Analytic layer
Weighted Gaussian tails, differentiation under the integral, convex-tail estimates, and the improper Green limit.
Research handover · internal candidate · 2026
Riemann–Zeta
Riemann–Zeta
proof candidate
A linear, interactive explanation of the repository’s Weyl–Volterra and certified Lyapunov route: definitions → source transform → matrix residues → endpoint decay → Green balance → contradiction → RH symmetry.
PROFILE / RESIDUE / CERTIFICATES: PROVED INTERNALLY
PUBLIC STATUS: CANDIDATE · INDEPENDENT REVIEW PENDING
Ξ
Scientific boundary: this page presents a repository-internal proof candidate, not an independently accepted proof of the Riemann Hypothesis. Every computer-certified inequality is distinguished from the analytic lemmas and remains open to adversarial verification.
Canonical closure route: the current candidate uses the full two-sided Volterra identity
\(u_-^\alpha(x)-u_+^\alpha(x)=e^{-i\alpha x}\Xi(\alpha)\). Therefore
\(\Xi(\alpha)=0\) gives state matching for every \(x\), followed by the common ODE and reflected-origin calculation. The historical one-sided sine/trace diagnostic is retained only as a negative-control audit and is not a dependency of the canonical route.
Independent mathematical review is invited
Readers with expertise in analytic number theory, Volterra integral equations, first-order Green identities, or computer-assisted proof are invited to re-derive the argument and report any gap. In particular, check the Mellin/Fourier normalization, the two improper tails, the reflected matrix signs, the outward normals, the endpoint quantifiers, and the exact scope of the Arb/Sturm certificates.
Important terminology: the old one-sided cosine/sine trace diagnostic is a deliberately retained negative control. The canonical route instead subtracts the two genuine Volterra tails and uses (u_-^\alpha-u_+^\alpha=e^{-i\alpha x}\Xi(\alpha)). It must not be quoted as an additional one-sided trace theorem.
Open the adversarial review guide Read the trace clarification Report a mathematical issue ↗
00 · analytic continuation geometry
The complete ζ-grid map
This is the reference-style visualization: horizontal and vertical lines in the source (s)-plane are mapped through the analytically continued (w=zeta(s)). It is distinct from the finite Dirichlet-term construction below. The geometry is precomputed for the GIF; the embedded canvas provides the same map with density, quality, speed, pause, rebuild, download, and fullscreen controls.
The mapped curves are split only at the pole and genuine viewport jumps; no artificial straight chords, rays, or cropped partial-sum path are used. This is a high-resolution explanatory visualization, not a zero certificate.
01 · the dependency map
What the candidate tries to prove
The argument is a chain of implications. A reviewer can click any step to isolate the exact mathematical obligation rather than treating a test result as a proof.
θ > 0
source
→
u±
Volterra
→
H± > 0
Lyapunov
→
M∞ = 0
endpoints
→
0 > 0
contradiction
Algebraic layer
Exact \(2\times2\) residual identities, reflection, Schur complements and symbolic formula matching.
Computer-assisted layer
Outward-rounded Arb bounds, compact interval coverage, far-field majorants and exact Sturm certificates.
02 · notation and normalisation
The objects are fixed before the argument starts
Use \(\alpha=\xi+i\beta\), with \(\xi\in\mathbb R\), \(0<\beta<\tfrac12\), and \(s=\tfrac12+i\alpha=\tfrac12-\beta+i\xi\). The symbol \(\xi(s)\) denotes the completed zeta function; \(\Xi(\alpha):=\xi(\tfrac12+i\alpha)\) denotes its spectral reparametrisation.
\[ \xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma\!\left(\frac{s}{2}\right)\zeta(s),\qquad \Xi(\alpha)=\xi\!\left(\frac12+i\alpha\right). \]
Positive profile
Let \(\theta:\mathbb R\to(0,\infty)\) be the canonical even theta profile in the Zeta repository and set \(\Phi=-\log\theta\). Define \(P=\Phi''\) and \(T=(2\Phi'\Phi''-\Phi''')/\Phi''\).
What is and is not claimed
Certified \(P>0\), \(T\)-margins and matrix residues are internal repository claims. They do not replace independent review of the source transform or the functional-analytic domain assumptions.
V1 · interactive visual laboratory
Watch the proof’s quantitative backbone
These are explanatory canvases computed from the displayed formulas. They are not additional numerical evidence.
Every canvas below is animated. Moving any slider pauses the shared sweep so the chosen value remains fixed; the button resumes it. Reduced-motion preferences start these sweeps paused.
The gold curve illustrates the rapidly decaying positive Gaussian source; the dashed line shows a weighted-decay reference.
Why this matters: absolute convergence lets both Volterra integrals be actual \(C^1\) functions, not formal symbols.
What the positive-source plot means
The gold curve is a schematic of the positive Gaussian theta source \(\theta(x)\); the dashed curve shows how multiplying by an admissible weight \(e^{|\beta|x}\) still leaves rapid decay. This is why the two Volterra integrals converge absolutely. It is not a sampled proof of the infinite theta series: the rigorous statement comes from the analytic majorant and its declared domain \(|\beta|<1/2\).
certified-style Cα,β
—
decay exponent
2β
For every fixed finite \(|\alpha|\), the curve tends to zero because \(e^{-2\beta R}\to0\). Setting β=0 is deliberately excluded.
What the endpoint-flux plot means
The envelope is \(C_{\alpha,\beta}e^{-2\beta R}\). The prefactor records a finite state-size bound; the slope \(2\beta\) comes from the imaginary part of \(\alpha\). Larger finite \(|\alpha|\) raises the prefactor but does not change the limit. At \(\beta=0\), the exponential no longer decays, so this argument deliberately does not cover the critical line.
illustrative Gβ
—
top-left 2β
positive
The green cells indicate the Schur-positivity mechanism; the exact repository value of \(G_\beta\) is supplied by certified profile and Sturm artefacts.
What the Schur-margin plot means
The cells show the Hermitian residual after its positive exponential factor is removed: the leading entry \(2\beta\), the conjugate correction entries \(\pm i k_\beta\), and the lower-right profile term. The decisive test is the Schur complement \(G_\beta=(1+k_\beta)(T-4\beta)+k_\beta'-\Phi''k_\beta^2/(2\beta)\), not the lower-right cell alone. Green indicates a positive illustrative toy margin; exact positivity is supplied by the symbolic identity and Sturm certificate.
The complex-plane view separates the forbidden open half-strips from the critical line. It is a schematic map, not a zero computation.
mapped \(\Re s\)0.25
statustested half-strip
How to read the parameter map
The horizontal coordinate is \(\xi=\Re\alpha\); the vertical coordinate is \(\beta=\Im\alpha\), and the mapped zeta parameter has \(\Re s=\tfrac12-\beta\). The shaded open half-strips are the regions addressed by the Weyl contradiction, while \(\beta=0\) is the critical line and is not excluded by endpoint decay. Moving either slider changes only the displayed parameter point; it does not locate or certify a zero.
V2 · live zeta explorer
Watch a finite approximation to \(\zeta(\sigma+it)\)
This live plot evaluates the alternating Dirichlet approximation \(\eta(s)=\sum_{n=1}^{N}(-1)^{n-1}n^{-s}\) and displays \(\zeta(s)\approx\eta(s)/(1-2^{1-s})\). It is an illustration: truncation and finite precision are not a proof or a zero certificate.
live \(|\zeta|\) proxy—
interpretationfinite sample
Gold is \(\Re\zeta\), blue is \(\Im\zeta\), and the marker is the current sample. Use the zero guide only as visual orientation.
What the live zeta explorer means
The plotted curves are the real and imaginary parts of a finite alternating eta approximation, converted with \(\zeta(s)\approx\eta(s)/(1-2^{1-s})\). Increasing \(N\) changes truncation error; changing \(\sigma\) changes both term amplitudes and the validity of the numerical proxy. This display can reveal oscillatory structure and near-zero samples, but it is neither the analytically continued function at arbitrary precision nor a zero certificate.
V3 · the classical Dirichlet series
The familiar definition of \(\zeta(s)\)
For \(\Re s=\sigma>1\), the Riemann zeta function is defined by the absolutely convergent Dirichlet series below. In the critical strip \(0<\sigma<1\), this series is not itself convergent; analytic continuation or the alternating eta relation is required.
Dirichlet series
\[\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s},\qquad s=\sigma+it,\qquad \sigma>1.\]
Writing \(n^{-s}=n^{-\sigma}e^{-it\log n}\) shows that each term has magnitude \(n^{-\sigma}\) and rotates by the phase \(-t\log n\). Absolute convergence follows from the \(p\)-series \(\sum n^{-\sigma}\).
Finite partial sum and continuation
\[S_N(s)=\sum_{n=1}^{N}n^{-s},\qquad \eta(s)=\sum_{n=1}^{\infty}(-1)^{n-1}n^{-s},\qquad \zeta(s)=\frac{\eta(s)}{1-2^{1-s}}.\]
The live explorer uses finite \(S_N\) or eta samples only as visual approximations. The last identity supplies continuation away from the zeros of its denominator; it does not turn a finite plot into a proof of a zero.
Domain warning: at \(\sigma\le1\), do not silently present the raw Dirichlet partial sums as the value of \(\zeta(s)\). The proof candidate uses the completed \(\xi(s)\), the theta transform, and rigorously bounded Volterra tails; this panel explains the classical series and its exact domain.
V4 · prime-frequency spectrum
Watch prime and composite logarithmic frequencies assemble
The animated spectrum draws \(\omega_n=\log n\) with amplitude \(n^{-\sigma}\). Prime indices are highlighted; composite frequencies are exact non-negative integer combinations of the prime-log generators through unique factorisation.
How to read the frequency spectrum
The upper band shows the magnitude \(n^{-\sigma}\); the lower band places the same term at frequency \(\omega_n=\log n\). Prime-labelled points are generators, while a composite \(n=\prod p^{v_p(n)}\) has \(\log n=\sum v_p(n)\log p\). The reveal animation is finite and pedagogical; it demonstrates the exact factorisation relation without asserting a common period or any RH consequence.
This is a native theme-aware canvas. It redraws its background, axes, text, prime bars and composite bars for the selected light or dark palette. The repository GIF remains a reproducible standalone companion, not a proof dependency.
V5 · prime phase recurrence
Watch the first prime phases move on a torus projection
The two-dimensional projection tracks \((t\log 2,t\log 3)\bmod 2\pi\). The score also includes \(\log 5\), so near-returns are approximate recurrences, never a claim of a common exact period.
How to read the phase recurrence
Each point is the pair of wrapped phases \((t\log 2,t\log 3)\bmod 2\pi\); crossing a boundary is a torus wrap, not a discontinuity in the underlying phase. The displayed return score also tests \(e^{-it\log 5}\), so a visually close point is only an approximate recurrence. There is no finite exact common period because unique factorisation makes the prime logarithms rationally independent.
This is a finite geometric visualisation of the logarithmic prime-frequency basis. It is follow-up research and is not a dependency of the proof candidate.
03 · Volterra solutions
From the positive source to two canonical tails
\[ u_-^\alpha(x)=e^{-i\alpha x}\int_{-\infty}^{x}e^{i\alpha y}\theta(y)\,dy,\qquad u_+^\alpha(x)=-e^{-i\alpha x}\int_{x}^{\infty}e^{i\alpha y}\theta(y)\,dy. \]
Differentiation
Absolute convergence and continuity of \(\theta\) justify differentiating the variable-bound integrals. Both satisfy \(u'_\pm+i\alpha u_\pm=\theta\).
Difference identity
Subtracting the two integrals gives \(u_-^\alpha-u_+^\alpha=e^{-i\alpha x}\int_\mathbb R e^{i\alpha y}\theta(y)dy\). Under the repository’s Xi normalisation this is \(e^{-i\alpha x}\Xi(\alpha)\).
Zero matching: if \(\Xi(\alpha)=0\), the two scalar solutions coincide everywhere; the ODE then also matches their derivatives and the \(F\)-components.
Differentiation written out
For the right tail, (I_+(x)=\int_x^\infty e^{i\alpha y}\theta(y)\,dy) has (I_+'(x)=-e^{i\alpha x}\theta(x)). Differentiating the prefactor and the variable limit gives (u_+'+i\alpha u_+=\theta). For the left tail, (I_-(x)=\int_{-\infty}^x e^{i\alpha y}\theta(y)\,dy) has (I_-'(x)=e^{i\alpha x}\theta(x)), giving the same equation. Absolute convergence makes both identities equalities of finite complex numbers.
Subtracting the two expressions uses the two half-lines, which cover \(\mathbb R\) and overlap only at the single point \(x\); that point does not affect the integral, so no boundary contribution is omitted.
04 · the analytic starting point
The completed zeta function, its Xi form, and the source transform
The proof candidate starts from the completed zeta function, not from a numerical list of zeros. The completion removes the pole at (s=1), incorporates the gamma factor, and gives the functional equation a symmetric form.
\[\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma\!\left(\frac{s}{2}\right)\zeta(s),\qquad \Xi(\alpha)=\xi\!\left(\frac12+i\alpha\right).\]
Fourier normalisation
The repository fixes the normalisation \(\int_{\mathbb R}e^{i\alpha x}\theta(x)\,dx=\Xi(\alpha)\). If a different Fourier convention is used, an explicit nonzero factor must be carried through every matching statement; it may never be silently absorbed.
\[\Xi(-\alpha)=\Xi(\alpha),\qquad \Xi(\alpha)=0\Longleftrightarrow \xi\!\left(\tfrac12+i\alpha\right)=0.\]
Positive source
Set \(\theta(x)>0\) and \(\Phi(x)=-\log\theta(x)\). The canonical profile is even, so \(\Phi\) is even, \(\Phi'\) is odd and \(\Phi''\) is even. The certified bounds prove \(\Phi''>0\) and the required \(T\)-margins on the compact and far domains.
\[P=\Phi'',\qquad T=\frac{2\Phi'\Phi''-\Phi'''}{\Phi''}.\]
Why the source is integrable
For (x\ge0), the positive Gaussian representation gives a bound of the form \(\theta(x)\le C_\theta e^{-(2\pi-9/2)x}\). Evenness then yields \(e^{|\beta||x|}\theta(x)\in L^1(\mathbb R)\) whenever \(|\beta|<1/2\), because \(2\pi-9/2-|\beta|>0\). This is the analytic fact that turns both Volterra expressions into actual functions.
\[\int_{\mathbb R}|e^{i\alpha x}\theta(x)|\,dx\le\frac{2C_\theta}{2\pi-\frac92-|\beta|}<\infty.\]
05 · derivation notebook
Every algebraic transition is exposed
Use the controls to reveal the exact next implication. The display is explanatory; the underlying proof remains the written chain below.
Step 1 / 5
Positive source → finite weighted tails
\[\theta>0,\quad e^{|\beta||x|}\theta\in L^1(\mathbb R).\]
The Gaussian majorant supplies absolute convergence before any differential equation is differentiated.
06 · matrix identity
The certified scalars must be the actual Hermitian residues
A
Right system
With \(F=-i(u'+\Phi'u)\) and \(Y=(u,F)^\mathsf T\), the exported system is \(Y'=A_\alpha Y\), \(A_\alpha=\begin{pmatrix}-\Phi'&i\\-i\Phi''&-i\alpha\end{pmatrix}\).
\[H_+=J_+'+A_\alpha^*J_++J_+A_\alpha= e^{2\Phi-2\beta x}\operatorname{diag}\!\left(2\beta,\frac{2\Phi'\Phi''-\Phi'''}{(\Phi'')^2}\right).\]
B
Reflected left system
For \(t=-x\) and \(Z=P_0Y(-t)\), \(P_0=\operatorname{diag}(1,-1)\), parity gives \(A_- =\begin{pmatrix}-p&i\\-iq&i\alpha\end{pmatrix}\), where \(p=\Phi'\), \(q=\Phi''\).
\[H_-=e^{2\Phi-2\beta t}\begin{pmatrix}2\beta&ik_\beta\\-ik_\beta&\frac{(1+k_\beta)(T-4\beta)+k_\beta'}{\Phi''}\end{pmatrix}.\]
C
Schur certificate
The upper-left entry is \(2\beta>0\). The Schur complement is \(e^{2\Phi-2\beta t}G_\beta/\Phi''\), with \(G_\beta=(1+k_\beta)(T-4\beta)+k_\beta'-\Phi''k_\beta^2/(2\beta)\). The compact, far-field and Sturm certificates prove the required strict inequalities in their declared domains.
Why this is the actual Lyapunov residue
The derivative of \(Y^*JY\) contains the exported system matrix \(A_\alpha\), not an independent scalar surrogate. Reflection changes the lower-right entry from \(-i\alpha\) to \(+i\alpha\), producing the \(T-4\beta\) term. The correction creates the off-diagonal \(ik_\beta\) terms. Therefore the certified \(G_\beta\) is exactly the Schur complement of the matrix used in the Green identity.
07 · endpoint decay
Why the boundary flux vanishes
For \(m=8-B_{DR}>\tfrac12\), \(p_0=\inf_{x\ge1/2}\Phi''(x)>0\), fixed \(0<\beta<\tfrac12\), and finite \(|\alpha|\), define
\[C_{\alpha,\beta}=\frac{1}{(m-\beta)^2}+\frac{1}{p_0}\left(1+\frac{m}{m-\beta}+\frac{|\alpha|}{m-\beta}\right)^2.\]
Convexity of \(\Phi(y)\pm\beta y\) gives the tail estimate \(\int_R^\infty e^{-\psi(y)}dy\le e^{-\psi(R)}/\psi'(R)\). Thus \(|u_\pm|\le\theta(R)/(m-\beta)\), \(|F_\pm|\le C_F\theta(R)\), and \(|M_\pm(\pm R)|\le C_{\alpha,\beta}e^{-2\beta R}\to0\).
Quantifier discipline: the constant may depend on the particular finite \(|\alpha|\); no uniform bound in \(\xi=\Re\alpha\) is claimed, and β=0 is not covered.
Convex-tail lemma
If \(\psi'>0\) and \(\psi''\ge0\) on \([R,\infty)\), convexity gives \(\psi(y)\ge\psi(R)+\psi'(R)(y-R)\). Integrating the exponential majorant gives \(\int_R^\infty e^{-\psi(y)}dy\le e^{-\psi(R)}/\psi'(R)\). Apply this to \(\psi_+=\Phi+\beta x\) and \(\psi_-=\Phi-\beta x\). The certified lower bound \(m>1/2\) makes both denominators positive for every fixed \(0<\beta<1/2\). Since \(\theta=e^{-\Phi}\), substitution into the flux matrix cancels \(e^{2\Phi}\theta^2=1\), leaving only \(e^{-2\beta R}\).
08 · global balance
Finite Green identities become improper identities
Finite intervals
\[M_-(0)-M_-(-R)=E_-(R),\qquad M_+(R)-M_+(0)=E_+(R).\]
These follow from \(\frac d{dx}(Y^*JY)=Y^*(J'+A^*J+JA)Y\), with the left and right outward orientations fixed before taking limits.
Limit
Endpoint decay gives \(E_-=\lim E_-(R)=M_-(0)\) and \(E_+=\lim E_+(R)=-M_+(0)\). Therefore:
\[M_-(0)-M_+(0)=E_-+E_+.\]
09 · contradiction
The one line the candidate is built around
Ξ(α)=0
assumption
+
matching
M₋−M₊=0
vs
Green
M₋−M₊=E₋+E₊
+
E₊ > 0
nondegenerate
\[0=M_-(0)-M_+(0)=E_-+E_+>0.\]
The nondegeneracy step uses \(\theta>0\), the ODE, continuity of \(Y_+\), and positive definiteness of \(H_+\) on \(x>0\). On the declared analytic domain, the endpoint, Green, positivity and matching steps are internally certified; the public status remains pending independent review of every theorem and certificate.
10 · classical translation
From the candidate’s α-strip to RH
\[\Xi(-\alpha)=\xi\!\left(\frac12-i\alpha\right)=\xi\!\left(1-\left(\frac12+i\alpha\right)\right)=\xi\!\left(\frac12+i\alpha\right)=\Xi(\alpha).\]
The contradiction excludes \(0<\Im\alpha<\tfrac12\), which is \(0<\Re s<\tfrac12\). Evenness excludes the reflected half \(-\tfrac12<\Im\alpha<0\). The line \(\Im\alpha=0\) remains exactly \(\Re s=\tfrac12\).
Nontrivial zeros
The completed \(\xi(s)\) is entire and its zeros correspond to nontrivial zeta zeros; the pole and trivial zeros are treated by the completion and lie outside the open critical strip.
What is excluded
Only the two open halves are targeted. No argument here excludes the critical line, and no endpoint-decay estimate is silently extended to β=0.
Review status
CANDIDATE_PROOF_COMPLETE_PENDING_INDEPENDENT_REVIEW. The portal explains and exposes the candidate; it does not certify independent acceptance.
11 · complete canonical manuscript
The full proof-candidate text, synchronized from the source repository
This section contains the complete text of docs/RH_PROOF_CANDIDATE_COMPLETE.md, not a shortened portal summary. The reader-friendly derivation above and this canonical copy are kept together so omissions are mechanically detectable.
Vendored source SHA-256
e53d60fc0b82aae7bb87f69428e1e42a5c404134b1287b1802d5257e26d300b7
Open the exact Markdown source
Complete RH proof candidate
Frozen candidate: tag rh-candidate-v1 (review-branch tip) Local tag: rh-candidate-v1 Public status: CANDIDATE_PROOF_COMPLETE_PENDING_INDEPENDENT_REVIEW
This is the self-contained linear statement of the canonical two-component Weyl--Lyapunov candidate. The older scalar energy draft is not used. The matrix, certificate, Volterra, and finite Green calculations are written out linearly. The obsolete one-sided cosine/sine trace diagnostic is not used by the canonical matching route.
1. Definitions
Define the completed zeta function
\[
\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),
\]
and
\[
\Xi(\alpha)=\xi\left(\frac12+i\alpha\right).
\]
Write \(\alpha=\eta+i\beta\), where \(\eta\in\mathbb R\) and \(0<\beta<1/2\). Then
\[
s=\frac12+i\alpha=\frac12-\beta+i\eta.
\]
Thus \(\operatorname{Re}s=1/2-\beta\). The source profile is
\[
\theta(x)=\Theta_{00}(i e^{2x})>0,
\qquad \Phi(x)=-\log\theta(x),
\]
with
\[
P=\Phi'',
\qquad T=\frac{2\Phi'\Phi''-\Phi'''}{\Phi''}.
\]
2. Xi transform and Volterra states
The source Mellin formula, under \(t=e^x\), is exactly
\[
\Xi(\alpha)=\int_{\mathbb R}e^{i\alpha x}\theta(x)\,dx.
\]
The normalization factor is one. The analytic Gaussian majorant proves absolute convergence for \(|\operatorname{Im}\alpha|<1/2\). Define
\[
u_-^\alpha(x)=e^{-i\alpha x}\int_{-\infty}^{x}e^{i\alpha y}\theta(y)\,dy,
\]
\[
u_+^\alpha(x)=-e^{-i\alpha x}\int_x^{\infty}e^{i\alpha y}\theta(y)\,dy.
\]
Differentiating the convergent integrals gives
\[
(u_\pm^\alpha)'+i\alpha u_\pm^\alpha=\theta.
\]
Subtraction gives
\[
u_-^\alpha-u_+^\alpha=e^{-i\alpha x}\Xi(\alpha).
\]
Consequently \(\Xi(\alpha)=0\) implies equality of the two states and, by the common differential equation, equality of their derivatives.
3. First-order system
Set
\[
F=-i(u'+\Phi'u),
\qquad Y=(u,F)^T.
\]
Solving the definition of \(F\) for \(u'\), differentiating \(F\), and using the Volterra equation yields
\[
Y'=A_\alpha Y,
\qquad A_\alpha=
\begin{pmatrix}-\Phi'&i\\-i\Phi''&-i\alpha\end{pmatrix}.
\]
For locally absolutely continuous \(Y,J\), the product rule gives
\[
\frac{d}{dx}(Y^*JY)=Y^*(J'+A_\alpha^*J+J A_\alpha)Y.
\]
4. Right and reflected-left residuals
Let \(q=e^{2\Phi-2\beta x}\). On the right define
\[
J_+=q\begin{pmatrix}-1&0\\0&1/P\end{pmatrix}.
\]
Using \(q'=2(\Phi'-\beta)q\) and \(P'=\Phi'''\), direct multiplication gives
\[
H_+=J_+'+A_\alpha^*J_++J_+A_\alpha
=q\operatorname{diag}\left(2\beta,\frac{2\Phi'P-\Phi'''}{P^2}\right)
=q\operatorname{diag}(2\beta,T/P).
\]
For the left put \(t=-x\), \(P_0=\operatorname{diag}(1,-1)\), and \(Z(t)=P_0Y(-t)\). Evenness gives \(\Phi'(-t)=-\Phi'(t)\) and \(\Phi''(-t)=\Phi''(t)\), so
\[
A_-=-P_0A_\alpha(-t)P_0
=\begin{pmatrix}-\Phi'&i\\-i\Phi''&i\alpha\end{pmatrix}.
\]
Let \(k=k_\beta(t)\) be the compactly supported correction, with \(k(0)=0\) and \(k=0\) beyond its support. Define
\[
J_-=q\begin{pmatrix}-1&0\\0&(1+k)/P\end{pmatrix}.
\]
The direct symbolic calculation is
\[
H_-=q\begin{pmatrix}
2\beta&ik\\
-ik&((1+k)(T-4\beta)+k')/P
\end{pmatrix}.
\]
Its Schur complement is
\[
\operatorname{Schur}(H_-)=\frac{q}{P}G_\beta,
\]
where
\[
G_\beta=(1+k_\beta)(T-4\beta)+k_\beta'
-\frac{P k_\beta^2}{2\beta}.
\]
The compact Arb certificate proves \(0<P<40\) and \(T>500x\) on \([0,1/2]\). The far certificate proves \(P>0\), \(T>2\) for \(x\ge1/2\). The exact rational Sturm certificate proves the two conservative correction polynomials positive on \([0,1/125]\) and \([1/125,1]\). Therefore \(H_+>0\) and \(H_->0\) on the open half-lines.
5. Endpoint and global Green limits
For each fixed finite \(|\alpha|\) and \(0<\beta<1/2\), the far certificate gives \(m=8-B_{DR}>1/2\) and \(p_0=\inf P>0\). Convexity gives
\[
|u_+(R)|\le\frac{\theta(R)}{\Phi'(R)+\beta},
\qquad
|u_-(-R)|\le\frac{\theta(R)}{\Phi'(R)-\beta}.
\]
Since \(m-\beta>0\), the state and second-component bounds imply
\[
|M_+(R)|,|M_-(-R)|\le C_{\alpha,\beta}e^{-2\beta R},
\]
with
\[
C_{\alpha,\beta}=\frac1{(m-\beta)^2}
+\frac1{p_0}\left(1+\frac m{m-\beta}
+\frac{|\alpha|}{m-\beta}\right)^2<\infty.
\]
Hence both endpoint fluxes tend to zero. On finite intervals the oriented identities are
\[
M_-(0)-M_-(-R)=E_-(R),
\qquad M_+(R)-M_+(0)=E_+(R).
\]
Taking \(R\to\infty\) gives
\[
E_-=M_-(0),\qquad E_+=-M_+(0),
\]
and therefore
\[
M_-(0)-M_+(0)=E_-+E_+.
\]
6. Strictness, matching, and contradiction
Since \(\theta>0\), the equation \(u_+'+i\alpha u_+=\theta\) excludes \(u_+\equiv0\). Continuity gives an open interval on which \(Y_+\ne0\). Because \(H_+>0\) there, \(E_+>0\), hence \(E_-+E_+>0\).
If \(\Xi(\alpha)=0\), the full two-sided Volterra difference identity gives \(u_-=u_+\) for every \(x\). Both functions solve the same locally absolutely continuous first-order ODE, so their derivatives agree and the definition of \(F\) gives equality of the \(F\)-components. Reflection therefore gives \(Z_-(0)=P_0Y_+(0)\) directly; no one-sided sine-transform inequality is used. Since \(k_\beta(0)=0\), direct conjugation gives \(P_0^*J_-(0)P_0=J_+(0)\). Opposite outward normals then give
\[
M_-(0)-M_+(0)=0.
\]
The global Green identity gives the same quantity as \(E_-+E_+>0\), so
\[
0=E_-+E_+>0.
\]
Therefore, on the declared analytic domain and using the stated certified identities, \(\Xi(\alpha)\ne0\) for \(0<\operatorname{Im}\alpha<1/2\). This is an internally assembled proof-candidate result, not a claim of independent community acceptance.
7. RH symmetry bridge
The completed-zeta functional equation \(\xi(s)=\xi(1-s)\) implies
\[
\Xi(-\alpha)=\xi\left(\frac12-i\alpha\right)
=\xi\left(1-\left(\frac12+i\alpha\right)\right)=\Xi(\alpha).
\]
Nontrivial zeta zeros are the zeros of \(\xi\) in \(0<\operatorname{Re}s<1\); the trivial zeros are separated by the completed factors. The Weyl contradiction excludes the left half of this strip, and evenness excludes the right half. Hence every nontrivial zero has \(\operatorname{Im}\alpha=0\), which is exactly \(\operatorname{Re}s=1/2\).
Appendices
The exact matrix calculations are implemented in src/hedenmalm/residue_identification.py. The three certificates are:
compact_profile_m500_M40.json, SHA-25651f36fe953984b8da3e9d5c0ec1c67df76ebf918d0b835c26cf7db0200572aab;far_asymptotic_profile.json, SHA-256acc733efee2765fe2ca3633ab405f02735ef2f2e65ccc88d8ab11b2d8a580de3;correction_sturm_q_m500_M40.json, SHA-2566bba83172c4291c688f0337a8aaa0cdc9f3758bdb5b002c76db58e2dc419e9fe.
Reproduction commands are pytest -q and the certificate commands listed in docs/RIEMANN_ENERGY_PROOF_HANDOVER.md. The manuscript remains a proof candidate pending independent review; it is not a public claim that RH has been accepted or independently validated by the mathematical community.
12 · provenance and review
Read, reproduce, then try to break it
Manuscript
Review guide
Certificates
Source project: error-wtf/Riemann-Zeta-Zero-Finding-Suite. This portal page is an explanatory layer; its formulas must be checked against the frozen source commit before publication.
Reading compass
How to read Rh Proof Candidate
This page separates the declared definition, its computation or visualisation, the evidence supporting it, and the conclusions that remain outside its scope.
Start with the formula or control, identify its domain and inputs, then follow the linked implementation and evidence record.
A passing identity, numerical limit, plot or comparison supports only the stated relation under its recorded domain and provenance.
It is not automatically an independently verified theorem or a proof beyond the explicit claim boundary.
Foundational synthesis
Definitions, evidence and limitations stay linked
The portal keeps the frozen canonical manuscript and proof-status composition map as the authoritative mathematical sources, while software verification remains separate from independent acceptance.
Canonical source
Use the frozen canonical manuscript and proof-status composition map as the authoritative mathematical source for the formula and domain shown on this page.
Evidence class
Read tests, convergence, reference compatibility and independent replication as different evidence classes.
Boundary
Every result retains its domain, limits and explicit non-claim so a visual or passing assertion is not over-promoted.