CanonicalPostingClosure
plain-language theorem explainer
Packages a named certificate that a multilevel level sequence closes under primitive posting: level 0 plus level 1 equals level 2, with the hierarchy additive recurrence derived from posting extensivity rather than assumed raw. Anyone citing the T5–T6 self-similarity bridge or the canonical routes that force φ from uniform scale and growth will use this object. It is a Prop-structure definition with a uniqueness (Subsingleton) instance; constructors build it from a closure equality or an explicit posting operation.
Claim. For a nontrivial multilevel composition $M$ with uniform successive ratios (no free scale) and growth ratio $M_1/M_0>1$, a canonical posting-closure certificate is a proposition asserting: (i) primitive closure $M_0+M_1=M_2$; (ii) the additive recurrence $M_2=M_1+M_0$ used by the hierarchy theorem; (iii) that this additive equality is exactly the one forced by the posting-extensivity theorem from the hierarchy ratio data and the primitive closure. Any two such certificates coincide.
background
The Unified Forcing Chain module aims to derive T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T5 pins the unique cost $J$; T6 forces the golden ratio $\varphi$ as the self-similar fixed point of a discrete ledger hierarchy.
A NontrivialMultilevelComposition supplies only a positive level sequence. It does not carry a posting or composition operation, so one cannot read off $M_2=M_1+M_0$ from the bare structure. Hierarchy forcing already yields a common successive ratio and growth orientation under the no-free-scale and ratio-greater-than-one hypotheses. Posting extensivity then converts a primitive closure equality into that additive recurrence.
This structure is the first theorem-facing certificate for the missing operation: it states closure in the natural order $M_0+M_1=M_2$ and records that the additive form used downstream is exactly the one forced by extensivity, not an ad hoc rewrite.
proof idea
Definitional packaging, not a derived theorem. The three fields are Prop components: primitive posting closure, the swapped additive recurrence, and a proof-equality identifying that recurrence with PostingExtensivity.closure_forces_additive applied to the level map, positivity, and the ratio data from HierarchyForcing.hierarchy_forced. A separate Subsingleton instance proves any two certificates are definitionally equal (allEq by rfl), so the certificate is unique once $M$ and the two scale hypotheses are fixed. Downstream constructors (from a raw closure equality, a local posting operation, or a seed posting) inhabit the structure; this declaration only names the interface.
why it matters
In the forcing chain this sits on the T5→T6 bridge: self-similarity in a discrete multilevel ledger forces $\varphi$. Downstream, canonical_posting_closure_forces_phi and the uniform-growth variants take such a certificate (plus scale uniformity) and conclude $\varphi$ is forced. Constructor theorems build the certificate from primitive closure, an explicit posting map, or canonical uniform-scale and growth seeds, so the bridge no longer depends on a bare equality hypothesis.
The module’s stronger claim is a complete inevitability chain from cost foundation through T6 ($\varphi$ forced) onward to the eight-tick octave and $D=3$. Replacing raw equalities by named canonical certificates is how that chain stays auditably forced rather than merely compatible. The Subsingleton instance ensures the certificate does not introduce choice or multiple inequivalent closures at fixed parameters.
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