graduation

hypermath | formal-universality | dictionary

What it is

graduation is the formal criterion that must be discharged before the next layer is licensed to proceed. Each layer file contains a graduation block that:

1. Lists all necessity constraints (NCs) — conditions that must be FORM before the next layer may proceed
2. Lists all dispensability constraints (DCs) — conditions whose absence is acceptable at this layer
3. Provides a formal derivation that all NCs are satisfied
4. Emits a graduation claim that, when closed, licenses L(k+1)

What it is not

• Not optional. The graduation is not advisory. A higher layer that builds on unmet NCs violates the foundational contract of the system.
• Not the same as completing all work. Graduation requires all NCs to be FORM. Additional open work (FRAME items with known higher-layer dependencies) does not block graduation — that work belongs to the higher layers.

What it clarifies

Clarifies why layer files contain self-contained verification machinery: the graduation is not checked by an external tool. It is derived from within the file itself. The file IS its own graduation checker.

Clarifies the layer co-necessity structure: L0 ↔ L1 form a co-necessary pair because their graduation criteria are mutually grounding. L0 cannot fully close without L1, and L1 requires L0 to be graduated first.

Concrete example

-- L0 graduation (Section VII):
graduation L0 -> L1:
  NC 1: form-is-derivable      FORM
  NC 2: prop-is-derivable      FORM
  NC 3: apply-is-defined       FORM
  NC 4: ground-is-defined      FORM
  close: graduation(L0 -> L1)  FORM
-- All 4 NCs present. L1 is licensed.

Visualization

A bridge-building inspection. Before the bridge can carry traffic, inspectors verify structural minimums: load capacity, foundation depth, material grade. Graduation is the inspection sign-off. The bridge does not open on a promise. It opens on a closed derivation.

Speculation

Whether graduation can be automated — whether the deriver can synthesize the graduation block from the accumulated derivation state — is OPEN. Automated graduation would mean the layer structure is self-maintaining: no human needs to manually author the transition. Composing graduation with discharge and the derivation matrix D: graduation at layer k is a batch-discharge over all NCs of layer k, verifiable as a column-completeness check on D. This would make graduation a decidable property.