[pinterbanget]

Summary

This adds research/core-row-operator-lift/. It answers the open question of jwr404's #4515. HALO's tensor bound needs no bound on ‖P_sec − P_global‖₂ off the core rows, because HALO only keeps core outputs and the lift holds there.

Claim type: a written lemma with exact rational controls. No new κ.

Lemma

Let Π_i be the kept-row projection on axis i. Assume ‖B_i − P_sec‖ < ε, ‖P_glob − C_i‖ < ε, HALO's row identity Π_i P_sec = Π_i P_glob, and ‖B_i‖, ‖C_i‖ ≤ M. Then

‖Π_i(B_i − C_i)‖₂ < 2ε,    ‖⊗Π_iB_i − ⊗Π_iC_i‖₂ < 2g·M^{g−1}·ε,    ⊗Π_iB_i = (⊗Π_i)(⊗B_i).

The key fact is ‖ΠX‖₂ ≤ ‖X‖₂, by the Gram identity XᵀX − (ΠX)ᵀ(ΠX) = ((I−Π)X)ᵀ((I−Π)X). The result is slightly sharper than #4508, since no +ε enters the base.

Exact controls

• The Gram certificate and the two-error split, on 40 random rational cases.
• The mixed-product rule and the exact restricted telescope, for g = 2, 3.
• Separate the shared-row sum from the operator 2-norm (no new κ) #4515's two examples: they fail the unrestricted lift but satisfy the restricted one.
• HALO's phrases are pinned.

Evidence

python3.12 -B scripts/research_package.py --package research/core-row-operator-lift --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/core-row-operator-lift
  PASS_REGISTERED_RESEARCH_PACKAGE research/core-row-operator-lift outcome=completed   (<1 s, stdlib only)

Head: 2280cd7a4c8238232f641cc0249280dfcae6e8c7. Base: canonical main f9f85c1e.

Credit: James Reed (jwr404, #4488–#4515) and HALO (#4100 notes). Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code