Markdown Tangent Categories | ICML 2026 KET Demo Console

Why this lane exists

The tutorial already has IC, KET, and UDL as conceptual pillars. Markdown tangent categories explain the common object on which the agentic workflows operate: skills, tickets, causal notes, plan graphs, guardrails, and workflow specs are not just text strings. They are typed artifacts with local admissible edits.

IC rolelocal variation, obstruction, and gluing

KET roletransport across contexts by extension and pullback

UDL roleadmit behavior after rollout and validation

Tangent rolemake edits into local directions on structured artifacts

The object is typed anchored Markdown

Raw Markdown is too syntactic. The object used by the tutorial systems is parsed, anchored, typed, and connected to validation semantics.

M = (A, P, tau, sigma)

Anchors

Stable identifiers for sections, instructions, examples, schemas, plan nodes, citations, tool contracts, and evidence spans.

Types

Block roles such as instruction, guardrail, example, schema, tool contract, plan node, plan edge, and admission rule.

Semantics

Observation functors send artifacts to rollout traces, validation scores, causal bundles, plan rewards, or Scylla/TICKET admission records.

Tangent structure

For each artifact \(M\), the edit fiber \(E(M)\) contains local admissible edits. The tangent object pairs the current artifact with a direction of change.

T M = (M, E(M))
T f(M, delta) = (f(M), Df_M(delta))

Projectionp(M, delta) = M

Zero edit0(M) = (M, 0)

Additiondelta_1 + delta_2 for compatible edits

Flip / liftsecond-order bookkeeping for interacting directions

Agentic workflow optimization

Artifactskill, plan, ticket, claim

Tangent editlocal admissible direction

Rolloutleft-Kan style extension

Pullbackvalidation and obstruction

Admit / repairupdate, reject, or quarantine

This is the general frame. SkillOpt and BASKET/ROCKET instantiate it differently: one edits Markdown skills, the other repairs plan artifacts.

SkillOpt instance

SkillOpt treats a Markdown skill as the object and proposes bounded edits. Odyssey supplies structured rollout rows: TICKET admission, DB obstruction energy, GT graph similarity, and KET transfer score.

Open the UDL causal-claim lane

BASKET / ROCKET instance

BASKET constructs plan objects from text evidence. ROCKET chooses tangent repairs over plan graphs: insert an action, add an edge, promote a motif, retrieve evidence, or stop.

Open the 10-K evidence lane

IC / BRACKET audit

Two edit fields can fail to commute. The residual becomes an IC obstruction: missing context, guardrail, subskill, mechanism, or sufficient statistic.

Open the IC drilldown

Defensible technical claim

The claim is not that arbitrary text is smooth. The claim is that typed Markdown plus anchored admissible edits forms an additive bundle with functorial transport, which is enough to use tangent-category language for agentic workflow optimization.

TICKET label

TICKET(markdown_tangent_category_framework) records this lane as a tutorial framework object, not an empirical run. It glues IC obstruction language to KET/UDL workflow transport and explains why SkillOpt and BASKET/ROCKET belong in the same family.

FROM tutorial_frameworks SLICE BY object("typed_markdown_artifacts") TICKET BY target_foundry