IBA2-012: full soft power with logarithmic cap penalty

reviews/iba2_012_log_cap_transfer_20261006/NOTE.md

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+# IBA2-012: full soft power with an optimal logarithmic cap penalty
+
+Object: IBA2-012-LOG-CAP-TRANSFER-20261006-v1.
+Dylan Roy — delegated AI mathematical work. Actual author OpenAI / GPT-6 Astra Pro,
+`iba2-log-cap-transfer-20261006-r7`, pickup main#259/6008206382. Author-exposed
+continuation of FR/#319/#324; organizational-independence credit0; scientific effect NONE.
+**Conditional author-side proof candidate. Nonauthor mathematical review required.**
+No prior proof, acceptance record, scientific register or formal target changes.
+
+## 1. Sources, new scope and ancestry
+
+Repository `d6g8k5htny-coder/Math-`. Source cut:
+`b25cac0bcbb4032d624f14795257d9578c90d105`.
+
+| Tag | Path | Complete Git blob | Consumed interface |
+|---|---|---|---|
+| FR | `reviews/iba2_012_finite_radius_20261005/PROOF.md` | `5ed684c9bb446efc9bef5f3980717c6dc039abfd` | Sections3–5/8, original finite-radius mixed-threshold event bound with each fixed derivative-norm power |
+| SH | `frontiers/c6_sharpened_20260929/PROOF.md` | `70ba19726a9114bfceae04d021095e6c8eed5026` | Sections3–4 and section5 BEFORE its first-moment assembly: measurable cap Psi, original-law tail(5.1), and fixed moments |
+| P | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | `dfed3b8d318a3ab1950957f393307733a4bef3f2` | (4.1), (5.1)–(5.5): uniform derivative and W/r^2 moments, and full normalizer Z>=z_*r^2 |
+| CT | `reviews/iba2_012_count_transfer_20261006/PROOF.md` | `752bc9da5e49af3d0724d80c91cffc0dde35a2d8` | Provenance/comparison; section3 only for the OPTIONAL minimum with its fixed-order bound |
+
+FR's spectral integration and CT's count insertion/abstract no-constant-loss
+obstruction are already recorded. This note does not duplicate those results or
+replace #324's localized-moment addendum. It uses an ADDITIONAL quantitative input:
+SH's tail for the analytic critical-count cap. The preceding chat derived a
+logarithmic improvement but did not publish or execute it; this packet supplies
+full source bindings, a dyadic exact-control sharpness family and finite tests.
+
+SH's cap statement precedes its later use of G_d/I5 to assemble factorial moments.
+The proof of LT below does not consume that first-moment assembly or C6 Palm
+Theorem Q. This is a dependency distinction at the quoted interface, NOT a new
+independent proof of SH's cap, its multidimensional Rouche/Bezout, covariance,
+complexification or protected-cover premises. Their exact stated scope and
+unresolved ancestral qualifications remain. No old author-side header, merge or
+green workflow is converted into independent mathematical acceptance here.
+
+All SH first-moment theorem qualifications remain in its source; only the cap-tail
+interface is imported. A defect in that tail invalidates its use below. P's
+congruence explanation is not used as a new equality: only the quoted uniform
+moments and full-normalizer estimate are consumed. No inverse-Hessian bound or
+extra soft-sector normalizer is introduced.
+
+## 2. Setting and statement
+
+Fix d>=3, torus side L>0, observation radius R>=1, a compact birth interval and
+compact positive gap marks 0<k_-<=k<=k_+. Frames range over O(d), with uniform
+constants on that compact set. The field, pins, open between-pin height window,
+typed maximum/saddle weight W_r, original Gaussian pin law Q_r and normalizer
+Z_r=E_Q W_r are EXACTLY those of FR and P. Write
+
+    P_r=Q_r^W,     D_r=W_r/Z_r,     dP_r=D_r dQ_r.
+
+Here D_r is a density, not a determinant or a derivative. N counts all critical
+points in the between-pin window on the torus, pins removed. N_R counts the
+subset in the Rr-ball. SH supplies a measurable Psi dominating the TOTAL critical
+count, so 0<=N_R<=N<=Psi. Its cover is fixed independently of r and of the soft
+thresholds. Use FR's C4 norm control K>=1; do not identify this K with another
+source's differently named cover radius, derivative constant or gap interval.
+
+For 1<=q<=d-2, let h_j be FR's ordered eigenvalues of the negative midpoint
+transverse Hessian. Choose deterministic ordered thresholds
+
+    0<=eta_2<=...<=eta_(q+1)<=1/2,     0<r<=1/2.
+
+Define
+
+    S={h_j<=eta_j for 2<=j<=q+1},    E={N_R>0} intersect S,
+    Xi=product_(j=2)^(q+1)(eta_j+r)^(j+2),
+    beta=q(q+7)/2.
+
+Then r^beta<=Xi<=1. Midpoint eigenvalues may be negative, zero thresholds are
+allowed, no unmentioned lower bound on other eigenvalues is assumed, and q=d-2
+includes the all-soft transverse case. FR-mixed states, for each fixed a>=0,
+
+    E_(P_r)[K^a 1_E] <= C_a r^3 Xi.                    (F)
+
+For sufficiently small r put
+
+    u=log(1/r),      Lambda(r)=u/log u.
+
+**Theorem LT.** Conditional on the quoted interfaces, for every fixed a>=0 and
+s>0 there are C<infinity and r_*>0 such that throughout the fixed parameter set,
+
+    E_(P_r)[K^a N^s 1_E]
+       <= C r^3 Xi Lambda(r)^(d s),                   (LT)
+    E_(P_r)[K^a N_R^s 1_S]
+       <= C r^3 Xi Lambda(r)^(d s).                   (LT-local)
+
+Constants can depend on a,s,q,d,L,R, the fixed cover and the compact marks; no
+numerical field constant or radius is evaluated. This is full soft POWER with a
+logarithmic penalty, NOT the constant loss-free estimate O(r^3 Xi).
+
+## 3. Proof: a cutoff and four-factor Holder under the original law
+
+### 3.1 Exactly the moments and tail used
+
+SH(5.1) gives constants c,C0,x0>0, uniform in the stated pin parameters, with
+
+    Q_r(Psi>x)<=C0 exp[-c x^(1/d) log x], x>=x0.        (T)
+
+It implies uniformly bounded E_Q Psi^t for every fixed t>0 by tail integration.
+P(4.1) gives the same fixed-order moment property for K. P(5.3)'s following
+moment statement and (5.5) give
+
+    ||D_r||_(L4(Q_r))
+      = ||W_r/r^2||_4 / (Z_r/r^2) <= C.               (M)
+
+In particular E_Q K^(4a) and E_Q Psi^(4s) are bounded. All these orders are
+FIXED, independent of r. No joint independence or joint conditional moment
+estimate is assumed. The density has integral1 and the full Z is used once.
+
+### 3.2 Charge both sides of the cutoff
+
+For any T>=max(1,x0), split the nonnegative expectation at Psi=T. Since N<=Psi,
+
+    E_P[K^a N^s 1_E; Psi<=T]
+      <= T^s E_P[K^a 1_E] <= C_a T^s r^3 Xi.           (1)
+
+For the other branch, drop E and return to Q:
+
+    E_P[K^a N^s 1_E; Psi>T]
+      <= E_Q[D_r K^a Psi^s 1_{Psi>T}]
+      <= ||D_r||_4 ||K^a||_4 ||Psi^s||_4
+                            Q_r(Psi>T)^(1/4)
+      <= C_(a,s) exp[-(c/4) T^(1/d) log T].            (2)
+
+The four conjugate reciprocal exponents sum to1. The quarter in (2) must be
+retained when selecting the cutoff. No separate count-moment theorem, random
+matrix inverse, witness conditioning or uncharged exceptional branch appears.
+
+### 3.3 A uniform cutoff and tail absorption
+
+Take A>=max(1,4(beta+4)/c), fixed, and
+
+    T=A^d (u/log u)^d.
+
+Reduce r_* so u>=e^2 and T>=max(1,x0). If t=log u>=2, then log t<=t/2: the
+function t/2-log t is positive at2 and nondecreasing for t>=2. Thus
+
+    log T=d(log A+log u-loglog u)>=(d/2)log u>=log u,
+    T^(1/d) log T >= A u.
+
+The last inequality uses d>=2; our field application has d>=3. Consequently
+
+    exp[-(c/4)T^(1/d)log T]
+       <= exp[-(beta+4)u]=r^(beta+4)<=r^3 Xi.           (3)
+
+The last step uses Xi>=r^beta and r<=1. At eta_j=0 it remains valid; there is
+no missing diagonal layer. Since T^s=A^(ds)Lambda(r)^(ds) and Lambda(r)>=1,
+adding (1)–(3) proves LT. For s>0,
+
+    N_R^s 1_S=N_R^s 1_E<=N^s 1_E,
+
+which proves LT-local. For s=0 use FR directly with the NONEMPTY event E; do not
+reinterpret N_R^0 1_S as that event. This completes the ordinary proof.
+
+## 4. Consequences and comparison with the recorded fixed-order bound
+
+For eta_j=r^alpha_j with fixed alpha_2>=...>=alpha_(q+1)>0, set
+
+    B=sum_(j=2)^(q+1)(j+2)min(alpha_j,1)>0.
+
+The same finite-r saturation as FR gives
+
+    E_P[K^a N_R^s 1_S]
+       =O(r^(3+B) Lambda(r)^(ds))=o(r^3).             (4)
+
+Examples at eta_j=r:
+
+| d | q | count power s | bound |
+|---|---|---|---|
+| 3 | 1 | 1 | O(r^7 Lambda^3) |
+| 3 | 1 | 2 | O(r^7 Lambda^6) |
+| 4 | 2 | 1 | O(r^12 Lambda^4) |
+
+With the CT interface additionally assumed at one fixed integer p>s, the two
+separately proved estimates may be combined:
+
+    E_P[K^a N_R^s 1_S]
+      <= C r^3 min{Xi^(1-s/p), Xi Lambda(r)^(ds)}.      (5)
+
+The first estimate can be better for very slowly shrinking thresholds. LT ALONE
+does not show negligibility for every deterministic Xi->0; one needs
+Xi Lambda^(ds)->0. CT supplies the broader qualitative statement under its own
+inputs. We do not send p to infinity or claim constants uniform in p.
+
+For each fixed integer s>=1, (N_R)_s<=N_R^s. Therefore deleting configurations in
+S from the r^-3-rescaled positive ordered-s-tuple intensity changes its total
+variation mass by at most C Xi Lambda^(ds), with the same K^a-weighted upper bound
+for tuple observables bounded by K^a. The convention is total mass for a positive
+measure, not half the L1 norm of probabilities. This concerns only the deleted
+sector. There is no assertion of a full point-process limit, location density,
+normalized factorial Palm law, local nonempty lower bound or elder selection.
+
+## 5. Sharpness for the NUMERICAL interfaces, not for the actual field
+
+The logarithmic factor cannot be replaced by o(Lambda^(ds)) solely from the event,
+density-moment and cap-tail inequalities used above. The following one family
+works for every fixed count order simultaneously. A sequence r_j suffices to
+refute any proposed uniform inequality; a continuous interpolation is unnecessary.
+
+Fix d>=3. For integers j>=2 put
+
+    n=2^j, r_j=2^(-n), v_j=floor(n/j)>=2, M_j=v_j^d.
+
+On an abstract three-atom probability space set Q_j=P_j. The ordinary atom has
+probability r_j^3 and count N_R=N=2. The rare atom has probability r_j^7 and
+count N_R=N=M_j. The remaining probability 1-r_j^3-r_j^7 has count0.
+Set W_j=r_j^2, Z_j=r_j^2, D_j=1, K=3 and Psi=N+2. These give a normalized
+density and all required W/r^2 and K moments. Set h_1=r_j/2 throughout; h_2=r_j
+on the rare atom and h_2=2 otherwise. For d>3 take h_3=...=h_(d-1)=3.
+These are abstract ordered marks, not an actual field Hessian or derivative norm.
+
+### 5.1 Event bounds for EVERY threshold
+
+For q=1 and every eta in [0,1/2], E={N_R>0,h_2<=eta} is empty if eta<r_j and
+consists only of the rare atom otherwise. Hence
+
+    E_P[K^a 1_E] <= 3^a r_j^3 (eta+r_j)^4.
+
+This holds for all a>=0. For q>=2 the extra mark h_3=3 makes E empty throughout
+the allowed thresholds. Thus no untested off-diagonal threshold is hidden.
+
+### 5.2 Uniform cap tail with an explicit numerical constant
+
+At t>=4 the ordinary cap value4 and the zero-count cap value2 contribute nothing
+to {Psi>t}. The tail is either0, or exactly r_j^7 when t<M_j+2. In the latter case,
+
+    t^(1/d)<=v_j+1<=3v_j/2,
+    log t<=d log(v_j+1)<=d(j+1)log2,
+
+using M_j+2<=(v_j+1)^d. Since v_j<=n/j and j>=2,
+
+    t^(1/d)log t <= (3/2)d n[(j+1)/j]log2
+                  <=3d n log2.
+
+Therefore, for every j>=2 and t>=4,
+
+    Q_j(Psi>t) <= exp[-(1/d)t^(1/d)log t],             (6)
+
+because its nonzero value is exp[-7n log2]. Thus (T) holds with C0=1,c=1/d,x0=4,
+not just a vaguely fitted tail. All fixed Psi moments consequently are uniformly
+finite. Also, for each fixed real p>0,
+
+    E[N^p]/r_j^3=2^p+2^(-4n)v_j^(dp)<=C_p.
+
+The final boundedness follows from exponential decay against any fixed polynomial;
+all factorial upper bounds follow. Even an added global first/second lower scale
+holds: P(N_R>0)>=r_j^3 and E[(N)_2]>=2r_j^3.
+
+### 5.3 Match the logarithmic order
+
+At eta=r_j,
+
+    E[N_R^s;S]/[r_j^3(2r_j)^4]=v_j^(ds)/16.           (7)
+
+Now Lambda(r_j)=n log2/(j log2+loglog2). The elementary inequalities
+1/2<log2<1 imply, for j>=2,
+
+    n/j <= Lambda(r_j) <= 2n/j,
+    n/(2j) <= v_j <= n/j.
+
+Hence Lambda(r_j)/4<=v_j<=Lambda(r_j), and (7) lies between
+4^(-ds)Lambda(r_j)^(ds)/16 and Lambda(r_j)^(ds)/16. It is not
+O(1) and not o(Lambda(r_j)^(ds)). The same model works for each fixed s>0.
+
+This establishes numerical-interface sharpness. It does NOT instantiate the
+periodic Gaussian law, density on Hessian space, analytic sample paths, Rouche
+construction of Psi, independence of a Gaussian residual or geometric relations
+between N and K. Those structures might imply a stronger result for the actual
+field. A uniform joint sector-conditioned count estimate would be additional
+information, not a consequence of this abstract bound. No actual-field lower
+asymptotic or impossibility theorem is claimed.
+
+## 6. Finite controls, execution and review boundaries
+
+`test_log_cap.py` exercises12 exact finite methods against `log_cap_check.py`.
+The initial missing-implementation run produced12 assertion failures and no test
+errors. The companion uses Fraction/integer arithmetic only. It checks mixed
+powers, the tail quarter/cutoff coefficient, a dyadic sharpness family, correlated
+four-factor Holder cases, the event-retaining truncation, one normalizer, zero
+threshold absorption, remote-only exclusion, and elementary factorial domination.
+It does NOT evaluate log at floating precision or numerically certify a Gaussian
+constant; the transcendental cutoff and infinite limits are the written arguments.
+
+Five intentional wrong variants must fail for their intended assertion reasons:
+M1 omits the cutoff's Holder-quarter compensation; M2 uses (d-1)s rather than ds;
+M3 drops the common radial r^3; M4 removes the divisor j in the spike scale;
+M5 drops the full normalizer. Unexpected exceptions are errors, not mathematical
+rejections. Normal/optimized results and exact source identities are recorded in
+separate receipts. The original failures remain preserved.
+
+Smallest nonauthor review: quoted law/scope alignment, (M), four-factor bound(2),
+cutoff and absorption(3), plus ALL-threshold/event/tail and order comparison in
+section5. Check the optional CT premise in(5) rather than conflating it with LT's
+inputs. The whole numerical-interface countermodel must remain explicitly
+non-Gaussian. No duplicate review of P/FR/CT ancestors is requested.
+
+General IBA2-012 and IBA2-009, escaping marks or observation radius, higher-jet
+sector exhaustion, remote-only events, lifetime integration, actual conditional
+count law, full process convergence, independent formal alignment and scientific
+acceptance remain outside this note. No source from #319/#324 is rewritten.

reviews/iba2_012_log_cap_transfer_20261006/README.md

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+# Logarithmic cap transfer — IBA2-012
+
+This additive conditional proof uses FR's finite-radius multi-soft event bound,
+SH's quantitative original-law critical-count cap tail and P's moments/full
+normalizer to obtain full soft power with a logarithmic penalty:
+
+    E[K^a N_R^s; S] <= C r^3 Xi [log(1/r)/loglog(1/r)]^(d s).
+
+This is NOT O(r^3Xi) with a uniform constant. An explicit dyadic abstract family
+satisfies the numerical tail/event/moment bounds and attains this logarithmic
+order. It is not an actual Gaussian-field counterexample. Read NOTE.md before
+using the formulas. Existing #319 count transfer and #324 input-reduction work
+are credited and unchanged.
+
+## Reproduce finite controls
+
+Python3.11+ standard library only, from this directory:
+
+```sh
+python3 -B -S test_log_cap.py
+python3 -B -O -S test_log_cap.py
+python3 -B -S test_log_cap.py --mutant M1
+python3 -B -S test_log_cap.py --mutant M2
+python3 -B -S test_log_cap.py --mutant M3
+python3 -B -S test_log_cap.py --mutant M4
+python3 -B -S test_log_cap.py --mutant M5
+```
+
+The baseline reports12 methods with empty errors/failures and exit0. Repeat the
+mutants under -O: each must exit1 with its intended assertion failures and no
+errors. Invalid labels exit2. Unittest writes diagnostics to stderr; empty
+baseline stderr is NOT required. Deterministic JSON stdout is the comparison
+surface. A crash or permission failure is never a successful negative control.
+These are finite exact controls, not a simulation, Lean replay or proof of the
+arbitrary-law/analytic inputs. Existing general CI does not automatically discover
+this packet's standalone tests.
+
+## Scope and offline agents
+
+SOURCES.json binds complete directly quoted sources. Preserve original files and
+save outputs into a new outbox, not inside a sealed packet. The accompanying
+private chat delivery includes these source texts and the source/execution audit;
+it is a supplement, not a new complete repository or laptop backup. Existing
+large archives and the corrected R5 installer stay unchanged. No laptop install,
+local inference or toolchain-cache completeness is implied.
+
+Nonauthor mathematical review and separate eligible integration remain distinct
+from these tests and current-candidate required checks. No scientific status,
+parent theorem, formal target, workflow, or historical review is promoted.