# Notes

Project memory for: **What Does Random Feedback Cost? (Feedback Alignment
capacity & operator-gap analysis)** — target AAAI-27.

## Start here (the writing pipeline)

| doc | role |
|---|---|
| [41_paper_plan.md](41_paper_plan.md) | **the plan**: venue rules, narrative, section/page budget, theorem & figure numbering, related-work list, schedule, risks |
| [36_evidence_ledger.md](36_evidence_ledger.md) | **the numbers**: every claim with its corrected headline figure, claim discipline, booby traps |
| [40_reproduction_manifest.md](40_reproduction_manifest.md) | **the runs**: claim -> exact command -> output dir -> note; everything not listed is not citable |
| [37_wrapup_corrections.md](37_wrapup_corrections.md) | audit record: e0 sampler de-circularized; closed-form comparison matched (0.977 retired -> 0.933/0.982) |

## Authoritative note per result

| result | note | status |
|---|---|---|
| T1 Beta alignment law + capacity cost | 01 (statements/proofs), 02 (validation) | proven + validated |
| P1 multilayer additivity, Theta(Ln^2), Gamma(L,1) | 01, 02 | proven + validated |
| T2 prior-free minimax 1/D | 01, 02 | proven + validated |
| P2 random-subspace null model (and its honest failure as a predictor) | 26 (theorem), 25 (failure) | proven; overpredicts real FA |
| T3 initial erosion = hidden BP speed share | 29 | proven + validated (synthetic 0.4378/0.4403; MNIST note 38) |
| P3 exact Gaussian e0 (one hidden layer) | 32 + fix in 37 | proven + validated (real backward) |
| T4 closed-form soft-ramp gap law | 34 + correction in 37 | proven under A1+A2; corr 0.933 matched / 0.982 ensemble |
| early-velocity estimator (+ bracket, bias structure) | 15, 16, 17 | no-fit, snapshot-conditional; MAE 0.0019, corr 0.999 |
| dense soft ramp (refutes phase transition) | 22 | measured, 352 traj |
| finite-time validity chain (T=5 local; oracle K(t)) | 10, 13 | measured |
| MNIST real-data validation | 38 | measured |
| teacher-task test gap (sign flip) | 39 | measured |
| positioning (Song-Xu-Lafferty; effective-capacity lit; methods for K_t) | 27, 24, 30 | reference |
| proof-vs-estimator-vs-fit discipline | 28 | reference |
| appendix proof source material | 33 | reference (framing partly stale) |

## Historical chain (chronological; superseded items marked)

Phase 1 — static geometry (validated, still paper-grade):
- 00 project overview — **superseded framing** (banner added); 01 theory
  notes — T1-T3 statements and proofs; 02 experiment notes — static
  validation record (Beta/Gamma/minimax/overlap) + early trajectory probes;
  03 paper outline — **superseded** (banner added).

Phase 2 — capacity-margin era (Adam sweeps; superseded by SGD work):
- 04 downstream design; 05 random-task results; 06 peak-shift analysis —
  all **superseded** by the Jacobian/NTK view of 07.

Phase 3 — operator view and finite time:
- 07 BP capacity theory (the pivot to tangent operators); 08 SGD T=50
  validation (corr 0.97, 2x scale -> diagnosis); 09 fixed-width probe;
  10 **T=5 local exactness** (paper-grade); 11 finite-time roadmap;
  12 error decomposition (BP-too-strong/FA-too-weak); 13 **oracle K(t)
  diagnostic** (paper-grade: drift is the whole error); 14 scalar predictors
  fail (negative, kept); 15 **the estimator** (paper-grade); 16 **stress
  grid** (paper-grade); 17 **bias structure** (paper-grade); 18 contribution
  roadmap.

Phase 4 — the phase-transition story dies:
- 19/20/21 kink-era analyses — **refuted** by longer training; 22 **dense
  T=30000 soft ramp** (paper-grade); 23 soft-erosion revision.

Phase 5 — exact theory of the burden:
- 24 literature + gap-theory positioning; 25 hard-k overprediction
  (negative, kept); 26 null-model theorem + claim discipline; 27
  Song-Xu-Lafferty positioning; 28 proof-vs-fit status; 29 **initial-moment
  theorem T3**; 30 methods survey for K_t dynamics; 31 derivative
  extrapolation blows up (negative, kept); 32 **e0 Gaussian validation**;
  33 complete markdown draft (appendix source); 34 **closed-form law T4**;
  35 consolidated snapshot.

Phase 6 — wrap-up for AAAI-27 (2026-06-09):
- 36 evidence ledger; 37 corrections; 38 **MNIST validation**; 39 **teacher
  test gap**; 40 reproduction manifest; 41 paper plan.

## Note conventions

- Prefer formulas and precise claims over prose-only ideas.
- Mark unproven statements as `Conjecture` or `Hypothesis`.
- Mark results ready for paper text as `Candidate theorem/proposition/corollary`.
- Keep experiment ideas close to the metric they test.
- When a note depends on an assumption, write the assumption explicitly.
- Never edit a historical note's results in place: append a dated correction
  block and record the fix in a new note (as done in 34/35/37).