IBA2-012: finite-radius cluster multi-soft bounds, including unequal scales

reviews/iba2_012_finite_radius_20261005/PROOF.md

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+# IBA2-012: a finite-radius additional-soft cluster estimate
+
+**Object:** IBA2-012-FINITE-R-CLUSTER-20261005-R3-v1.1.  
+**Author:** OpenAI / GPT-6 Astra Pro, continuation session `iba2-012-publication-audit-20261005-r3`, acting for Dylan Roy.  
+**Disposition:** NEW AUTHOR-SIDE PROOF CANDIDATE, conditional on the exact imported source interfaces below; nonauthor mathematical review required.  
+**Scientific effect:** NONE. No existing status, theorem, proof, register, review, flag or prize is changed. This is an additive publication of the preceding local proposal, not an accepted theorem or a review of its own mathematics. The author is exposed to the predecessor audit and other agents' reports; independence credit is zero.
+
+## 1. What this adds and what it does not
+
+SC §4 already establishes qualitative exclusion of further soft transverse eigenvalues from the fixed-radius, rescaled nonempty cluster measure. Codex `/root`'s main#259 comments 6003533425 and 6003547392 isolate and quantify the corresponding **limiting** measure: q additional hard eigenvalues in `(0,eta]` cost `eta^[q(q+7)/2]`. That work is credited, not reclaimed.
+
+A limiting estimate does not by itself allow `eta = r`. The present proposal proves an original-law **finite-r** bound, including norm tails, so this substitution becomes valid at fixed positive compact marks and fixed observation radius. It is separate from the preceding author's small-k cusp estimate (main#259/6004452810): that estimate concerns a different normalization and range of marks. The present argument does not consume that new, not-yet-reviewed lemma.
+
+This is not an exhaustive IBA2-012 closure, a rate of convergence of the full cluster distribution, a sharp lower bound, an all-mark result, or an elder matching theorem. No uniformity is claimed as the observation radius, dimension, volume, or marks escape their specified sets.
+
+## 2. Exact bindings and consumed interfaces
+
+Repository: `d6g8k5htny-coder/Math-`. Source cut:
+
+`04f47774bace871e3ad7dedc409e04f0280dfdd7`.
+
+| Tag | Path | Complete Git blob | Consumed material |
+|---|---|---|---|
+| P | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | `dfed3b8d318a3ab1950957f393307733a4bef3f2` | §3 (3.5); §4 full-field regression and conditional moments; §5 endpoint bounds and full normalizer (5.5); §6 typed determinant inequality (6.2); §7 Lebesgue spectral Jacobian |
+| SC | `frontiers/spectral_cluster_closure_20260929/PROOF.md` | `16c56821b52fd76b0be791622b9c3809eafde75a` | §3 deterministic vector exclusion (11); §4 is the qualitative predecessor, not a quantitative premise |
+
+The consumed interfaces were re-read at this source cut and have the same complete Git blobs as the preceding e31b6ad8 and c08e8359 cuts. The core finite-radius proof is retained; v1.1 adds the unequal-scale corollary below and publication/reproduction details. The new proof does not need C6's all-order factorial moment theorem, remote kernels, the cubic classifier, or a full-variation convergence rate. A source-interface read is not represented as a fresh independent review of all ancestors.
+
+Immutable sources:
+- https://github.com/d6g8k5htny-coder/Math-/blob/04f47774bace871e3ad7dedc409e04f0280dfdd7/imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md
+- https://github.com/d6g8k5htny-coder/Math-/blob/04f47774bace871e3ad7dedc409e04f0280dfdd7/frontiers/spectral_cluster_closure_20260929/PROOF.md
+
+## 3. Precise statement
+
+Fix `d >= 3`, `m=d-1`, `L>0`, a compact birth set B, a compact positive mark interval `[k_-,k_+]` with `k_->0`, and a fixed `R>=1`. Use P's exact variance-one periodized field, original Gaussian pin law Q, typed maximum/saddle weight W, and **full** normalizer Z. All frames are allowed, with constants uniform on the compact frame space.
+
+Write `A_o = D_y^2 f(0)` and order the eigenvalues of `-A_o` as
+
+    h_1 <= h_2 <= ... <= h_m.
+
+The h_i may be negative at finite r. Let N_R count additional critical points in the physical ball of radius Rr about the midpoint, with heights strictly between the pin heights, excluding the pins. Choose `K >= 1` as a fixed dimension-dependent multiple of `1+||f||_{C^4}` large enough to bound the operator derivatives in P and SC.
+
+For every fixed real `a>=0`, and every integer `1<=q<=m-1`, set
+
+    beta_q = 4q + q(q-1)/2 = q(q+7)/2.
+
+There are `C<infinity` and `r_*>0` such that, for `0<r<=r_*`, `0<=eta<=1`, and the fixed parameter ranges above,
+
+    E_{Q^W}[K^a ; N_R>0, h_{q+1}<=eta]
+        <= C r^3 (eta+r)^beta_q.                       (FR)
+
+The event does not require h_1,...,h_{q+1} to be positive. Hence it also covers the narrower SC strip with positive hard eigenvalues, or `|h_{q+1}|<=eta`.
+
+The q=0 version is the usual fixed-R `O(r^3)` occurrence bound. For d=2 there is no additional transverse direction, so the q>=1 assertion is empty.
+
+## 4. Proof
+
+**Charts and measurability.** Reduce r_* so r<=1 and the closed radius-Rr ball and its surrounding cylinder embed in one torus chart. For fixed pins and marks, the event N_R>0 is Borel: exhaust the punctured closed ball by compact sets at distance at least 1/n from the pins and require the critical value to lie in [b-kr^3+1/n,b-1/n]. Existence of a zero gradient satisfying these closed conditions is closed in C^1, since its location lies in a fixed compact set; the countable union is precisely the nonempty event. Eigenvalue and norm conditions are Borel as well. No extra zero-count moment theorem is needed.
+
+### 4.1 Original-law moment and density facts
+
+On typed support, set `B_M=-D_y^2f(M)>0`, with ordered eigenvalues
+
+    0<lambda_1<=...<=lambda_m.
+
+P (3.5) gives a common upper bound `C exp(-c||B_M||_F^2)` for its density on symmetric-matrix Lebesgue space. P (4.2) and its bounded regression coefficients imply, for every fixed s,
+
+    E_Q[K^s | B_M] <= C_s (1+||B_M||_F)^s.             (1)
+
+This is conditioning on the whole endpoint transverse matrix, not asserting independence of its eigenvalues. The bounded compact pin target is included in C_s. P §5 gives `Z>=z_*r^2` and `W/r^2<=CK^(2d)`.
+
+### 4.2 Convert near occurrence into a first-eigenvalue width
+
+Let
+
+    E_large = {rK > k_-/(4R)}.
+
+Off E_large, SC (11) applies. If there is any additional critical point in the ball, there is one in the surrounding SC cylinder. The contrapositive of its exclusion criterion gives
+
+    h_1 <= (8RK+56RK^2/k)r <= C_R rK^2.
+
+The Hessian Lipschitz bound and the min-max eigenvalue inequality give, for every i,
+
+    |lambda_i-h_i| <= CrK.
+
+Therefore, on `{N_R>0,h_{q+1}<=eta}` outside E_large,
+
+    lambda_1 <= C_R rK^2,
+    lambda_{q+1} <= eta+CrK.                           (2)
+
+This step is the additional localization not supplied by a bare determinant-layer statement. It uses no hard-eigenvalue inverse.
+
+### 4.3 Retain all endpoint determinant factors
+
+P (6.2), with `M3<=K` and a larger fixed constant, yields
+
+    W/r^2 <= C K^2 prod_{i=1}^m lambda_i(lambda_i+c rK).
+                                                               (3)
+
+In particular the saddle factor is retained even where a transverse eigenvalue is smaller than r. Replacing `lambda_i+c rK` by `lambda_i` would be invalid in that range.
+
+Split `K` into bands `[t,2t)`, `t=2^j`, j>=0. Let
+
+    delta=c_1 rt,
+    A_t=c_2 rt^2,
+    B_t=eta+c_3 rt,
+
+where constants are enlarged so that `delta<=B_t` and (2) is contained in `lambda_1<=A_t`, `lambda_{q+1}<=B_t` on that band.
+
+By (1), for an arbitrarily large fixed s,
+
+    P_Q(K>=t | B_M) <= C_s t^(-s)(1+||B_M||)^s.         (4)
+
+After bounding `K^(a+2)` by `(2t)^(a+2)`, conditioning on B_M, and multiplying its density, the remaining polynomial from (4) must be absorbed **before** dropping any small-eigenvalue Gaussian factor:
+
+    (1+||B_M||)^s exp(-c||B_M||^2)
+        <= C_s exp(-(c/2)||B_M||^2).                   (5)
+
+The constant may depend on the chosen s. Crucially, no power of t depending on s remains. This ordering prevents a circular choice of the tail moment.
+
+### 4.4 Spectral integration with two different widths
+
+Put `p=q+1` and `v_p=p(p-1)/2`. The Lebesgue spectral Jacobian is the Vandermonde times finite angular measure, independent of whether the Gaussian density is rotationally invariant.
+
+On the actual ordered integration domain, all of the first p eigenvalues are at most B_t. Their internal Vandermonde is therefore at most `B_t^v_p`. Every cross difference between a soft and a remaining hard eigenvalue is bounded by the hard eigenvalue; absorb these and all hard-hard differences into a fixed polynomial in the remaining eigenvalues. The hard determinant factors obey
+
+    lambda_i(lambda_i+delta) <= C t(1+lambda_i)^2,
+
+because r<=1 and t>=1. The hard Gaussian integrals are thus at most `C t^(m-p)`.
+
+Only after those bounds are imposed on the actual domain, extend the nonnegative remaining soft integrals to independent intervals: lambda_1 in `[0,A_t]` and each of the other q soft variables in `[0,B_t]`. This extension remains valid even if A_t>B_t: the Vandermonde estimate was made before extension, not asserted on the enlarged domain.
+
+The first integral is
+
+    int_0^{A_t} x(x+delta) dx
+      = A_t^3/3+delta A_t^2/2 <= C r^3 t^6.             (6)
+
+Each other integral is at most
+
+    int_0^{B_t} x(x+delta) dx <= (5/6)B_t^3,             (7)
+
+using delta<=B_t. The numerical 5/6 may be absorbed into C. No dimension-dependent singular integral remains.
+
+Since
+
+    3q+v_p = 3q+q(q+1)/2 = beta_q,
+    B_t <= C t(eta+r),
+
+the contribution of the jth band, after dividing by the full Z and using (3)-(7), is at most
+
+    C_s r^3(eta+r)^beta_q
+       t^[a+8+(m-p)+beta_q-s].                         (8)
+
+Choose an integer `s>a+9+(m-p)+beta_q`. The dyadic sum converges. Its constant depends on the fixed parameters, q and a, but not on eta or r. This proves (FR) off E_large.
+
+### 4.5 The large-norm branch is paid for, not discarded
+
+For any M>0, the full normalizer and original Q moments give
+
+    E_{Q^W}[K^a ; E_large]
+      <= C E_Q[K^(a+2d); K>k_-/(4Rr)]
+      <= C_M r^M.                                     (9)
+
+Choose an integer M>=3+beta_q. Since eta+r>=r and r<=1,
+
+    r^M <= r^(3+beta_q) <= r^3(eta+r)^beta_q.
+
+Adding (9) proves (FR), including eta=0. This finishes the analytic argument.
+
+## 5. Quantitative consequences
+
+For eta>=r, (FR) is `O(r^3 eta^beta_q)`. For eta<=r, the estimate saturates at `O(r^[3+beta_q])`. There is no claim of an extra eta power below the perturbation scale r.
+
+| Additional soft directions q | Total transverse corank addressed | beta_q | Bound at eta=r |
+|---:|---:|---:|---:|
+| 1 | 2 | 4 | O(r^7) |
+| 2 | 3 | 9 | O(r^12) |
+| 3 | 4 | 15 | O(r^18) |
+| 4 | 5 | 22 | O(r^25) |
+
+Rows require `d>=q+2`. The exponent equals the compact p-soft count `3p+p(p-1)/2` at p=q+1, but now the count has an original-law proof with a nonempty spatial event and all derivative tails retained.
+
+More generally, for eta=r^alpha with alpha>0,
+
+    E_{Q^W}[K^a ; N_R>0, h_{q+1}<=r^alpha]
+       = O(r^[3+beta_q min(alpha,1)]).                  (10)
+
+This is a joint event/derivative-weight estimate, not a rate for arbitrary test functions of the entire cluster process. Exact-count events `N_R=j` are subevents and inherit the same upper bound. Inserting a factor `N_R^j` would require a separate count-moment argument; it is not silently included.
+
+The estimate does not control an effective quartic-zero sector with a hard transverse Hessian, repeated nonzero hard eigenvalues, remote collisions, the limit R->infinity, or k->0. Those need their own source-bound statements. The preceding small-k cusp lemma and this fixed-positive-k cluster lemma cannot be combined into an all-mark theorem by relabeling constants.
+
+## 6. Why limiting mass alone cannot justify the diagonal
+
+For 0<r<1, the nonnegative measures on (0,1)
+
+    dmu_r(h) = [h^3 + 2r 1_{[r/2,r]}(h)] dh
+
+converge in total variation to `h^3 dh`, and share an integrable majorant. The limit has `mu_0([0,eta])=eta^4/4`. Nevertheless,
+
+    mu_r([0,r])=r^4/4+r^2,
+
+which is not O(r^4). This is an abstract measure obstruction, not a realization in the Gaussian field. It shows precisely why the new original-law estimate is necessary; neither nullity, dominated convergence, nor a quantitative limiting measure alone is enough.
+
+## 7. Adversarial self-audit and review request
+
+This is author verification, not a nonauthor verdict. The following failure points were checked in the written argument:
+
+1. **Endpoint versus midpoint:** use the ordered-eigenvalue Lipschitz bound for every index; no shared eigenbasis is presumed.
+2. **Two widths:** the first width is O(rt^2), others O(eta+rt). Make the Vandermonde bound before enlarging the domain.
+3. **Correlations:** use conditional norm moments given the whole matrix, not independence of K or eigenvalues.
+4. **Moment choice:** absorb the conditional polynomial into the Gaussian before dropping small-variable decay, so increasing s genuinely makes the dyadic sum converge.
+5. **Large derivatives:** bound in original coordinates with the full normalizer, choose M after beta_q, and retain eta=0.
+6. **Finite-r saddle factor:** keep lambda_i+delta, so no false improvement is asserted below r.
+7. **No extra counting weight:** N_R>0 is an event only. No unexplained determinant or factorial moment is added.
+8. **Scope:** fixed d,L,R and positive compact marks throughout; all status changes withheld.
+
+The smallest useful nonauthor review is the chain SC(11) -> (2), the matrix-conditioned dyadic integration (3)-(8), and exception absorption (9), against the two exact blobs. No full re-audit of C6/remote ancestors is required for this proposition because they are not consumed.
+
+The accompanying standard-library exact checker tests the exponent ledger and soft polynomial integrals, including degree-changing negative controls. These are algebra controls, not formal verification of Gaussian regression or the continuum exclusion argument.
+
+## 8. Unequal soft scales: the same finite-radius argument
+
+**Corollary FR-mixed.** Keep every hypothesis of (FR). Fix `q` with `1<=q<=m-1`, put `p=q+1`, and choose deterministic thresholds `0<=eta_2<=...<=eta_p<=1`. Then, with C independent of all these thresholds and r,
+
+    E_{Q^W}[K^a; N_R>0, h_j<=eta_j for every 2<=j<=p]
+        <= C r^3 prod_{j=2}^p(eta_j+r)^(j+2).           (FR-mixed)
+
+The remaining eigenvalues have no hard lower bound. The all-soft case `p=m` is included. This is still a nonempty event bound, not a count-moment or convergence-rate theorem.
+
+*Proof.* In each norm band `[t,2t)`, keep `lambda_1<=A_t=c_2 r t^2` from (2) and set `B_{j,t}=eta_j+c_3 rt`, increasing c_3 so `delta<=B_{j,t}`. The actual ordered domain satisfies `lambda_j<=B_{j,t}`. Bound each internal Vandermonde factor by its larger eigenvalue, so
+
+    prod_{1<=i<j<=p}(lambda_j-lambda_i)
+        <= prod_{j=2}^p B_{j,t}^(j-1).
+
+Make this bound on the original ordered domain, before extending lambda_1 to `[0,A_t]` and the remaining soft variables to `[0,B_{j,t}]`. The first integral remains `O(r^3 t^6)`; each other determinant integral is `O(B_{j,t}^3)`. Thus the product is
+
+    O(r^3 t^6 prod_{j=2}^p B_{j,t}^(j+2)).
+
+Each `B_{j,t}<=C t(eta_j+r)`, while `sum_{j=2}^p(j+2)=beta_q`. Conditional-moment absorption, hard Gaussian integration and the dyadic sum are exactly (4)-(8), with the same moment choice and no inverse eigenvalue. The large-norm estimate (9), with `M>=3+beta_q`, is absorbed because `prod(eta_j+r)^(j+2)>=r^beta_q`. This proves the result, including zero thresholds.
+
+For positive exponents with `eta_j=r^alpha_j` (ordered consistently),
+
+    contribution = O(r^[3+sum_{j=2}^p(j+2)min(alpha_j,1)]).
+
+For example `q=2`, `eta_2=r^(1/2)` and `eta_3=r^(1/4)` give `O(r^(25/4))`; equal thresholds recover (FR). The factors have different powers: exchanging two exponents is not innocuous. This corollary adds hierarchical softening to the equal-width slice, not a general classification of arbitrary effective-jet or collision sectors.
+
+## 9. External reconnaissance boundary
+
+Consensus Primary again returned its monthly quota error in this continuation; no Consensus paper was retrieved or used. Public primary metadata for Armentano–Azais–Leon, arXiv:2304.07424, confirms that the current landing page is v4, while the project cites v3. No version substitution is made here. Azais–Delmas, arXiv:1911.02300v3, studies isotropic critical-point correlations; it is not a replacement for this exact-periodic determinant-weighted pin law. This proposition imports only P and SC as listed above and makes no literature novelty claim.
+
+## 10. Publication and reproduction
+
+Run `python -B -S test_finite_radius_cluster.py` and `python -B -O -S test_finite_radius_cluster.py` from this packet. The twelve test methods include the retained nine equal-scale controls and three mixed-scale checks. They are exact finite algebra tests, not a simulation, Lean replay or proof of the imported Gaussian interfaces. The newly added helper tests were run before implementing the helpers and failed for their two missing definitions; their successor run is recorded separately. No shared workflow is changed and the existing repository CI does not automatically run this standalone packet. Nonauthor analytic review and applicable current-candidate hosted checks remain required before integration.

reviews/iba2_012_finite_radius_20261005/README.md

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+# Finite-radius additional-soft cluster bounds
+
+Author-side mathematical candidate for IBA2-012. Scientific effect **NONE**.
+Actual performer: OpenAI / GPT-6 Astra Pro, continuation `iba2-012-publication-audit-20261005-r3`, acting for Dylan Roy. This is the same author continuation as the preceding local proof, not an independent review.
+
+For fixed dimension, torus side, observation radius and compact positive marks, PROOF.md proves the proposed bound
+
+    E_{Q^W}[K^a; N_R>0, h_{q+1}<=eta] <= C r^3 (eta+r)^{q(q+7)/2}.
+
+It also proves an unequal-scale refinement, with factors (eta_j+r)^{j+2}. The result is a finite-radius event/derivative-weight estimate. It does not establish a full cluster convergence rate, count-weighted factorial bound, all-mark statement, or general IBA2-012 closure. The separate small-k cusp proposal is not consumed.
+
+## Reproduce
+
+From this directory:
+
+```sh
+python -B -S test_finite_radius_cluster.py
+python -B -O -S test_finite_radius_cluster.py
+```
+
+Each runs 12 test methods and prints an exact algebra ledger. Normal and optimized stdout must agree. The tests check finite polynomial integration, exponent accounting, domain-extension bounds, large-norm absorption algebra, the limiting-measure counterexample, and unequal-scale accounting. Degree-changing variants are tested within the suite. No Lean execution, random-field simulation or analytic proof acceptance follows from these tests.
+
+SOURCES.json records the three other packet files by bytes, SHA-256 and Git blob, and the exact P/SC interfaces. Its own blob is recorded in the PR. Only these four files are introduced. No shared workflow, formal manifest, parent proof, status, scientific flag, review or prize is changed. Existing repository CI does not execute this standalone suite automatically.
+
+## Review and integration
+
+Pickup: main#259 comment6005046599. Credit for the limiting additional-hard slice remains with comments6003533425/6003547392. The preceding local archive has SHA-256 f2bf763ea204fc293d64fbae2b9eb68e7e37102e66ce438cc859d2465296d562.
+
+Nonauthor review should cover SC(11) to endpoint localization, P's conditional moments/full normalizer, Vandermonde bounds before domain extension, norm-polynomial absorption before discarding Gaussian decay, the retained saddle r term, and the unequal-scale refinement. Preserve fixed d,L,R and k_->0 throughout. The author will not self-merge. Applicable exact-candidate checks, ownership reconciliation and a guarded integration remain required; a green check is not a theorem disposition.