# Reading roadmap: The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8

The paper tries to contradict the existence of a rightmost possible zero in an entire Hecke L-function family. It constructs a completed cubic-theta sum with two exact representations: reflection gives a small direct bound, while Poisson summation exposes a principal Mellin signal containing the reciprocal of the target L-function. Bounds for the other rows would continue that reciprocal across the possible zeros. A balanced first stage claims a zero-free half-plane to the right of 11/12; prime compensation and two additional moment inductions claim 7/8. The full Hecke family is needed because Poisson introduces twists, even when the final target is zeta. These are the manuscript’s proposed steps, not a completed verification of its difficult inputs.

Source SHA-256: `42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3`. Indexed 766,316 characters.

Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates.

## §§1–2 — continuation

Replace nonvanishing with a uniform reciprocal-continuation problem: a direct bound and a common power saving approximation force every reciprocal through the rightmost family zeros.

Claims: thm:main, lem:continuation-criterion

## §§3–6 — base reflection

Build one completed cubic-theta probe and bound it through its reflected form; balanced scales give the low Z^(1/4+epsilon) estimate.

Claims: prop:completed-reflection, lem:unmarked-completed-moment, lem:balanced-additive-norm, prop:balanced-low

## §§7–11 — first high

Poisson-expand the same probe, isolate its principal reciprocal signal, count troublesome nonprincipal rows through saturated inverse witnesses and the sextic sieve, then close 11/12.

Claims: lem:local-euler, prop:detector-witness, lem:sextic-large-sieve, prop:sextic-row-count, prop:balanced-high, thm:eleven-twelfths

## §§12–16 — compensated low

Insert independent selected prime slots and asymmetric scales; obtain a new reflected low bound and an exact holomorphic local correction with controlled errors.

Claims: cor:marked-reflection, lem:reflected-energy, prop:probe-low, prop:probe-errors

## §17 — marked inverse

Prove the selected prime inverse moment by a canonical invariant, terminal reflected energy and two masked Poisson transforms; initialize it and amplify the no-slot estimate.

Claims: lem:marked, lem:canonical-moment, lem:finite-seminorm-propagation, lem:masked-poisson, lem:inverse-amplification

## §18 — plain fourth

Prove the plain-witness fourth moment by decreasing effective width; center equal-product-scale comparisons so fixed inducing-character main terms cancel.

Claims: lem:plain, lem:centered-coefficient-invariant, lem:centered-lattice-cancellation

## §19 — refined counts

Turn the two simultaneous witnesses and independent short prime slots into complementary inverse/plain row counts, with explicit slot capacity and supply margins.

Claims: lem:bin-prime-bound, prop:detector-counts

## §20 — final closure

Balance the complementary counts, certify a small positive endpoint margin, reserve all losses before choosing target dependent heights, and apply continuation plus quadratic transfer.

Claims: lem:high-bin, lem:balanced-endpoint, prop:parameter-order, lem:continuation-criterion, prop:hecke-dirichlet-transfer

## Internal claim index

### Theorem 1.1 — `thm:main`

Source lines 106–112.

All finite-order Hecke L-functions over Q(sqrt(-3)) and all Dirichlet L-functions are claimed zero-free to the right of 7/8; principal poles are allowed.

- `prop:parameter-order` (proof_dependency; source_audit_annotation): Final high bound and principal contraction verify the continuation criterion at 7/8. Source: paper.tex lines 16455–16459.
- `lem:continuation-criterion` (proof_dependency; source_audit_annotation): Continuation contradicts beta*>7/8. Source: paper.tex lines 16455–16460.
- `prop:hecke-dirichlet-transfer` (proof_dependency; source_audit_annotation): Extend primitive Hecke result to imprimitive Hecke and Dirichlet functions. Source: paper.tex lines 16459–16464.
- `external:MilneCFT` (proof_dependency; source_audit_annotation): Finite-order idelicHecke characters have no nontrivial continuous complex-place part. Source: paper.tex lines 50–58.

### Corollary 1.2 — `cor:nonresidues-square-roots`

Source lines 325–338.

Deduce a polylogarithmic bound for the least quadratic nonresidue and a deterministic algorithm for square roots modulo a prime from the claimed zero-free half-plane.

- `external:BIMS` (proof_dependency; source_audit_annotation): Fixedhalf-plane implies polylogarithmic least quadratic nonresidue. Source: paper.tex lines 339–347.
- `external:Galbraith` (proof_dependency; source_audit_annotation): Tonelli–Shanks once a deterministic nonresidue is found. Source: paper.tex lines 349–360.

Candidate references: `thm:main` (candidate_proof_dependency); `external:TaoNonresidues` (external_reference); `external:BIMS` (external_reference); `external:Galbraith` (external_reference).

### Proposition 2.1 — `lem:continuation-criterion`

Source lines 400–433.

Continuation from a common signal

A common positive saving between a bounded probe and a reciprocal-L Mellin signal continues every target reciprocal across the family zero supremum.


Candidate references: `eq:signal-euler-contract` (candidate_proof_dependency); `eq:common-residue` (candidate_proof_dependency); `eq:uniform-saving` (candidate_proof_dependency).

### Theorem 3.1 — `thm:eleven-twelfths`

Source lines 532–538.

The $11/12$ half-plane

The balanced base probe first establishes the weaker 11/12 family bound, providing the bootstrap for Part II.

- `prop:balanced-high` (proof_dependency; source_audit_annotation): Common low/high savings close the first continuation argument. Source: paper.tex lines 6783–6798.
- `lem:local-euler` (proof_dependency; source_audit_annotation): Provide holomorphic nonzero principal correction. Source: paper.tex lines 6783–6786.

Candidate references: `lem:local-euler` (candidate_proof_dependency); `eq:shared-principal-nonvanishing` (candidate_proof_dependency); `prop:balanced-high` (candidate_proof_dependency); `lem:continuation-criterion` (candidate_proof_dependency); `prop:hecke-dirichlet-transfer` (candidate_proof_dependency).

### Lemma 4.1 — `lem:fixed-numerator-ray`

Source lines 646–675.

Fixed numerators give ray characters

Identify a fixed-numerator sixth-power residue symbol with a finite-order ray character through its Kummer extension, retaining the original excluded primes.

- `external:MilneCFT` (proof_dependency; source_audit_annotation): Kummerdescription for fixed-numerator residuecharacters. Source: paper.tex lines 676–699.

Candidate references: `external:MilneCFT` (external_reference).

### Lemma 4.2 — `lem:prime-gauss-identities`

Source lines 766–776.

Prime Gauss identities

Establish the prime Gauss-sum identities and absolute-value-one normalizations used to convert cubic coefficients into sextic notation.

### Lemma 4.3 — `lem:quadratic-four-term`

Source lines 841–870.

Quadratic four-term formula

Evaluate the quadratic Gauss sum through four residue classes modulo two; its values supply the fixed reciprocity phase modulo four.


Candidate references: `eq:quadratic-four-term` (candidate_proof_dependency).

### Lemma 4.4 — `lem:fixed-gauss-phase`

Source lines 934–973.

Sextic reciprocity and the fixed Gauss phase

Record sextic reciprocity and a fixed ray-class Gauss phase so later transforms can track phases separately from moving residue symbols.

- `external:DR` (proof_dependency; source_audit_annotation): Cubic reciprocity on primary coprime inputs. Source: paper.tex lines 994–1003.

Candidate references: `eq:reciprocity-four-class` (internal_reference); `lem:prime-gauss-identities` (candidate_proof_dependency); `eq:G-quadratic-refinement` (candidate_proof_dependency); `eq:signal` (candidate_proof_dependency); `external:DR` (external_reference).

### Lemma 4.5 — `lem:smooth-calculus`

Source lines 1123–1211.

Smooth calculus

Separate joint smooth norm profiles by logarithmic Fourier inversion and control Sobolev bounds, Mellin tails, and norm-twist height costs.


Candidate references: `eq:late-fourier-tail` (internal_reference); `eq:pointwise-mellin-tail` (internal_reference); `eq:parameter-sobolev` (candidate_proof_dependency); `eq:late-fourier-tail` (candidate_proof_dependency); `eq:pointwise-mellin-tail` (candidate_proof_dependency); `eq:joint-gaussian-slice` (candidate_proof_dependency).

### Lemma 4.6 — `lem:gaussian-annular`

Source lines 1288–1316.

Gaussian annular decomposition

Split the noncompact logarithmic Gaussian into smooth annular profiles whose weighted seminorms remain summable.


Candidate references: `lem:smooth-calculus` (internal_reference); `eq:pointwise-mellin-tail` (internal_reference); `lem:smooth-calculus` (candidate_proof_dependency).

### Lemma 4.7 — `lem:kernel-seminorms`

Source lines 1347–1390.

Finite seminorms for Fourier and Mellin kernels

Bound Fourier and Mellin kernels through finitely many weighted smooth seminorms, including annular pullbacks and separated height parameters.

### Lemma 4.8 — `lem:hecke-strip-growth`

Source lines 1425–1435.

Hecke strip growth

Give polynomial conductor and height bounds on a fixed Hecke L-function strip, removing the principal pole before applying the same bound.

- `external:GZ` (proof_dependency; source_audit_annotation): Functional equation plusgammafactor conductor scaling for finite-order Hecke L-functions. Source: paper.tex lines 1436–1458.

Candidate references: `eq:hecke-growth` (candidate_proof_dependency); `external:GZ` (external_reference).

### Lemma 4.9 — `lem:logarithmic-control`

Source lines 1531–1553.

Logarithmic control

A zero-free buffered disk yields subpower bounds for L and its reciprocal, plus logarithmic derivative control.


Candidate references: `lem:hecke-strip-growth` (candidate_proof_dependency); `eq:disk-control` (candidate_proof_dependency).

### Lemma 4.10 — `lem:deleted-euler-factors`

Source lines 1602–1622.

Deleted Euler factors

Bound finite deleted Euler products and their reciprocals in a positive half-plane; include their radical explicitly when they vary with the scale.

### Proposition 5.1 — `prop:completed-reflection`

Source lines 1676–1882.

Completed cubic reflection

Transform the whole completed index nb^3 while retaining all fixed and moving nonunit zeros; expose three fixed cusp coefficient families and explicit active-prime factors.

- `external:DR` (proof_dependency; source_audit_annotation): Cubic supplementarylaw makes the fixed mask periodic modulo9. Source: paper.tex lines 2001–2010.

Candidate references: `old-eq:4.3` (candidate_proof_dependency); `eq:reflection-infinity-coefficient` (candidate_proof_dependency); `eq:reflection-cubic-sextic-conversion` (candidate_proof_dependency); `eq:reflection-local-fourier` (candidate_proof_dependency); `eq:reflection-translated-theta` (candidate_proof_dependency); `eq:reflection-cusp-fraction` (candidate_proof_dependency); `eq:reflection-multiplier` (candidate_proof_dependency); `par:reflection-marks` (candidate_proof_dependency); `eq:reflection-cusp-expansion` (candidate_proof_dependency); `eq:reflection-direct-mellin` (candidate_proof_dependency); `eq:reflection-reflected-mellin` (candidate_proof_dependency); `eq:reflection` (candidate_proof_dependency); `eq:pointwise-mellin-tail` (candidate_proof_dependency); `eq:reflection-additive-phase` (candidate_proof_dependency); `eq:reflection-row-phase` (candidate_proof_dependency); `eq:reflection-kernel` (candidate_proof_dependency); `eq:reflection-annular-fourier` (candidate_proof_dependency); `eq:reflection-test-seminorm` (candidate_proof_dependency); 1 more in the JSON/interactive graph.

### Lemma 5.2 — `lem:reflection-row-sectors`

Source lines 2470–2507.

Fixed ray sectors for nonzero rows

Reduce arbitrary nonzero element rows to finitely many fixed ray sectors, separating primes in the fixed excluded set from the moving good part.


Candidate references: `lem:fixed-numerator-ray` (internal_reference); `prop:completed-reflection` (internal_reference); `lem:fixed-gauss-phase` (candidate_proof_dependency); `eq:reflection-row-reduction` (candidate_proof_dependency); `lem:fixed-numerator-ray` (candidate_proof_dependency).

### Lemma 5.3 — `lem:reflection-common-profile`

Source lines 2540–2567.

Common annular kernel profile

Represent the reflected annular kernel by one joint smooth profile common to every row in a block, with a controlled Fourier coefficient measure.


Candidate references: `prop:completed-reflection` (internal_reference); `eq:reflection-kernel` (candidate_proof_dependency); `eq:reflection-common-profile` (candidate_proof_dependency); `lem:smooth-calculus` (candidate_proof_dependency); `eq:reflection-common-measure` (candidate_proof_dependency).

### Lemma 5.4 — `lem:lattice-kernel-tail`

Source lines 2618–2629.

Lattice kernel tails

Turn rapid decay of a reflected kernel into a summable power bound for the remote Eisenstein-lattice frequencies.


Candidate references: `eq:reflection-kernel` (candidate_proof_dependency).

### Lemma 5.5 — `lem:completed-quadratic-reduction`

Source lines 2680–2730.

Quadratic reduction for completed indices

Reduce the quadratic sieve on squarefree-times-cube indices without dropping their nonunit masks.

- `external:GL` (proof_dependency; source_audit_annotation): Quadratic residue-family norm bound(U+V)(UV)^epsilon; paper checks family hypotheses. Source: paper.tex lines 2638–2671.

Candidate references: `eq:completed-quadratic-reduction` (candidate_proof_dependency); `eq:completed-quadratic-norm` (candidate_proof_dependency).

### Definition 5.6 — `def:unmarked-reflected-block`

Source lines 2888–2951.

An unmarked reflected block

Specify an unmarked reflected block with frozen local data, a common cusp sector, and actual row and completed-index annuli before estimating its norm.


Candidate references: `prop:completed-reflection` (internal_reference); `old-eq:4.3` (internal_reference); `eq:reflection` (internal_reference).

### Lemma 5.7 — `lem:unmarked-reflected-block`

Source lines 2956–3002.

Unmarked reflected block

Separate the unmarked block into row and column factors and bound its energy while preserving the common profile and original masks.


Candidate references: `prop:completed-reflection` (candidate_proof_dependency); `eq:unmarked-structural-block` (candidate_proof_dependency); `eq:unmarked-actual-annuli` (candidate_proof_dependency); `lem:reflection-common-profile` (candidate_proof_dependency); `eq:reflection-common-minkowski` (candidate_proof_dependency); `eq:completed-quadratic-norm` (candidate_proof_dependency); `eq:unmarked-reflected-energy` (candidate_proof_dependency).

### Lemma 5.8 — `lem:unmarked-completed-moment`

Source lines 3138–3169.

Unmarked completed-row moment

Bound the original completed theta-row mean square by the reflected unmarked energy.


Candidate references: `prop:completed-reflection` (internal_reference); `lem:reflection-row-sectors` (candidate_proof_dependency); `eq:actual-residual-row-dyad` (candidate_proof_dependency); `def:unmarked-reflected-block` (candidate_proof_dependency); `eq:reflection` (candidate_proof_dependency); `eq:unmarked-central-mellin` (candidate_proof_dependency); `lem:unmarked-reflected-block` (candidate_proof_dependency); `eq:unmarked-row-width` (candidate_proof_dependency); `eq:unmarked-powerful-radical` (candidate_proof_dependency); `eq:unmarked-reflected-energy` (candidate_proof_dependency); `eq:unmarked-completed-moment` (candidate_proof_dependency); `prop:completed-reflection` (candidate_proof_dependency).

### Lemma 6.1 — `lem:planar-additive-sieve`

Source lines 3497–3515.

Planar additive large sieve

Bound an additive polynomial over a planar lattice ball by the ball size plus the inverse square of the frequency separation.

### Lemma 6.2 — `lem:balanced-additive-norm`

Source lines 3587–3603.

The balanced additive norm

Reduced additive Gauss fractions have controlled spacing and coefficient mass; their mean square costs (Q+Y^2)/Y.


Candidate references: `lem:planar-additive-sieve` (candidate_proof_dependency).

### Proposition 6.3 — `prop:balanced-low`

Source lines 3672–3683.

Balanced low estimate

Combine the completed-row energy and additive mean square by Cauchy–Schwarz to obtain the balanced Z^(1/4+epsilon) bound.

- `lem:unmarked-completed-moment` (proof_dependency; source_audit_annotation): Control the mean-square completed thetafactor. Source: paper.tex lines 3686–3697.
- `lem:balanced-additive-norm` (proof_dependency; source_audit_annotation): Control the additivefactor uniformly in its twist height. Source: paper.tex lines 3701–3719.

Candidate references: `eq:low-separated` (candidate_proof_dependency); `lem:unmarked-completed-moment` (candidate_proof_dependency); `lem:balanced-additive-norm` (candidate_proof_dependency).

### Lemma 7.1 — `lem:local-euler`

Source lines 4010–4068.

The complete local identity

Factor each sixth-power-free Poisson row into scalar Hecke L-functions and a holomorphic Euler correction; the principal row contains the target reciprocal.


Candidate references: `eq:local-full-series` (internal_reference); `eq:probe-high` (internal_reference).

### Lemma 8.1 — `lem:buffered-bins`

Source lines 4281–4323.

Buffered zero-free bins

Assign buffered zero-free rectangles to each row family and obtain pointwise controls.


Candidate references: `lem:logarithmic-control` (candidate_proof_dependency); `lem:hecke-strip-growth` (candidate_proof_dependency); `external:GZ` (external_reference).

### Lemma 8.2 — `lem:detector-dyads`

Source lines 4385–4418.

Pointwise dyadic estimates

Keep pointwise dyadic polynomial estimates and admissible row-dependent profile parameters with fixed polynomial height costs.


Candidate references: `old-eq:1.1a` (internal_reference); `old-eq:1.1` (internal_reference); `lem:smooth-calculus` (candidate_proof_dependency); `lem:buffered-bins` (candidate_proof_dependency); `eq:pointwise-mellin-tail` (candidate_proof_dependency); `eq:late-fourier-tail` (candidate_proof_dependency); `old-eq:6.2` (candidate_proof_dependency); `old-eq:6.3` (candidate_proof_dependency).

### Proposition 8.3 — `prop:detector-witness`

Source lines 4510–4548.

Two saturated witnesses

A selected zero yields two simultaneous saturated Dirichlet-polynomial witnesses, inverse and plain, sharing one presentation and twist height.


Candidate references: `lem:detector-dyads` (internal_reference); `lem:buffered-bins` (internal_reference); `eq:hecke-growth` (candidate_proof_dependency); `eq:detector-note` (candidate_proof_dependency); `lem:detector-dyads` (candidate_proof_dependency); `eq:saturated-witness` (candidate_proof_dependency); `lem:smooth-calculus` (candidate_proof_dependency); `external:MP` (external_reference); `external:GM` (external_reference).

### Lemma 9.1 — `lem:sextic-large-sieve`

Source lines 4707–4726.

Sextic large sieve

Establish the squarefree sextic large-sieve bound K+D+(KD)^(2/3) in the primary convention with original zeros.

- `external:BGL` (proof_dependency; source_audit_annotation): Higher-order norm-recursion interfaces checked in the paper for sextic symbols. Source: paper.tex lines 4727–4736.

Candidate references: `lem:fixed-gauss-phase` (candidate_proof_dependency); `eq:sextic-sieve-unit-class` (candidate_proof_dependency); `eq:sextic-sieve-pair-gauss` (candidate_proof_dependency); `eq:sextic-sieve-pair-poisson` (candidate_proof_dependency); `eq:sextic-sieve-transform-bounds` (candidate_proof_dependency); `eq:sextic-sieve-pair-values` (candidate_proof_dependency); `eq:sextic-sieve-pair-norm` (candidate_proof_dependency); `lem:planar-additive-sieve` (candidate_proof_dependency); `eq:sextic-sieve-norm-initial` (candidate_proof_dependency); `eq:sextic-sieve-norm-monotone` (candidate_proof_dependency); `eq:sextic-sieve-inductive-bound` (candidate_proof_dependency); `eq:sextic-sieve-norm-power` (candidate_proof_dependency); `eq:sextic-sieve-norm-reflection` (candidate_proof_dependency); `eq:sextic-sieve-norm-recurrence` (candidate_proof_dependency); `eq:sextic-sieve-norm-shell` (candidate_proof_dependency); `eq:sextic-large-sieve` (candidate_proof_dependency); `external:BGL` (external_reference); `external:GL` (external_reference).

### Proposition 9.2 — `prop:sextic-row-count`

Source lines 5181–5214.

Sextic-sieve row envelope

Apply the sextic sieve to frozen powerful parts and squarefree residual rows to count large inverse-witness rows.

- `lem:sextic-large-sieve` (proof_dependency; source_audit_annotation): Apply the sieve to a fixed squarefree residual row-factor. Source: paper.tex lines 5298–5339.
- `prop:detector-witness` (proof_dependency; source_audit_annotation): Use each row’s saturated inverse-witness lower bound. Source: paper.tex lines 5215–5234.

Candidate references: `lem:buffered-bins` (internal_reference); `lem:detector-dyads` (internal_reference); `prop:detector-witness` (internal_reference); `lem:smooth-calculus` (internal_reference); `prop:detector-witness` (candidate_proof_dependency); `eq:sextic-actual-row-annulus` (candidate_proof_dependency); `lem:sextic-large-sieve` (candidate_proof_dependency); `eq:sextic-witness-spike` (candidate_proof_dependency); `lem:smooth-calculus` (candidate_proof_dependency); `eq:sextic-stratum-fixed-moment` (candidate_proof_dependency); `eq:sextic-stratum-cardinality` (candidate_proof_dependency); `eq:sextic-stratum-moment` (candidate_proof_dependency); `eq:sextic-stratum-envelope` (candidate_proof_dependency); `eq:sextic-row-count` (candidate_proof_dependency).

### Definition 10.1 — `def:stage-high-data`

Source lines 5582–5651.

Data for an exact high representation

Specify the common contract for an exact high representation: one independently defined physical probe, a row Euler correction, and the precise Mellin identity and normalization.


Candidate references: `sec:probe` (internal_reference); `eq:general-probe` (internal_reference); `eq:probe-poisson-identity` (internal_reference); `eq:stage-high-identity` (internal_reference).

### Lemma 10.2 — `lem:external-contour-tails`

Source lines 5672–5712.

External integrated and trace tails

Integrate rapidly decaying external kernels against polynomial arithmetic bounds, including tail and trace estimates on contour joins.


Candidate references: `eq:external-arithmetic-majorant` (candidate_proof_dependency); `eq:external-integrated-tail` (candidate_proof_dependency); `eq:external-trace-tail` (candidate_proof_dependency).

### Lemma 10.3 — `lem:fixed-bin-contour`

Source lines 5808–5835.

Contour transformation for a fixed bin

Move the Mellin contours of one fixed row bin only after its rows and zero-free buffers have been fixed; track the resulting heights and errors.


Candidate references: `def:stage-high-data` (internal_reference); `lem:buffered-bins` (internal_reference); `eq:stage-high-identity` (internal_reference); `sec:detector` (candidate_proof_dependency); `eq:stage-high-identity` (candidate_proof_dependency); `lem:logarithmic-control` (candidate_proof_dependency); `lem:deleted-euler-factors` (candidate_proof_dependency); `eq:stage-all-height-majorant` (candidate_proof_dependency); `lem:external-contour-tails` (candidate_proof_dependency); `eq:external-arithmetic-majorant` (candidate_proof_dependency); `lem:buffered-bins` (candidate_proof_dependency).

### Lemma 10.4 — `lem:bin-contour-accounting`

Source lines 6021–6116.

A retained integral and its exponent

Move only a fixed retained bin and account for its exponent, row count and height factors within a common high representation.


Candidate references: `def:stage-high-data` (internal_reference); `lem:buffered-bins` (internal_reference); `eq:stage-central-contours` (internal_reference); `eq:stage-central-bound` (internal_reference); `lem:external-contour-tails` (internal_reference); `lem:smooth-calculus` (internal_reference); `lem:fixed-bin-contour` (candidate_proof_dependency); `eq:stage-height-allocation` (candidate_proof_dependency); `eq:stage-central-bound` (candidate_proof_dependency); `lem:buffered-bins` (candidate_proof_dependency); `eq:stage-central-contours` (candidate_proof_dependency); `eq:stage-high-exponent` (candidate_proof_dependency); `eq:stage-central-contribution` (candidate_proof_dependency).

### Lemma 10.5 — `lem:principal-residue-interface`

Source lines 6174–6223.

Extraction of the principal signal

Extract the target Mellin signal from the principal row and bound its unresidued contour remainders.

- `external:TZ` (proof_dependency; source_audit_annotation): Chebotarev detects a nontrivial sexticFrobenius class, separating the sole principal numerator row. Source: paper.tex lines 5500–5516.

Candidate references: `def:stage-high-data` (internal_reference); `eq:shared-principal-nonvanishing` (internal_reference); `eq:stage-high-identity` (internal_reference); `lem:external-contour-tails` (candidate_proof_dependency); `eq:external-two-dimensional-kernel` (candidate_proof_dependency); `eq:principal-correction-bound` (candidate_proof_dependency); `lem:logarithmic-control` (candidate_proof_dependency); `lem:deleted-euler-factors` (candidate_proof_dependency); `eq:principal-remainder-bound` (candidate_proof_dependency); `eq:principal-correction-residue` (candidate_proof_dependency); `lem:local-euler` (candidate_proof_dependency); `eq:shared-principal-nonvanishing` (candidate_proof_dependency).

### Lemma 10.6 — `lem:outer-row-tails`

Source lines 6342–6402.

Outer row norms

Handle small and large physical row norms separately using global or absolute contour lines.


Candidate references: `def:stage-high-data` (internal_reference); `eq:stage-high-identity` (internal_reference); `lem:principal-residue-interface` (candidate_proof_dependency); `lem:logarithmic-control` (candidate_proof_dependency); `sec:detector` (candidate_proof_dependency); `lem:deleted-euler-factors` (candidate_proof_dependency); `lem:external-contour-tails` (candidate_proof_dependency); `eq:hecke-growth` (candidate_proof_dependency); `eq:small-correction-bound` (candidate_proof_dependency); `eq:small-row-bound` (candidate_proof_dependency); `eq:small-row-sum` (candidate_proof_dependency); `eq:stage-all-height-majorant` (candidate_proof_dependency); `eq:large-correction-bound` (candidate_proof_dependency); `eq:large-row-bound` (candidate_proof_dependency); `eq:large-row-sum` (candidate_proof_dependency).

### Lemma 11.1 — `lem:late-height-closure`

Source lines 6520–6559.

Late choice of height and external order

Fix all retained orders first; choose a target-dependent height power and external tail order while preserving a common final saving.


Candidate references: `lem:detector-dyads` (internal_reference); `eq:late-height-hypothesis` (internal_reference); `lem:continuation-criterion` (internal_reference); `eq:late-height-hypothesis` (candidate_proof_dependency).

### Proposition 11.2 — `prop:balanced-high`

Source lines 6582–6593.

Balanced high estimate

Sum central, principal and outer contributions to approximate the balanced signal with common saving 1/4800.

- `prop:balanced-low` (proof_dependency; source_audit_annotation): Supply the low hypothesis before applying the late height closure. Source: paper.tex lines 6689–6697.
- `lem:late-height-closure` (proof_dependency; source_audit_annotation): Convert finite height losses into the final common high saving. Source: paper.tex lines 6692–6697.
- `prop:sextic-row-count` (proof_dependency; source_audit_annotation): The central row-count loss budget is uniformly fixed. Source: paper.tex lines 6617–6625.

Candidate references: `sec:shared-analytic-estimates` (candidate_proof_dependency); `eq:balanced-central-contribution` (candidate_proof_dependency); `prop:sextic-row-count` (candidate_proof_dependency); `lem:detector-dyads` (candidate_proof_dependency); `lem:fixed-bin-contour` (candidate_proof_dependency); `prop:balanced-low` (candidate_proof_dependency); `lem:late-height-closure` (candidate_proof_dependency).

### Proposition 11.3 — `prop:hecke-dirichlet-transfer`

Source lines 6705–6714.

Quadratic transfer

Transfer a Hecke family half-plane through the quadratic norm Euler-product factorization, checking poles at one separately.

### Lemma 13.1 — `lem:ray-prime-normalizer`

Source lines 7006–7018.

Fixed-ray prime normalizer

Fixed-ray-class prime counting makes each prime normalizer asymptotic and positive, without needing an error exponent uniform in the conductor.

- `external:TZ` (proof_dependency; source_audit_annotation): Fixed-ray prime normalizer asymptotic, with fixed conductor before P→infinity. Source: paper.tex lines 7019–7040.

Candidate references: `eq:ray-prime-normalizer` (candidate_proof_dependency); `external:TZ` (external_reference).

### Lemma 13.2 — `lem:prime-power-fourier`

Source lines 7055–7066.

Prime-power Fourier sums

Evaluate local prime-power Fourier sums, retaining the nonunit zeros and the Ramanujan-sum branch when the exponent is divisible by six.

### Lemma 13.3 — `lem:full-correlation`

Source lines 7081–7133.

Full correlation and common factors

Compute finite Fourier correlations with full local supports and common factors retained.


Candidate references: `eq:correlation-lift` (internal_reference); `eq:correlation-lift` (candidate_proof_dependency).

### Lemma 13.4 — `lem:complete-support-correlation`

Source lines 7196–7211.

Complete-common-support correlation

Keep a common nonprimitive modulus during complete support extraction and identify the induced child character.


Candidate references: `old-eq:2.11` (candidate_proof_dependency).

### Corollary 14.1 — `cor:marked-reflection`

Source lines 7361–7416.

Completed reflection with whole-index marks

Apply the completed reflection to product-form marks on the whole index, preserving branch compatibility and row-slot phases.

- `prop:completed-reflection` (proof_dependency; source_audit_annotation): Expand zero masks and apply the completed reflection to each fixed summand. Source: paper.tex lines 7417–7424.

Candidate references: `prop:completed-reflection` (internal_reference); `eq:marked-completion` (internal_reference); `prop:completed-reflection` (candidate_proof_dependency); `eq:marked-reflection` (candidate_proof_dependency); `eq:marked-reflection-phase` (candidate_proof_dependency).

### Lemma 14.2 — `lem:hybrid-energy`

Source lines 7510–7541.

A quadratic--cubic norm bound

Combine quadratic and cubic character norms while explicitly resolving the coupled masks.

- `external:HB` (proof_dependency; source_audit_annotation): Cubic squarefree large-sieve norm U+V+(UV)^(2/3), arbitrary complex coefficients. Source: paper.tex lines 7485–7508.

Candidate references: `old-eq:4.6b` (candidate_proof_dependency); `eq:completed-quadratic-reduction` (candidate_proof_dependency); `eq:hybrid-mark-mask-reduction` (candidate_proof_dependency); `eq:hybrid-after-quadratic-reduction` (candidate_proof_dependency); `eq:cubic-large-sieve` (candidate_proof_dependency).

### Lemma 14.3 — `lem:reflected-energy`

Source lines 7747–7805.

Reflected energy

Estimate the marked reflected row energy uniformly over admissible moving supports and frozen labels.

- `cor:marked-reflection` (proof_dependency; source_audit_annotation): Transform whole-index marks inside the original active-tuple sum. Source: paper.tex lines 7819–7828.
- `lem:hybrid-energy` (proof_dependency; source_audit_annotation): After phase cancellation, apply the quadratic–cubic bound to fixed data. Source: paper.tex lines 7954–7979.

Candidate references: `eq:marked-completion` (internal_reference); `old-eq:4.2` (internal_reference); `eq:actual-residual-row-dyad` (internal_reference); `eq:unmarked-structural-block` (candidate_proof_dependency); `cor:marked-reflection` (candidate_proof_dependency); `eq:marked-reflection` (candidate_proof_dependency); `old-eq:4.5` (candidate_proof_dependency); `lem:reflection-common-profile` (candidate_proof_dependency); `lem:lattice-kernel-tail` (candidate_proof_dependency); `eq:reflection-common-minkowski` (candidate_proof_dependency); `eq:reflection-actual-annuli` (candidate_proof_dependency); `eq:reflection-pair-phase` (candidate_proof_dependency); `old-eq:4.6a` (candidate_proof_dependency); `lem:hybrid-energy` (candidate_proof_dependency); `old-eq:4.6` (candidate_proof_dependency).

### Lemma 15.1 — `lem:probe-row-norm`

Source lines 8111–8123.

The compensated completed-row norm

Bound the compensated completed row using reflected marked energy at its actual annular lengths.

- `lem:reflected-energy` (proof_dependency; source_audit_annotation): Bound each actual retained annular block. Source: paper.tex lines 8196–8209.

Candidate references: `lem:gaussian-annular` (candidate_proof_dependency); `lem:smooth-calculus` (candidate_proof_dependency); `lem:reflection-row-sectors` (candidate_proof_dependency); `lem:reflected-energy` (candidate_proof_dependency); `old-eq:4.6` (candidate_proof_dependency); `eq:reflection-common-minkowski` (candidate_proof_dependency); `old-eq:4.3` (candidate_proof_dependency); `old-eq:4.7` (candidate_proof_dependency).

### Proposition 15.2 — `prop:probe-gram`

Source lines 8340–8362.

A quantitative additive Gram bound

Obtain the quantitative additive Gram bound used with the asymmetric compensated scales.


Candidate references: `sec:probe` (internal_reference); `eq:correlation-lift` (candidate_proof_dependency); `old-eq:2.11` (candidate_proof_dependency); `eq:local-F` (candidate_proof_dependency); `lem:fixed-numerator-ray` (candidate_proof_dependency); `eq:gram-periodic-poisson` (candidate_proof_dependency).

### Proposition 15.3 — `prop:probe-low`

Source lines 8564–8577.

The compensated low estimate

Combine row energy and additive Gram estimates to bound the compensated physical probe directly.

- `lem:probe-row-norm` (proof_dependency; source_audit_annotation): Cauchy–Schwarz pairs row energy with additive Gram. Source: paper.tex lines 8605–8618.
- `prop:probe-gram` (proof_dependency; source_audit_annotation): Supply additivefactor bound at asymmetric lengths. Source: paper.tex lines 8599–8613.

Candidate references: `eq:compensated-probe-definition` (internal_reference); `old-eq:5.1` (internal_reference); `eq:low-separated` (candidate_proof_dependency); `prop:probe-gram` (candidate_proof_dependency); `lem:probe-row-norm` (candidate_proof_dependency); `old-eq:5.10` (candidate_proof_dependency); `eq:part-II-mellin-target` (candidate_proof_dependency).

### Proposition 16.1 — `prop:probe-errors`

Source lines 8852–8906.

Dynamic local errors and conductor allocation

Split selected local factors into main/error pieces on the dynamic region and allocate each ramified conductor deficit only once.

- `external:GZ` (proof_dependency; source_audit_annotation): Reflect a local nonprincipal numerator while allocating selected conductor deficits. Source: paper.tex lines 9007–9023.

Candidate references: `old-eq:5.13c` (internal_reference); `lem:local-euler` (candidate_proof_dependency); `old-eq:5.13c` (candidate_proof_dependency); `eq:holomorphic-selected-tuple` (candidate_proof_dependency); `eq:dynamic-compensated-decomposition` (candidate_proof_dependency); `old-eq:5.13b` (candidate_proof_dependency); `lem:fixed-numerator-ray` (candidate_proof_dependency); `old-eq:5.13d` (candidate_proof_dependency); `old-eq:5.13e` (candidate_proof_dependency); `external:GZ` (external_reference).

### Lemma 16.2 — `lem:compensated-absolute-local-bounds`

Source lines 9101–9132.

Absolute local tuple bounds

Retain full selected factors on the small/absolute lines where the central main/error split is unavailable.


Candidate references: `sec:compensation` (internal_reference); `eq:unselected-local-product` (candidate_proof_dependency); `old-eq:5.13b` (candidate_proof_dependency); `old-eq:5.13c` (candidate_proof_dependency); `eq:absolute-local-tuple-bound` (candidate_proof_dependency).

### Lemma 17.1 — `lem:marked`

Source lines 9250–9270.

Marked inverse moment

Bound an inverse witness times independent disjoint prime slots when r+2z<m and 2r+8z<3m with fixed positive margins.

- `lem:canonical-moment` (proof_dependency; source_audit_annotation): Initialization verifies canonical puncture and length margins, then applies its bound. Source: paper.tex lines 12279–12297.
- `lem:masked-poisson` (proof_dependency; source_audit_annotation): One masked initial transform reduces the inversemoment. Source: paper.tex lines 11729–11744.

Candidate references: `old-eq:4.2` (candidate_proof_dependency); `eq:inverse-log-fourier` (candidate_proof_dependency); `eq:initial-overlap-polynomial` (candidate_proof_dependency); `eq:inverse-fourier-moment` (candidate_proof_dependency); `lem:masked-poisson` (candidate_proof_dependency); `eq:inverse-raw-tail` (candidate_proof_dependency); `eq:initial-residual-polynomial` (candidate_proof_dependency); `eq:initial-formal-component` (candidate_proof_dependency); `eq:initial-common-extraction` (candidate_proof_dependency); `eq:inverse-slot-priority` (candidate_proof_dependency); `eq:initial-canonical-lengths` (candidate_proof_dependency); `eq:initial-outer-weight` (candidate_proof_dependency); `eq:initial-root-kernel` (candidate_proof_dependency); `eq:initial-separated-identity` (candidate_proof_dependency); `eq:weighted-cauchy` (candidate_proof_dependency); `eq:canonical-energy` (candidate_proof_dependency); `eq:inverse-initial-row` (candidate_proof_dependency); `old-eq:4.13` (candidate_proof_dependency); 8 more in the JSON/interactive graph.

### Lemma 17.2 — `lem:canonical-moment`

Source lines 9296–9341.

Canonical marked estimate

Close a restricted canonical family under two masked Poisson transforms; puncture size and 3M+6z0<4(N+V) are part of its invariant.

- `lem:reflected-energy` (proof_dependency; source_audit_annotation): Supply terminal canonical-energy ranges. Source: paper.tex lines 9880–9903.
- `lem:masked-poisson` (proof_dependency; source_audit_annotation): First and second masked transforms produce admissible child families. Source: paper.tex lines 10173–10187.
- `cor:indexed-seminorm-propagation` (proof_dependency; source_audit_annotation): Fix propagated orders through the finite canonical induction. Source: paper.tex lines 9789–9799.

Candidate references: `old-eq:4.2` (internal_reference); `cor:indexed-seminorm-propagation` (candidate_proof_dependency); `sec:moment-propagation` (candidate_proof_dependency); `eq:inverse-old-label-fibre` (candidate_proof_dependency); `eq:marked-completion` (candidate_proof_dependency); `lem:reflected-energy` (candidate_proof_dependency); `eq:actual-residual-row-dyad` (candidate_proof_dependency); `eq:common-width-charge` (candidate_proof_dependency); `eq:inverse-fixed-invariant` (candidate_proof_dependency); `old-eq:4.6` (candidate_proof_dependency); `old-eq:4.9` (candidate_proof_dependency); `eq:marked-completion-inversion` (candidate_proof_dependency); `eq:canonical-block` (candidate_proof_dependency); `eq:canonical-energy` (candidate_proof_dependency); `eq:inverse-ratio-bounds` (candidate_proof_dependency); `lem:smooth-calculus` (candidate_proof_dependency); `lem:kernel-seminorms` (candidate_proof_dependency); `eq:normalized-kernel-profile` (candidate_proof_dependency); 41 more in the JSON/interactive graph.

### Lemma 17.3 — `lem:finite-seminorm-propagation`

Source lines 9366–9442.

Finite seminorm propagation

Propagate finite smooth and height orders along a finite recursive sequence rather than assuming derivative orders stay bounded.

### Corollary 17.4 — `cor:indexed-seminorm-propagation`

Source lines 9486–9532.

Indexed finite propagation

Extend finite propagation to the normalized label measures used in the induction.


Candidate references: `lem:finite-seminorm-propagation` (internal_reference); `eq:indexed-exponent-defect` (candidate_proof_dependency); `lem:finite-seminorm-propagation` (candidate_proof_dependency); `eq:indexed-coefficient-measure` (candidate_proof_dependency).

### Lemma 17.5 — `lem:masked-poisson`

Source lines 9722–9744.

Masked primitive Poisson

Give exact finite Poisson summation for a primitive character with a fixed mask, including principal restoration.


Candidate references: `eq:masked-poisson` (candidate_proof_dependency).

### Lemma 17.6 — `lem:inverse-amplification`

Source lines 12362–12390.

Sixth-power amplification

Amplify an unmarked inverse polynomial on sixth-power-free rows to longer column lengths.


Candidate references: `lem:marked` (internal_reference); `eq:raw-moment` (candidate_proof_dependency); `lem:smooth-calculus` (candidate_proof_dependency); `eq:inverse-scale-supremum` (candidate_proof_dependency); `eq:amplified-note` (candidate_proof_dependency).

### Lemma 18.1 — `lem:plain`

Source lines 12531–12580.

Fourth moment with short prime factors

Control a fourth moment of two plain witnesses and short prime slots, with a dynamic zero-free parameter and exclusion of a finite inducing-character family.

- `thm:eleven-twelfths` (proof_dependency; source_audit_annotation): Part II uses the 11/12 bootstrap for the available dynamic zero-family range; the actual plain parameter is kappa=2beta*-1. Source: paper.tex lines 6810–6834.
- `lem:full-correlation` (proof_dependency; source_audit_annotation): Secondtransform explicitly uses finite Fourier correlation. Source: paper.tex lines 13603–13635.
- `lem:complete-support-correlation` (proof_dependency; source_audit_annotation): Supportextraction is performed on the complete modulus. Source: paper.tex lines 13681–13697.
- `lem:centered-coefficient-invariant` (proof_dependency; source_audit_annotation): Exceptionalrow extraction preserves the same centered coefficient. Source: paper.tex lines 14535–14541.
- `lem:centered-lattice-cancellation` (proof_dependency; source_audit_annotation): Quantify the remaining common-character rectangle cancellation. Source: paper.tex lines 13039–13052.
- `lem:logarithmic-control` (proof_dependency; source_audit_annotation): Obtain prime bounds above the dynamic zero supremum. Source: paper.tex lines 12831–12845.
- `external:TZ` (proof_dependency; source_audit_annotation): Fixedfield prime pool of Z^(ell*+o(1)) size, disjoint from live short slots. Source: paper.tex lines 13450–13475.

Candidate references: `old-eq:1.2` (internal_reference).

### Lemma 18.2 — `lem:centered-coefficient-invariant`

Source lines 13953–13982.

Common coefficient under complete extraction

Preserve one common centered coefficient under complete divisor/support extraction.


Candidate references: `eq:centered-coefficient-form` (internal_reference).

### Lemma 18.3 — `lem:centered-lattice-cancellation`

Source lines 14545–14597.

Masked rectangle cancellation

Cancel the equal-product-scale centered rectangles on the exceptional inducing-character rows.


Candidate references: `old-eq:2.18a` (internal_reference); `old-eq:2.18d` (candidate_proof_dependency); `old-eq:2.18e` (candidate_proof_dependency); `old-eq:2.18f` (candidate_proof_dependency).

### Lemma 19.1 — `lem:bin-prime-bound`

Source lines 15015–15026.

Prime bound in a bin

Control each prime amplitude inside a buffered row bin with finite height/profile costs.


Candidate references: `lem:detector-dyads` (internal_reference); `lem:logarithmic-control` (candidate_proof_dependency); `eq:pointwise-mellin-tail` (candidate_proof_dependency); `eq:late-fourier-tail` (candidate_proof_dependency).

### Proposition 19.2 — `prop:detector-counts`

Source lines 15185–15221.

Row counts from the witnesses

Combine saturated witnesses with inverse/plain capacities and available short prime slots to get refined row counts.

- `lem:marked` (proof_dependency; source_audit_annotation): Inversewitness selected prime capacity is read from its two inequalities. Source: paper.tex lines 15224–15243.
- `lem:plain` (proof_dependency; source_audit_annotation): Plainwitness selected prime capacity uses its affine inequality and zero-slot endpoint. Source: paper.tex lines 15224–15243.
- `lem:inverse-amplification` (proof_dependency; source_audit_annotation): Handle long inverse witnesses and the zero-capacity endpoint. Source: paper.tex lines 15245–15271.
- `prop:detector-witness` (proof_dependency; source_audit_annotation): Saturated pair supplies common r,m and rowwise witness lower bounds. Source: paper.tex lines 15224–15231.

Candidate references: `prop:detector-witness` (internal_reference); `lem:inverse-amplification` (internal_reference); `lem:plain` (internal_reference); `prop:detector-witness` (candidate_proof_dependency); `lem:marked` (candidate_proof_dependency); `lem:plain` (candidate_proof_dependency); `lem:inverse-amplification` (candidate_proof_dependency); `eq:amplified-note` (candidate_proof_dependency); `eq:no-slot-inverse-count` (candidate_proof_dependency); `eq:selected-counts` (candidate_proof_dependency); `eq:plain-zero-loss-capacity` (candidate_proof_dependency).

### Remark 19.3 — `remark:line:15448`

Source lines 15448–15468.

Unselected inverse witnesses beyond the Part II bin ceiling

Explain that the unselected inverse-witness count can be reused beyond the Part II bin ceiling when the same detector and amplification hypotheses hold.


Candidate references: `eq:no-slot-inverse-count` (internal_reference); `prop:detector-witness` (internal_reference); `lem:inverse-amplification` (internal_reference).

### Lemma 20.1 — `lem:high-bin`

Source lines 15699–15758.

A high exponent for one row bin

Translate a refined row bin count and local correction decomposition into the compensated high exponent.

- `prop:probe-errors` (proof_dependency; source_audit_annotation): Decompose the local correction only on retained contours. Source: paper.tex lines 15770–15792.
- `lem:bin-contour-accounting` (proof_dependency; source_audit_annotation): Provide fixed-bin contour move and complete exponent account. Source: paper.tex lines 15759–15773.

Candidate references: `lem:buffered-bins` (internal_reference); `lem:detector-dyads` (internal_reference); `lem:bin-prime-bound` (internal_reference); `eq:stage-height-allocation` (internal_reference); `sec:amplitudes` (internal_reference); `sec:selection` (internal_reference); `old-eq:6.6` (internal_reference); `lem:bin-contour-accounting` (candidate_proof_dependency); `eq:part-II-bin-ceiling` (candidate_proof_dependency); `sec:selection` (candidate_proof_dependency); `eq:stage-height-allocation` (candidate_proof_dependency); `prop:probe-errors` (candidate_proof_dependency); `eq:dynamic-compensated-decomposition` (candidate_proof_dependency); `lem:deleted-euler-factors` (candidate_proof_dependency); `old-eq:5.13e` (candidate_proof_dependency); `old-eq:5.13d` (candidate_proof_dependency); `eq:stage-central-bound` (candidate_proof_dependency); `eq:stage-high-exponent` (candidate_proof_dependency); 2 more in the JSON/interactive graph.

### Lemma 20.2 — `lem:balanced-endpoint`

Source lines 15994–16005.

Compensated endpoint certificate

Certify a strictly positive numerical margin at the balanced endpoint by an explicit completed-square polynomial identity.

- `prop:detector-counts` (proof_dependency; source_audit_annotation): Its two affine row counts are balanced at the selected cutoff. Source: paper.tex lines 15949–15992.

Candidate references: `eq:target-cutoff` (internal_reference); `eq:common-high-exponent` (candidate_proof_dependency); `eq:balanced-endpoint-certificate` (candidate_proof_dependency).

### Proposition 20.3 — `prop:parameter-order`

Source lines 16196–16218.

Order of choices

Choose geometry, slots, losses and margins before the target; retained orders before height/tail orders; conclude a family-common positive high saving.

- `lem:high-bin` (proof_dependency; source_audit_annotation): Combine retained bins with outer ranges and principal remainders. Source: paper.tex lines 16432–16451.
- `lem:balanced-endpoint` (proof_dependency; source_audit_annotation): Lemma20.2 certifies the ideal endpoint margin; the following exponent comparison adds the allocated losses and compares it with the rightmost-zero scale. The final proof uses this endpoint inequality to reserve common margins. Source: paper.tex lines15994–16108 and16340–16366.
- `prop:probe-low` (proof_dependency; source_audit_annotation): Proposition15.3 is first divided by the common positive normalizer to obtain the normalized low estimate, which the final order-of-choices proof applies with lossDelta/2. Source: paper.tex lines15633–15647 and16369–16372.
- `lem:late-height-closure` (proof_dependency; source_audit_annotation): Choose target-dependent tau then external N after retained orders, preserving common sigma. Source: paper.tex lines 16444–16452.
- `lem:principal-residue-interface` (proof_dependency; source_audit_annotation): Principalremainder estimates contribute separately retained margins. Source: paper.tex lines 16357–16368.

Candidate references: `eq:part-II-bootstrap` (internal_reference); `old-eq:5.14` (candidate_proof_dependency); `prop:detector-counts` (candidate_proof_dependency); `lem:plain` (candidate_proof_dependency); `eq:compensated-probe-definition` (candidate_proof_dependency); `eq:principal-scalar` (candidate_proof_dependency); `sec:principal-signal` (candidate_proof_dependency); `lem:detector-dyads` (candidate_proof_dependency); `sec:tails` (candidate_proof_dependency); `eq:shared-principal-nonvanishing` (candidate_proof_dependency); `prop:probe-errors` (candidate_proof_dependency); `lem:high-bin` (candidate_proof_dependency); `lem:principal-residue-interface` (candidate_proof_dependency); `eq:part-II-normalized-low` (candidate_proof_dependency); `sec:selection` (candidate_proof_dependency); `eq:stage-height-allocation` (candidate_proof_dependency); `eq:late-height-hypothesis` (candidate_proof_dependency); `lem:late-height-closure` (candidate_proof_dependency).

## External inputs and references

### [1] BaierBansal — `external:BaierBansal`

Source lines 16466–16474.

S. Baier and A. Bansal, The large sieve with power moduli for $\mathbb Z[i]$, International Journal of Number Theory 14 (2018), no. 10, 2737--2756. \href{https://doi.org/10.1142/S1793042118501658}{doi:10.1142/S1793042118501658}; \href{https://arxiv.org/abs/1802.08964}{arXiv:1802.08964}.

Comparison with an existing planar/number-field Poisson proof; the paper gives its own Eisenstein-lattice bound.

Role: **background**. paper.tex lines 146–150; bibliography[1] at lines 16466–16474

- `manuscript` (background; source_audit_annotation): Comparison with an existing planar/number-field Poisson proof; the paper gives its own Eisenstein-lattice bound. Source: paper.tex lines 146–150.

Candidate references: `sec:introduction` (external_reference); `sec:probe` (external_reference).

### [2] BIMS — `external:BIMS`

Source lines 16474–16482.

V. Bhargava, G. Ivanyos, R. Mittal, and N. Saxena, Irreducibility and deterministic $r$-th root finding over finite fields, Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation (ISSAC 2017), ACM, 2017, 37--44. \href{https://doi.org/10.1145/3087604.3087620}{doi:10.1145/3087604.3087620}; \href{https://www.cse.iitk.ac.in/users/nitin/papers/irred-and-rth-root.pdf}{author manuscript}.

Downstream least-nonresidue application, not an input to the main zero-free theorem.

Role: **used**. paper.tex lines 343–347; bibliography[2] at lines 16474–16482

- `cor:nonresidues-square-roots` (proof_dependency; source_audit_annotation): Fixedhalf-plane implies polylogarithmic least quadratic nonresidue. Source: paper.tex lines 339–347.

Candidate references: `cor:nonresidues-square-roots` (external_reference).

### [3] BCSS — `external:BCSS`

Source lines 16482–16490.

K. Billington, M. Cheng, J. Schettler, and A. I. Suriajaya, The average number of Goldbach representations and zero-free regions of the Riemann zeta-function, International Journal of Number Theory 21 (2025), no. 2, 289--316. \href{https://doi.org/10.1142/S1793042125500150}{doi:10.1142/S1793042125500150}; \href{https://arxiv.org/abs/2306.09102}{arXiv:2306.09102}.

Definition/context of the quasi-Riemann hypothesis.

Role: **background**. paper.tex lines 77–81; bibliography[3] at lines 16482–16490

- `manuscript` (background; source_audit_annotation): Definition/context of the quasi-Riemann hypothesis. Source: paper.tex lines 77–81.

Candidate references: `sec:introduction` (external_reference).

### [4] BGL — `external:BGL`

Source lines 16490–16498.

V. Blomer, L. Goldmakher, and B. Louvel, $L$-functions with $n$-th-order twists, International Mathematics Research Notices 2014 (2014), no. 7, 1925--1955. \href{https://doi.org/10.1093/imrn/rns257}{doi:10.1093/imrn/rns257}; \href{https://arxiv.org/abs/1112.1650}{arXiv:1112.1650}.

Norm-recursion method supporting the manuscript’s internally proved sextic large sieve.

Role: **used**. paper.tex lines 133–137; bibliography[4] at lines 16490–16498

- `lem:sextic-large-sieve` (proof_dependency; source_audit_annotation): Higher-order norm-recursion interfaces checked in the paper for sextic symbols. Source: paper.tex lines 4727–4736.

Candidate references: `sec:introduction` (external_reference); `sec:sextic-row-count` (external_reference); `lem:sextic-large-sieve` (external_reference).

### [5] Clay — `external:Clay`

Source lines 16498–16503.

Clay Mathematics Institute, Riemann Hypothesis, Millennium Prize Problems, \url{https://www.claymath.org/millennium/riemann-hypothesis/}.

Critical-line Riemann-hypothesis context.

Role: **background**. paper.tex lines 117–121; bibliography[5] at lines 16498–16503

- `manuscript` (background; source_audit_annotation): Critical-line Riemann-hypothesis context. Source: paper.tex lines 117–121.

Candidate references: `sec:introduction` (external_reference).

### [6] DFDS — `external:DFDS`

Source lines 16503–16509.

C. David, A. de Faveri, A. Dunn, and J. Stucky, Non-vanishing for cubic Hecke $L$-functions, preprint, arXiv:2410.03048v2 (2026). \href{https://arxiv.org/abs/2410.03048}{arXiv:2410.03048}.

Related cubic-family central-point nonvanishing result.

Role: **background**. paper.tex lines 152–156; bibliography[6] at lines 16503–16509

- `manuscript` (background; source_audit_annotation): Related cubic-family central-point nonvanishing result. Source: paper.tex lines 152–156.

Candidate references: `sec:introduction` (external_reference).

### [7] Dirichlet — `external:Dirichlet`

Source lines 16509–16519.

P. G. L. Dirichlet, \emph{Beweis des Satzes, da\ss{} jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enth\"alt}, Abhandlungen der K\"oniglichen Preussischen Akademie der Wissenschaften zu Berlin (1837), 45--81. English translation by R. Stephan, \href{https://arxiv.org/abs/0808.1408}{arXiv:0808.1408}.

Historical introduction of character L-series.

Role: **background**. paper.tex lines 85–89; bibliography[7] at lines 16509–16519

- `manuscript` (background; source_audit_annotation): Historical introduction of character L-series. Source: paper.tex lines 85–89.

Candidate references: `sec:introduction` (external_reference).

### [8] DR — `external:DR`

Source lines 16519–16526.

A. Dunn and M. Radziwi\l\l, Bias in cubic Gauss sums: Patterson's conjecture, Annals of Mathematics 200 (2024), no. 3, 967--1057. \href{https://doi.org/10.4007/annals.2024.200.3.3}{doi:10.4007/annals.2024.200.3.3}; \href{https://arxiv.org/abs/2109.07463}{arXiv:2109.07463}.

Unconditional arithmetic laws, theta automorphy, cusp expansions and Bessel formulas; conditional prime asymptotic separately excluded.

Role: **used**. paper.tex lines 122–126; bibliography[8] at lines 16519–16526

- `lem:fixed-gauss-phase` (proof_dependency; source_audit_annotation): Cubic reciprocity on primary coprime inputs. Source: paper.tex lines 994–1003.
- `prop:completed-reflection` (proof_dependency; source_audit_annotation): Cubic supplementarylaw makes the fixed mask periodic modulo9. Source: paper.tex lines 2001–2010.
- `manuscript` (excluded_input; source_audit_annotation): The introduction explicitly excludes the GRH-conditional prime asymptotic while importing only unconditional cusp formulas. Source: paper.tex lines120–128. DRnative source lines213–221 show the GRH hypothesis of its main prime asymptotic.

Candidate references: `sec:introduction` (external_reference); `lem:fixed-gauss-phase` (external_reference); `sec:completed-reflection` (external_reference); `prop:completed-reflection` (external_reference).

### [9] FG — `external:FG`

Source lines 16526–16533.

J. B. Friedlander and D. A. Goldston, Sums of three or more primes, Transactions of the American Mathematical Society 349 (1997), no. 1, 287--310. \href{https://doi.org/10.1090/S0002-9947-97-01652-8}{doi:10.1090/S0002-9947-97-01652-8}.

Uniform Dirichlet quasi-Riemann formulation.

Role: **background**. paper.tex lines 80–84; bibliography[9] at lines 16526–16533

- `manuscript` (background; source_audit_annotation): Uniform Dirichlet quasi-Riemann formulation. Source: paper.tex lines 80–84.

Candidate references: `sec:introduction` (external_reference).

### [10] Galbraith — `external:Galbraith`

Source lines 16533–16541.

S. D. Galbraith, Mathematics of Public Key Cryptography, Cambridge University Press, 2012. \href{https://doi.org/10.1017/CBO9781139012843}{doi:10.1017/CBO9781139012843}. Section and algorithm references use the author's extended and corrected version 2.0, \href{https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch2.pdf}{Chapter 2}.

Downstream deterministic square-root algorithm.

Role: **used**. paper.tex lines 355–359; bibliography[10] at lines 16533–16541

- `cor:nonresidues-square-roots` (proof_dependency; source_audit_annotation): Tonelli–Shanks once a deterministic nonresidue is found. Source: paper.tex lines 349–360.

Candidate references: `cor:nonresidues-square-roots` (external_reference).

### [11] GZ — `external:GZ`

Source lines 16541–16549.

P. Gao and L. Zhao, Moments of quadratic Hecke $L$-functions of imaginary quadratic number fields, Journal of Number Theory 209 (2020), 359--377. \href{https://doi.org/10.1016/j.jnt.2019.09.002}{doi:10.1016/j.jnt.2019.09.002}; \href{https://arxiv.org/abs/1707.00091}{arXiv:1707.00091}.

Ordinary finite-order Hecke functional equation.

Role: **used**. paper.tex lines 141–145; bibliography[11] at lines 16541–16549

- `lem:hecke-strip-growth` (proof_dependency; source_audit_annotation): Functional equation plusgammafactor conductor scaling for finite-order Hecke L-functions. Source: paper.tex lines 1436–1458.
- `prop:probe-errors` (proof_dependency; source_audit_annotation): Reflect a local nonprincipal numerator while allocating selected conductor deficits. Source: paper.tex lines 9007–9023.

Candidate references: `sec:introduction` (external_reference); `lem:hecke-strip-growth` (external_reference); `lem:buffered-bins` (external_reference); `prop:probe-errors` (external_reference).

### [12] GZSextic — `external:GZSextic`

Source lines 16549–16557.

P. Gao and L. Zhao, Moments and one level density of sextic Hecke $L$-functions, Functiones et Approximatio Commentarii Mathematici 70 (2024), no. 1, 7--28. \href{https://doi.org/10.7169/facm/2057}{doi:10.7169/facm/2057}; \href{https://arxiv.org/abs/2201.01885}{arXiv:2201.01885}.

Related recorded Eisenstein sextic sieve form, only cited for comparison.

Role: **background**. paper.tex lines 138–142; bibliography[12] at lines 16549–16557

- `manuscript` (background; source_audit_annotation): Related recorded Eisenstein sextic sieve form, only cited for comparison. Source: paper.tex lines 138–142.

Candidate references: `sec:introduction` (external_reference).

### [13] GL — `external:GL`

Source lines 16557–16565.

L. Goldmakher and B. Louvel, A quadratic large sieve inequality over number fields, Mathematical Proceedings of the Cambridge Philosophical Society 154 (2013), no. 2, 193--212. \href{https://doi.org/10.1017/S0305004112000370}{doi:10.1017/S0305004112000370}; \href{https://arxiv.org/abs/1112.1642}{arXiv:1112.1642}.

Quadratic Hecke-family large-sieve theorem plus recursion framework comparisons.

Role: **used**. paper.tex lines 131–135; bibliography[13] at lines 16557–16565

- `lem:completed-quadratic-reduction` (proof_dependency; source_audit_annotation): Quadratic residue-family norm bound(U+V)(UV)^epsilon; paper checks family hypotheses. Source: paper.tex lines 2638–2671.

Candidate references: `sec:introduction` (external_reference); `sec:completed-reflection` (external_reference); `lem:sextic-large-sieve` (external_reference); `sec:inverse` (external_reference).

### [14] GM — `external:GM`

Source lines 16565–16572.

L. Guth and J. Maynard, New large value estimates for Dirichlet polynomials, Annals of Mathematics 203 (2026), no. 2, 623--675. \href{https://doi.org/10.4007/annals.2026.203.2.6}{doi:10.4007/annals.2026.203.2.6}; \href{https://arxiv.org/abs/2405.20552}{arXiv:2405.20552}.

Zero-density contrast and a comparison for classical zero detection; no imported density bound used to exclude all zeros.

Role: **background**. paper.tex lines 101–105; bibliography[14] at lines 16565–16572

- `manuscript` (background; source_audit_annotation): Zero-density contrast and a comparison for classical zero detection; no imported density bound used to exclude all zeros. Source: paper.tex lines 101–105.

Candidate references: `sec:introduction` (external_reference); `prop:detector-witness` (external_reference).

### [15] Hadamard — `external:Hadamard`

Source lines 16572–16580.

J. Hadamard, Sur la distribution des z\'eros de la fonction $\zeta(s)$ et ses cons\'equences arithm\'etiques, Bulletin de la Soci\'et\'e Math\'ematique de France 24 (1896), 199--220. \href{https://doi.org/10.24033/bsmf.545}{doi:10.24033/bsmf.545}.

Historical prime-number theorem/nonvanishing on Re(s)=1.

Role: **background**. paper.tex lines 90–94; bibliography[15] at lines 16572–16580

- `manuscript` (background; source_audit_annotation): Historical prime-number theorem/nonvanishing on Re(s)=1. Source: paper.tex lines 90–94.

Candidate references: `sec:introduction` (external_reference).

### [16] HBQuadratic — `external:HBQuadratic`

Source lines 16580–16586.

D. R. Heath-Brown, A mean value estimate for real character sums, Acta Arithmetica 72 (1995), no. 3, 235--275. \href{https://doi.org/10.4064/aa-72-3-235-275}{doi:10.4064/aa-72-3-235-275}.

Recursive Poisson moment framework; the marked recursion is proved internally.

Role: **background**. paper.tex lines 140–144; bibliography[16] at lines 16580–16586

- `manuscript` (background; source_audit_annotation): Recursive Poisson moment framework; the marked recursion is proved internally. Source: paper.tex lines 140–144.

Candidate references: `sec:introduction` (external_reference); `sec:inverse` (external_reference).

### [17] HB — `external:HB`

Source lines 16586–16592.

D. R. Heath-Brown, Kummer's conjecture for cubic Gauss sums, Israel Journal of Mathematics 120, part A (2000), 97--124. \href{https://doi.org/10.1007/s11856-000-1273-y}{doi:10.1007/s11856-000-1273-y}.

Cubic large-sieve inequality; its recursive framework also cited for comparison.

Role: **used**. paper.tex lines 134–138; bibliography[17] at lines 16586–16592

- `lem:hybrid-energy` (proof_dependency; source_audit_annotation): Cubic squarefree large-sieve norm U+V+(UV)^(2/3), arbitrary complex coefficients. Source: paper.tex lines 7485–7508.

Candidate references: `sec:introduction` (external_reference); `sec:marked` (external_reference); `sec:inverse` (external_reference).

### [18] Hecke — `external:Hecke`

Source lines 16592–16602.

E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. Erste Mitteilung, Mathematische Zeitschrift 1 (1918), 357--376; Zweite Mitteilung, Mathematische Zeitschrift 6 (1920), 11--51. \href{https://doi.org/10.1007/BF01465095}{doi:10.1007/BF01465095} (first paper).

Historical character zeta analytic theory.

Role: **background**. paper.tex lines 92–96; bibliography[18] at lines 16592–16602

- `manuscript` (background; source_audit_annotation): Historical character zeta analytic theory. Source: paper.tex lines 92–96.

Candidate references: `sec:introduction` (external_reference).

### [19] Huxley — `external:Huxley`

Source lines 16602–16608.

M. N. Huxley, The large sieve inequality for algebraic number fields, Mathematika 15 (1968), no. 2, 178--187. \href{https://doi.org/10.1112/S0025579300002540}{doi:10.1112/S0025579300002540}.

Comparison to classical planar/number-field large sieve; direct proof supplied.

Role: **background**. paper.tex lines 145–149; bibliography[19] at lines 16602–16608

- `manuscript` (background; source_audit_annotation): Comparison to classical planar/number-field large sieve; direct proof supplied. Source: paper.tex lines 145–149.

Candidate references: `sec:introduction` (external_reference); `sec:probe` (external_reference).

### [20] Kubota — `external:Kubota`

Source lines 16608–16615.

T. Kubota, On automorphic functions and the reciprocity law in a number field, Lectures in Mathematics, Department of Mathematics, Kyoto University, no. 2, Kinokuniya Book-Store, Tokyo, 1969. \url{http://hdl.handle.net/2433/84907}.

Metaplectic setting.

Role: **background**. paper.tex lines 120–124; bibliography[20] at lines 16608–16615

- `manuscript` (background; source_audit_annotation): Metaplectic setting. Source: paper.tex lines 120–124.

Candidate references: `sec:introduction` (external_reference).

### [21] MP — `external:MP`

Source lines 16615–16623.

J. Maynard and K. Pratt, Half-isolated zeros and zero-density estimates, International Mathematics Research Notices 2024 (2024), no. 19, 12978--13014. \href{https://doi.org/10.1093/imrn/rnae191}{doi:10.1093/imrn/rnae191}; \href{https://arxiv.org/abs/2206.11729}{arXiv:2206.11729}.

Comparison for the truncated-inverse zero-detector mechanism.

Role: **background**. paper.tex lines 154–158; bibliography[21] at lines 16615–16623

- `manuscript` (background; source_audit_annotation): Comparison for the truncated-inverse zero-detector mechanism. Source: paper.tex lines 154–158.

Candidate references: `sec:introduction` (external_reference); `prop:detector-witness` (external_reference).

### [22] MilneCFT — `external:MilneCFT`

Source lines 16623–16628.

J. S. Milne, Class Field Theory, course notes, version 4.03, 6 August 2020. \url{https://www.jmilne.org/math/CourseNotes/CFT.pdf}.

Finite-order Hecke/ray-class identification and Kummer character construction.

Role: **used**. paper.tex lines 55–59; bibliography[22] at lines 16623–16628

- `thm:main` (proof_dependency; source_audit_annotation): Finite-order idelicHecke characters have no nontrivial continuous complex-place part. Source: paper.tex lines 50–58.
- `lem:fixed-numerator-ray` (proof_dependency; source_audit_annotation): Kummerdescription for fixed-numerator residuecharacters. Source: paper.tex lines 676–699.

Candidate references: `sec:introduction` (external_reference); `lem:fixed-numerator-ray` (external_reference).

### [23] OpenAICubicFirstMoment — `external:OpenAICubicFirstMoment`

Source lines 16628–16635.

OpenAI, An unconditional first moment for cubic Gauss sums, OpenAI Math Release preprint \href{https://github.com/openai/math/blob/main/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf}{OAI:An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026}, 2026.

Theorem 1.1 of the independent cubic first moment manuscript is explicitly not used.

Role: **excluded**. paper.tex lines 125–129; bibliography[23] at lines 16628–16635

- `manuscript` (excluded_input; source_audit_annotation): The introduction explicitly says this independent first-moment theorem is not used. Source: paper.tex lines124–128.

Candidate references: `sec:introduction` (external_reference).

### [24] Patterson — `external:Patterson`

Source lines 16635–16642.

S. J. Patterson, A cubic analogue of the theta series, Journal f\"ur die reine und angewandte Mathematik 296 (1977), 125--161. \href{https://doi.org/10.1515/crll.1977.296.125}{doi:10.1515/crll.1977.296.125}.

Original theta/coefficient calculations underlying DR; the paper says its formal input remains DR.

Role: **background**. paper.tex lines 120–124; bibliography[24] at lines 16635–16642

- `manuscript` (background; source_audit_annotation): Original theta/coefficient calculations underlying DR; the paper says its formal input remains DR. Source: paper.tex lines 120–124.

Candidate references: `sec:introduction` (external_reference); `sec:completed-reflection` (external_reference).

### [25] Riemann — `external:Riemann`

Source lines 16642–16650.

B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Gr\"osse, Monatsberichte der K\"oniglichen Preussischen Akademie der Wissenschaften zu Berlin (1859), 671--680. Manuscript, transcription, and translation at \url{https://www.claymath.org/library/historical/riemann/r.html}.

Historical connection of zeta zeros and primes.

Role: **background**. paper.tex lines 87–91; bibliography[25] at lines 16642–16650

- `manuscript` (background; source_audit_annotation): Historical connection of zeta zeros and primes. Source: paper.tex lines 87–91.

Candidate references: `sec:introduction` (external_reference).

### [26] TaoNonresidues — `external:TaoNonresidues`

Source lines 16650–16658.

T. Tao, The Elliott--Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture, Algebra \& Number Theory 9 (2015), no. 4, 1005--1034. \href{https://doi.org/10.2140/ant.2015.9.1005}{doi:10.2140/ant.2015.9.1005}; \href{https://msp.org/ant/2015/9-4/ant-v9-n4-p09-p.pdf}{publisher PDF}.

Vinogradov least nonresidue conjecture contextual citation.

Role: **background**. paper.tex lines 331–335; bibliography[26] at lines 16650–16658

- `manuscript` (background; source_audit_annotation): Vinogradov least nonresidue conjecture contextual citation. Source: paper.tex lines 331–335.

Candidate references: `cor:nonresidues-square-roots` (external_reference).

### [27] TZ — `external:TZ`

Source lines 16658–16665.

J. Thorner and A. Zaman, A unified and improved Chebotarev density theorem, Algebra \& Number Theory 13 (2019), no. 5, 1039--1068. \href{https://doi.org/10.2140/ant.2019.13.1039}{doi:10.2140/ant.2019.13.1039}; \href{https://arxiv.org/abs/1803.02823}{arXiv:1803.02823}.

Fixed-extension/Frobenius prime counting and nontrivial sextic detection; its classical zero-free region statement is background.

Role: **used**. paper.tex lines 97–101; bibliography[27] at lines 16658–16665

- `lem:principal-residue-interface` (proof_dependency; source_audit_annotation): Chebotarev detects a nontrivial sexticFrobenius class, separating the sole principal numerator row. Source: paper.tex lines 5500–5516.
- `lem:ray-prime-normalizer` (proof_dependency; source_audit_annotation): Fixed-ray prime normalizer asymptotic, with fixed conductor before P→infinity. Source: paper.tex lines 7019–7040.
- `lem:plain` (proof_dependency; source_audit_annotation): Fixedfield prime pool of Z^(ell*+o(1)) size, disjoint from live short slots. Source: paper.tex lines 13450–13475.

Candidate references: `sec:introduction` (external_reference); `sec:shared-analytic-estimates` (external_reference); `lem:ray-prime-normalizer` (external_reference); `sec:plain` (external_reference).

### [28] ValleePoussin — `external:ValleePoussin`

Source lines 16665–16676.

C.-J. de la Vall\'ee Poussin, Recherches analytiques sur la th\'eorie des nombres premiers. Premi\`ere partie: La fonction $\zeta(s)$ de Riemann et les nombres premiers en g\'en\'eral, Annales de la Soci\'et\'e scientifique de Bruxelles 20, deuxi\`eme partie (1896), 183--256. Reprinted in Recherches analytiques sur la th\'eorie des nombres premiers, Hayez, Brussels, 1897, \url{https://books.google.com/books?id=7e0GAAAAYAAJ}.

Historical prime-number theorem/nonvanishing on Re(s)=1.

Role: **background**. paper.tex lines 90–94; bibliography[28] at lines 16665–16676

- `manuscript` (background; source_audit_annotation): Historical prime-number theorem/nonvanishing on Re(s)=1. Source: paper.tex lines 90–94.

Candidate references: `sec:introduction` (external_reference).

## Coverage and limits

{"mode": "native_latex_index", "claims": 67, "sections": 20, "external_references": 28, "source_bibliography_entries": 28, "unresolved_bibliography": 0, "unresolved": 0}

- Reference edges are not certified proof dependencies. Edge direction is prerequisite → consumer.
- Numbers reflect simple source counters; labels and line anchors are authoritative.
- Extractive statements and proof openings are marked; they are not reconstructed proof gists.
- Macros, included files and custom counters require an expanded source or annotated fallback.