# O_617 Bridge Sub-Investigation - Deep Dive Status: current, corrected 2026-05-22. Audience: collaborators, paper-side writers, future coding agents. Companion: `docs/isotrophy/kfacet/kfacet_v03h_writeup.md`. Receipt: `results/isotrophy/k-facet-v03-O617-deep-dive/deep_dive_receipt.json`. Tooling: `scripts/o617_deep_dive.py` (one-shot, no integration). ## One-Line Read > `O_617` is a clean opposite-strict catalog row. Its quarantine is not caused > by an admission weakness; it is caused by a real bridge direction > that falls outside the orbit's valid `D3` representation at the admitted > kernel boundary. This corrects the first deep-dive attribution. Sorting by canonical sigma3 residual made every opposite-strict row look weak. The orientation-aware admission residual for `O_617` is `1.01e-8` (`to_closure = 1.105`), which is catalog-normal. The old `1.62e-1` value is the canonical residual and is only a diagnostic for an opposite-strict row; it is not the row's admission residual. The bridge audit still stands: at `k_dim = 8`, admitting the bridge vector makes the restricted representation defective (`sigma3^3 - I ~= 3.96e-2` and an odd `E(1)` residual). The cause is not upstream catalog admission. It is a structural boundary direction that the row's `D3` action does not carry as a valid real standard irrep. ## Probe Summary ### Probe 1 - Geometric Anatomy Of `v_bridge` The bridge singular vector has mixed `(12)` parity: ```text (12)-symmetric position norm: 7.48e-1 (12)-antisym. position norm: 1.13e-1 (12)-symmetric velocity norm: 9.36e-2 (12)-antisym. velocity norm: 6.47e-1 ``` It is mostly symmetric in positions and antisymmetric in velocities. That mixed parity is a warning sign for the `D3` projection, which expects coherent behavior under `F_beta = (12) x R_pi x tau`. ### Probe 2 - `sigma3^3` Closure At Kernel Floors ```text On k=7 kernel: 1.181e-5 (clean) On k=8 kernel: 4.07e-2 (defective bridge admitted) ``` The unambiguous k=7 kernel is clean. The order-3 failure appears only when the bridge vector is admitted into the kernel. ### Probe 3 - Monodromy Spectrum ```text Eigenvalue |.| range: [0.8125, 1.2307] Near unit circle (||.| - 1|<1e-2): 16 of 18 Near +1 (|lambda-1|<1e-2): 12 of 18 ``` The spectrum is marginal/elliptic with reciprocal pairing as expected for a symplectic monodromy. The bridge is not explained by a visible Jordan defect. ### Probe 4 - Constraint Tangency ```text cos(v_bridge, grad H) = 4.02e-10 cos(v_bridge, grad |L|) = -2.26e-9 ``` The bridge vector is tangent to the `(E, |L|)` level set. It is physically valid as an infinitesimal direction; the defect lives in the representation, not in a conservation-law violation. ### Probe 5 - Orientation-Aware Sigma3 Admission The strict-21 receipt splits as: ```text 13 canonical-strict rows 8 opposite-strict rows ``` `O_617` is opposite-strict: ```text O_617 admission orientation: opposite O_617 admission residual: 1.010e-8 O_617 admission to closure: 1.105 O_617 canonical residual: 1.616e-1 (diagnostic only) ``` This is the correction. The row is not weakly admitted. It belongs normally to the opposite-orientation strict catalog. ### Probe 6 - Catalog Metadata ```text Label: O_{617}(1) Stability: U Period: 129.276643 z0=0.250595, vx=0.318838, vy=0.551458, vz=0.000723 ``` Standard unstable row in the `m_3 = 1` slice; no inertia-degeneracy flag. ## Synthesis `O_617` is a clean opposite-strict choreography whose bridge-admitted kernel contains a real near-kernel direction that the orbit's `D3` representation does not act on irreducibly. The mechanism is: 1. The row is admitted normally under the opposite sigma3 orientation. 2. At the adaptive floor, `k_dim = 7` is clean and gives no standard sector. 3. The bridge singular value (`7.8359e-4`) is physically meaningful and tangent to the conserved-invariant surface. 4. Admitting that vector at `k_dim = 8` breaks the restricted order-3 relation at about 4% and yields `T(2)+S(6)+E(1)`. 5. `E(1)` is not a valid real standard `D3` block, so the bridge audit correctly classifies the row as `defective_E_block_confirmed`. The honest label is therefore **structural bridge outside the valid D3 representation**, not admission weakness. ## WHY-Dive Addendum The follow-up WHY-dive refines the bridge diagnosis again. The bridge direction is not a loose quasi-kernel vector. It is a near-sign-isotypic direction: ```text Rayleigh lambda at v_bridge: 0.999999 ||(M-I)v|| / ||v||: 7.84e-4 sigma3 v projection onto v_bridge: 0.9998 sigma3 v in k=8 fraction: 1.0000 : -0.9999997 ||F_beta v + v|| / ||v||: 7.79e-4 F_beta^2 v - v: 0.0 sigma3^3 v - v: 7.06e-2 ``` So `v_bridge` tries to live in the sign `S` sector, not the trivial `T` sector. `F_beta` acts as `-I` on the bridge direction, `F_beta^2` closes exactly, and `sigma3` sends it almost exactly back to itself, but with a sub-clean scalar drift. That drift accumulates over the three sigma3 actions and leaves `sigma3^3 v` outside the `1e-3` relation floor. The character projector cannot keep the approximate direction purely in `S`; the bookkeeping residue appears as the defective odd `E(1)` block. The corrected WHY outcome is: ```text bridge_approx_sign_isotypic ``` This is still a quarantine. It is not a valid standard `D3` sector and not countable evidence for `Gamma_i`. But the causal label is now sharper: **O_617 sits at the edge of the sign sector, not outside D3 in a featureless way.** ## Catalog-Wide Separator Addendum The catalog-wide isotypic-edge separator (`scripts/catalog_near_t_separator.py`) checks the same signed isotypic diagnostic across all 21 strict G.2 rows from existing receipts. It confirms the WHY-dive is row-unique: ```text clean_T: 42 clean_S: 116 clean_E: 0 edge_T: 0 edge_S: 2 (O_617 bridge contamination) edge_E_as_T: 0 edge_E_as_S: 0 edge_E_other: 1 (O_617 bridge contamination) ``` Receipt: `results/isotrophy/k-facet-v03-near-T-separator/separator_manifest.json`. The load-bearing parts are `edge_S_rows = [617]`, `edge_E_other_rows = [617]`, and `clean_E_rows = []`. All 20 other rows are pure clean `T(2)+S(5 or 6)` with no edge class of any kind. The `O_617` projector readout overcounts one physical bridge direction (`T+S+E = 9` while `k_dim = 8`) because the underlying group action on the bridge-admitted kernel is only approximate: the bridge is caught mostly by `S` (near `F_beta = -I`) and weakly by `E` as contamination. The separator nevertheless gives an independent check that no catalog direction exhibits standard `E`-rotation behavior and that `O_617` is the only row with a structural edge direction. ## Implications - The v0.3h catalog result remains: 20 clean structural zeros plus one named quarantine. - `O_617` is not counterevidence against the `Gamma_i` audit chain. - A tighter sigma3 catalog-reconstruction sweep is no longer the natural next test for this row; its admission residual is already at integration scale. - The WHY-dive's `bridge_approx_sign_isotypic` diagnosis is row-unique across the strict catalog, not a generic phenomenon. - The open research question is what symmetry, near-symmetry, or boundary mechanism the bridge direction actually carries. ## Out Of Scope - Promoting this script to a general `kfacet-row-anatomy` subcommand. - Extending the bridge audit to non-strict rows or supplementary-B. - Naming the bridge mechanism beyond the present defective-D3 diagnosis.