Turn the Erdős 625 roadmap into a reader-first Version 2 manuscript pass

[SamPetkov]

Purpose

This is a stacked exposition PR on top of #46. PR #46 supplies the corrected proof roadmap; this PR turns that roadmap into a manuscript-level prose revision with copy-ready front matter, section transitions, terminology, and an explicit current-audit boundary.

It does not modify the frozen canonical TeX and does not claim that the Section VIII closure theorem, normalized second moment, or Erdos625Statement has been proved.

Files

• 625/proofs/CANONICAL_V2_READER_FIRST_PROSE_REVISION.md
• 625/arxiv/READER_FIRST_FRONT_MATTER_V2.tex

Main editorial change

The proof is presented from the first page as three separate layers:

location
  ordinary root versus signed four-size root

existence
  one normalized signed second moment

amplification
  positive-probability seed to high probability.

Every technical object in Sections VIII--IX is then introduced only through the aggregate estimate it controls.

What the prose pass changes

• supplies an audit-safe current abstract and a separate post-closure publication abstract;
• rewrites the introduction around the exact factor 2^k and the three proof layers;
• states the single normalized-second-moment target before any overlap bookkeeping;
• standardizes the dictionary for profile blocks, block atoms, support, full multiplicities, deficits, partial physical matchings, endpoint tables, bare skeletons, and residual attachments;
• makes the one-global-denominator point explicit:

  (n)_(J+H)/(n)_J = (n-J)_H <= n^H;
  

• replaces the old near/middle prose by the all-deficit physical-fibre route;
• replaces the cycle-walk narrative by injective restriction outside the exposed matching;
• supplies copy-ready transitions for every major proof stage;
• separates mathematical proof, exact certificate, Lean theorem, and CI evidence;
• moves superseded Section VIII--IX machinery and theorem-name inventories out of the proposed main narrative.

Current mathematical boundary

The PR keeps the remaining seam explicit:

attained canonical high skeleton
  <-> block support + admissible deficits + local partial physical matchings

with exact aggregate weight preservation. The Version 2 theorem box and stronger coefficient remain labelled conditional until that finite reindexing theorem is closed in #45 or a successor audit branch.

Proposed Version 2 order

1. Introduction and theorem
2. Phase notation and one-class asymptotics
3. Ordinary-colouring location
4. Signed four-size location
5. Exact signed-overlap identity
6. Partial diagonals and canonical high support
7. Endpoint reference transport
8. Physical high-cell fibres and all deficits
9. Residual attachment by matching restriction
10. Positive-probability seed and amplification
11. Final event intersection
12. Appendices for exact certificates, formalisation maps, and superseded machinery

Validation

The branch is exactly two commits and two files ahead of #46. No canonical manuscript, PDF, theorem statement, Lean source, formalisation DAG, verifier, bibliography, or audit status is changed.

The TeX fragment is intentionally not included automatically. It contains an explicit warning separating the audit-safe abstract from the post-closure publication abstract. A future integration PR must compile and render the selected version, synchronize theorem-status language, and rerun the focused Section VIII--IX and root workflows.

Review focus

Please review:

• whether the three-layer opening makes the proof legible before the technical sections;
• whether the object dictionary matches the exact physical-fibre formalisation;
• whether the proposed Section VIII text avoids any objectwise full-completion claim;
• whether the Section IX injection is stated at the right level of generality;
• whether the current proof gap and conditional coefficient are impossible to miss;
• whether any superseded machinery is still needed in the Version 2 main narrative.