Run 10: unary guards and terminal geometry

Date: 23 September 2026

Contents

Result

The unary paper uses four different certificates for four different kinds of change. They should not be collapsed into one informal notion of equivalent pointer syntax.

Move	What may change	Required certificate	What is preserved
`D`	one descriptor spelling	same physical endpoint	every prefix figure
`A`	an old pointer becomes fresh `SELF`	old target is a literal physical self-loop and has the same recursive residual	every prefix figure
`G`	unsupported label or descriptor	absence from replay-closed causal support	the final physical replay and figure
`T`	adjacent `(epsilon,0)` and `(0,epsilon)`	the two candidate carrier labels agree	the final figure, but not necessarily every prefix figure

This typing is the reason the proof can combine physical rewriting with behavioral equivalence without testing pointer identity inside the scaffold. It also explains why literal self-looping in A is stronger than merely presenting a recursive behavior.

The absorption guard is necessary

Suppose a fresh row points to an old row whose behavior is recursive. That is not enough for one elementary absorption. The old row may itself point farther down to the actual loop carrier. Replacing the fresh pointer immediately by SELF can then skip the physical refactoring needed by the local coherence diagrams.

The paper therefore requires the old target q to satisfy e(q)=q. Once this literal guard holds, identity on the old prefix together with the fresh-row and q pair is a bisimulation. The replacement preserves the strong trace.

The focused checker found 33,060 small cases where an old target presents a recursive behavior but is not a literal self-loop, alongside 103,242 legal literal absorptions. This is a useful stress test of the guard. It is not a proof of the parameter-uniform theorem.

Temporal transport and the base

A temporal slide uses three labelled rows x,y,z. It exchanges the two descriptors on y,z only when the labels of candidate carriers x,y agree. The final suffix then sees the same labelled behavior even though the physical carrier time moves by one. Earlier prefix figures can change; 40,978 of the 68,584 checked slides did so.

The distinguished time-zero base is unlabelled. It cannot satisfy the candidate-label comparison and cannot be crossed by a slide. This is not an omitted boundary case. It is what separates the carrier coordinates:

for a missing or base terminal, a first retained carrier may occur at time 1, giving ideals of [m-1] x [n-m];
for a recursive terminal, the omitted physical self-loop carrier lies below the first retained finite carrier, so time 1 is unavailable, giving ideals of [m-1] x [n-m-1];
with no retained carrier, the recursive case is a singleton.

The checker verified the corresponding binomial counts for 144 parameter pairs through length 12. More importantly, every checked D, A, G, and T move preserved the terminal kind: missing, base, and recursive do not collapse into one case.

Inherited behavioral assumption

The unary argument inherits equality of full unfoldings, equivalently bisimilarity, from the causal paper. It does not require the false statement that a native radius-m view equals the port-word endpoint table through the same radius. Run 07’s one-layer repair is therefore sufficient: all-depth behavioral equivalence survives, and the unary guards use physical endpoints separately where physical information is needed.

Coherence evidence

The pinned original verification suite enforces the same boundaries. Its source checks:

exact R/R critical signatures A/T and D/T;
exactly three non-strict garbage handoffs;
the raw A/T hexagon without quotienting its endpoint by garbage; and
a homology falsifier showing that A/T cells cannot simply be deleted.

The historical harness records 147,448 exhaustive histories and 12,914 raw A/T overlaps. Run 10 inspected that source and added a separate focused checker for move soundness, terminal preservation, the base boundary, and the carrier-coordinate counts. The historical suite was not rerun from its old repository environment in this Site workspace.

Cross-paper disposition

No new mismatch was found.

The unary theorem uses the causal paper’s surviving all-depth behavioral equivalence, not its incorrect same-radius proof sentence.
It does not use the causal batch compiler or the direct sequence contract.
It does not enter the independent real-time transfer proof.

R05 is complete. This audit does not justify changing the unary theorem or any of the other three circulated papers. The review ledger should be retained for a future coordinated numbered edition, but no edition work is started by this result.

Executable evidence

check_unary_guards.py checked 124,828 histories, including 288,292 endpoint detours, 103,242 literal absorptions, 68,584 temporal slides, and 1,940,484 garbage-coordinate changes. It also checked the three terminal classes and the carrier rectangles through length 12.

Primary source inspected:

Unary manuscript and verification tools at source commit f69e26f41703c0b93cd5a3315b38cd9d96c8641d.