For the actual canonical complete Weil operator, put a=log(lambda).
There are explicit source-defined physical subspaces G_a with
[ \dim G_a=2\lfloor a/(2D)\rfloor+1, \qquad |\mathsf W_\lambda P_{G_a}| \le C a e^{-\lambda/100}. ]
Here D=ceil(4*pi*sqrt(M2)) is a single fixed integer, where
M2=||(1+|x|)^2 phi||_2 and phi is the ordinary-unit-normalized
Gaussian co-Poisson radical already present in lemma v115-source-mass.
C and the eventual threshold are not numerically certified. The exponent
is deliberately conservative. It is an asymptotic analytic estimate, not
an effective finite-window certificate.
The full proof is in new_section.tex. Its essential quantitative steps are:
1/6. No growing source-to-physical transfer is assumed.1/2. Centers stay in [-a/2,a/2]. The Gram stays above I/2.lambda^2, the two poles, and the archimedean multiplier. Ordinary
column bounds become a growing-block operator bound only after the
verified Gram normalization.It follows that removal of any subspace of dimension below dim G_a
leaves a unit vector with absolute form value at most C a e^{-lambda/100}.
Thus positive polynomial or uniform lower gaps fail after removal of
o(log lambda) directions. This is an upper ceiling on a proposed positive
gap, not a lower bound on the form.
Antisymmetric pairs of translated columns give floor(a/(2D)) independent
odd near-radical vectors. Every one is orthogonal to all even image columns.
Therefore the same ceiling applies to the physical G2.6 complement when
its deep and plunge images are even, however large their total rank.
If odd images are also included, the assertion requires that their number
of independent constraints be smaller than that odd block dimension.
Compact resolvent then implies at least dim G_a eigenvalues, including
the stated odd count, in [-2 epsilon_a,2 epsilon_a]. Their signs are not
determined. No G2 sign gap was closed.
The continuum cancellation is valid on all complex physical tests:
[ 2|c\rangle\langle c|-2|s\rangle\langle s|-K_a^+=K_a^-. ]
Consequently the original arithmetic beta_a represents the whole form,
not just its even restriction. The even and odd signed concentration
inequalities remain independent blocks. Only the even block loses the
actual even source. The resulting complete lower error is
[ \max{\eta_a^{odd},\eta_a^{even}+2\epsilon_a/\sqrt3}. ]
The odd block is not proved positive. A pole-only countermodel already
shows why parity transfer cannot be automatic: its even form is
nonnegative with many exact null sources, but its odd eigenvalue is
-(sinh(a)-a). This scoped countermodel is not the arithmetic Weil form.
Register ID: G2.6-RAD-02, a quantitative growing-block obstruction. Classification: continuation and strengthened no-go, not a new live positivity mechanism; worldwide novelty unresolved and not claimed.
The nearest project antecedent is the v1.14 research-log entry under “Adversarial work, rejected shortcuts, and research findings.” It uses even derivatives of exactly the same Gaussian radical and smooth cuts to exclude a fixed gap after any fixed-rank removal. That entry explicitly does not establish a growing-rank rate.
The additional ingredient here is a uniform physical Riesz Gram bound
for a growing lattice of translates, together with a complete operator
residual uniform over that lattice. It gives rank proportional to
log(lambda), an explicit exponential scale, and an odd-block statement
that survives arbitrarily many even image columns. It is not a renamed
Schur complement, coordinate change, cutoff tuning, or tighter scalar
prime-norm estimate. The prime absolute bound is used on an exponentially
small exterior radical remainder, not on arbitrary vectors to infer sign.
It does not duplicate the structured inverse, finite-rank metric, dyadic pair-norm, shared-channel/common-Gram, Picone, negative-only Schatten, or primitive transport sign proposals. None of those mechanisms supplies the sign here. CAP-01b remains the live conditional sign target.
Primary literature: Connes–Consani, Spectral triples and zeta-cycles, Section 3, supplies the classical radical and explains near-radical prolate sources: published article, original preprint. Connes–Consani–Moscovici’s Zeta Spectral Triples is the operator framework already used by the manuscript. Zhu’s compact-window study, v2 separately treats the two pole signs and fixed-window parity certificates. The present extension reuses these classical structures; no priority claim is made for radical localization or small Weil eigenvalues.
Everything acts in ordinary L^2(-a,a) after zero extension, with the
manuscript’s first-slot-linear pairing. The cutoff columns are in
C_c^infinity(-a,a), so operator-domain membership is direct. The
orthogonal projector is G(G*G)^(-1)G*. No Fourier cutoff is imposed.
The original finite sampler and its explicit graph defect are unchanged.
No new endpoint comparison is asserted; the new columns vanish near
the actual endpoints. The old repaired source and its boundary normalization
are not replaced by these columns.
The gap-free estimate remains conditional on the signed arithmetic bound
[ q_a[f]\ge-\eta_a|f|^2, \quad f\perp G_a,\quad \eta_a\to0. ]
Alternatively keep the actual rank-one source and establish both parity blocks of CAP-01b with errors tending to zero. Small residuals do not prove either inequality. The new growing projection may be useful but is not registered as a new coercivity mechanism without an independent sign estimate for its remaining complement.
An independent adversarial agent reviewed the analytic proof, its source
dilation, all-prime tail estimate, Gram factors, explicit spacing, and
spectral-count deduction. See adversarial_review.md.
check_growing_radical.py checks Gaussian moments, Poisson symmetry and
complex mixed-parity pole cancellation. Its values are ordinary floating
point diagnostics, not interval certificates. Its approximate spacing
D=15 is not used as an exact constant by the proof.
The previous primitive-transport diagnostic remains only a finite-window failure to demonstrate decay; it is not a proved arithmetic cofinal no-go. The rigorous new obstruction is the growing-block estimate above.
No lambda=8 computation was attempted. G2 and RH remain open.