KEYBOOK / 1,000 KEYS PER PAGE

Follow one scalar through every layer.

Compare adjacent stages, inspect the XOR signature, and test the exact curve mirror k ↔ n−k.

PAGE
LINE
SCALAR BITS
MIRROR X
EXACT MIRROR + ANCHOR NAVIGATION

Navigate from a reference anchor

k = a + dn − k = (n − a) − d

Choose an anchor a and an exact offset d. Flip the book side to mirror both your anchor and your current position; the offset changes sign.

Current book side

Reference anchor a
Current position k
Offset from anchor d · scalars

Mirrored book side

Mirrored anchor n − a
Mirrored position n − k
Mirrored offset −d · scalars
QR mask directions

Steps stop at the first and last valid scalar.

Waiting for the current scalar…

Navigation equations and exact mirror scalar

Curve equation
A = aG; Q = (a + d)G = A + dG
Q′ = −Q = (n − a − d)G = −A − dG

Book coordinates
page(k) = 1 + floor((k − 1) / 1000)
line(k) = 1 + ((k − 1) mod 1000)
k = 1000 × (page − 1) + line

Apply these page and line equations to n − k for its exact destination. Flipping twice returns to the same anchor, scalar, page and line. The final page contains 336 valid lines.

Equation for the optimum layers
s(k) = +1 for masks 0 → 7
s(k) = −1 for masks 7 → 0
s(k) = 0 for all other mask pairs
s(n − k) = −s(k)

Here the masks are those selected for SEC(kG) and SEC((n − k)G), in that order, using the lab’s fixed QR settings. This is a QR classification; it is not the curve’s y-coordinate. Opposite-mask navigation visits the nearest recorded hit in the chosen scalar direction with the other nonzero sign.

Mirror scalar · hexadecimal
Mirror scalar · decimal
Generator order n · decimal

The mirror equation holds for every valid scalar. Recorded optimum hits are reference points selected by their QR masks. Their spacing does not yet predict the next hit or an unknown key’s page. Navigation stays within 1 ≤ k < n and never wraps silently.

SEC 1: curve arithmetic and point encoding

SAVED REFERENCE ANCHOR

One reproducible point in the book.

The Pair Lab example is saved in hexadecimal. Use its verified point and page to check navigation and curve transformations. Its QR pairing score does not assign probabilities to other pages.

Verifying the reference point…

SCALAR · HEXADECIMAL
SCALAR · DECIMAL
PAGE · 1,000 KEYS
LINE
COMPRESSED PUBLIC POINT
FORWARD CHAIN

Six QR projections of the same key

Each square is a standards-based QR code using fixed M error correction. Hover to inspect; click a card to pin it.

ADJACENT-STAGE ANALYSIS

XOR / difference matrix

changed bit same bit
THE CLEAN INVARIANT

Curve mirror laboratory

(n−k)G = −kG. The X coordinate is identical, Y is reflected across the field, and the compressed SEC prefix flips between 02 and 03. HASH160 then destroys that visible relation.

X coordinates
SEC XOR distance
HASH160 XOR distance
SEC QR masks
Native changed modules
Exact-pair coverage
Eight-neighbor coverage
Pair components
Same-mask control

Derive a scalar to measure its mirror pairing.

PAIR CEILING EXPERIMENT

Find the most completely paired mirror signature.

The experiment first exhausts all 36 unordered QR-mask combinations. It then searches a bounded scalar interval for keys whose natural masks reach the proven ceiling.

CURRENT THEORETICAL CEILINGBuilding the mask atlas…
MASK SIGNATURES36
BEST MASK PAIR
ALL-PAIR SIGNATURES
KNOWN OPTIMUM0x743186…53218
LIVE EXPERIMENT

Ready

TARGET 0 ↔ 7
CHECKED0 / 1,001
MEAN RATE0 / sec
OPTIMUM HITS0
BEST COVERAGE
BEST VERIFIED CANDIDATENo candidate measured.

Gold marks exact isolated pairs, red marks singletons, and cyan marks larger connected components.

Highest pair scores in this bounded scan
RankScalarMasksChangedExact pairsExact-pair coverageNeighbor coverageSingletons
Start a bounded scan to rank candidate scalars.
FIND NATURAL OPTIMUMS

All optimum hits

Every optimum hit from the current scan is kept here in discovery order. Pause, resume, or close this window without losing hits. Export before starting a new scan or reloading the page.

No scan started.

All hits · 100 per page · export includes every page
HitScalar (hex / decimal)Keybook page / lineMasksExact pairs / coverageAction
Start a scan to collect optimum hits.
PAGE · LINE · MASK DIRECTION

The 3D book

Each dot is an optimum hit. Page and line locate it in the book; mask direction places it on one of two layers. Drag to rotate, or use the rotation controls.

Your recorded scan is ready.

Recorded scalars from the selected source.

7 → 00 → 70 direction changes

X: page offset · Y: line 1–1,000 · Z: mask direction

Page zero in this view:

The path joins recorded hits in scalar order. Non-hits and untested scalars are omitted.

Inspect all 36 mask-pair templates
Complete fixed-setting feasibility atlas
RankMasksChangedExact pairsExact-pair coverageNeighbor coverageSingletonsLarge-cluster cells
Building atlas…

Interpretation boundary: under lowercase compressed-SEC hex, QR byte mode, error correction M, and version 5, no scalar can produce 100% isolated pairs. A scalar changes which masks the encoder selects; it does not change the signature belonging to that mask pair.

What this instrument can—and cannot—show

It can reveal exact transformations that survive from one representation to the next, including the secp256k1 point-negation symmetry.

It cannot reverse HASH160 or infer a private page from an address image. A visually close final matrix is not “warmer” cryptographically.