B.10 Paper IX: simultaneous coefficient and parameter deformation
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Author: Komorebi P. — 8 September 2026.
This block combines coefficient perturbation with prescribed parameter motion. The original solved Jacobian controls existence, local uniqueness, displacement, and the first-order error. The four new mathematical modules compile. The fresh verification report and independent audit record the authoritative release status and exact source revisions.
B.10.1 The theorem
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Let \(p\) be any prime, \(R=\mathbb Z_p\), and \(Q=\mathbb Q_p\). Let
Here \(e,k\ge0\) are arbitrary finite dimensions. There are as many solved coordinates as equations. Define the actual derivative blocks at the original root:
Suppose, for natural numbers \(s,M\), that
For every \(v\in R^k\), set
Then there is exactly one \(y\in y_0+p^{s+1}R^e\) with
It satisfies
and there is an integral vector \(\varepsilon\in p^{2M-3s}R^e\) such that, in \(Q^e\),
The same original exponent \(s\) appears throughout. No inverse or rank assumption on the perturbed Jacobian is imposed. The statement includes arbitrary integral p-adic coefficients and centers, the dyadic prime, and zero dimensions.
The full-point theorem constructs \(b=(y,\tau)\) and also proves
Parameter equality \(b_T=\tau\) is exact. Uniqueness concerns every root in the stated solved-coordinate ball with these parameters, not only roots obtained by the construction. No global uniqueness is claimed.
B.10.2 Actual augmented system
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Define polynomial systems with \(e+k\) equations and variables by
Both are actual polynomials over \(R\). Direct evaluation proves
Actual polynomial partial differentiation gives
These identities are proved from evaluation and partial derivatives, rather than supplied as additional hypotheses.
Choose the integral matrix \(A=p^sJ^{-1}\). The existing inverse-certificate equivalence gives
Put \(c=p^s\) and define an integral block matrix
Then
The second equality uses \(JA=cI_e\). The generic block proof works over any commutative ring. The p-adic specialization also explicitly proves that \(B\) is invertible over \(Q\) and that \(p^sB^{-1}\) is integral. There is no denominator loss from appending the parameter equations.
B.10.3 Construction, error, and uniqueness
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Apply the previously checked weighted coefficient-perturbation construction to \(\mathcal F,\mathcal H,a,C\). Its hypotheses follow from the actual augmented identities and the two-sided certificate. It constructs \(b\), proves the full displacement bound \(p^{M-s}\), and proves uniqueness throughout \(a+p^{s+1}R^{e+k}\). The augmented root equivalence gives the required original equations and exact parameter values.
The checked integral first-order estimate gives
The decisive block action is
Projecting the solved coordinates therefore gives
The existing rational-conversion theorem turns \(p^{M-s}A\) into \(p^MJ^{-1}\) and retains an integral error vector. This proves the displayed first-order formula with the original derivative blocks.
For any competing solved-coordinate root \(z\in y_0+p^{s+1}R^e\), the joined point \((z,\tau)\) lies in the augmented uniqueness ball: \(M>2s\) implies \(M\ge s+1\), so \(\tau-t_0=p^Mv\in p^{s+1}R^k\). Full-point uniqueness gives \((z,\tau)=b\), hence \(z=y\). This is the additional comparison needed to obtain uniqueness in exactly the ball stated in the solved-coordinate theorem.
B.10.4 Checked controls and precision boundary
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For \(p=2\), \(s=0\), \(F(Y,T)=Y+T\), \(H=1\), \(a=0\), and \(v=1\), both original blocks satisfy \(J=K=1\). At \(M=1\), the exact parameter is \(\tau=2\), and the unique solved value is \(y=-4\). The combined correction is \(-2(1+1)=-4\), with zero error.
The sign controls use \(M=2\): the actual augmented root is \((y,\tau)=(-8,4)\), and the required error ideal is \(16\mathbb Z_2\). Omitting \(Kv\) predicts \(-4\), with error \(-4\notin16\mathbb Z_2\). Reversing its sign predicts \(0\), with error \(-8\notin16\mathbb Z_2\). Both failures are checked. At binary depth \(M=1\), the reversed-sign difference would lie in the allowed error ideal, so that shallower example alone would not distinguish the sign. This control-design limitation was identified and corrected before the release gate.
For parameter-only motion, use
A checked transfer from the prior scalar quadratic root constructs an even \(y\) with
Thus the error exponent \(2M-3s=2\) is attained even when only parameters move. No uniform extra error digit can be added to the general theorem. This example establishes that limitation; it does not assert optimality at every choice of \(p,s,M\).
The inherited controls continue to record the necessity of the strict weighted threshold and other earlier scope boundaries. General rationally rank-deficient systems are outside the inverse hypothesis.