Oct 5, 2026 · Oct 5, 2026
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+# Scope erratum: the normalized cap bound is a compact-mark statement
+
+**Object:** C7-NORMALIZED-SCOPE-ERRATUM-20261005-v1.  
+**Date:** 2026-10-05. **Scientific effect: NONE** — no theorem promotion or
+scientific-register change.  
+**Actual author:** OpenAI / GPT-6 Astra Pro, session
+`github-rules-and-closure-round3-20261005`, acting for Dylan Roy as delegated AI
+work. This erratum does not change the attribution of the original Claude proof
+or the independent audit. Organizational-independence credit: 0.
+
+## Reading rule and exact source
+
+Read [the retained original PROOF.md](PROOF.md) together with this erratum when
+using its Section 3 **Remark (normalized form)**, lines 134–136 at the source
+identified below. The original file remains byte-for-byte unchanged. Its
+unrestricted-positive-`k` normalized reading is withdrawn by this correction;
+the replacement is the compact-mark statement in the next section.
+
+Source snapshot: Math- commit
+`8fb6d41d201c7695cc23cfdd235f420be204a0c7`.
+
+| Source at that commit | Git blob identity | Consumed slice |
+|---|---|---|
+| `frontiers/c7_total_bounded_20260929/PROOF.md` | `28748b086ec6761ef467dc67cba475fbdf8b7455` | Theorem K2; Section 3 normalized remark; Section 4's unnormalized assembly |
+| `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | `dfed3b8d318a3ab1950957f393307733a4bef3f2` | P Section 1, (5.4)–(5.5), and the cap implication in Sections 7–8 |
+| `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` | `213594d6ca6a86fb938110f4d166d9ce275a02d0` | The existing mandatory reading correction for P Section 5 |
+
+These are Git blob identifiers, not SHA-256 digests or proof certificates. The
+parent's Section 5 is consumed **with its existing congruence erratum**, not with
+the uncorrected displayed scaling factor.
+
+## Exact replacement for the normalized remark
+
+> **Remark (normalized form, compact marks only).** Fix `d >= 2`, `L > 0`, a
+> nonempty compact birth set `B` in the real line, and a fixed gap interval
+> `K = [k_-, k_+]` with `0 < k_- <= k_+ < infinity`. There are constants
+> `r_* > 0` and `C_{B,K,d,L} < infinity`, which may depend on all these fixed
+> choices, such that, for every `b in B`, every `k in K`, every orthogonal
+> frame, and every `0 < r <= r_*`,
+>
+>     0 <= 1 - p_r <= Q_r^W(G_r^c) <= C_{B,K,d,L} (r/k)^3.
+>
+> Here `Q_r^W = (W_r/Z_r) Q_r` and `Z_r = E_Q W_r` are the actual typed
+> weighted law and its full normalizer. This follows from K2 and the
+> compact-parameter lower bound P (5.5), as detailed below. It does not assert
+> a constant or radius uniform as `k_-` tends to zero or `k_+` tends to
+> infinity. The condition `r <= c k` alone does not replace these compact-mark
+> restrictions. Corollary T does not use this normalized remark.
+
+## Derivation and all constant dependencies
+
+Retain K2, with its original constants `C_K`, `N_K` and radius `r_0`, depending
+on `d,L` only:
+
+    E_Q[(W_r/r^2) 1_{G_r^c}]
+        <= C_K (r^3/k) (1 + |b| + k)^{N_K},
+        when 0 < r <= min(k, r_0).
+
+By P (5.4)–(5.5), on the fixed compact set of birth marks, gap marks and frames
+there are `z_* > 0` and `r_Z > 0` such that
+
+    E_Q[W_r/r^2] = Z_r/r^2 >= z_*  for 0 < r <= r_Z.
+
+This is a compact-set lower bound, **not** a globally uniform relative error
+estimate. Its justification uses the continuity and strict positivity of the
+contact cone moment, compactness of the parameters, and the parent's uniform
+convergence on that compact set. The qualitative convergence supplies a radius;
+no numerical value of that radius is claimed here.
+
+Let `P_* = 1 + max_{b in B}|b| + k_+`, and choose
+
+    r_* = min(r_0, r_Z, k_-, 1).
+
+For every parameter in the replacement statement, K2 and the normalizer floor
+both apply. Consequently
+
+    Q_r^W(G_r^c)
+      = E_Q[(W_r/r^2) 1_{G_r^c}] / E_Q[W_r/r^2]
+      <= (C_K P_*^{N_K}/z_*) r^3/k
+      =  (C_K P_*^{N_K}/z_*) k^2 (r/k)^3
+      <= (C_K P_*^{N_K} k_+^2/z_*) (r/k)^3.
+
+Thus one may take `C_{B,K,d,L} = C_K P_*^{N_K} k_+^2/z_*`. The lower-normalizer
+direction and the upper-gap dependence are explicit. On the parent's generic
+typed support the deterministic cap implies elder pairing, so
+`1 - p_r <= Q_r^W(G_r^c)`. This uses the existing cap/mark interface; it is not a
+new proof of that interface. No independence between `W_r` and `G_r` is assumed.
+
+## Why the former unrestricted reading must not be used
+
+[Audit IBA2-001 in main#259](https://github.com/d6g8k5htny-coder/main/issues/259)
+identifies the false quantifier extension. Its exact-periodic example has
+`d=2`, axial orientation, `b=0`, and `k(r)=r^(-2)`: the cap-failure probability
+under the typed weighted law tends to one, whereas `(r/k)^3=r^9` tends to zero.
+That analytic counterexample is the audit's work, not a new execution or
+independent reconstruction claimed by this erratum. This erratum supplies the
+replacement statement and the compact-window derivation above.
+
+In particular, `z_0=(6k)^2 E[det(A_0)^2 1_{A_0<0}]` does not make the entire
+coefficient independent of the marks or extend compact-parameter convergence
+to an unbounded moving sequence of gap targets. Nor does a derivative identity
+alone control the normalized expectation uniformly on such a sequence.
+
+Cap failure is the failure of a **sufficient criterion**. It is not the same as
+failure of elder pairing. The audit's counterexample does not assert that
+`1-p_r` tends to one.
+
+## Preserved results and review boundary
+
+The all-mark **unnormalized** K1/K2 statements and their proofs are unchanged.
+Corollary T's Section 4 continues to use the pin density multiplied by the
+weighted numerator, K1/K2 with their target-growth factors, the actual lower
+integration cutoff, and the near/far split. It does not divide by a uniform
+all-mark normalizer. No new review or acceptance of that complete argument is
+claimed here.
+
+`PROOF.md`, `SOURCE_FILES.json`, `RESULTS.json`, `exponent_check.py`, all original
+source pins, and every workflow and formal package file are retained unchanged.
+The new README is a reading guide, not another scientific-status register.
+
+Nonauthor review of this correction must check the compact quantifiers, the
+normalizer division, the `k_+^2` factor and the original-source preservation.
+Review and integration dispositions belong to the source-bound PR record;
+this file itself does not award acceptance. The broader audit coverage items
+in main#259 remain separate from the correction of IBA2-001.
@@ -0,0 +1,20 @@
+# C7 total rejected density: source and scope correction
+
+Read [PROOF.md](PROOF.md) together with the [normalized-form scope
+erratum](ERRATUM_NORMALIZED_FORM.md). The Section 3 normalized cap-probability
+remark is valid in the corrected reading only on fixed compact birth and
+strictly positive compact gap intervals; do not quote it as uniform over every
+positive gap mark.
+
+The original proof, its unnormalized K1/K2 estimates, Corollary T, checker,
+results and source inventory are retained unchanged. The erratum identifies
+its exact source and supplies the compact-window derivation. It neither
+promotes the original proof candidate nor resolves the broader audit coverage
+items in [main#259](https://github.com/d6g8k5htny-coder/main/issues/259).
+
+[Source inventory](SOURCE_FILES.json) · [Exact checker](exponent_check.py) ·
+[Recorded results](RESULTS.json).
+
+Original proof: Anthropic Claude. Additive erratum and reading guide:
+OpenAI / GPT-6 Astra Pro, session `github-rules-and-closure-round3-20261005`,
+Dylan Roy — delegated AI work. Scientific effect: **NONE**.