Elder lower event in every fixed dimension: one-soft-direction Schur chart and sharp rejected moments

.github/workflows/elder-dimension-lift.yml

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+name: Elder dimension-lift exact controls
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/elder_dimension_lift_20260928/**'
+      - '.github/workflows/elder-dimension-lift.yml'
+  push:
+    branches: [main]
+    paths:
+      - 'frontiers/elder_dimension_lift_20260928/**'
+      - '.github/workflows/elder-dimension-lift.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  finite-dimension-lift:
+    runs-on: ubuntu-latest
+    timeout-minutes: 10
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Exact sources, both modes and intended semantic failures
+        run: |
+          set -euo pipefail
+          python -B -S frontiers/elder_dimension_lift_20260928/verify.py --output "${RUNNER_TEMP}/dimension-lift-${GITHUB_RUN_ID}-${GITHUB_RUN_ATTEMPT}"
+          git diff --exit-code

frontiers/elder_dimension_lift_20260928/README.md

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+# Dimensional lift of the direct elder-failure lower event
+
+Author-side candidate; nonauthor analytic review required. Scientific effect NONE.
+
+[PROOF.md](PROOF.md) extends the original planar cubic/path mechanism to every
+fixed d>=2 using a physical Hessian Schur chart and a determinant-one shear.
+Only one raw scalar band has width O(r); stable transverse modes stay O(1).
+The proposed direct result is 1-p_r>=c r^3 under the original full determinant
+tilt. Two additional index-(d-1) window critical points persist after correcting
+the stable coordinates by O(r^2).
+
+With separately named LP upper/density premises: the r^3 order and compact-mark
+ell^(2/3) density loss are sharp in every fixed dimension. The rejected-pair
+inverse-moment threshold is p<5/3; its endpoint divergence is logarithmic.
+This does not assert any new all-height/global factorial upper bound, uniform-in-d
+constant, numerical cutoff, unrestricted-mark difference or legacy q/p identity.
+
+Run the complete source/finite-control verification:
+
+    python -B -S verify.py --output /path/to/new-output
+    python -B -O -S verify.py --output /path/to/other-new-output
+
+The output directory must be new and outside this source directory. Each runner
+checks exact membership/bytes/SHA256/Git blob identities, executes both Python
+modes, and requires every semantic negative to fail the intended assertion with
+valid JSON and no execution errors. The 18 finite tests and 10 semantic mutants
+are not proofs of the Gaussian uniformity or inverse-function arguments.
+
+[Source map](SOURCE_MAP.json) separates direct inputs from conditional corollary
+inputs. [Reconnaissance](RECONNAISSANCE.md) records the limited literature read
+and Consensus quota limitation. No frozen source or review is rewritten.

frontiers/elder_dimension_lift_20260928/RECONNAISSANCE.md

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+# Limited primary-source reconnaissance — 28 September 2026
+
+Scope: derivative geometry and critical-point correlations near collisions;
+not a claim of novelty, first discovery, or an exhaustive priority audit.
+
+Consensus was called for Gaussian critical points, higher-dimensional Hessian
+Schur complements and elder persistence. It returned its monthly 30-search quota
+error with an October 1 reset. No result was invented or cited from that call.
+
+The following primary-source abstracts/publisher records were inspected with web
+search. No external theorem is used as an unexamined replacement for this proof:
+
+1. Jean-Marc Azaïs and Céline Delmas, *Mean number and correlation function of
+   critical points of isotropic Gaussian fields and some results on GOE random
+   matrices*, arXiv:1911.02300; Stochastic Processes and their Applications 150
+   (2022), 411-445, DOI 10.1016/j.spa.2022.04.013.
+   https://arxiv.org/abs/1911.02300
+   This already studies adjacent-index short-range correlation effects in the
+   unconditioned isotropic setting. Such adjacency by itself is not a novelty
+   claim for this project. It does not by its abstract establish this exact
+   pair-conditioned full-tilt elder obstruction or its compact-window loss.
+2. Louis Gass and Michele Stecconi, *The number of critical points of a Gaussian
+   field: finiteness of moments*, Probability Theory and Related Fields 190
+   (2024), 1167-1197, DOI 10.1007/s00440-024-01273-5.
+   https://doi.org/10.1007/s00440-024-01273-5
+   The record addresses finite moments under jet nondegeneracy. Its conclusions
+   are not automatically uniform at this coalescing conditioned pair, and no
+   global O(r^3) first moment or factorial upper bound is imported from it.
+
+The actual positive-spectrum, pin-frame and uniform-regression ingredients are
+read from the exact LP source in SOURCE_MAP.json. The new construction is proved
+thereafter with a raw physical-jet chart, not an assumed isotropic GOE law or
+independent soft eigenvector. Schur complementation, congruence, Taylor bounds,
+and nondegenerate-zero stability are classical; the proof writes out their role.
+The generic peer-review and governing status boundaries remain separate.

frontiers/elder_dimension_lift_20260928/RESULTS.json

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+{
+  "tests_per_mode": 18,
+  "semantic_mutants_per_mode": 10,
+  "identical_output_pairs": 11,
+  "all_intended_failures": true,
+  "scientific_effect": "NONE",
+  "gaussian_analysis_verified_by_code": false
+}

frontiers/elder_dimension_lift_20260928/SOURCE_FILES.json

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+{
+  "schema": 1,
+  "scientific_effect": "NONE",
+  "files": [
+    {
+      "path": "PROOF.md",
+      "bytes": 23971,
+      "sha256": "b529fe3780e1909014b3cb144b9a74156d976bbdc19b6711d0c5b1a1da9e5669",
+      "git_blob": "7303bd791a68a1139251f0f6e403a9f7cc89b006"
+    },
+    {
+      "path": "README.md",
+      "bytes": 1771,
+      "sha256": "64950f041cee272079f4dd5e04dcf2293f5ed6d8c3ee231df8817d2d10fce2ac",
+      "git_blob": "716c275954910521102d1e154ef526360f19d908"
+    },
+    {
+      "path": "RECONNAISSANCE.md",
+      "bytes": 2214,
+      "sha256": "d6f0bb97f4555ebb44b06df5d35d12f90fb746bbef45fb2053b8294d6621ed28",
+      "git_blob": "dd56b21f2da20be803f431357c5de49e6efbf452"
+    },
+    {
+      "path": "RESULTS.json",
+      "bytes": 205,
+      "sha256": "57dba071afd0e53e1317f165b5170dd0a69a456fef41d50b2f775d33ffd123f5",
+      "git_blob": "457fcb8bc2bcd84ad0b25c67be8869fefb9681b8"
+    },
+    {
+      "path": "SOURCE_MAP.json",
+      "bytes": 2106,
+      "sha256": "8cdedee5300ecc64bc5e38ea94d9bf263f0d87c4b75ccba67fb108edcf7013b1",
+      "git_blob": "713cad893e3d922c971a866f9b6accd89758f6d1"
+    },
+    {
+      "path": "lift.py",
+      "bytes": 5514,
+      "sha256": "e17fa8384b6d5e516223499efff26354da0d84de9ba3e9ea3d96cfe75373ee9d",
+      "git_blob": "03d6827ac66afeb04f70153108aa195668e58a22"
+    },
+    {
+      "path": "test_lift.py",
+      "bytes": 6109,
+      "sha256": "6901d40a216db622ee6cf82dc2092cdd7cd7dae261ea81b2c00fbb72a6f4a02b",
+      "git_blob": "c3ff861519c6b522ad8ca049c2f1aebecebae623"
+    },
+    {
+      "path": "verify.py",
+      "bytes": 4910,
+      "sha256": "f79ba38b27c40ec86b5d2bbec4f5c9891fbe27c62ba328b6d58769b6f04469dd",
+      "git_blob": "7bbd0612ebae56c8b05c97d22142f0086a3549cd"
+    }
+  ]
+}

frontiers/elder_dimension_lift_20260928/SOURCE_MAP.json

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+{
+  "scientific_effect": "NONE",
+  "source_repository": "d6g8k5htny-coder/Math-",
+  "source_commit": "d39dbbc25c25a7bc3f6c15b21f957a7fcac30615",
+  "inputs": [
+    {
+      "id": "LP",
+      "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+      "bytes": 40261,
+      "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7",
+      "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
+      "direct_use": "Exact model, original pins/tilt/selector, Sections 2-4 positive spectrum and contact regression; Section 8 generic locus/maximin.",
+      "conditional_corollary_use": "Theorem A upper bound and Sections 9-12 compact-mark density ledger/moments; not required for the new direct lower proof."
+    },
+    {
+      "id": "EL",
+      "path": "frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md",
+      "bytes": 10229,
+      "sha256": "8ce05a43bde8e0d718f09e2da1ffe75d3ec5893489254afdbf5b58b602cdba0c",
+      "git_blob": "aaefc8da9668e819591dd9db0b34442394f2362e",
+      "use": "Unchanged planar cubic/path and lower event; basis of d=2 and credited mechanism, not an assumed d>2 proof."
+    },
+    {
+      "id": "LOCAL",
+      "path": "frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md",
+      "bytes": 13100,
+      "sha256": "e6272e8709835ca6290fecb7034fa84832dde08c3c4cc990a48b8019e89e7c8d",
+      "git_blob": "b5c18aac5529c5f975ee5bb056b304debafc6697",
+      "use": "Planar normal-form and extra-saddle data. Its planar global first-moment upper estimate is NOT imported for d>2."
+    }
+  ],
+  "new_obligations": [
+    "Complete physical-jet covariance floor in every fixed dimension",
+    "Nonlinear raw Schur/shear chart with unit Jacobian and one shrinking coordinate",
+    "Conditional C4 control on raw jets before choosing the spatial shear",
+    "Full-dimensional endpoint Schur bounds and anisotropic saddle persistence"
+  ],
+  "pending_pair_distance_sources_consumed": false,
+  "historical_blocked_code_consumed_or_retried": false,
+  "nonauthor_acceptance": false,
+  "governing_status_changed": false
+}

frontiers/elder_dimension_lift_20260928/lift.py

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+"""Exact finite helpers for the one-soft-transverse-direction construction.
+
+No Gaussian continuum claim follows from these computations. Standard library only.
+"""
+from fractions import Fraction as F
+from itertools import product
+from math import factorial
+
+MUTANT=None
+
+def eye(n): return [[F(int(i==j)) for j in range(n)] for i in range(n)]
+def tr(A): return [list(c) for c in zip(*A)]
+def mm(A,B):
+    return [[sum((x*y for x,y in zip(row,col)),F(0)) for col in zip(*B)] for row in A]
+
+def solve(A,b):
+    n=len(A)
+    if n==0 or len(b)!=n or any(len(row)!=n for row in A):
+        raise ValueError('nonempty square linear system required')
+    W=[[F(x) for x in row]+[F(b[i])] for i,row in enumerate(A)]
+    for j in range(n):
+        p=next((i for i in range(j,n) if W[i][j]),None)
+        if p is None: raise ValueError('singular matrix')
+        W[j],W[p]=W[p],W[j]
+        pivot=W[j][j]; W[j]=[x/pivot for x in W[j]]
+        for i in range(n):
+            if i!=j:
+                v=W[i][j]; W[i]=[x-v*y for x,y in zip(W[i],W[j])]
+    return [row[-1] for row in W]
+
+def det(A):
+    n=len(A)
+    if not n:return F(1)
+    W=[[F(x) for x in row] for row in A]; out=F(1)
+    for j in range(n):
+        p=next((i for i in range(j,n) if W[i][j]),None)
+        if p is None:return F(0)
+        if p!=j:W[j],W[p]=W[p],W[j];out=-out
+        pivot=W[j][j];out*=pivot
+        for i in range(j+1,n):
+            z=W[i][j]/pivot
+            for k in range(j+1,n):W[i][k]-=z*W[j][k]
+    return out
+
+def rank(A):
+    W=[[F(x) for x in row] for row in A]
+    if not W:return 0
+    h=0
+    for j in range(len(W[0])):
+        p=next((i for i in range(h,len(W)) if W[i][j]),None)
+        if p is None:continue
+        W[h],W[p]=W[p],W[h];pivot=W[h][j]
+        for i in range(h+1,len(W)):
+            z=W[i][j]/pivot
+            if z:
+                for k in range(j,len(W[0])):W[i][k]-=z*W[h][k]
+        h+=1
+        if h==len(W):break
+    return h
+
+def chart(D,v,sigma):
+    q=solve(D,v);n=len(D)
+    sign=-1 if MUTANT=='wrong-schur-sign' else 1
+    a=F(sigma)+sign*sum((x*y for x,y in zip(v,q)),F(0))
+    A=[[a,*v]]+[[v[i],*D[i]] for i in range(n)]
+    S=eye(n+1)
+    for i in range(n):S[i+1][0]=F(0) if MUTANT=='omit-shear' else -q[i]
+    return A,S
+
+def multiindices(d,degree,upto=False):
+    return [a for a in product(range(degree+1),repeat=d) if sum(a)<=degree] if upto else [a for a in product(range(degree+1),repeat=d) if sum(a)==degree]
+
+def monomial(x,a):
+    out=1
+    for u,p in zip(x,a):out*=u**p
+    return out
+
+def jet_labels(d):
+    zero=(0,)*d; e=[tuple(int(i==j) for j in range(d)) for i in range(d)]
+    plus=lambda a,b:tuple(x+y for x,y in zip(a,b))
+    ex=e[0]; two=plus(ex,ex);three=plus(two,ex)
+    U=[zero,ex,two,three]
+    for j in range(1,d):U += [e[j],plus(ex,e[j])]
+    J=[a for a in multiindices(d,3,upto=True) if a not in U]
+    if MUTANT=='duplicate-cubic-pin':J[-1]=three
+    return U,J
+
+def _pmul(P,Q):
+    R={}
+    for a,c in P.items():
+        for b,e in Q.items():
+            k=tuple(x+y for x,y in zip(a,b));R[k]=R.get(k,F(0))+c*e
+    return {a:c for a,c in R.items() if c}
+
+def cubic_change(d,q):
+    """Map third-derivative coordinates by f -> f(x,z,w-q z); pure xxx stays fixed."""
+    labels=multiindices(d,3);M=[[F(0) for _ in labels] for _ in labels]
+    unit=[tuple(int(i==j) for j in range(d)) for i in range(d)]
+    substitutions=[{u:F(1)} for u in unit]
+    for j,v in enumerate(q):substitutions[j+2][unit[1]]=-v
+    def mf(a):
+        out=1
+        for x in a:out*=factorial(x)
+        return out
+    for j,a in enumerate(labels):
+        P={(0,)*d:F(1,mf(a))}
+        for i,power in enumerate(a):
+            for _ in range(power):P=_pmul(P,substitutions[i])
+        for i,b in enumerate(labels):M[i][j]=P.get(b,F(0))*mf(b)
+    return labels,M
+
+def rare_power(d):return d-1 if MUTANT=='rare-every-direction' else 1
+def weight_power(d):return 2*d if MUTANT=='all-directions-soft' else 4
+def band_width(r,k,epsilon):return 2*epsilon*k*(1 if MUTANT=='drop-band-jacobian' else r)
+
+def plane(X,Z,k,s,a,beta,c):
+    lin=0 if MUTANT=='drop-transverse-linear' else F(1,4)
+    value=2*k*X**3-F(3,2)*k*X-k/2+s*Z*Z/2+a*(X*X-lin)*Z/2+beta*X*Z*Z/2+c*Z**3/6
+    dx=6*k*X*X-F(3,2)*k+a*X*Z+beta*Z*Z/2
+    dz=s*Z+a*(X*X-lin)/2+beta*X*Z+c*Z*Z/2
+    return value,[dx,dz]
+
+def blocks(A,C,D):
+    return [list(A[i])+list(C[i]) for i in range(len(A))]+[list(row)+list(D[i]) for i,row in enumerate(tr(C))]
+
+def schur(A,C,D):
+    invC=tr([solve(D,row) for row in C]);Q=mm(C,invC)
+    if MUTANT=='ignore-cross-schur':Q=[[F(0) for _ in A] for _ in A]
+    return [[A[i][j]-Q[i][j] for j in range(len(A))] for i in range(len(A))]
+
+def endpoint_model(d,r,k):
+    stable=1 if MUTANT=='positive-stiff' else -1
+    def diag(vals):return [[v if i==j else F(0) for j in range(d)] for i,v in enumerate(vals)]
+    return diag([-6*k*r,-k*r/2]+[F(stable)]*(d-2)),diag([6*k*r,-5*k*r/2]+[F(stable)]*(d-2))
+
+def inertia(A):
+    """Exact Descartes signs for a real-rooted characteristic polynomial of a symmetric matrix."""
+    n=len(A);coef=[F(1)];B=eye(n)
+    for k in range(1,n+1):
+        AB=mm(A,B);c=-sum(AB[i][i] for i in range(n))/k
+        coef.append(c);B=[[AB[i][j]+(c if i==j else 0) for j in range(n)] for i in range(n)]
+    asc=list(reversed(coef));z=0
+    while z<n and asc[z]==0:z+=1
+    signs=[v*(-1)**i>0 for i,v in enumerate(asc) if v]
+    neg=sum(a!=b for a,b in zip(signs,signs[1:]))
+    return neg,z
+
+def loss_powers():
+    return (F(-1,3)+(0 if MUTANT=='omit-loss-factor' else 1),F(-2,3)-1)