# Riemann Path (ii) / triple — desk vacuity pre-check & decision

## Header

- Receipt id: `2026-05-30_pathii_triple_desk_precheck`
- Purpose: before any Probe 01b build, decide whether the residual Category-B
  question ("is there a finite-group / representation-sector reading we are
  prematurely dismissing?") converts into **real cause to email a professional**,
  or closes conservatively. Concretely: is the natural triple statistic
  **forced-vacuous** (an identity / relabeling / baseline-pinned null) or
  **genuinely non-forced** (a real observable the GUE baseline does not fix)?
- Method: Q2-style symmetry analysis of every proposed triple construction,
  cross-checked against the two bridge notes and the Q1/Q3 control results.
- Inputs: `REPRESENTATION_BRIDGE_NOTES.md` §"Path (ii)",
  `NONLINEAR_PAIR_CORRELATION_BRIDGE_NOTES.md` §"Hook 2 / Hook 3",
  `2026-05-30_category_a_controls_summary.md` (Q1/Q3 results).
- Type: desk decision (no run).

## The decision in one line

**FORCED-VACUOUS / STANDARD across every runnable triple construction.** The
email is **not earned** by Path (ii). Do not build Probe 01b. The only
non-forced direction is an unsolved construction problem whose own admission
gate predicts vacuity — research, not a 10-minute expert question.

## Why — every triple construction, adjudicated

| Construction | Verdict | Mechanism |
| --- | --- | --- |
| **C3 cyclic statistical triple** `ρ:(s_n,s_{n+1},s_{n+2})→(s_{n+1},s_{n+2},s_n)` | relabeling (`R-NL-NEG-B`) | ρ fabricates a cyclic adjacency between `s_{n+2}` and `s_n` that a **linearly ordered** sequence does not have. The real shift the process admits is stationarity `→(s_{n+1},s_{n+2},s_{n+3})`, not ρ. Even a *nonzero* result measures a symmetry the data lacks → meaningless. |
| **S₃ rep-theory pure-zero triple (ii-a)** elementary symmetric `(e_1,e_2,e_3)` | forced-trivial | symmetric polynomials are S₃-invariant *by definition* → every row in the trivial sector → no "absent-by-construction" target. This is the **Q2 identity, one dimension up.** |
| **Genuine 3-point reversibility statistic** (the honest triple analog of the Probe 05 S2 null) | baseline-pinned (`R-NL-NEG-A`) | the sine-kernel correlation functions depend only on `|x_i−x_j|` → reflection-symmetric at **all orders**, so any n-tuple reversibility observable has its expectation pinned exactly as the pair statistic did. Bogomolny–Keating/Berry arithmetic corrections enter even-in-distance at accessible order. **Reinforced empirically:** the Q3 sweep to 100k ordinates showed the pair reversibility null *deepening* (|D|→0.001), i.e. no emerging irreversibility a higher-order statistic could resolve. |
| **Genuine 3-point correlation magnitude** | standard RMT (`R-NL-NEG-A`) | this is the Rudnick–Sarnak n-level correlation the Track-A literature already computes; binning it (Hook 3) adds vocabulary, no "absent-by-construction" sector. No structural-zero to earn. |
| **S₃ prime-side selector (ii-b / ii-c)** `(γ_n, 1−γ_n, γ_{n'})` | non-forced — but not email-able | the *only* construction that escapes triviality, and it does so **only by importing the linear explicit formula** (Q2 territory — identity/linearity nulls) **plus an arbitrary selector `n'`** whose degrees of freedom the receipt inherits. Its own pre-registered admission gate requires proving the selector is "not already a known explicit-formula device in disguise" — i.e. the gate itself predicts vacuity. |

## What this means for the email

The residual-B question is answered **in-house, conservatively**:

- The two natural triple sectors are vacuous by **named mechanisms** (fabricated
  adjacency; symmetric-polynomial triviality), not by hope.
- Genuine 3-point statistics are pinned to the GUE baseline by the **same
  reflection-invariance** that produced the Q1-confirmed pair null — and the Q3
  100k deepening-null is direct evidence against a hidden higher-order
  irreversibility.
- The lone non-forced direction (ii-b/c) is **unbounded construction research**
  (define a principled, non-arbitrary, falsifier-equipped prime-conjugate
  selector that survives a Front-A "known device" check) — not something an
  expert can yes/no in a paragraph, and predicted vacuous by its own gate.

So Path (ii) yields **clarity, not cause**: the clarity that there is no earned
Riemann email from the representation-sector direction. This is "outcome 1"
(clean conservative closure), reached by argument rather than assumed.

## Disposition

- **Probe 01b: NOT BUILT.** The triple horizon is closed at desk level for all
  runnable constructions.
- **Residual-B rep-sector ask: retired** to "checked; vacuous/standard by named
  mechanisms," not escalated to outreach. The bounded-null synthesis can absorb
  this as a strengthened conservative call.
- **Reopen condition (research, not control):** a defended prime-conjugate
  selector for (ii-b/c) with named falsifiers for selector arbitrariness that
  passes the Front-A "not a known explicit-formula device" check. Until such a
  selector exists, there is nothing to probe and nothing to email.
- Guards intact: `R-NL-NEG-B` (no fabricated-symmetry rescue), `R-NL-NEG-D`
  (no silent S₂→S₃ upgrade), `R-C1-NEG-B` (no identity-zero laundering).

## Cross-references

- `2026-05-30_category_a_controls_summary.md` — Q1/Q3 controls this leans on.
- `NONLINEAR_PAIR_CORRELATION_BRIDGE_NOTES.md` — Hook 2 (C3) / Hook 3 adjudication.
- `REPRESENTATION_BRIDGE_NOTES.md` — Path (ii-a/b/c) constructions.