The Transition Table

The periodic table works because it has two independent organizing axes — period (row) and group (column) — plus a third deeper property, orbital type, that creates the block structure. The Transition Table has the same architecture: tier (row), decision class (column), and coupling class (block). The coupling class is the quantum number analog. It describes the relationship of a transition to prior state — the property that determines which other elements behave like it.

Elements that share a coupling block are chemically analogous: they require the same kind of prior state to exist, they fail in the same way when that prior state is absent, and they advance by the same kind of legitimacy check. A DENY at T2 and a DENY at T8 share a block because both are denials of a declared-instruction attempt — even though they sit at different tiers and look different on the surface.

In the periodic table, the f-block (lanthanides and actinides) is displaced from its natural position in the grid and shown as a separate row beneath the main table. This is not a layout convenience — it reflects that f-block elements bridge two periods and don't fit cleanly into the contiguous tier structure.

The Replay coupling class has the same property. Replay — reconstructs consequence state but cannot reverse consequence — appears at two non-contiguous tiers: T3 (evidence/receipt, where replay first becomes meaningful) and T11 (controlled mutation, where the ability to reconstruct how a state came to exist is a precondition for execution). These are structurally the same coupling class separated by the tiers between them. They belong together but cannot be placed contiguously in the main grid without distorting the tier axis.

The Replay block is therefore shown as a displaced section beneath the main table — exactly as the f-block is shown in chemistry.

Three consecutive blocks share a structural family: they each define a coupling class where the entity may observe, propose, or package — but explicitly cannot install, execute, or claim authority.

B5 Discovery   — observes, models, compares, classifies, proposes. Does not install.
B6 Packet      — portable evidence of a proposed governed transition. Not authority.
B7 Install-Plan— candidate transition. Not installation authority.

This triad corresponds to Tiers T7–T10 in the main table — the range where external reasoning enters (LLM Adapter Gate), context packets are admitted (KnowledgeVault), and mutation proposals are produced. All three blocks share the same core rule: the output of this coupling class may inform a transition but cannot authorize one.

This is why LLM output, context packets, and install plans share a structural family even though they look different on the surface. They are all authority-denial coupling classes — the formalism treats them as the same block type with different instantiations.

The two most fundamental results from the Stage 1–31 proof are:

B2 Data:      Same data does not imply same continuation admissibility.
B3 Composite: Local allow plus local allow does not imply composite allow.

These are not just rules — they are the core claim that makes the Transition Table necessary at all. If same data implied same admissibility, a lookup table would suffice. If local allows composed into composite allows, governance could be decomposed into independent checks. Neither is true.

B2 (T1–T2) establishes that the same input in a different context or at a different time can have a different admissibility outcome. B3 (T3–T4) establishes that passing N individual checks does not prove the composite transition is admissible. Both blocks occupy the lowest-consequence tiers — they are foundational, not advanced.

This means the non-composability results are not edge cases. They are present at every tier above T2 as implicit constraints — the table cannot be read without them.

The Commit-Time coupling class — admissibility must be resolved at the binding moment — occupies T5–T6 (candidate preparation and gate enablement). This is structurally deliberate: by T5 the system is producing candidates that could affect real state, and by T6 it is operating within an active gate that has a declared STOP condition.

The Commit-Time block is the point where the Transition Table transitions from descriptive to enforcement. Before this block, the table records what the system has proven. At and after this block, the table controls what the system may do next.

The B4 column in the visual table should be the densest — because commit-time admissibility applies at every higher tier as well. Every populated cell at T7 and above implicitly carries a Commit-Time constraint even when assigned to a later block.

The DENY and FAIL-CLOSED decision classes must be present at every tier without exception. A tier that lacks a DENY element has no answer to "what happens when this transition is refused." A tier that lacks a FAIL-CLOSED element has no answer to "what happens when this transition is unsafe but the reason is unknown."

This makes every empty cell in the DENY and FAIL-CLOSED columns a formalism gap — a predicted element, not a legitimate absence. The visual table marks these as mandatory predictions. They are the highest-priority cells to formalize.

In the periodic table analogy: every period must contain a noble gas (full outer shell, stable, no further bonding). DENY and FAIL-CLOSED are the noble gases of the Transition Table — every tier row must end with them.