canonical_seed_post_index_unique
plain-language theorem explainer
Any natural number carrying a seed-posting index certificate must equal 2. Hierarchy-forcing arguments cite this to pin the seed index before proving additive size laws on multilevel compositions. The proof is a one-line simp that discharges the certificate field against the definition of the canonical seed-posting index.
Claim. If a natural number $n$ is certified as a seed-posting index (that is, $n$ equals the canonical seed-posting index), then $n = 2$.
background
The Unified Forcing Chain module derives the full T-1 through T8 stack as inevitabilities from the Recognition Composition Law plus normalization and calibration. Within that chain, hierarchy posting needs a fixed seed index: the level at which the two primitive seed layers are posted into the next composite level.
CanonicalSeedPostIndex is a propositional certificate on a candidate natural post01. Its sole field asserts equality with the named constant for the canonical seed-posting index. That constant is definitionally 2; the present theorem packages the identification so downstream proofs never reopen the definition.
The local setting is seed-level bookkeeping for nontrivial multilevel compositions: once the index is forced, size laws (additivity of the two seed levels) can be stated without a free parameter.
proof idea
Term-mode one-liner. The hypothesis supplies the structure field eq_two : post01 = canonical_seed_post_index. A single simpa step unfolds the definition of the canonical seed-posting index (which is 2) and rewrites the goal to that equality. No auxiliary lemmas are invoked beyond definitional reduction.
why it matters
Parent use is canonical_seed_posting_forces_closure: given a canonical seed-posting operation on a nontrivial multilevel composition, the seed-level posting identity rewrites, via this uniqueness theorem, to the concrete additive law $M.levels,0 + M.levels,1 = M.levels,2$. That closure step is the size half of seed posting; the index half is isolated here so the next certificate can derive additivity from RCL and posting-potential composition without re-proving uniqueness.
In the forcing chain this sits under ledger and recognition structure (T3–T4 territory) feeding discrete hierarchy bookkeeping that later supports the eight-tick octave (T7) and $D=3$ (T8). It does not itself force $\varphi$ or $J$; it only locks the seed rung to 2 so those later steps see a rigid posting skeleton.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.