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A proof has an architecture.

The paid packets use the typography, problem boxes, diagrams, and complete-solution standard of the Lyran coursebooks. This public page demonstrates the learning sequence without exposing an enrolled-student artifact or private answer key.

Sample theory proposition

Symmetric cubics from power sums

Let a monic cubic have roots r₁, r₂, r₃. If the first two power sums are known, then

e₂ = ((r₁+r₂+r₃)² − (r₁²+r₂²+r₃²))/2.

The packet derives this identity, identifies the exceptional information still needed to reconstruct the cubic, and shows how a root value supplies it.

Sample proposed exercise

Finish the reconstruction.

A monic cubic has root sum 3 and root-square sum 5. One root is 0. Determine the polynomial, and justify why the data determine it uniquely.

Workshop prompt: compute the second elementary symmetric sum first; then use the known root to recover the remaining product without guessing the other roots.

Inside each enrolled packet

Designed for live use and later proof-writing.

Six core exercisesSelected from rights-cleared A0 source blocks, with every editorial clarification recorded and harder extensions after the core sequence.

Provenance manifestPrinted number and page, source anchor, statement and solution hashes, and distribution status are recorded privately.

Complete review solutionsReleased only after the final live stream for the title, with exceptional cases and equality conditions made explicit.

A new 60 + 10 examinationFour graduated core problems and one genuinely harder bonus, followed by personal feedback and one correction round.

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