The height obstruction to lifting

Formal target

Let $E/\mathbb F_p$, let $\widetilde E/\mathbb Q$ have good reduction $E$ at $p$, and let $\widetilde P\in\widetilde E(\mathbb Q)$ reduce to $P\in E(\mathbb F_p)$.

The proposed target is a lower bound $\widehat h(\widetilde P)=\Omega(p^c)$ for an explicit $c>0$, uniformly over the lifts satisfying the constraints needed by a xedni-calculus attack.

Operational target for this repository

• Decide whether the literal uniform statement is tenable.
• Measure one- and multi-point lifts at toy primes, with the height convention and lift-sampling rule stated explicitly.
• Separate data by the number of prescribed reductions and by Mordell-Weil rank when a rank routine is available.
• Fit a growth law with stored residuals and state a falsifiable conjecture.
• State precisely what additional hypotheses and height-theoretic input a proof of an attack-relevant theorem would require.

Falsifiers

• A proof of the constrained lower bound with an explicit exponent resolves the stated target positively.
• A construction satisfying the actual attack constraints with heights polynomial in $\log p$ refutes the proposed obstruction in that model.
• A low-height construction outside the attack constraints refutes only the literal uniform statement and identifies a missing hypothesis.

Current disposition

PROVED The literal uniform statement is false: NOTES.md constructs $O(\log p)$ single-point lifts and, under an explicit row-rank condition, $O_k(\log p)$ simultaneous lifts for $k\leq4$.

PROVED The construction does not impose rational dependence or total rank below $k$, so the attack-relevant refinement is not resolved.

CITED The primary 2000 failure analysis uses a conditional bound on relation coefficients together with finite-group counting rather than the lower bound stated in the prompt.

State in five lines

PROVED The literal $\Omega(p^c)$ height target is false. PROVED Single points and full-row-rank tuples with $k\leq4$ admit $O_k(\log p)$ lifts. EMPIRICAL: 144 variants, relation bound 8 SG-08 found 99 finite relations but only two rational two-torsion relations. PROVED Bounded non-detection is not a Mordell-Weil independence certificate. PROVED SG-09 failed: A002 is not a reproduction and supplies no accepted dependency rate.

What is established

• PROVED The balanced direct short lift has canonical height $O(\log p)$.
• PROVED The five-coefficient linear lift has height $O_k(\log p)$ when its constraint matrix has row rank $k$ modulo $p$.
• EMPIRICAL: three LMFDB values lib/heights.py agrees within $2\cdot10^{-6}$.
• EMPIRICAL: six primes, three trials each General least-norm logarithmic slopes are $4.543,6.983,8.643,10.308$ for $k=1,2,3,4$.
• EMPIRICAL: exact p=257 enumeration The paper's Experiment C probability $1/65$ is reproduced.
• EMPIRICAL: 108 k=2,3,4 variants No rational relation through coefficient bound eight was found.
• CITED Jacobson et al. obtain conditional failure from bounded relation coefficients, not a $p^c$ height lower bound for selected lifts.

Failed attempt A002

• CITED The target was Table 3's 317 dependent cases in 100,000 $p=17$ Experiment A runs.
• PROVED The source does not fix the sampling/tie-breaking distribution over short projective and coefficient-lattice vectors.
• EMPIRICAL: local environment on 2026-07-14 The LiDIA/SIMATH and 2-descent pipeline is unavailable.
• PROVED code/reproduce_xedni_p17.py is a diagnostic prototype only; do not report or compare a rate from it.
• CONDITIONAL: original code or complete sampling specification plus equivalent 2-descent Reuse the validated lattice and model-conversion components.

Active thread

PROVED A001 corrects the formulation but does not force rational dependence. A002 is dead. The remaining problem is a dependence-conditioned structural theorem or a genuinely faithful historical reproduction.

Next action

Do not rerun A002 as evidence. Resolve Q018 by obtaining the original sampling/2-descent pipeline before reopening SG-09; otherwise work from the exact SG-08 bounded-relation table.

Invariants

• Use the LMFDB/Sage non-normalized canonical height.
• Keep rank_status=unavailable until an actual rank computation is run.
• Say no relation through bound 8, never independent, for SG-08 negatives.
• State the coefficient-lift sampling rule; least Euclidean norm is a construction bias.
• Do not infer a positive power exponent from the six-prime range.
• Do not call A002 a reproduction or use its prototype output as evidence.

Files that matter

• NOTES.md: proofs, measurements, SG-08 audit, and A002 failure boundary.
• attempts/A001-explicit-small-lifts.md: successful formulation correction.
• attempts/A002-p17-experiment-reproduction.md: dead attempt and post-mortem.
• code/analyze_lift_relations.py: exact bounded-relation audit.
• data/analyze_lift_relations_b5-7-9-11_B8_allv_20260714_{rows,summary}.csv: SG-08 output.
• code/reproduce_xedni_p17.py: failed-attempt prototype, reusable components only.
• refs/jacobson-et-al2000.md: primary-source result and assumptions.
• OPEN_QUESTIONS.md Q018: exact condition for reopening SG-09.

What I would tell my replacement

PROVED Small simultaneous lifts are easy at fixed $k$; the missing xedni ingredient is rational dependence compatible with finite inputs that exclude useful small relations. The exact audit sharpens that distinction, while A002 must remain recorded as a failure unless its missing reproduction inputs are supplied.