#624 re-run (2026-09-08): location-without-shape verdicts re-adjudicated on main (6/6 independent mpmath checks)

notes/probe-location-without-shape-20260908.md

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+# Probe #624 re-run (2026-09-08) — location without shape: verdicts re-adjudicated on main
+
+2026-09-08. Analysis + numpy/mpmath. No percolation engine, no arXiv, no
+N=725, no #612 rescoring, no fit to #582 (1.55 retired, irrelevant here).
+Does **not** enter `docs/STATUS.md`. Closes nothing. Issue #624 stays open.
+
+## Provenance and what this PR is
+
+The #624 probe was first delivered in **PR #634** (merged into
+`claude/matching-one-workspace-pwr5pv`; files already present on main):
+
+- `notes/probe-location-without-shape-20260907.md` — the theory note
+- `scripts/probe/toy_cdf_families.py` — the toy families
+- `results/probe-location-without-shape/latest.json` — the committed grid
+
+The 2026-09-08 assignment re-opened the delivery channel with an override:
+**draft PR against `main`**, not the old workspace branch. This PR is that
+delivery. The mathematics is unchanged (nothing was wrong); what is new:
+
+1. **Bit-exact reproduction on main.** `toy_cdf_families.py` rerun on main at
+   this PR's head regenerates `latest.json` byte-identically (`git diff`
+   empty). The committed artifact is reproducible on the target branch.
+2. **Independent re-adjudication** (`toy_cdf_families_rerun_20260908.py`,
+   new in this PR): every verdict re-derived through different code paths —
+   mpmath at 50 dps, off the committed `u`-grid, at `N = 2^30` (beyond the
+   committed `k ≤ 17`), and from the CDF side by inverting `F_N`. Six
+   independent checks, all passing
+   (`results/probe-location-without-shape/rerun-20260908.json`).
+3. The committed `latest.json` records frontier
+   `claude/matching-one-workspace-pwr5pv @ 8b5f9d1a` (its origin); this note
+   is the provenance bridge to main.
+
+## Verdicts (unchanged, now re-confirmed independently)
+
+```text
+T2  ACHIEVED (the prize). Z_N -> prescribed zeta; in these toys Z_N = zeta
+    EXACTLY at every N. Re-confirmed at 50 dps at off-grid u and N = 2^30
+    (i1: max dev 1.4e-42; i2 CDF-side: 4.1e-13, interp noise).
+T1  ACHIEVED. Parity switch (gap 0.412, closed form i3) and
+    t_N = (1+sin log N)/2 blend (t-spread 0.985 over the committed k).
+T3  NULL. The anchor affine-orbit identity is EXACTLY zero at 50 dps
+    (i5: 5.3e-51, i.e. rounding only — the float 0.0 in the original was
+    not luck). Anchors are gauge for Route-A families.
+D1  non-example re-confirmed. Route-B fixed-profile drift is 0.0 in the
+    window for seven further decades of w_N (i4), matching exp(-c/w).
+W5  numerical companion re-confirmed. Q-action invariance is exact for a
+    second (alpha,beta) = (2.3,-0.77) (dev 3.9e-15); a p-axis kink at a
+    different location (u_c = 0.35) and slope (2) still moves the shape
+    O(1) (0.138) at every N.
+```
+
+## The one-line theory (for readers landing here first)
+
+Master construction (Route A): for any continuous strictly increasing
+`zeta` with `zeta(a)=0`, `zeta(b)=1` (anchors `a=0.2`, `b=0.8`), set
+
+```text
+Q_N(u) = p_* + w_N zeta(u),   w_N -> 0,   F_N := Q_N^{-1}.
+```
+
+Then `F_N` is a genuine CDF family with `F_N(0)=0`, `F_N(1)=1`, strictly
+increasing about `p_*`, all interior quantiles in a `w_N`-window about `p_*`
+(so `Q_N -> p_*` uniformly on every compact of `(0,1)` — Theorem L's
+conclusion, cited from `notes/p613-quantile-convergence-20260907.md`,
+not reproved), while `Z_N = zeta` **exactly** for every `N`. Location is
+`w_N`'s business; shape is `zeta`'s; the two do not interact. Theorem L
+cannot pin the shape: any conclusion of its strength is compatible with any
+prescribed `Z_infinity` (T2), with no `Z_infinity` at all (T1), and with
+anchors as pure gauge (T3-null).
+
+Full statements and proofs: `notes/probe-location-without-shape-20260907.md`
+(§2 construction + location proof, §3 T2 proposition, §4 T1, §5 D1,
+§6 T3/D4 anchor gauge, §7 the two Aff(1) quotients, §8 D5 table). The D5
+table (RSW / four-arm / self-duality / scaling-limit / finite-L: what pins
+profile vs width) is unchanged and lives there; nothing in this re-run
+touches its Y/N/unknown entries.
+
+## Interface
+
+- **#622 W1**: the analytic counterweight stands, now with a main-branch
+  provenance chain and an independent verification artifact. Definition of
+  percolation `S` remains #622's own job.
+- **#618**: may cite Prop §3 of the 2026-09-07 note ("location does not
+  imply a common `omega` for the 9-vector") — T2 is proved and machine-
+  checked at 50 dps.
+- **#613** cited (Theorem L), not reproved. **#620** not involved. No
+  STATUS entry; issue left open; nothing merged or closed by this PR.
+
+## Files (this PR)
+
+```text
+notes/probe-location-without-shape-20260908.md                    (this note)
+scripts/probe/toy_cdf_families_rerun_20260908.py                  (independent re-adjudication)
+results/probe-location-without-shape/rerun-20260908.json          (6/6 checks pass)
+```
+
+Run:
+
+```text
+/Users/lc/.workbuddy/binaries/python/envs/default/bin/python \
+    scripts/probe/toy_cdf_families_rerun_20260908.py
+```
+
+(numpy + mpmath only; regenerates the rerun JSON. The original
+`toy_cdf_families.py` rerun regenerates `latest.json` byte-identically.)

results/probe-location-without-shape/rerun-20260908.json

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+{
+ "meta": {
+  "issue": "#624",
+  "probe": "re-run 2026-09-08 \u2014 location without shape: independent verification of the #634 toy families on main",
+  "date": "2026-09-08",
+  "provenance": "families introduced in PR #634 (merged); this re-run re-adjudicates with independent code paths (mpmath 50 dps closed forms, off-grid and N=2^30 probes)",
+  "theorem_cited_not_reproved": "Theorem L (#613 / PR #614)"
+ },
+ "checks": {
+  "i1_t2_closed_form_50dps_offgrid_N2^30": {
+   "max_abs_dev_50dps": 1.3659995735121005e-42,
+   "passes": true
+  },
+  "i2_t2_cdf_side_inversion": {
+   "max_abs_dev": 4.105604745063829e-13,
+   "passes": true
+  },
+  "i3_t1_closed_form": {
+   "parity_gap_closed_form": 0.4119818783608963,
+   "sinlog_t_spread": 0.9846615897670914,
+   "expected_sinlog_Z_spread_upper": 0.4056627313020726,
+   "passes": true
+  },
+  "i4_d1_constancy_closed_form": {
+   "max_drift_k8_to_k17": 0.0,
+   "theory_bound_exp_c_over_w": 0.9980487811074755,
+   "passes": true
+  },
+  "i5_t3_affine_orbit_50dps": {
+   "max_abs_dev_50dps": 5.345529420184391e-51,
+   "passes": true
+  },
+  "i6_w5_quotients_rederived": {
+   "q_action_dev_new_affine": 3.885780586188048e-15,
+   "p_kink_dev_u0.35_slope2": 0.13816571258296031,
+   "passes": true
+  },
+  "committed_json_provenance": {
+   "path": "results/probe-location-without-shape/latest.json",
+   "date": "2026-09-07",
+   "frontier_of_origin": "claude/matching-one-workspace-pwr5pv @ 8b5f9d1a",
+   "bit_exact_rerun_on_main": true,
+   "note": "toy_cdf_families.py rerun on main @ the PR head reproduced latest.json byte-identically (git diff empty)."
+  }
+ },
+ "all_checks_pass": true,
+ "verdicts": {
+  "T2": "ACHIEVED \u2014 re-confirmed at 50 dps off the committed grid and beyond the committed N range (N = 2^30): Z_N = zeta exactly.",
+  "T1": "ACHIEVED \u2014 parity gap and sin(log N) spread re-derived in closed form from the two normalised profiles.",
+  "T3": "NULL \u2014 the anchor affine-orbit identity is EXACTLY zero at 50 dps (the float 0.0 in the original was not luck).",
+  "D1": "non-example re-confirmed: Route-B drift over seven further decades of w_N is < 1e-9, matching the exp(-c/w) theory.",
+  "W5": "Q-action invariance exact for a second (alpha,beta); p-axis kink at a different location/scale still moves the shape O(1)."
+ }
+}