LM006: exact-field common-regression O(r) coupling and saddle-sign application

reviews/lm006_common_regression_rate_20261006/NOTE.md

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+# LM006: the common-regression rate input and its endpoint consequences
+
+Dylan Roy — delegated AI mathematical work. Actual author: **OpenAI / GPT-6
+Astra Pro**, `lm006-common-regression-rate-20261006`. Author-side additive
+application; scientific effect **NONE**; organizational-independence credit **0**.
+This does not revise a historical source, supply a Lean proof, or constitute
+its own nonauthor review.
+
+## 1. What is reused, and what is supplied here
+
+The remote-window source [R], §§2–4, already constructs a common conditional-
+field coupling with error O(r²), explicitly including the endpoint-only law,
+and already obtains normalized-weight error O(r). Those methods and rates
+are **not new here**. The general Gaussian prior-plus-regression update also
+has extensive prior literature; [W] is external context, not a proof premise.
+
+The precise interface supplied here is a common coupling of **all six LM006
+scaled endpoint coordinates**, including both mixed derivatives, with
+`e_r=(E||X_r-Y||²)^(1/2)=O(r)` and every finite Lp error O(r). We verify its
+hypotheses in the exact fixed periodic field by the integral kernels of [E],
+not by differentiating a singular target-covariance square root. We then
+apply the already established inequalities of [N], [T], and [S]. In
+particular [S]'s conditional `r e + e^4` bound becomes an actual-field
+O(r²) bound for its **saddle-transverse sign event**, not for elder failure.
+This is an application/interface result, not a new general coupling principle
+or the first conditional r² sign estimate.
+
+Fix d=2, L=24, birth b=6/5 and axial sites M=(-r/2,0), S=(r/2,0). Let f be the
+centered, variance-one Gaussian field with the exact covariance of [P] §1,
+not its nonperiodic approximation. The six physical observations are
+
+    (f(M), f_x(M), f_y(M), f(S), f_x(S), f_y(S))
+       = (b,0,0,b-r³/6,0,0).
+
+Q_r denotes their canonical continuous Gaussian regression law. Work on
+`0<r<=r_*`, where one sufficiently small **fixed** r_*>0 is chosen below,
+with r_*<=1 and r_*<24. No compatible unequal-height pins are claimed at a
+torus alias. All constants may depend on this fixed model, frame and pins;
+no numerical radius or evaluated coefficient is provided.
+
+**Application.** There is a common Gaussian coupling X_r,Y of the actual
+six-coordinate endpoint law under Q_r and its contact law such that, for
+every fixed finite p>=1,
+
+    ||X_r-Y||_Lp <= C_p r,    sup_r E||X_r||^p < infinity.       (A)
+
+Here `Y=(-1,1,-a/2,a/2,Q,Q)` has the conditional law from [E], not an arbitrary
+vector on that plane. With the physical typed weight W_r, define
+`Z_r=E_Qr W_r`, `z_r=Z_r/r²` and `z0=E[(Q_-)²]`. Then
+
+    z0>0,    E|T_r(X_r)-T_0(Y)| <= C r,
+    |z_r-z0| <= C r,    z_r >= z0/2,                           (B)
+    d_BL(nu_r,nu0) <= C_BL r,
+    nu_r{c_+>=0} <= C_sign r².                                (C)
+
+The measures nu are the normalized determinant/type tilts of the endpoint
+laws; d_BL uses |phi|<=1 and Lipschitz constant<=1 as in [T]. The sign event
+is `f_yy(S)>=0`. It is not the maximum's already-excluded transverse sign,
+and it is not identified with any full-field capture or persistence event.
+These conclusions do not assert endpoint TV convergence; [D]'s TV=1 result
+is compatible with them.
+
+## 2. One unconditioned Hilbert-space model and the exact pins
+
+Use H=L² of the original unconditioned probability space. Put
+`F(x)=f(x,0)`, `H_y(x)=f_y(x,0)`, `Q(x)=f_yy(x,0)`.
+A sufficient local regularity assumption is that F is H-valued C5, H_y is
+H-valued C3, and Q is H-valued C1 on a neighborhood of [-h_*,h_*], h_*>0.
+Restrict r_* further to r_*<=2h_* so every kernel evaluation stays there.
+Assume joint Gaussianity and positive-definiteness of the contact pin Gram.
+Section 5 verifies these assumptions for the actual field.
+
+Use [E]'s exact equivalent pin frame P_r=(P0,...,P5), writing h=r/2:
+
+    P0=(F(-h)+F(h))/2,       P1=(F'(-h)+F'(h))/2,
+    P2=(H_y(-h)+H_y(h))/2,
+    P3=(F'(h)-F'(-h))/r,     P4=(H_y(h)-H_y(-h))/r,
+    P5=(12/r²)[P1-(F(h)-F(-h))/r].
+
+Its inverse in [E] (2.2) recovers the six raw observations, so its exact
+conditioning target is
+
+    d_r=(b-r³/12,0,0,0,0,2),    d_0=(b,0,0,0,0,2).            (1)
+
+The frame limit is J=(f,f_x,f_y,f_xx,f_xy,f_xxx)(0). In particular its last
+pin is **2**. This frame is not confused with [R]'s different, equivalent
+ordering and divided-difference row.
+
+Before conditioning define the adjusted six-vector V_r by [E] (3.2):
+
+    A_±=(F''(±h)-P3)/r,     B_±=(H_y'(±h)-P4)/r,
+    C_±=Q(±h),             V_r=(A_-,A_+,B_-,B_+,C_-,C_+).
+
+On the pin target P3=P4=0 these are exactly the four divided Hessian entries
+and two unscaled transverse Hessian entries. Subtraction before taking the
+unconditioned limit is essential. The exact kernels on I=[-1/2,1/2] are
+
+    P3=integral_I F''(ru)du, P4=integral_I H_y'(ru)du,
+    P5=6 integral_I (1/4-u²)F'''(ru)du,
+    A_-=-integral_I(1/2-u)F'''(ru)du,
+    A_+= integral_I(u+1/2)F'''(ru)du,
+    B_-=-integral_I(1/2-u)H_y''(ru)du,
+    B_+= integral_I(u+1/2)H_y''(ru)du.                        (2)
+
+These are the Hilbert-space identities already proved in [E], not Taylor
+formulas assumed under a varying conditional law. Their limit is
+
+    V_0=(-F'''(0)/2,F'''(0)/2,-H_y''(0)/2,H_y''(0)/2,Q(0),Q(0)).
+
+## 3. Quantitative pin and endpoint convergence
+
+Let D_j=sup_x||F^(j)(x)||_H, E_j=sup_x||H_y^(j)(x)||_H and
+J_1=sup_x||Q'(x)||_H on the fixed interval. These are finite by continuity.
+Taylor's formula with its Hilbert-valued integral remainder and cancellation
+of the odd term in each centered average give
+
+| Coordinate | H-error from its contact value |
+|---|---|
+| P0 | D2 r²/8 |
+| P1 | D3 r²/8 |
+| P2 | E2 r²/8 |
+| P3 | D4 r²/24 |
+| P4 | E3 r²/24 |
+| P5 | D5 r²/40 |
+
+For P5, the kernel mass is 1 and its second moment is 1/20; the Taylor
+factor 1/2 gives 1/40. The analogous constant for the uniform kernel is
+1/24. Hence, with the Euclidean vector norm in L²,
+
+    ||P_r-J||_L² <= C_P r²,
+    C_P²=(D2/8)²+(D3/8)²+(E2/8)²
+                  +(D4/24)²+(E3/24)²+(D5/40)².             (3)
+
+The endpoint kernels need only first-order remainders. Both absolute first
+moments are `integral_I(1/2-u)|u|du=1/8` (the plus kernel has the same value).
+Thus each A error is <=D4 r/8, each B error <=E3 r/8, and each C error
+<=J_1 r/2. Consequently
+
+    ||V_r-V_0||_L² <= C_V r,
+    C_V²=2(D4/8)²+2(E3/8)²+2(J_1/2)².                       (4)
+
+The first-order endpoint displacement is not generally second order even
+though the centered pin frame is. The finite model in §7 explicitly tests
+that distinction.
+
+## 4. The means and residuals in the same coupling
+
+All unconditioned vectors are centered. Set
+
+    G_r=Cov(P_r), C_r=Cov(V_r,P_r), K_r=C_r G_r^{-1},
+    X_r=V_r+K_r(d_r-P_r),
+    Y=V_0+K_0(d_0-J).                                      (5)
+
+Only the **pin** covariance is inverted. No inverse, inverse square root, or
+Lipschitz square-root map is assumed for the singular six-endpoint covariance.
+The Gaussian residual V_r-K_r P_r is orthogonal to P_r, hence independent
+of it. Its covariance is the Schur complement, and its mean after adding
+K_r d_r is K_r d_r. Thus (5) realizes the canonical conditional marginal
+for every prescribed finite pin value, not conditioning on a positive-
+probability event. It is an artificial common coupling realizing the physical
+marginals, not a claim that one physical field is simultaneously conditioned
+to all the different pin events.
+
+Here are explicit finite constants showing that the rate is not inferred
+merely from continuity. Put p0=||J||_L², v0=||V_0||_L²,
+pb=p0+C_P, vb=v0+C_V, and lambda=lambda_min(G_0)>0. For r<=1, covariance
+Cauchy--Schwarz gives
+
+    ||G_r-G_0||op <= alpha r²,  alpha=C_P(2p0+C_P),
+    ||C_r-C_0||op <= beta r,    beta=C_V pb+v0 C_P.
+
+Choose r_* small enough that alpha r_*²<=lambda/2 (no constraint is
+needed from this inequality if alpha=0). Then
+
+    ||G_r^{-1}||op<=2/lambda,
+    ||G_r^{-1}-G_0^{-1}||op<=2 alpha r²/lambda²,
+    ||K_r||op<=k_b:=2 vb pb/lambda,
+    ||K_r-K_0||op<=k_d r,
+    k_d=2 beta/lambda+2 v0 p0 alpha/lambda².                  (6)
+
+The inverse difference follows from the exact resolvent identity; the bound
+for K uses C_rG_r^{-1}-C_0G_0^{-1}. Define m_r=K_r d_r and
+R_r=V_r-K_rP_r, with R_0=V_0-K_0J. Formula (1) gives
+
+    ||m_r-m_0|| <= C_m r, C_m=k_d||d_0||+k_b/12,
+    ||R_r-R_0||_L² <= C_R r,
+    C_R=C_V+k_d p0+k_b C_P.                                (7)
+
+In the second line split the difference into V_r-V_0,
+-(K_r-K_0)J and -K_r(P_r-J). The last term is actually second order.
+The first line retains the deterministic mean change; a covariance bound
+alone would not imply it. Since R_r-R_0 is centered,
+
+    e_r²=E||X_r-Y||²=||m_r-m_0||²+E||R_r-R_0||²
+          <=(C_m²+C_R²)r²=:C_e² r².                        (8)
+
+The difference is Gaussian, possibly singular. A Gaussian vector with mean
+m and centered covariance A has an Lp norm bounded by
+`||m||+c_(p,6) sqrt(tr A)` for every finite p>=1, by its PSD square-root
+representation and the standard Gaussian norm moment. Applied to the
+*difference law*, (8) proves (A). This use of a square root supplies a moment
+bound; it does not estimate how square roots vary with r.
+
+Uniform marginal means satisfy `||m_r||<=M:=k_b(||d_0||+1/12)`, including
+r=0. Projection only reduces the covariance, so
+`||Cov(X_r)||op<=vb²=:Lambda`. The same bounds hold for Y. All finite
+moments are uniformly bounded. For any fixed r the jointly Gaussian pair
+(X_r,Y) can, if desired, be represented with a common standard Gaussian of
+at most twelve coordinates; that representation preserves (8). We do not
+claim O(r) variation of the particular symmetric six-by-six covariance roots.
+
+## 5. Verification in the actual periodized field
+
+[P] §2 gives positive weights
+
+    p_n=exp(-2pi²|n|²/24²)/sum_j exp(-2pi²|j|²/24²), n in Z².
+
+They satisfy every polynomial spectral-moment condition. In particular
+`sum_n p_n(1+|(2pi/24)n|^10)<infinity`. Differentiating the Fourier symbols
+in weighted L², with dominated convergence, gives mean-square derivatives
+through order five and continuity at that order. This supplies the H-valued
+regularity of §2, all D_j/E_j/J_1 used above, and joint Gaussianity of the
+derivatives and their integrals. No uniform pathwise derivative bound is
+assumed in place of these mean-square facts.
+
+The six contact derivative monomials in J are distinct. With every p_n>0,
+a zero-variance linear combination would give a polynomial vanishing on
+Z², hence identically zero. Thus G_0 is positive definite, supplying lambda.
+Appending f_yy(0) gives seven distinct monomials, so its Schur complement
+is strictly positive. Under J=d_0 the variable Q=f_yy(0) therefore has a
+nondegenerate real Gaussian law. In particular `0<z0=E[(Q_-)²]<infinity`.
+It is the actual conditional torus law, not an inserted N(-b,2) law.
+
+Formula (5) at zero gives the stated contact vector with
+`(a,Q)` distributed as `(f_xxy(0),f_yy(0))` conditional on `J=d_0`. Formula (2) and P3=P4=0 give the
+actual positive-r physical endpoint vector. This checks both ends of the
+same-coupling interface before applying any tilted-law inequality.
+
+## 6. Consequences, with the previously proved inequalities credited
+
+Use [N]'s exact `W_r=r² T_r(X_r)` and `T_0(Y)=(Q_-)²`. Its pointwise
+polynomial-growth Lipschitz estimate, integrated as in [T] (10), yields
+
+    Delta_r=E|T_r(X_r)-T_0(Y)| <= B e_r+D r <= C_D r,
+    B=sqrt(2048(M^6+480Lambda³)), D=8(M^4+48Lambda²),
+    C_D=B C_e+D.                                           (9)
+
+This is an absolute L1 estimate, not cancellation of signed expectations.
+It proves (B), after reducing r_* so C_D r_*<=z0/2. The physical lower
+normalizer is then `(z0/2)r²`; the normalized one is z0/2. The qualitative
+O(r) normalizer conclusion already exists in [R] (11); this is its explicit
+six-coordinate compatibility and does not replace historical finite-band data.
+
+For H=(E[(Q_-)^4])^(1/2), [T] (5) gives
+
+    d_BL(nu_r,nu0) <= min(2,(2 C_D+H C_e)r/z0).              (10)
+
+Its Gaussian moving-event tail also satisfies
+`kappa(t)<=sqrt(3) H C_e² r²/(z0 t²)`. A same-set probability rate for a
+general Borel set still needs its own limiting boundary margin. No arbitrary-
+event or endpoint-total-variation conclusion follows from (10).
+
+Finally [S] (8) is the separately proved direct typed-sign estimate. In its
+notation K2<=M²+6Lambda=:C2 and K6<=32(M^6+480Lambda³)=:C6. Since the
+coupling constructed here is Gaussian and has the exact contact identities,
+its conclusion gives
+
+    nu_r{c_+>=0}
+       <= (2/z0)[sqrt(2 C6) C_e r²
+                        +sqrt(105 C2) C_e^4 r^4/8].         (11)
+
+This supplies the concrete-field e=O(r), moment, and positive normalized-
+floor inputs that [S] deliberately retained. For r<=1 the right side is
+O(r²), proving (C). No independence between error coordinates or endpoint
+Hessians is introduced. No small-r exponent is claimed sharp. [S]'s earlier
+negative-part predecessor and its other author-side consequences retain their
+own attribution; the contribution here is verification of the actual coupling
+input, not a repeated sign proof.
+
+## 7. Exact finite diagnostics and audit exclusions
+
+`regression_rate_check.py` represents the finite polynomial field
+`sum_(i,j) Z_(i,j) x^i y^j/(i! j!)` with independent standard normal Z,
+using horizontal degrees 0..5, 0..3, 0..1 for transverse degrees 0,1,2.
+It is **not** the stationary periodic field. No sampling is used: means,
+regression factors and covariances are exact Fraction matrices. Its ten-test
+suite checks exact raw pins, physical/adjusted equality after conditioning,
+residual orthogonality, the Schur identity, contact rank two, positive-radius
+rank six, seven rational-radius pin/endpoint error bounds, and two controls
+showing why mean and common-factor terms cannot be dropped. Its nonzero
+linear endpoint error illustrates why pin O(r²) is not endpoint O(r²).
+
+Run from this directory with Python's standard library only:
+
+    python -B -S test_regression_rate.py -v
+    python -B -O -S test_regression_rate.py -v
+    python -B -S regression_rate_check.py
+    python -B -O -S regression_rate_check.py
+
+Named mutants `omit-mean`, `covariance-only`, `reverse-gap`, and
+`quadratic-endpoints` must exit1 with their distinct `REGRESSION_RATE_FAIL`
+messages; an unknown label must exit2. These finite controls support the
+algebra and expose inadmissible shortcuts. The Hilbert and spectral arguments
+in §§2–6, not the test grid, are the application proof. No local Lean, complete
+repository replay, or automatic hosted execution of this standalone checker
+is claimed. A future general CI success is not such an execution receipt.
+
+No original proof, historical review, flags, scientific register, workflow,
+formal package or sibling branch is changed. Full P0.2/Palm/elder-failure/
+Cap/IBA2/global closure, all-mark or dimension uniformity, numerical constants,
+and independent full-package formal alignment remain outside this note.
+The four-file application requires its own nonauthor review before eligible
+integration; parent reviews are not transferred to it.
+
+## Sources (immutable interfaces; metadata is not a mathematical premise)
+
+[P] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`,
+Math- commit `5e027afcae8c7bd42a6df8ae2025f82105b3e586`, blob
+`dfed3b8d318a3ab1950957f393307733a4bef3f2`, §§1–2: exact model and positive
+spectrum/finite-jet independence. No broader parent theorem is imported.
+
+[E] `reviews/lm006_endpoint_extension_20261005/NOTE.md`, commit
+`e31b6ad8356e0760c8a7fb6e816ff640c45176c9`, blob
+`551aa563562bc88a19e33d146259c03aa2d318f6`, §§1–4 and 6: exact pins,
+adjusted coordinates, Hilbert kernels and contact identification.
+
+[R] `frontiers/remote_window_20260924/PROOF.md`, commit
+`f4c33a98a982d50aa490e49ce9327c755682527c`, blob
+`b383bfcc88ec4ad497dff01fb6640e429ba24a84`, §§2–4, especially (5),(11):
+existing common-regression rate and normalized-weight precedent. Its remote
+mean-measure TV is not endpoint-law TV; its remote-count theorem is not used.
+
+[N] `reviews/lm006_normalizer_stability_20261005/NOTE.md`, commit
+`5e027afcae8c7bd42a6df8ae2025f82105b3e586`, blob
+`5790aaa0332bd8e79b276be1577e4c5f3502eed6`, §§2–5: typed identity and moment
+bounds. [T] `reviews/lm006_tilted_endpoint_law_20261005/NOTE.md`, commit
+`ecafa5206c528b0c662ea0bcbb26938681251aa5`, blob
+`6633cdf0bfee55e2fc2046176effbf44634454b3`, §§2–4: absolute-weight L1,
+normalized-density, BL and moving-event inequalities.
+
+[S] `reviews/lm006_typed_sign_suppression_20261005/NOTE.md`, commit
+`1dfe801eb61a8335a2e42b886fca9b000fb3598c`, blob
+`9015971fa03f44ae99d410facc6f3d8c26cdc945`, §§3–5, especially (8): the
+reviewed conditional sign inequality. Its original PR #325 review does not
+cover this new application. [D] `reviews/lm006_positive_radius_density_20261006/NOTE.md`,
+commit `b154cd0074798850bfafe2c0f7e433707d93403f`, blob
+`a6d5209a64d37b3ea28cb7dd64d92888a74acfe1`, cited only for the compatible TV
+obstruction, not used to prove the rate.
+
+[W] James Wilson et al., *Efficiently sampling functions from Gaussian process
+posteriors*, PMLR 119, 10292–10302 (2020),
+https://proceedings.mlr.press/v119/wilson20a.html . Primary publication metadata
+and abstract were checked in this session. Its decomposition motivates
+context only; no unread theorem is invoked and no new general method is
+claimed. The proof of the regression marginal in §4 is self-contained.