Oct 5, 2026 · Oct 5, 2026
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+# IBA2-009: a half-power matching layer invisible to compact cusp parity
+
+**Object:** IBA2-009-MATCHING-20261005-v1. **Date:** 2026-10-05.
+**Actual author:** OpenAI / GPT-6 Astra Pro, session
+`github-rules-and-closure-round5-20261005`, Dylan Roy — delegated AI work.
+**Scientific effect: NONE.** Author-side analytical audit contribution, requiring
+nonauthor review; no historical verdict, proof, flag or register is changed.
+Organizational-independence credit: 0. This is not a claim of informational
+blindness: the original audit's coverage map and repository discussion were read.
+
+## 1. Result, limits, and source pin
+
+The compact-to-global implication in main#259 IBA2-009 needs a genuine matching
+estimate. Here are three concrete outputs:
+
+1. An exact formula for the actual rejected-density contribution in the moving
+   band `a <= k^3/r <= b`: it has an `ell^(1/2)` prefactor and an explicitly
+   weighted rescaled-kernel integral.
+2. An abstract kernel family with a nonzero half-power contribution that is
+   invisible in every fixed compact cusp window and at every fixed positive
+   gap. It obeys the selected upper bounds below, including the C7 K2 scale.
+3. Sufficient weighted-tail conditions for a correctly matched residual to be
+   `o(ell^(1/2))`, stating exactly what compact parity does not supply.
+
+**Not proved:** that the Gaussian field has this nonzero profile, that its
+half-power coefficient exists or vanishes, or that all its other probabilistic
+interfaces admit the abstract family. This is not a counterexample to Theorem N
+or to the persistence law. It is a counterexample to inferring global absence
+from the listed estimate-level premises alone. IBA2-009 remains open for the
+actual field. A positive contribution from one moving band alone also does not
+establish the coefficient of a full asymptotic expansion after other regions and
+lower-order terms are matched.
+
+All source reads use Math- commit
+`dcd2a886e322738324a745adcad12fd3735bd2c5`:
+
+| Source | Exact path / Git blob | Scope read and consumed |
+|---|---|---|
+| N | `frontiers/elder_cusp_parity_20261002/PROOF.md` / `16a1db0658918057ff741df888c13702763d9919` | Statement, Corollary N-prime and its section 6 proof; section 7 Remark 1. Selected section 5 bookkeeping was read, not re-audited. |
+| T | `frontiers/third_order_rate_20261001/PROOF.md` / `110ed33a57902bef11873807e722fde3888b59fb` | Section 0 kernel definitions, E3-plus/R-plus remainders and W-plus small-gap window. |
+| K | `frontiers/c7_total_bounded_20260929/PROOF.md` / `28748b086ec6761ef467dc67cba475fbdf8b7455` | K1/K2 and section 4 unnormalized marked-intensity identity and Gaussian pin-density envelope. |
+
+N already explicitly limits its no-linear-term theorem to fixed compact
+`kappa=k/r` intervals. Its Remark 1 distinguishes the two uncontrolled ends and
+says that even a hypothetical uniform `O(r^2+k^2)` estimate would not finish the
+global half-power question. This report quantifies a specific obstruction to
+that inference; it does not attribute an unrestricted claim to N.
+
+## 2. Exact moving-band formula for the actual nonnegative kernel
+
+Use the source conventions, without dividing by the full normalizer:
+
+    A_rej(r,b,k,u) = 12 pi_r(v_r) E_Q[(W_r/r^2)(1-e)],
+    B_r(k) = integral_b integral_u A_rej(r,b,k,u) d sigma(u) db >= 0,
+    nu_rej^near(ell) = integral_0^r0 r^(-2) B_r(ell/r^3) dr.
+
+This is K section 4 in the separation variable. The cap implication
+`1-e <= 1_{G_r^c}`, K2 and the Gaussian pin-density envelope give
+
+    0 <= B_r(k) <= C r^3/k,       0 < r <= min(k,r0), 0 < k <= 1.       (2.1)
+
+Integration over birth and direction, and absorption of polynomial gap growth
+on `0<k<=1`, are included in C. Fix `0<a<b<infinity`. Set
+
+    y = k^3/r = ell^3/r^10,
+    r_ell(y) = ell^(3/10) y^(-1/10),
+    k_ell(y) = ell^(1/10) y^(3/10),
+    J_ell(y) = k_ell(y) B_{r_ell(y)}(k_ell(y))/r_ell(y)^3.             (2.2)
+
+For all sufficiently small ell (depending on a,b,r0), the whole band satisfies
+`r<=min(k,r0)` and `k<=1`. Substitution of (2.2), including `dr`, gives exactly
+
+    nu_[a,b](ell)
+      = ell^(1/2)/10 * integral_a^b J_ell(y) y^(-3/2) dy.              (2.3)
+
+Indeed `r/k = ell^(1/5)y^(-2/5)` and
+`|dr/dy| = ell^(3/10)y^(-11/10)/10`. By (2.1), `0<=J_ell<=C` on
+this fixed band. Therefore this actual band contribution is `O(ell^(1/2))`.
+If `J_ell -> J` almost everywhere on [a,b], dominated convergence yields
+
+    ell^(-1/2) nu_[a,b](ell) -> (1/10) integral_a^b J(y)y^(-3/2)dy.    (2.4)
+
+More generally convergence of that weighted integral suffices. A uniform
+`J_ell -> 0` proves a little-o contribution from this band. K2 supplies a
+bounded J, not a vanishing J. Establishing its actual limit is a sharply
+localized analytic task; the present report does not assume a limit exists.
+
+The band has
+
+    r ~ ell^(3/10),  k ~ ell^(1/10),  kappa ~ ell^(-1/5).              (2.5)
+
+Thus k tends to zero while kappa tends to infinity. Neither fixed-positive-k
+Taylor expansion nor compact-kappa parity describes the passage uniformly.
+
+## 3. An explicit estimate-level obstruction
+
+First omit a harmless gap cutoff. Let
+
+    h(t) = (t-1)^4(2-t)^4 for 1<=t<=2, and 0 otherwise,
+    psi(y) = h(sqrt(y)),
+    H(r,kappa) = r^2 kappa^2 psi(r^2 kappa^3)
+                 = k^2 psi(k^3/r),              k=kappa r.          (3.1)
+
+Here H is a **scaled** kernel perturbation; its raw kernel is `D=r^2 H`.
+The polynomial bump is C^3, is nonnegative, and is bounded by 1. The support
+is `1<=k^3/r<=4`.
+
+### 3.1 Why the usual estimates do not see it
+
+- At fixed kappa, (3.1) is even in r. On every fixed compact kappa interval it
+  is identically zero for all sufficiently small |r|. In particular it has
+  zero linear term and satisfies compact `O(r^2)` estimates.
+- At any fixed k>0 it is identically zero once `r<k^3/4`. It changes no
+  coefficient in a fixed-k asymptotic expansion.
+- Globally, `0<=H<=k^2<=r^2+k^2`. It therefore meets even the hypothetical
+  quadratic bound discussed in N Remark 1.
+- On `k<=1`, `H<=k<=r(1+kappa)`, the CE-plus-plus scale.
+- On its support, `D=r^2 k^2 psi <=4r^3/k`, the K2 scale, with the original
+  `r<=k` condition satisfied whenever k<=1. The raw-versus-scaled distinction
+  and the lifetime Jacobian are essential here.
+- If `k<=r<=1`, then `k^3/r<=r^2<=1`, so D is zero. The W-plus small-gap
+  window is unaffected by this perturbation.
+
+The family can respect positivity and a Gaussian-envelope-type upper bound.
+Choose an even smooth cutoff theta(k) with `0<=theta<=1`, theta=1 for
+`|k|<=1/2`, and theta=0 for `|k|>=1`, and replace H,D by theta(k)H,theta(k)D.
+As an explicit abstract nonnegative split on `0<r<=1`, take
+
+    A_cand(r,k) = k^2 theta(k),
+    A_rej(r,k)  = r^2 k^2 theta(k) psi(k^3/r),
+    A_eld(r,k)  = A_cand(r,k)-A_rej(r,k).                            (3.2)
+
+Then `0<=A_rej<=A_cand`, and `A_eld>=0`. These raw kernels obey a K1-sized
+majorant; on the compact k support they also obey a polynomial times
+`exp(-c k^2)` majorant, after enlarging the constant. Multiplication by a
+normalized Gaussian birth envelope and normalized angular measure preserves
+these statements. This is only an abstract kernel construction, not a
+realization as Gaussian critical points, an elder mark, or a Palm law.
+
+### 3.2 Exact half-power mass
+
+For ell sufficiently small that the whole support is inside `0<r<=r0` and
+`k<=1/2`, theta=1 on the support. It suffices to take
+`ell <= min(2^(-16), r0^(10/3))`, with `0<r0<=1`. Put
+`t=ell^(3/2)/r^5`; then `k^3/r=t^2` and
+
+    integral r^(-2) A_rej(r,ell/r^3) dr
+      = integral ell^2 r^(-6) h(ell^(3/2)/r^5) dr
+      = ell^(1/2)/5 * integral_1^2 (t-1)^4(2-t)^4 dt
+      = ell^(1/2)/3150.                                           (3.3)
+
+The integration range is `[ell^(3/10)2^(-1/5),ell^(3/10)]`.
+The coefficient is exact: after v=t-1, the bump integral is
+`integral_0^1 v^4(1-v)^4 dv = 1/630`.
+
+C^3 regularity is not the reason this occurs. Replacing h by any nonzero,
+nonnegative C-infinity bump supported inside (1,2), bounded by 1, preserves
+all the estimates and produces coefficient `(1/5)integral h>0`.
+The explicit rational coefficient (3.3) is reserved for the polynomial h;
+no closed-form value is asserted for the smooth variant.
+
+An added half-power correction is smaller than the existing
+`O(ell^(3/7))` remainder and `O(ell^(4/9)log(1/ell))` upper scale. It is
+therefore not excluded merely by those error orders. The model's perturbation
+is zero on the small-kappa region; it does not purport to reconstruct the
+sources' complete leading laws or all cross-region matching identities.
+
+## 4. A sufficient global passage, with the required subtraction explicit
+
+Let E_ell(kappa) denote a **correctly matched residual**: the known fold, cusp,
+and common terms must already have been subtracted with no double counting.
+Constructing this residual is itself part of the missing analysis. It is NOT
+permissible simply to label the raw difference in N.1 as this residual:
+that difference can still contain the fold contribution to the ell^(1/3)
+coefficient when integrated globally.
+
+Suppose the residual density has the exact representation
+
+    R(ell) = ell^(1/4)/4 * integral_0^infinity
+               E_ell(kappa) kappa^(-5/4) d kappa + F(ell),           (4.1)
+
+where the integrand is extended by zero below the physical lower cutoff
+`kappa=ell/r0^4`, and `F(ell)=o(ell^(1/2))` accounts for separately controlled
+far/matching remainders. Assume on every fixed `0<a<b<infinity`,
+
+    integral_a^b |E_ell(kappa)| kappa^(-5/4)d kappa = O_(a,b)(ell^(1/2)).
+
+This is the type of compact control furnished by an O(r^2) estimate after
+`r=(ell/kappa)^(1/4)`, provided the subtraction also has that property. A
+sufficient missing tail condition is
+
+    lim_(a down to 0, b up to infinity) limsup_(ell down to 0)
+      ell^(-1/4) * [integral_0^a + integral_b^infinity]
+      |E_ell(kappa)| kappa^(-5/4)d kappa = 0.                        (4.2)
+
+Split (4.1) into those two tails and [a,b], divide by ell^(1/2), first let
+ell tend to zero, then exhaust the interval. The compact part is O(ell^(1/4))
+after division, and (4.2) controls the tails. Hence `R(ell)=o(ell^(1/2))`.
+
+For a stronger convenient sufficient estimate, an all-domain bound
+
+    |E_ell(kappa)| <= C ell^(1/2) kappa^(-1/2) M(kappa),
+    integral_0^infinity M(kappa) kappa^(-7/4)d kappa < infinity
+
+gives an O(ell^(3/4)) integral in (4.1). For example the integral condition
+holds for nonnegative, locally integrable M with M=O(kappa^p) near 0, p>3/4, and
+M=O(kappa^q) near infinity, q<3/4. These are sufficient assumptions, not
+established properties of the actual Gaussian kernel.
+
+The obstruction (3.1) violates (4.2): its normalized weighted mass stays
+positive while its support escapes to kappa=infinity. Pointwise convergence
+and compact parity thus cannot replace quantitative control of these tails.
+
+## 5. Reproduction, coverage and next mathematical target
+
+Run `python3 -B -S boundary_layer_check.py`, and repeat with `-O`.
+The deterministic JSON reports 25 moving-layer, 30 compact-cusp, 25 fixed-gap,
+20 small-gap and 3 exact-integral cases. The explicit finite checks verify
+identities and inequalities, not the Gaussian realization or the general
+claims by sampling; the proofs are sections 2–4 above.
+
+Negative controls (each must exit 1 with its named failure):
+
+| Variant | Deliberate error | Failure |
+|---|---|---|
+| M1 | k^3/r^2 instead of k^3/r | MOVING_LAYER_ARGUMENT |
+| M2 | omit r^2 between raw and scaled kernels | LIFETIME_JACOBIAN |
+| M3 | omit the 1/5 substitution factor | INTEGRATION_FACTOR_ONE_FIFTH |
+| M4 | ignore the moving support at fixed kappa | COMPACT_CUSP_VANISHING |
+| M5 | replace the half-power x^5 by x^6, ell=x^10 | HALF_POWER_SCALING |
+
+Example: `python3 -B -S boundary_layer_check.py --mutant M2`.
+Invalid arguments exit 2. All fourteen CLI cases were run in normal/optimized
+modes, and the positive JSON was byte-identical. A separate seven-test local
+contract suite passed both modes after its expected missing-source failure.
+No local Lean execution is claimed. Existing hosted repository checks are
+separate and do not automatically execute this newly added standalone script.
+
+**Actionable target:** determine the actual nonnegative profile in (2.2) on a
+fixed positive y-band, rather than extrapolating fixed-kappa parity. Uniform
+vanishing would remove that band's half-power mass; a nonzero limiting profile
+would require explicit incorporation into global matching. Neither outcome is
+assumed. Both ends of the matched residual in (4.2) still require control before
+a global absence statement. Historical IBA2-009, IBA2-010, IBA2-012 and all
+scientific acceptance fields remain unchanged by this contribution.
@@ -0,0 +1,34 @@
+{
+  "scientific_effect": "NONE",
+  "scientific_acceptance": false,
+  "source_commit": "dcd2a886e322738324a745adcad12fd3735bd2c5",
+  "sources": [
+    {
+      "path": "frontiers/elder_cusp_parity_20261002/PROOF.md",
+      "blob": "16a1db0658918057ff741df888c13702763d9919",
+      "scope": "statement; section 6; section 7 Remark 1"
+    },
+    {
+      "path": "frontiers/third_order_rate_20261001/PROOF.md",
+      "blob": "110ed33a57902bef11873807e722fde3888b59fb",
+      "scope": "section 0: definitions and stated remainder/window bounds"
+    },
+    {
+      "path": "frontiers/c7_total_bounded_20260929/PROOF.md",
+      "blob": "28748b086ec6761ef467dc67cba475fbdf8b7455",
+      "scope": "K1/K2; section 4 rejected intensity and pin-density envelope"
+    }
+  ],
+  "files": [
+    {
+      "path": "REPORT.md",
+      "bytes": 12716,
+      "sha256": "570a8bf7fdd78e56e6edd04d6a1f7431312daa5839c05a210aa72a76336dc27a"
+    },
+    {
+      "path": "boundary_layer_check.py",
+      "bytes": 4892,
+      "sha256": "58ba3205d390b685344e19c989efe2ef90fd7837237b47ed8a7c1eef8ac49385"
+    }
+  ]
+}