$$\eta(X^{\mathrm{al}}):=\min_{p\text{ good}}\bigl(\rho(X_p^{\mathrm{al}})-\rho(X^{\mathrm{al}})\bigr)$$
Consider
$$\Pi_{\mathrm{jump}}(X):=\{p\text{ good}:\rho(X_p^{\mathrm{al}})>\rho(X^{\mathrm{al}})+\eta(X^{\mathrm{al}})\}$$
Is this set infinite? What is its density?
What about
$$X/\Bbb{Q}:\quad\gamma(X,B):=\frac{\#\{p\leq B:p\in\Pi_{\mathrm{jump}}(X)\}}{\#\{p\leq B:p\text{ prime}\}}\quad\text{as }B\rightarrow\infty\quad ?$$
NEEDS APPROVAL: [s14-m01] Recommendation: Keep the displayed minimum definition before Pi_jump on slide 18; recall sharpness from slide 16.
Source: Charles 2014, Thm. 1.
NEEDS APPROVAL: [s14-m02] Recommendation: Keep the source questions and displays on slide 18.
Source: V:L559-576.
NEEDS APPROVAL: [s14-m07] Recommendation: Keep the quoted candidate wording, formulas and arrangement on this slide.
Exact candidate text: "Jumping Picard ranks" | "$$\eta(X^{\mathrm{al}}):=\min_{p\text{ good}}\bigl(\rho(X_p^{\mathrm{al}})-\rho(X^{\mathrm{al}})\bigr)$$" | "Consider" | "$$\Pi_{\mathrm{jump}}(X):=\{p\text{ good}:\rho(X_p^{\mathrm{al}})>\rho(X^{\mathrm{al}})+\eta(X^{\mathrm{al}})\}$$" | "Is this set infinite? What is its density?" | "What about" | "$$X/\Bbb{Q}:\quad\gamma(X,B):=\frac{\#\{p\leq B:p\in\Pi_{\mathrm{jump}}(X)\}}{\#\{p\leq B:p\text{ prime}\}}\quad\text{as }B\rightarrow\infty\quad ?$$"
Speaker notes proposed: "$\eta$ is the minimum excess. Odd characteristic-zero rank forces an increase but does not imply $\eta=1$. Which primes exceed the minimum? Is that set infinite? What is its density?" | "Charles will compute this minimum and prove its attainment. The counting function here uses rational primes; over a number field, count places by norm." | "Charles's theorem identifies the forced minimum: zero in the CM or even-dimensional case, and the endomorphism-field degree in the totally real odd-dimensional case."
SETTLED mathematical source: V:L559-576; Charles 2014, Thm. 1. Exact teaching arrangement still needs approval.
Allocation after the move: this candidate is on slide 18. The active Charles note points back to slide 16; the earlier finding is retained in s14-m90; Charles is now on slide 16.
PROPOSED (2026-09-14): [s14-m10] Recommendation: Charles is on slide 16, before the separate eta definition on slide 18, as authorized by the reorder instruction. The author said "Maybe have that in slide 16, and then eta definition is natraul". Current allocation: ORDER-astra.md section 4 and artifacts/plan/lecture1.md. The historical quoted suggestion is retained.
NEEDS APPROVAL: [s14-m90] The historical speaker note said "Charles will compute this minimum and prove its attainment." That future reference is stale after the move: Charles computes the minimum on slide 16. Its case formula identifies eta as 0 or d. Applied s17-c01: the active note now points back to Charles on slide 16; the quoted sentence is retained as history.