Far elder density: living bars have flat components, ν_eld^far(ℓ) = O(ℓ^N) for every N (author-side candidate; scientific effect NONE)

.github/workflows/far-elder-flat-ridge.yml

@@ -0,0 +1,86 @@
+name: Far elder flat-ridge exact controls
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/far_elder_flat_ridge_20260930/**'
+      - '.github/workflows/far-elder-flat-ridge.yml'
+      # bound sources (consumed and cited_only in SOURCES.json): a change to any of them must re-run the identity check
+      - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md'
+      - 'imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md'
+      - 'reviews/d1_section9_borel_repair_20260925/REPAIR.md'
+      - 'reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md'
+      - 'frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md'
+      - 'frontiers/c7_zero_gap_limit_20260929/PROOF.md'
+      - 'frontiers/unrestricted_selection_difference_20260929/PROOF.md'
+      - 'reviews/c7_nonvanishing_openai_20260929/REVIEW.md'
+      - 'frontiers/far_elder_rate_20260930/PROOF.md'
+      - 'frontiers/remainder_vanishing_20260930/PROOF.md'
+      - 'frontiers/remainder_rate_20260930/PROOF.md'
+      - 'frontiers/elder_third_order_20261001/PROOF.md'
+      - 'frontiers/third_order_rate_20261001/PROOF.md'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  exact-controls:
+    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: Verify sources, replay both modes, reject mutants
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S - <<'PY'
+          import hashlib, json, pathlib, subprocess, sys
+          root = pathlib.Path('frontiers/far_elder_flat_ridge_20260930').resolve()
+          manifest = json.loads((root / 'SOURCES.json').read_text())
+          present = sorted(p.name for p in root.iterdir())
+          expected_names = sorted([e['path'] for e in manifest['files']] + ['SOURCES.json'])
+          if present != expected_names:
+              raise ValueError('packet tree differs from manifest: ' + repr(present))
+          for entry in manifest['files']:
+              path = root / entry['path']
+              if path.is_symlink():
+                  raise ValueError('symlink: ' + entry['path'])
+              data = path.read_bytes()
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']:
+                  raise ValueError('packet file identity mismatch: ' + entry['path'])
+          for entry in manifest['consumed'] + manifest['cited_only']:
+              data = pathlib.Path(entry['path']).read_bytes()
+              framed = b'blob ' + str(len(data)).encode() + b'\0' + data
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or hashlib.sha1(framed).hexdigest() != entry['git_blob']:
+                  raise ValueError('merged source drifted: ' + entry['path'])
+          script = str(root / 'flat_ridge_check.py')
+          expected = (root / 'RESULTS.json').read_bytes()
+          for flags in (['-B', '-S'], ['-B', '-O', '-S']):
+              out = subprocess.run([sys.executable, *flags, script], check=True, capture_output=True, timeout=600).stdout
+              if out != expected:
+                  raise RuntimeError('output differs from RESULTS.json under ' + ' '.join(flags))
+              for m in manifest['mutants']:
+                  rc = subprocess.run([sys.executable, *flags, script, '--mutant', m], capture_output=True, timeout=600).returncode
+                  if rc != 1:
+                      raise RuntimeError('mutant ' + m + ' not rejected')
+              rc = subprocess.run([sys.executable, *flags, script, '--mutant', 'M9'], capture_output=True, timeout=600).returncode
+              if rc != 2:
+                  raise RuntimeError('unknown mutant label not rejected with exit 2')
+          print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False}))
+          PY
+          git diff --exit-code
+      - name: Replay the published C89 v1.1 reviewer controls on the exact sources, both modes
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S frontiers/far_elder_flat_ridge_20260930/c89_replay.py
+          python -B -O -S frontiers/far_elder_flat_ridge_20260930/c89_replay.py
+          git diff --exit-code

frontiers/far_elder_flat_ridge_20260930/C89_V1_1_PROOF.md

@@ -0,0 +1,307 @@
+# Quantitative separated jets and a shrinking far-elder region
+
+Object: OA-QSF-20261002-C89-v1.1. Author: OpenAI / Codex, root01a0bbb5.
+Dylan Roy — delegated AI work. Personal reading PENDING; organizational
+independence 0; scientific effect NONE. This is an additive analytic candidate,
+not a change to a scientific status register. Review disposition belongs in
+REVIEW.md, separately from this author statement.
+
+## 0. Exact interface and result
+
+Fix d >= 2 and L > 0. Use the centered, variance-one periodized Gaussian field
+on X = R^d/(L Z^d) of [P] §1, with Fourier weights
+
+    a_n = exp(-2 pi^2 |n|^2/L^2) / sum_k exp(-2 pi^2 |k|^2/L^2) > 0.
+
+[P] is `source/P.md`, SHA256
+9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7,
+40261 bytes. [G] is `source/G.md`, Math- PR188 head
+5e0b31c8304e3a6ef96a5ee28c8faab5f9fdf799, blob
+0b089b894960b0ee53fc2885e054d73ab73472bd, SHA256
+967148439e32e8e5d6c06a12df5cc7c1eb2475fe6828e4aeb3748c48817a4cba,
+29110 bytes. Its complete source interface is preserved in
+`source/G_SOURCES.json`. We consume [P] §§1–2 and [G]'s canonical marked
+Kac–Rice definition (0.1), deterministic Lemmas 1–3, and conditional genericity
+§2(d). We do not consume Remark 6 or the OA-FD/OA-MT consequences, the SIDE24
+coefficient, a finite-radius RN certificate, or a near-pair asymptotic.
+The marked-measure construction retains [G]'s E2 repair and REC interpretation;
+this note is not a new proof of those imported interfaces.
+
+Write O_y=(f(0),grad f(0),f(y),grad f(y)), v=(b,0,b-ell,0), Q for Gaussian
+regression on O_y=v, H_0,H_y for the two Hessians, and
+
+    W = |det H_0 det H_y| 1{H_0 negative definite, index H_y=d-1}.
+
+Let e_y be the ordinary global superlevel H_0 elder mark, not a branch-adjacency
+mark. With p_y the density of O_y, define the canonical per-unit-spatial-volume
+far-elder lifetime density
+
+    F(ell,rho) = integral_{dist(0,y)>=rho} integral_R
+                 p_y(v) E_Q[W e_y] db dy.                         (0.1)
+
+Let J=(d+1)(d+2)/2, the number of independent derivatives of orders 0,1,2 at
+one site; a symmetric Hessian has d(d+1)/2 coordinates, not d^2. Put
+
+    rho_* = min(1,L/4),
+    A_N = 24J + 12dN + 12N(N+1)(d+1).
+
+**Theorem QSF.** For every integer q>=1 set N=2(q+1). There is a finite constant
+C_q=C(d,L,q), independent of ell,rho, such that
+
+    F(ell,rho) <= C_q rho^(-A_N) ell^(q+1),                       (0.2)
+
+whenever 0<rho<=rho_* and 0<ell<=rho^2/(64N^2). In particular:
+
+(a) For any fixed 0<beta<=1/A_N, F(ell,ell^beta)=O(ell^q) as ell decreases
+to zero. Constants and the sufficiently small cutoff may depend on d,L,q,beta.
+
+(b) For any fixed a>0, c>0, rho(ell)=c[log(e/ell)]^(-a), and every fixed
+integer q>=1, F(ell,rho(ell))=O(ell^q) as ell decreases to zero.
+
+These are bounds on an actual elder-marked far population with a specified
+moving spatial cutoff. They are not uniform in arbitrary shrinking cutoffs,
+in L, d or q. They do not establish the contact-scale region, witness uniqueness,
+the intermediate region inside rho(ell), or either direction of the full
+canonical-witness/persistence correspondence. There is no weighted-Palm
+normalizer Z_r in (0.1); dividing by one would change the statement.
+
+## 1. Constructive quantitative finite-jet rank
+
+**Lemma QJ.** Fix s>=1 distinct sites x_1,...,x_s with pairwise torus distance
+at least delta, where 0<delta<=1. Let T be the vector of all independent
+coordinate derivatives through order two at these sites. Then
+
+    lambda_min Cov(T) >= c(d,L,s) delta^[24(s-1)].                (1.1)
+
+The constant is strictly positive and finite. It can be constructed from
+finitely many positive Fourier weights; no sampled-covariance inference occurs.
+
+Proof. Put omega=2pi/L and
+
+    B_j(x)=sum_{k=1}^d [1-cos(omega(x_k-x_{j,k}))].
+
+Choose coordinate differences in [-L/2,L/2]. The elementary inequality
+sin t >= 2t/pi for 0<=t<=pi/2 gives
+
+    B_j(x_i) >= 8 dist(x_i,x_j)^2/L^2 >= 8delta^2/L^2.
+
+The sum of absolute Fourier coefficients of B_j is at most 2d. For each i set
+
+    Q_i(x)=product_{j!=i} [B_j(x)/B_j(x_i)]^2.
+
+It has value 1 at x_i, vanishes to order at least four at every other site,
+has Fourier support |n|_infinity<=2(s-1), and has Fourier coefficient l1 norm
+at most C delta^[-4(s-1)]. All its derivatives through order two at x_i have
+the same bound, with a different C depending only on d,L,s. Here and below
+C denotes a defined finite bound from the finite coefficient sum and its
+frequency multipliers, never a constant allowed to depend on delta or the sites.
+
+For a multiindex alpha with |alpha|<=2 let R_{i,alpha}(h) be the Taylor polynomial
+through total degree two of
+
+    h^alpha / (alpha! Q_i(x_i+h)).
+
+The reciprocal is legitimate since Q_i(x_i)=1. If Q_i(x_i+h)=1+U_1(h)+U_2(h)
++O(|h|^3), its reciprocal through degree two is 1-U_1+U_1^2-U_2. Thus every
+coefficient of R_{i,alpha} is bounded by C delta^[-8(s-1)]. Define the periodic
+local coordinates z_k(x)=sin(omega(x_k-x_{i,k}))/omega and the trigonometric
+polynomial
+
+    P_{i,alpha}(x)=Q_i(x) R_{i,alpha}(z(x)).
+
+Since z(x_i+h)=h+O(|h|^3), its jets satisfy
+
+    partial^gamma P_{i,alpha}(x_j) = 1{i=j,alpha=gamma}, |gamma|<=2. (1.2)
+
+At other sites the fourth-order zero of Q_i kills every derivative in question.
+The Fourier support is |n|_infinity<=2s, and its coefficient l1 norm is bounded
+by C_* delta^[-12(s-1)], uniformly over i,alpha and all separated configurations.
+More precisely, choose C_Q to bound the coefficient l1 norm of Q_i after
+factoring out delta^[-4(s-1)], and C_R to bound the sum of absolute monomial
+coefficients of every R_{i,alpha} after factoring out delta^[-8(s-1)]. The
+preceding derivative and reciprocal formulas construct both finite constants.
+Each z-monomial of total degree at most two has Fourier coefficient l1 norm
+at most C_Z=max(1,omega^(-2)). By the triangle and convolution inequalities,
+the valid choice is C_*=C_Q C_R C_Z, a product of bounds, not their maximum.
+
+Let a_* = min_{|n|_infinity<=2s} a_n >0. For real coefficients t_{i,alpha} set
+L_t=sum t_{i,alpha} partial^alpha|_{x_i}. Fourier expansion gives
+
+    V_t=Var(L_t f)=sum_n a_n |L_t exp(i omega n.x)|^2.
+
+Writing P_{i,alpha}=sum p_n exp(i omega n.x), weighted Cauchy–Schwarz and (1.2)
+give
+
+    |t_{i,alpha}|^2 <= V_t sum |p_n|^2/a_n
+                     <= V_t a_*^(-1) C_*^2 delta^[-24(s-1)].
+
+Summing over sJ coordinates proves (1.1), with c=a_*/(sJ C_*^2). For s=1
+the empty product Q_i=1 supplies the same argument with exponent zero.
+Every subvector has at least this covariance lower bound. If a subvector is
+split into V,Z, its conditional covariance also has this lower bound: for z,
+the Schur-complement variational formula minimizes Var(z.Z-a.V) over a, and
+(1.1) bounds every such variance below by c delta^[24(s-1)] |z|^2. QED.
+
+This is deliberately conservative; it is a usable uniform power, not a sharp
+collision exponent. The positivity of only the indicated finite cube suffices
+for this rank estimate. The smooth field and moment arguments below additionally
+use the summability of all Fourier modes of the stated Gaussian covariance.
+
+## 2. Two-pin weighted moments, with the separation tracked
+
+Let V_y=(O_y,H_0,H_y), with distinct symmetric Hessian coordinates. Its dimension
+is 2J. Uniformly for dist(0,y)>=rho and 0<rho<=rho_*, Lemma QJ with s=2 implies
+
+    lambda_min Sigma_y >= c rho^24,
+    lambda_max Sigma_y <= C.                                      (2.1)
+
+The upper bound follows from the fixed finite derivative variances. Thus at
+the slice v=(b,0,b-ell,0,H_0,H_y), 0<ell<=1, the full Gaussian density satisfies
+
+    p_{V_y}(v) <= C rho^(-24J)
+                    exp[-c(b^2+|H_0|^2+|H_y|^2)].                 (2.2)
+
+The determinant prefactor uses dimension 2J; the exponent uses the uniform
+upper eigenvalue bound, not the deteriorating lower bound. The slice retains
+the b coordinate and all independent Hessian coordinates, so its Euclidean norm
+dominates b^2+|H_0|^2+|H_y|^2 up to fixed coordinate-norm constants.
+
+Put K=1+sup_X ||D^2 f||_op. For every finite p>=1, Gaussian regression gives
+
+    E[K^p | V_y=v] <= C_p rho^(-12p) (1+|v|)^p.                  (2.3)
+
+Here is a direct uniform justification. Expand the real field into independent
+standard-normal sine/cosine coefficients, with summable square-root spectral
+amplitudes times (1+|n|)^2. For each standardized coefficient xi, its conditional
+mean has absolute value at most sqrt(v^T Sigma_y^(-1)v), by covariance
+Cauchy–Schwarz; its conditional centered variance is at most 1. Minkowski's
+inequality, the spectral sum, and (2.1) bound the conditional L^p norm of K by
+C_p(1+rho^(-12)|v|). The residual coefficients need not be independent of one
+another. Infinite sums converge in conditional L^p(C^2) by the same summable
+majorant. Gaussian regression, not conditioning on a positive-probability pin
+event, defines these laws at every target v.
+
+Consequently, for p>=1 (and with the evident p=0 version),
+
+    M_p(ell,rho) := integral_{dist(0,y)>=rho} integral_R
+                    p_y(v) E_Q[W K^p] db dy
+                 <= C_p rho^[-(24J+12p)].                         (2.4)
+
+Indeed, disintegrate further onto the Hessians: p_y(v) times the conditional
+Hessian density is exactly p_{V_y}(v,H_0,H_y). W is V_y-measurable and at most
+|det H_0 det H_y|, a polynomially bounded function of degree 2d. Apply (2.2)
+and (2.3), integrate the polynomial times the Gaussian in b,H_0,H_y, and bound
+the y volume by L^d. This preserves the full determinant weight and integrates
+over all birth heights. No unweighted-to-weighted probability substitution or
+inverse random Hessian moment is used.
+
+## 3. N-site bound, retaining the weighted event
+
+Use exactly [G] §3's shells
+
+    S_j={x: |dist(0,x)-j rho/(2N)|<rho/(8N)}, j=1,...,N.
+
+Every tuple (0,y,q_1,...,q_N), q_j in S_j, dist(0,y)>=rho, is separated by at
+least delta=rho/(4N). Apply Lemma QJ with s=N+2 to the full two-jets, then take
+the subvector (V_y,Z), where Z=(f(q_j),grad f(q_j))_{j=1}^N has m=N(d+1)
+coordinates. Its conditional density given V_y, uniformly over all values, is
+bounded by
+
+    C_N rho^[-12N(N+1)(d+1)].                                    (3.1)
+
+Therefore the conditional probability of |f(q_j)-b|<=3ell and
+|grad f(q_j)|<=3sqrt(K_0 ell) at all N sites is at most
+
+    C_N rho^[-12N(N+1)(d+1)] K_0^(dN/2) ell^[N(1+d/2)].          (3.2)
+
+For 0<ell<=rho^2/(64N^2) and K_0>=2, [G] Lemmas 1–3 force on the elder event
+with K<=K_0 a near-critical ball of volume omega_d(ell/K_0)^(d/2) in every
+shell. The pathwise product-volume bound [G](3.1), Tonelli, and (3.2) yield
+
+    integral integral p_y E_Q[W e_y 1{K<=K_0}]
+      <= C_N rho^[-12N(N+1)(d+1)] K_0^(dN) ell^N M_0
+      <= C_N rho^[-24J-12N(N+1)(d+1)] K_0^(dN) ell^N.            (3.3)
+
+The shell product volume is at most L^(dN); this bound is independent of rho.
+The tower step is valid because W is V_y-measurable, while (3.2) is uniform
+in the conditioning values. No measurable choice of a point on each sphere
+is needed: the forced lower volume is pathwise, and only measurable shell
+integrals enter Tonelli. The conditional generic locus and the actual elder
+topology are those explicitly imported from [G]; no claim that a local saddle
+must be the elder saddle is introduced.
+
+For the complement, Markov's inequality and (2.4) give
+
+    integral integral p_y E_Q[W e_y 1{K>K_0}]
+      <= K_0^(-p) M_p
+      <= C_p K_0^(-p) rho^[-24J-12p].                            (3.4)
+
+This is Markov applied inside the determinant-weighted integral, not an
+unweighted tail probability subsequently divided by a normalizer.
+
+## 4. Threshold, moving cutoff, and error order
+
+Take K_0=2 rho^(-12) ell^[-1/(2d)]. It is at least 2. Equations (3.3)–(3.4)
+become, for any integers N>=1 and p>=1,
+
+    F(ell,rho) <= C_N rho^(-A_N) ell^(N/2)
+                  + C_p rho^(-24J) ell^[p/(2d)].                (4.1)
+
+Choose N=2(q+1), p=2d(q+1). Since rho<=1 and A_N>=24J, (0.2) follows.
+This explicit loss of half the small-box exponent avoids any need for uniform
+Borell constants in this extension.
+
+For rho=ell^beta, (0.2) is C_q ell^[q+1-beta A_N], at most C_q ell^q when
+beta A_N<=1. Since A_N>2, beta<1/2, so ell^(1-2beta)<=1/(64N^2) eventually;
+rho<=rho_* also eventually. Thus the domain restriction is satisfied, rather
+than silently discarded. More generally rho=c ell^beta with fixed c>0 works,
+changing constants and the small cutoff.
+
+For rho=c[log(e/ell)]^(-a), (0.2) is
+
+    C_q c^(-A_N) ell^(q+1) [log(e/ell)]^(a A_N).
+
+The elementary bound ell[log(e/ell)]^B=O(1) for each fixed B, and
+ell[log(e/ell)]^(2a)->0, prove both the asserted order and the domain condition.
+q is chosen before the limit; no uniformity over q is claimed.
+
+For example d=2,q=1 gives J=6,N=4,A_N=960, and the conservative valid choice
+rho=ell^(1/960) yields F=O(ell). This is much farther out than the contact
+scale ell^(1/3), and is not a closure of that region.
+
+For any one fixed moving-cutoff function in (a) or (b), the expected number per
+unit volume of bars with 0<ell<=t and endpoint separation at least rho(ell)
+is the integral of F(ell,rho(ell)) and is O(t^(q+1)). This counts a lifetime-
+dependent selection. It does not differentiate the cutoff or identify that
+count with the different observable using the common cutoff rho(t).
+
+## 5. Limits, negative controls, and remaining obligations
+
+All bounds hold for the fixed finite torus and stated covariance. The finite
+Fourier construction is quantitative even as sites approach, but its constants
+depend on the fixed number of sites. The proof makes no interchange of L with
+ell, no diagonal choice of N, and no claim of sharp constants. Nonnegative
+integrands justify Tonelli; (2.4) supplies the needed finite dominating weighted
+moments in every displayed integral.
+
+The accompanying exact rational Fourier controls check cardinal second jets
+on finite configurations, the need for the square in Q_i, the need for the
+reciprocal Taylor correction, repeated-site degeneracy, and the exponent
+ledger. They do not prove the uniform analytic statements: those are proved
+above. A finite spectral lower bound is used because it analytically constructs
+a dual for every configuration, not because a finite sample happened to be
+nonsingular.
+
+Explicit non-promotions:
+- Fixed-rho [G] alone did not imply this result; (1.1), (2.4), and (3.1) are
+  the new quantitative bridges.
+- A bound on the far elder subset does not control rejected candidates or
+  identify the full persistence coefficient.
+- The positive-power cutoff allowed for a given q is restricted by A_N. It
+  cannot be set equal to the contact scale, and the same beta is not claimed
+  to give every power q simultaneously.
+- The logarithmic cutoff permits each fixed order separately; it leaves a
+  large inner/intermediate region unresolved.
+- Shrinking multiple-witness collision, actual field confinement for the
+  QSA1 window, the complete two-sided persistence bridge, and imported parent
+  hypotheses retain their own statuses. No dependency is erased by this note.