ζQRH / Method AtlasRESEARCH VIEW
THE ESTIMATES BEHIND THE ENDPOINT

See what moves
the boundary.

Trace each method from its estimates to the endpoint, and see exactly what it changed.

Explore the methods Loading source snapshot…
VERIFIED FRONTIER Framework verified
—
Reading the catalogue
—verified methods
—gain over 7/8 · ×10⁻⁶
—exact θ · smaller is stronger
THE NUMERICAL OUTCOME

An endpoint is the result. The proof has dimensions.

Improvement over 7/8, in millionths →
01 / METHOD COMPARISON

Which part of the proof changed?

Pick two methods. Click any cell for its evidence.

Dimensions
Changed Uses Retuned Foundation Not extracted
Methods and their documented changes by proof dimension
02 / HOW THE DIMENSIONS CONNECT

Follow the estimates to the endpoint.

Choose a node to trace its inputs and consumers.

Complete probe=Principal signal+Complementary sourceLOW controls the probe · HIGH controls the complement
Selected Required inputs Uses this result Method changes / retunes

Arrows mean “used by”. Every branch remains required.

03 / WORKED EXAMPLE

Comparing moment-capacity regions.

Standard marked moment vs Cycle61’s local affine moment (n₁ = n₂ = m). Length constraints only.

One example · Moment constraints
Standard · conditional Cycle61 · local gates Test point

Tap the plot to move the test point.

Standard length inequality—
Cycle61 local length gates—

Dashed box: Cycle61’s local window (0.41 ≤ m ≤ 0.44, 0.034 ≤ z ≤ 0.04); the orange line outside it is extrapolated.

ADJUST THE PARAMETERS

Test a length pair.

0.4050.445

Cycle61’s local window: 0.41–0.44.

0.70000.7500

0.0300.046

Move past a boundary to see which gate fails.

Standard maximum z · conditional—
Cycle61 maximum z · local—
Capacity difference at this m—

Formulas, assumptions, and sources
Standard: 2m + 6κz ≤ 1
Cycle61: 2m + (19312/4787)z ≤ 14200/14361 − 1/100000

The standard gate assumes β* ≥ 0.51, 2β* − 1 ≤ κ, and the documented source and slot restrictions. The affine example fixes source width M = 1 and moving width 0, and requires the local m and z box. Values are compared before display rounding. Passing these length gates alone does not certify a new endpoint.

Standard gate · B0.5 ↗Cycle61 · Theorem 2.2 ↗

Use the right chart for the question.

Matrix: where a method intervenes. Proof map: how it reaches the endpoint. Capacity: estimates in real units. Spider-plot area is not strength.

Back to comparison ↑
QUICK REFERENCE

Dimension guide

Keep your place. Explore a definition, then return to the proof.

These are different jobs in a proof, not interchangeable scores. Choose a dimension for its meaning, examples, and connection to the endpoint.

01Moment estimatesHow large are the arithmetic sums on average?

A moment averages the size of an arithmetic polynomial over a family of characters or rows. It can control many sums even when individual sums are large.

Σᵤ |Sᵤ,₁ Sᵤ,₂ Qᵤ|² ≲ U^(1+ε)

An improvement: a smaller energy exponent, or a larger admissible range of plain lengths and prime-slot length at the same exponent. The coefficients, masks, and heights must still be covered.

Example: compare the standard capacity 2m + 6κz ≤ 1 with an affine capacity 2m + Λz ≤ B. Its intercept B and local length window matter alongside Λ.

Feeds into: detector row counts and the weighted HIGH numerator. The proof map separates the inverse/Möbius moment from the paired plain moment.

Explore the worked capacity example ↗
02LOW estimateHow small can the complete physical probe be made?

LOW is the direct reflected-energy bound for the complete compensated physical probe. It uses the completed theta reflection, signed Gram bounds, and all marked or rescaled subsets.

An improvement: a smaller probe exponent, or a wider range of physical scales where that complete estimate holds. Every correction and normalization must still be paid.

Feeds into: the final comparison with the principal term. Better HIGH estimates alone do not suffice if the direct probe bound still exceeds the endpoint budget.

Global LOW is a proof estimate. It is not simply the contribution of small rows, and it does not mean zeros of small imaginary height.

03HIGH estimateHow small is the complementary source after isolating the principal term?

After an exact transformation isolates the principal term, HIGH bounds the complementary, zero-sensitive source. It combines selected-row counts with estimates for the actual physical numerator and the required reciprocal factors.

An improvement: fewer selected rows, a stronger weighted numerator average, or a better allocation of the available prime amplitudes. This argument has its own small, central, and tail row ranges.

Representative weighted factor: U^(1/4 + 3R/4 − qzₐ/2)

At fixed remaining parameters, reducing the row exponent R or increasing a legally available zₐ improves this factor. The full formula and its domain are in the HIGH proof gate.

Global HIGH is the complementary proof estimate. It is not simply a large-row contribution or a category of zero heights.

04Detector / densityHow many rows can support a selected zero?

A zero forces an inverse or plain polynomial witness to be large. Moment bounds then limit how many rows can have such witnesses. The shared-zero and common-frequency conditions connect the actual witnesses to that count.

Number of selected rows ≲ U^(R+ε)

An improvement: a smaller R under the same conditions, or a valid count on a larger parameter domain. Reusing an existing moment in the detector can improve this dimension without proving a new moment theorem.

Feeds into: HIGH, where the population of selected rows multiplies the cost of their numerator. The detector count and the numerator estimate must each be valid for the rows being used.

05Geometry / reservesHow should the proof’s scales and budgets be balanced?

Geometry chooses physical window sizes, reflected lengths, prime-slot allocations, and density splits. It balances LOW, HIGH, and the other branches against the same principal signal.

A reserve is a positive gap in an exponent budget. If a contribution costs U^(c−η) while its budget is U^c, the reserve η > 0 can pay specified small losses.

An improvement: a better simultaneous choice of parameters or sharper reserve bookkeeping. This can improve the endpoint while reusing existing arithmetic estimates.

In the recorded saturating geometry: θ = 11/12 − ℓ/4

This relation is conditional on that geometry’s full set of gates. Increasing ℓ is useful only while those gates remain valid. A smaller θ gives a stronger zero-free half-plane.

06Recovery / endpointDoes the estimate cover the complete theorem?

A bound for one retained sum or one local parameter box must be assembled into the complete analytic argument. Recovery pays the pieces introduced, shortened, or excluded during the transformations.

This includes original nonunit zeros and masks, Euler and ramified terms, residues, all relevant scale ranges, independent heights, derivative profiles, and discarded tails. Continuation and character transfer then reach the stated target family.

An improvement: extending a valid local estimate to every required input, reducing a real reconstruction loss, or completing a previously missing branch.

Feeds into: the unconditional endpoint statement. A component saving becomes an endpoint gain only after all required consumers and recovery steps are covered.

07VerificationWas the exact endpoint theorem checked?

Verification checks that the submitted proof establishes the claimed mathematical statement, with its exact definitions, target family, exact endpoint (rational or algebraic, as claimed), and permitted axioms.

Comparator checks the statement and its meaning against the canonical target. Lean, NanoDa, and con-ron check the proof when those kernel acceptances are reported by the receipt.

A change here: moving from a proposed or locally built result to the recorded checks and tracker acceptance. This is a categorical evidence status, not an additional arithmetic saving.

Inspect each linked receipt for its scope: a checked zeta statement alone does not establish an all-Dirichlet or all-Hecke claim.

How the estimates fit together.

In the recorded probe architecture, think of the identity probe = principal term + complementary source. LOW bounds the probe directly. HIGH bounds the complementary source. The principal, geometry, and recovery gates make those estimates imply the endpoint.

Prime amplitudes + selection: how much useful prime-slot amplitude is available, and which actual slots can be selected without exceeding a moment’s capacity.

Principal + floor + intermediate: the nonzero principal signal and the remaining scale or zero-bin branches. Their costs must be paid even after LOW or HIGH improves.

Source means the actual arithmetic expression being estimated. It is the input to one or more dimensions above; the linked papers and proof receipts are its evidence.

Follow the dependency graph ↗