then the training intervention is not determined by the represented initialization and scalar objective alone.

Proof.

If those two objects determined the intervention, equal inputs would give the same represented next state. The witnessed inequality contradicts that factorization.

Proposition proposition [ftip-00BE] and Example [ftip-00BF] instantiate its premise for one factored quadratic and zero-state Adam. It proves underspecification, not a capability difference.

proposition. A first-step defect falsifies a loss-only protocol specification [ftip-00BW]
✍️source

Proof.

Optimizer geometry in training interventions [ftip-00BS]

Definition. Objective-equivalent parameterizations [ftip-00BT]

Definition. Parameterization-stable training intervention [ftip-00BU]

proposition. Gauge equivariance gives parameterization stability [ftip-00BV]

Proof.

proposition. A first-step defect falsifies a loss-only protocol specification [ftip-00BW]
✍️source

Proof.

Example. The orthogonal gauge of an attention head [ftip-00BX]

Remark. What the attention-twin experiment observes [ftip-00BY]

Example. One loss quotient and two optimizer trajectories [ftip-00BZ]

Remark. Product symmetry and the scope of optimizer conclusions [ftip-00C0]

Remark. Continuous-flow conclusions and discrete optimization [ftip-00C1]

Example. A two-coordinate Adam defect witness [ftip-00BF]

proposition. Exact represented-matrix defect after one Adam step [ftip-00BE]

Proof.