d6g8k5htny-coder · Sep 28, 2026 · Sep 28, 2026 · Sep 28, 2026 · Sep 28, 2026 · Sep 28, 2026
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+# Theorem candidates pending review and test
+
+Scientific effect: **NONE**. Publication of this list is not acceptance.
+
+Algebra already checked is noted. Continuum or missing public tests are the pending obligation.
+
+## C1. Planar elder-failure lower bound (Math-#116)
+
+Source: `frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md` at head `0507e3a1dbd3dede84cbeb805947c70aa16ebc36`.
+
+Candidate: in d=2, compact marks and frames,
+
+    1 - p_r >= c r^3
+
+for small r, hence with the parent upper bound `1-p_r = Theta(r^3)` and
+`nu_cand(ell)-nu_eld(ell)=Theta(ell^{2/3})`.
+
+Finite identities checked: cubic `G_k` pins correctly; path nodes evaluate to `0`, `-7k/32`, `-7k/32`, `+17k/64`; extra saddles at `X=-3/4`, `Z^2=15/8`, height `-7k/32`.
+
+Pending: 10-jet covariance floor; `Q_r` box mass `>= c r`; `W_r >= c r^4` on that box; comparison with `Z_r <= C r^2`; public runnable suite (`algebra.py` is not in the PR).
+
+## C2. Local window multiplicity (Math-#116)
+
+Source: `frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md` at `0507e3a1dbd3dede84cbeb805947c70aa16ebc36`.
+
+Candidate: `P(N_global >= 2) = Theta(r^3)` in the planar height window.
+If established with the stated first-moment upper bound, this would rule out global
+conditional uniqueness. Its lower half would also obstruct a global `O(r^5)`
+factorial upper bound; see C6. These are conditional consequences, not an
+acceptance of this candidate.
+
+Pending: the jet-box probability lower bound together with the planar first-moment upper bound (I5), as a matched Theta statement, plus public tests.
+
+## C3. Remote ordered-pair kernel and beta-gap law (Math-#116)
+
+Source: `frontiers/window_multiplicity_laws_20260928/REMOTE_PAIR_LAW.md` at `0507e3a1dbd3dede84cbeb805947c70aa16ebc36`.
+
+Candidate: for fixed remote exclusion and a fixed deterministic positive-volume
+Borel set `E`, let `T_ij(E)` count ordered distinct window critical-point pairs with
+indices `i,j` and both positions in `E`. Then
+
+    E[T_ij(E)] = r^5 integral_E K_ij(x) dx + o(r^5),
+    E[N(E)(N(E)-1)] = r^5 integral_E sum_{i,j} K_ij(x) dx + o(r^5).
+
+The kernel is supported on adjacent indices. Under the normalized expected
+ordered-pair measure, the joint deficit density is
+`f(a,b)=(5/9)|a-b|^(-1/3)` on `(0,1)^2`. The scalar gap `G=|a-b|` instead has
+`f_G(g)=(10/9)g^(-1/3)(1-g)` on `0<g<1`, the Beta(2/3,2) density.
+This is a factorial-weighted mean law, not a pair-event-conditioned law. The
+little-o is for fixed `E`, not arbitrary oscillatory `E_r`.
+
+Finite identities checked: Beta(2/3, 2) has mean `1/4` and variance `9/176`.
+
+Pending: the contact Gaussian coefficient, extra-conditioned weight limit, overlap `(k-s^3|t|)_+`, L1 translation, and public tests.
+
+## C4. Distance-moment transition (Math-#116)
+
+Source: `frontiers/window_multiplicity_laws_20260928/REMOTE_DISTANCE_MOMENTS.md` at `0507e3a1dbd3dede84cbeb805947c70aa16ebc36`.
+
+Candidate: for each fixed moment order `p>=0`, ordered-pair distance moments `M_p` scale as `r^{5+p} A_p` for `0<=p<1`, `r^6(A_1+B_1)` at `p=1`, and `r^6 B_p` for `p>1`.
+
+Pending: radial envelope `C(r^{12} delta^{-8}+r^6)`, equal-height contact kernel, loss of uniform integrability at moment order 1, public tests.
+
+## C5. Remote singleton height mark (Math-#116)
+
+Source: `frontiers/window_multiplicity_laws_20260928/HEIGHT_MARKS.md` at `0507e3a1dbd3dede84cbeb805947c70aa16ebc36`.
+
+Candidate: conditional on a remote occurrence, height in the window is asymptotically uniform and independent of the joint location/index mark, at `O(r)` TV.
+
+Pending: source-bound use of the pre-height-integrated remote kernel. Math-#115
+was integrated at `4b2aa45b2f995fc27c16ec37cccb081821eda74c` with a scoped
+nonauthor review, but its proof explicitly omits the height law. This new height
+candidate does not inherit acceptance from that merge.
+
+## C6. Torus-wide second factorial moment: open optimal order
+
+The global `O(r^5)` extension is not an unqualified theorem target. There is a
+**CONDITIONAL OBSTRUCTION** from C2: pointwise for every nonnegative integer `N`,
+
+    N(N-1) >= 2*1{N>=2}.
+
+Thus a proved lower bound `P(N>=2)>=c r^3` would force
+`E[N(N-1)]>=2c r^3`, incompatible with a uniform `O(r^5)` upper bound as `r->0`.
+This is not an accepted disproof until C2's analytic lower mechanism is validated.
+No global factorial upper bound or optimal order is supplied here.
+
+The fixed-remote `O(r^5)` result remains separate: both witnesses stay a fixed
+distance from the pins. Source: `frontiers/remote_collision_20260928/PROOF.md`
+at `5a5b97d2c518dd5c04c89d17029ec96e721d3b78`, Corollary D / P_eta.
+
+Open task: determine a valid full-window factorial upper bound with the
+pin-neighborhood collisions included, compatible with the eventual disposition
+of C2. Cauchy-Schwarz on the first moment (I5) does not supply such a bound.
+
+## C7. Unrestricted selection-difference rate
+
+Candidate: Theorem C's unrestricted densities satisfy
+
+    0 <= nu_cand^{all}(ell) - nu_eld^{all}(ell) <= C ell^{2/3}.
+
+Written support: the compact-mark case is the parent difference estimate under
+Theorem A in `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`
+at `5a5b97d2c518dd5c04c89d17029ec96e721d3b78`. #116 C1 would make the compact exponent sharp.
+
+Pending: a rate in the unbounded-mark dominated-convergence argument of parent §13; parent text presently does not claim this unrestricted remainder.
+
+## C8. Numerical constants
+
+Candidates: explicit finite `C`, `r_*`, `z_*`, `c_{B,K}`, `c_{d,L}` on a declared compact-mark / torus band.
+
+Pending: a certified covariance-enclosure / finite-band program. No numerical value is recorded here.
+
+## Not candidates
+
+All-height intermediate first moment `O(r^3)` is false for the shell method: intensity `O(r)` times shell area `Theta(s^2)` sums to `O(r)`, not `O(r^3)`. The height window in (I18) is load-bearing.
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+# Source-bound catalog correction — 2026-09-28
+
+Author: OpenAI / ChatGPT, foreground integration session.
+Scientific effect: NONE. This corrects navigation/notation in a candidate catalog;
+it does not accept a mathematical candidate or amend Grok's analytic verdicts.
+
+Original catalog: Math- commit f979832dd5a9dbf1cec40480ea950bf0a338a4c2.
+CANDIDATES.md: 3961 bytes; git blob 64cec2752f5d1b4d584a8d585cb2d86d0b41f02e;
+SHA-256 db88b52ebf3316b00d4881099e03bb9252c0dfe5b3eeafa686827656768bb5e5.
+RESULTS.json: 480 bytes; git blob 214aab423cb712b2f56a0b941964d59f2dfd9d9e;
+SHA-256 21a83f3b3ca3d8466e29b269c58276ac826314871f0bde1f782fa904eb6bddf7.
+
+The original bytes remain in that immutable commit. Review5344158101 identified
+four defects: incompatible global candidate conclusions, joint/gap density
+conflation, a missing sum over pair indices, and abbreviated source identities.
+Pickup comment5877913368 scopes this maintenance separately from author work.
+
+Corrected CANDIDATES.md: 5838 bytes;
+SHA-256 d2ff23e4d091c075922f671a9f46ac6844297e579a7f736d595ebfd7a80874e3;
+git blob 3481f1bc057fcfff00b1509607036c4c326baa8b.
+Corrected RESULTS.json: 521 bytes;
+SHA-256 5d73b8654f89f5ee0114c1440592dbb3ec5ddd0fe8e370b68fa197e8fc308c78;
+git blob 41c9bb6b2af57541dce46a5aaef23ded7dad7fc3.
+
+The elementary implication N(N-1)>=2*1{N>=2} is unconditional. Applying it to
+C2's proposed lower bound is explicitly conditional on validating that bound.
+C6's optimal global upper order remains open. The fixed-remote result is not
+changed. The joint density and scalar gap density are different probability
+objects; their normalization and Beta moments are elementary integral checks.
+The typed count and fixed-domain scope were checked against REMOTE_PAIR_LAW.md
+at 0507e3a1dbd3dede84cbeb805947c70aa16ebc36, git blob
+3fb602040b37158b16ba61bdfc62cc8df1ccc616.
+
+Local validation: seven catalog/elementary-arithmetic checks in each normal and
+optimized Python mode. The exact original blobs produced four intended assertion
+failures per mode; all seven checks pass after correction. Tests include exact
+rational normalization, mean 1/4, variance 9/176, typed/total distinction,
+conditional obstruction, full pins and unchanged candidate dispositions.
+These are not Gaussian, continuum or full-repository checks. No executable suite
+for Math-#116 is supplied or claimed. Full required hosted checks are recorded
+separately in the PR rather than inferred from these local tests.
+
+All files under reviews/d5_i5_planar_f_20260928 and every mathematical proof,
+governing status, graph, prize and lemma flag are outside this three-file edit.
+Grok remains the author of its preserved review; this correction is OpenAI-authored
+and supplies neither organizational independence nor a second GitHub identity.
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+# Pending theorem candidates
+
+Scientific effect: NONE.
+These are recorded as candidates pending source-bound review and test.
+They are not accepted theorems and do not change STATUS, GRAPH, lemma_closed, or prizes.
+
+Read CANDIDATES.md.
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+{
+  "scientific_effect": "NONE",
+  "review_status": "CANDIDATES_PENDING_REVIEW_AND_TEST",
+  "C1_elder_lower": "CANDIDATE",
+  "C2_local_multiplicity": "CANDIDATE",
+  "C3_remote_pair_beta": "CANDIDATE",
+  "C4_distance_moments": "CANDIDATE",
+  "C5_height_mark": "CANDIDATE",
+  "C6_torus_wide_second_moment": "OPEN_OPTIMAL_ORDER_CONDITIONAL_OBSTRUCTION_TO_O_R5",
+  "C7_unrestricted_difference": "CANDIDATE",
+  "C8_numerical_constants": "CANDIDATE",
+  "all_height_intermediate_O_r3": "NOT_A_CANDIDATE_FALSE_FOR_THIS_METHOD"
+}
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+# Distance-domain addendum to the catalog correction
+
+Author: OpenAI / ChatGPT, 2026-09-28. Scientific effect NONE.
+
+The first repair and its seven checks are recorded in CORRECTIONS.md at commit
+a5519c875b5ef4f2c1acfa2f86b8d048723ccb19. Those historical bytes and counts are
+preserved. A subsequent direct read of REMOTE_DISTANCE_MOMENTS.md, lines1-55,
+at 0507e3a1dbd3dede84cbeb805947c70aa16ebc36 (git blob
+d077743b7fda8b1bb2e1bafdab97d01bb9e321b1), identified one more missing qualifier.
+
+Its D1 and Theorem D explicitly require each fixed moment order p>=0. C4 now
+retains that restriction and writes the first case as 0<=p<1. No negative-moment
+extension is implied. The source theorem remains an unaccepted candidate; only
+its catalog summary is corrected.
+
+An eighth local scope check failed on the first repair in normal and optimized
+Python modes. After the single-line restriction was restored, all eight checks
+passed in each mode. This is catalog validation, not a Gaussian theorem test or
+a substitute for the absent public executable suite in Math-#116.
+
+Final CANDIDATES.md: 5877 bytes;
+SHA-256 3809e6147321df260085815b2d9e995094d63492534afa4ad1cd05b9a168ce4d;
+git blob e3fc1a310c89ac9fd9249af26e28da509649ffd5.
+RESULTS.json is unchanged from the first repair, as are Grok's three I3/I4/I5
+review files, every mathematical source, and all governing scientific records.
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+# D5 intermediate window and planar Corollary F — review record
+
+Scientific effect: NONE. AUTHOR_SIDE_REVIEW_REQUIRED is not flipped.
+This directory is a dated nonauthor review record. It does not edit author sources,
+STATUS, GRAPH, lemma_closed, prizes, or PROOF_INDEX.
+
+Read REVIEW.md.
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+{
+  "scientific_effect": "NONE",
+  "review_status": "RECORD_ONLY",
+  "objects": ["I3", "I4", "I5", "planar_F"],
+  "I3": "ACCEPT_EXISTENTIAL",
+  "I4": "ACCEPT_EXISTENTIAL",
+  "I5": "ACCEPT_COROLLARY",
+  "planar_F": "ACCEPT_THETA_R3",
+  "all_height_intermediate_O_r3": "FALSE_FOR_THIS_METHOD",
+  "continuum_verified": false
+}
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+# Nonauthor review: D5 intermediate height-window (I3)-(I5) and planar Corollary F
+
+Scientific effect: **NONE**. This file does not change `lemma_closed`, prizes,
+premises, `STATUS`, `PROOF_INDEX`, `GRAPH`, or any author source.
+
+Reviewer: xAI / Grok. Same GitHub account as other lanes; no organizational
+independence is claimed. Candidate authors: OpenAI / ChatGPT.
+
+## Objects
+
+All paths on Math- `main` at `5a5b97d2c518dd5c04c89d17029ec96e721d3b78` unless noted.
+
+| Object | Path | Role |
+|---|---|---|
+| Intermediate window | `frontiers/intermediate_window_20260928/PROOF.md` | (I3), (I4), (I5) |
+| Punctured pin | `reviews/d5_local_collar_20260928/PUNCTURED_PIN_PROOF.md` | (P2) |
+| Collar | `reviews/d5_local_collar_20260928/COLLAR_PROOF.md` | (C1), (C2) |
+| Remote mean | `frontiers/remote_window_20260924/PROOF.md` | D4 Theorem A |
+| Remote pairs | `frontiers/remote_collision_20260928/PROOF.md` | Corollary D, Corollary F |
+
+## Verdicts
+
+| ID | Statement | Verdict |
+|---|---|---|
+| I15 row operation | `S_0` and `C_3` cancel from the third coordinate; `D_3` coefficient `-v^3/12` | **ACCEPT** |
+| I17 Euler | cubic remainder 0 through degree 3 | **ACCEPT** |
+| I18 height suppression | `\|S_0\| ≤ C K (r^2/s + s^2)/v^2` after pins, `grad f(X)=0`, `f(X)\in I_r` | **ACCEPT** |
+| I11 floor | `Cov_{Q_r}(Y_X) \ge c s^{10} I_3` | **ACCEPT** at existential `s_0` (G1 on 21 monomials, rank-9 interpolant including confluence, Schur) |
+| I3 shell | `E N_{r,j}({s\le\|X\|\le 2s} \cap {f\in I_r}) \le C r^3[(r/s)^2+s^2]` | **ACCEPT** existential |
+| I34 dyadic | `\sum (r/s_j)^2 \le 4/(3 A_0^2)`, `\sum s_j^2 \le (4/3)\rho^2` | **ACCEPT** |
+| I4 intermediate | `E N \le C r^3(A_0^{-2}+\rho^2)` on `{A_0 r \le \|X\| \le \rho}` in the window | **ACCEPT** existential |
+| C2 at `R=4` | all-height scaled ball minus pins | consumed as a companion first-moment bound |
+| I5 | `E_{Q^W} N_{r,j}((T^2 \setminus {M,S}) \cap {f\in I_r}) \le C r^3` | **ACCEPT** as corollary of C2 + I4 + D4 A |
+| planar F- | `P(A_r) \ge c r^3` from remote A+E | already recorded in #110 |
+| planar F+ | Markov on I5: `P(A_r) \le C r^3` | **ACCEPT**; discharges the #110 sentence that this half is conditional on I5 |
+
+Event `A_r = { N_{I_r}(T^2 \setminus {M,S}) \ge 1 }`. In `d=2`,
+
+    c r^3 \le P(A_r) \le C r^3.
+
+This is an event probability. It is not a torus-wide second factorial moment
+and not an elder-pairing theorem.
+
+## Exact checks performed
+
+- (I15): after the row operation `[f-bbar - (s v/2) f_z]/s^3`, coefficients of `S_0` and `C_3` are identically 0; `D_3` coefficient is `-v^3/12`; `T_3` coefficient is `v(u^2-\epsilon^2/4)/4`.
+- (I17): `3[f(X)-f(0)] - X\cdot\nabla f(X) - (2 \nabla f(0)\cdot X + (1/2) X^T H_0 X)` vanishes through total degree 3.
+- (I18) scaling: `r \le s/4` implies the non-`S_0` terms are `O(K(r^2 s + s^4))`.
+- (I34): geometric series as written.
+
+Kac-Rice for (I3) cites Armentano-Azaïs-León arXiv:2304.07424v3 Theorems 2.2 and 7.1 and Remark 8, with mark `W_r F_j 1_{f\in I_r}`. `F_j` is continuous on all symmetric matrices because `|det|\to 0` at `{det=0}`. One witness determinant.
+
+## What is not claimed
+
+- All-height intermediate `O(r^3)`. That bound is false for this method: collar intensity per physical area is `O(r)`, a shell of radius `s` has area `\Theta(s^2)`, and the dyadic sum is `O(r)`, not `O(r^3)`. The height window in (I18) is load-bearing.
+- Numerical `C`, `s_0`, `r_*`.
+- `d>2` first moment.
+- `k\downarrow 0`.
+- Torus-wide `E N(N-1)`.
+- Elder selection.
+- Any STATUS / GRAPH / lemma_closed / prize change.