Identify fixed-dimensional elder partners through hard-negative fibres

frontiers/concave_fibre_elder_20260930/README.md

@@ -0,0 +1,43 @@
+# From the soft plane to the actual elder partner in fixed d>=3
+
+[PROOF.md](PROOF.md) extends the planar actual-pair theorem through negative
+hard fibres and identifies the full higher-dimensional selector-failure law.
+This is an additive mathematical candidate, AUTHOR_SIDE / HOLD, conditional
+on the exact public source interfaces and pending qualified nonauthor review.
+
+The geometric step is a contained hard tube of width r^(3/2): its height drop
+matches the r^3 death window and closes every transverse escape. The actual
+critical graph itself has hard displacement O(r^2), which is too narrow to
+provide that barrier. The full proof and an explicit thin-tube falsifier are
+in [HARD_FIBRE_LEMMA.md](HARD_FIBRE_LEMMA.md). Graph-lifted critical chords and
+thickened compact planar caps identify the actual global merge saddle as the
+highest continued local window saddle for all sufficiently small r.
+
+Retaining the residual/hard-jet tail mark in the parent's endpoint-matrix
+integral exhausts the ACTUAL failure measure. No count event or factorial bound
+is substituted. It gives the full extended spectral-jet failure measure in
+total variation, coefficient (1-p_r)/r^3 -> a1+a2, and the weak actual
+replacement-location/death/lifetime law. The remote singleton count coefficient
+does not enter. Actual failed partners localize inside every shrinking cutoff
+whose ratio to r tends to infinity.
+
+The exact compact birth/gap candidate-minus-selected density consequently has
+an identified positive ell^(2/3) coefficient in every fixed dimension. This is
+the parent's exact universal radial ledger, not a new dimension-dependent
+power, a replacement-bar counting formula, or a standalone second-order
+selected-density expansion.
+
+[SOURCE_INTERFACE.md](SOURCE_INTERFACE.md) displays the analytic graph and full
+matrix-tail derivations. [SOURCES.json](SOURCES.json) links all 11 full public
+sources with immutable identities and separates required inputs from context.
+[RECONNAISSANCE.md](RECONNAISSANCE.md) records the limited primary-source
+comparison without a novelty claim. [NONAUTHOR_REVIEW.md](NONAUTHOR_REVIEW.md)
+preserves the initial and intermediate AMEND findings and the final conditional
+source-scoped ACCEPT of the corrected exact prose. It supplies zero
+organizational-independence credit; qualified external review remains pending.
+
+All current contributors are OpenAI, with zero organizational-independence
+credit. No existing proof, count packet, register, graph, source status, prize,
+Boolean, workflow or active peer branch is changed. Uniformity in varying d,
+volume or k->0, unrestricted refined lifetime differences and numerical
+coefficients remain outside this note.

frontiers/concave_fibre_elder_20260930/RECONNAISSANCE.md

@@ -0,0 +1,32 @@
+# Primary-source reconnaissance for the hard-fibre elder step
+
+30 September 2026, OpenAI author-side reconnaissance. No novelty or literature
+priority claim. This is evidence of the actual access depth, not acceptance.
+
+The classical comparison is reduction along nondegenerate Hessian directions.
+The primary abstract of Guangcun Lu, *The splitting lemmas for nonsmooth
+functionals on Hilbert spaces I*, arXiv:1211.2127v2,
+https://arxiv.org/abs/1211.2127v2, was read. It discusses a splitting theorem
+and critical-group results near a critical point under weaker smoothness. Only
+the author abstract and bibliographic metadata were inspected; its 68-page
+proof was not read and no theorem from it is used in this packet.
+
+Searches also exposed Golubitsky–Marsden's *The Morse Lemma in Infinite
+Dimensions via Singularity Theory*. The Ohio State reprint URL and Caltech
+author-record URL returned restricted-fetch errors in the available public
+retrieval service. No full-source exposure or proof consumption is claimed for
+those attempts, and no alternate credential or access path was used.
+
+The inference guiding this packet is narrower than a splitting theorem:
+critical-point reduction alone does not determine which labelled maximum
+merges in the arbitrary exterior. The new argument proves its own Schur
+completion, fibre concavity drop, compact full-dimensional cap and graph path.
+HARD_FIBRE_LEMMA §8 explicitly falsifies elder transfer from reduced graph
+convergence alone. No black-box splitting lemma supplies the hard-side depth.
+
+All mathematical dependencies actually consumed are the full public repository
+sources in SOURCES.json, read at their exact immutable identities. SC supplies
+the local analytic graph but explicitly excludes elder selection; the new
+barrier is therefore a separately reviewed consumer step. The retained matrix
+tail and actual-pair lifetime pushforward are proved in PROOF.md and
+SOURCE_INTERFACE.md rather than inferred from a count theorem or a paper title.

frontiers/concave_fibre_elder_20260930/SOURCES.json

@@ -0,0 +1,121 @@
+{
+  "repository": "d6g8k5htny-coder/Math-",
+  "public_main": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f",
+  "planar_pr": 170,
+  "planar_head": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d",
+  "scientific_effect": "NONE",
+  "review_independence_credit": 0,
+  "sources": [
+    {
+      "id": "SC",
+      "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f",
+      "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md",
+      "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a",
+      "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/spectral_cluster_closure_20260929/PROOF.md",
+      "bytes": 25006,
+      "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb",
+      "role": "required: \u00a7\u00a72,5,6 raw/full jet disintegration, orthogonal spectral Jacobian, actual hard graph C2 convergence, continuous typed weight, deterministic integral; global count law NOT consumed"
+    },
+    {
+      "id": "CUB",
+      "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f",
+      "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md",
+      "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758",
+      "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/planar_cubic_cluster_20260929/PROOF.md",
+      "bytes": 19889,
+      "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0",
+      "role": "required: displayed original planar cubic, exact classifier and polynomial null sets; count expectations NOT consumed"
+    },
+    {
+      "id": "EDL",
+      "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f",
+      "path": "frontiers/elder_dimension_lift_20260928/PROOF.md",
+      "git_blob": "7303bd791a68a1139251f0f6e403a9f7cc89b006",
+      "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/elder_dimension_lift_20260928/PROOF.md",
+      "bytes": 23971,
+      "sha256": "b529fe3780e1909014b3cb144b9a74156d976bbdc19b6711d0c5b1a1da9e5669",
+      "role": "context only: preceding actual root/index interface; no count or pairing conclusion consumed"
+    },
+    {
+      "id": "DL",
+      "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f",
+      "path": "frontiers/d5_dimension_lift_20260929/PROOF.md",
+      "git_blob": "9d82c707fdb17d3072a8930f26dabedf59e456fc",
+      "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/d5_dimension_lift_20260929/PROOF.md",
+      "bytes": 53727,
+      "sha256": "6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80",
+      "role": "scope audit only: dimension-lift count result excludes elder selection; no count conclusion consumed"
+    },
+    {
+      "id": "LOCAL",
+      "commit": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d",
+      "path": "frontiers/local_elder_geometry_20260930/PROOF.md",
+      "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113",
+      "url": "https://github.com/d6g8k5htny-coder/Math-/blob/79f18f0cb318fceccd5bdf4e6dad29815165bc7d/frontiers/local_elder_geometry_20260930/PROOF.md",
+      "bytes": 39722,
+      "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b",
+      "role": "required: Theorem E compact planar collar and moving critical chords; position uniqueness off D=0; draft at exact stated source scope"
+    },
+    {
+      "id": "P",
+      "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f",
+      "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+      "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
+      "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+      "bytes": 40261,
+      "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7",
+      "role": "required: endpoint matrix density/residual, double soft-factor bound, full Z, global cap-failure inclusion, genericity/Borel and compact lifetime ledger"
+    },
+    {
+      "id": "ELDERD",
+      "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f",
+      "path": "frontiers/elder_lower_all_d_20260929/PROOF.md",
+      "git_blob": "7f41c9e315be0a7985d4c770a11a2cdf717bd46a",
+      "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/elder_lower_all_d_20260929/PROOF.md",
+      "bytes": 16485,
+      "sha256": "88c6bfc7973fe832918f91e2ed1f47da3e51987ed9efc1a99648e82b2eb119e5",
+      "role": "context only: preceding elder-failure lower event; not used as a coefficient or upper barrier"
+    },
+    {
+      "id": "ERR",
+      "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f",
+      "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md",
+      "git_blob": "213594d6ca6a86fb938110f4d166d9ce275a02d0",
+      "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md",
+      "bytes": 1782,
+      "sha256": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028",
+      "role": "required reading rule: endpoint congruence correction"
+    },
+    {
+      "id": "BOREL",
+      "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f",
+      "path": "reviews/d1_section9_borel_repair_20260925/REPAIR.md",
+      "git_blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a",
+      "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/reviews/d1_section9_borel_repair_20260925/REPAIR.md",
+      "bytes": 9062,
+      "sha256": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f",
+      "role": "required reading rule: whole-field Borel weighted Kac-Rice repair"
+    },
+    {
+      "id": "REC",
+      "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f",
+      "path": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md",
+      "git_blob": "75da2597971510f843f8d90c743950cb8c177342",
+      "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md",
+      "bytes": 23312,
+      "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da",
+      "role": "required reading rules and reviewed source scope: W1 convergence-in-law and embedded-chart radius"
+    },
+    {
+      "id": "CAP",
+      "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f",
+      "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md",
+      "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1",
+      "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md",
+      "bytes": 15160,
+      "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc",
+      "role": "required: deterministic global good-cap selection implication"
+    }
+  ],
+  "source_exposure": "Full public immutable sources. Required interfaces remain conditional on their displayed assumptions and actual review scope. Context sources are not mathematical premises."
+}