Zero-radius MCA
\( \delta=0 \)
\( \varepsilon_{\rm mca}(C,\delta)=2/|F| \)
At zero distance the bad-line event is the two endpoint challenges where the affine line equals one of its defining source words.
trivial
Proximity Prize threshold board
Can you push RS MCA frontier?
Track the \(2^{-128}\) MCA and interleaved-list frontiers for \(\rho\in\{1/2,1/4,1/8,1/16\}\), \(k\le2^{40}\), and \(|\mathbb F|<2^{256}\). Each row states whether it is a \(\delta^*\) upper cap, proof record, candidate, or target, with field ledgers and non-claims visible by default.
Pay-per-bit view. If the prize is allocated pro-rata by soundness-gap bits, our rows are already measured in that currency: Paper D v13.2 gives at least +42 certified bits over \(2^{-128}\) in its cap range, the Cycle116/119 finite numerator gives +32.82 bits, and the exact tangent staircase supplies the structural 6/7 threshold gate. The best rows are those that combine auditable bit margin with a smaller certified unsafe radius.
MCA failure score
\[
S_{\rm mca}^{\rm fail}=128+\log_2\!\left(\frac{N_{\rm bad}}{q_{\rm line}}\right)
\]
List failure score
\[
S_{\rm list}^{\rm fail}=128+\log_2\!\left(\frac{L_{\rm lower}}{|\mathbb F|}\right)
\]
Prize rates4MCA and interleaved-list boards for each official rate
Threshold target2−128shown as MCA and list failure scores separately
Exact finite MCA gate506/507tangent-star classification for the \(F_{17^{32}}\) row
Largest failure margin+32.82Cycle116/119 bad-challenge numerator over \(17^{32}\)
Rate leaderboards
The frontier is a four-rate threshold board at \(2^{-128}\). Pick a Proximity Prize rate, then switch between the MCA/CA/line-MCA track and the interleaved-list track.
Proximity Prize regime
Default rows must sit inside this envelope before they count toward the public prize leaderboard.
✓ rate
✓ smooth domain
✓ k ≤ 2^40
✓ |F| < 2^256
✓ denominator
✓ endpoint
Code family
\(C=\operatorname{RS}[\mathbb F,\mathcal L,k]\)
\(\mathcal L\subseteq\mathbb F\) smooth
\(\rho\in\{1/2,1/4,1/8,1/16\}\)
Finite envelope
\(\varepsilon^*=2^{-128}\)
\(k\le2^{40}\)
\(|\mathbb F|<2^{256}\)
Threshold questions
\(\varepsilon_{\rm mca}(C,\delta_C^*)\le2^{-128}\)
\(|\Lambda(C^{\equiv m},\delta_C^*)|\le2^{-128}|\mathbb F|\)
Rows outside this box are useful experiments, but they are not counted in the default prize leaderboard.
Four primary boards and known thresholds
Rate is the top-level partition. The table opens each rate directly into its MCA or interleaved-list track.
| Rate ? | Capacity ? | Johnson approx. ? | MCA track ? | Interleaved-list track ? |
|---|
MCA vs list
The two tracks answer different threshold questions at the same target \(2^{-128}\). They should not be ranked together or used interchangeably.
MCA / CA / line-MCA track
Use this track for rows that bound \(\varepsilon_{\rm mca}(C,\delta)\) directly, count bad line or challenge slopes, or give an explicit bridge from CA, line-MCA, or curve-MCA to the same MCA-style failure event.
Failure lower-bound test: 2^128 * N_bad > q_line.
The denominator is the actual line/slope field \(q_{\rm line}\), not a generated-field or challenge-field proxy unless a theorem justifies the transfer.
Interleaved-list track
Use this track for rows that bound the list size of the interleaved Reed-Solomon code \(C^{\equiv m}\) at radius \(\delta\). The target is a list-size budget, not an MCA probability.
Failure lower-bound test: 2^128 * L_lower > |F|.
A list row does not become a protocol or MCA row unless a source proves the exact bridge to the primitive being used.
\[
\sup_{\mathbf U}|\Lambda(C^{\equiv\mu},\delta,\mathbf U)|
\le
\sup_U|\operatorname{Fib}_U(a)|
\]
Frontier progress
Every point is a proof record, candidate, or target with a row, radius, field denominator, predicate, and status. Smaller reserve means closer to capacity; higher margin means more certified failure mass. Rows that pass the Prize Box gates feed into the MCA track of the rate leaderboards.
Selected certificate
Click a chart point or leaderboard row to inspect the row, radius, proof status, and non-claims.
Proximity Prize benchmarks
Known MCA regimes from Open Problems in List Decoding and Correlated Agreement. They give the public comparison scale for the frontier rows.
Below unique decoding
\( \delta<\delta_{\min}(C)/2 \)
\( \varepsilon_{\rm mca}(C,\delta)\le O(n)/|F| \)
CA and MCA coincide in the unique-decoding regime, and Reed-Solomon codes have small MCA error up to this boundary.
ACFY25; BCIKS20
Approaching Johnson
\( \delta=J(\delta_{\min}(C))-\eta \)
\( \varepsilon_{\rm mca}(C,\delta)\le n\,\mathrm{poly}(1/\eta)/|F| \)
The strongest general Reed-Solomon MCA upper bounds reach the Johnson radius with polynomial dependence on the reserve.
BCHKS25; Hab25; BCGM25; BCIKS20
Approaching capacity
\( \delta\approx\delta_{\min}(C)-1/\Omega(\log n) \)
\( \varepsilon_{\rm mca}(C,\delta)\ge n^{\Omega(1)}/|F| \)
Near capacity, known lower bounds show MCA/CA error must grow polynomially in n over the field-size denominator.
BCHKS25; KK25; CGHLL26
Existing results
Recent proof notes, audits, counterexample families, and bridge results are pulled into a browsable result ledger with source links.
F1 syndrome-pencil normal form
PR #118 gives a Hankel-pencil incidence test for support-wise line explanations and a matching noncontainment criterion.
L2 interleaved codegree reduction
PR #108 proves a two-regime and recursive reduction from interleaved lists to base-code lists at agreements a and 2a-k.
Paper D MCA cap gets a CS25-free route
PR #115 isolates a simple-pole/deep-point averaging proof route for the headline MCA cap under Paper D's finite-field hypotheses.
Strict352 dyadic quotient-core proof record
The dyadic quotient-core packet proves LD_sw(C,a) ≥ 7 for every agreement 264 ≤ a ≤ 352, with 16 bad slopes at agreement 352.
Large-domain mixed-Weil energy floor
PR #119 adds a possible L1 route cut for large-domain prefix-dual energy, kept conditional pending the mixed Weil citation and constant check.
Cycle119/Cycle120 ABF-row obstruction
The active value remains 52,747,567,092 bad challenges over F_17^32; Cycle116 reaches the printed closed endpoint and Cycle119 gives the strict endpoint if finite inputs check out.
Standalone Cycle120 LDsw proof note
PR #105 adds a compact standalone note for the source-scoped lower bound LD_sw(RS[F_17^32,H,256],262) >= 52,747,567,092.
Exact MCA to line-decoding normalization
PR #102 records the support-wise identity epsilon_mca(C,delta) = LD_sw(C,ceil((1-delta)n))/|F| for the two-source object.
Extension-line lower floors
PR #103 strengthens the F1 warning: fixed-rate extension-line families create floors before extension-degree dilution.
M1 beta-pushforward spectral audit
PR #104 scans beta-pushforward rows and finds no hidden growing p^2 mass in the tested windows.
L1 quotient-budgeted locator route
PRs #99 and #106 add characteristic-zero prefix reductions, bad-prime proof-record machinery, and a full-list quotient proof program.
Sharp interleaved-list target
PR #107 sharpens L2: interleaved-list analysis should charge quotient-core mass diagonally and avoid the naive Cartesian-product numerator.
X1 bridges and Lean formalization stub
PR #101 adds deep-point/interleaved bridge notes, quotient-reduction notes, prize-path maps, and a small Lean scaffold.
Certificate calculator
Check exact lower-bound and upper-bound margins for MCA or interleaved-list rows, then inspect the legacy fixed \([512,256]\) certificate calculation.
Prize mode
Lower-bound rows clear the target iff \(2^{128}N_{\rm bad}>q_{\rm line}\) for MCA or \(2^{128}L_{\rm lower}>|\mathbb F|\) for list decoding. Upper-bound safe margins use the opposite numerator direction.
Fixed [512,256] MCA row
For this fixed row, \(\delta=1-a/512\), \(\sigma=a-k\), and \(\eta=1/2-\delta=\sigma/512\).
Admissibility gates
A certificate is only useful if row, predicate, sampler, radius convention, and event-retention semantics are made explicit.
✓
RS row
Explicit \(\mathrm{RS}[F,L,k]\), rate in the challenge set, smooth domain, and degree bound.
✓
Support-wise predicate
One support \(S\) explains the combination but not the two source words on the same support.
✓
Sampler
\(\gamma\) is sampled uniformly from the line field used in the denominator.
✓
Radius convention
Closed and strict support thresholds are shown separately to prevent endpoint ambiguity.
!
Event retention
Any endpoint, duplicate, quotient, charge, or retained-event filter must print the retained numerator.
!
Non-claims
Not ordinary list decoding, not protocol soundness, not an efficient attack, not exact \(\delta_C^*\).
Why this board exists
Paper B turns no-slack smooth-domain failures into a corrected slack ledger. The frontier asks which certified bad-slope sets survive at larger agreement slack.
Slack frontier
For \(C=\mathrm{RS}[\mathbb F_{17^{32}},H,256]\), a certificate at agreement \(a=256+\sigma\) proves failure at \(\delta=1-a/512\).
Beyond 507
The tangent-star refinement rules out a seventh finite-slope support-wise branch at \(a\ge507\). The adjacent and generalized ledgers now give high-agreement CA, projective, curve, and interleaved-list calculators beyond the \(F_{17^{32}}\) row; challenge-field, extension-lift, folding, query, and protocol losses still need explicit ledgers.
Replay before publicity
Each entry should identify whether the census was rerun, hash-bound, imported, locally checked, peer-reviewed, or only proposed.
Contributor artifacts
The board keeps claims, checkers, thresholds, audits, and failure modes next to their source notes so results can be inspected before they are reused.
Papers
The main references behind the frontier. Paper A explains the obstruction, Paper B rebuilds the reserve theory, Paper C tracks protocol ledgers, Paper D gives the universal field-size cap, and the towards-prize note packages the current threshold plan.
A
Paper A
Paper A: No-Slack Obstruction
Refutes the old no-slack support-wise line-MCA statement for smooth multiplicative Reed-Solomon domains.
B
Paper B
Paper B: Slack, Quotient, and Entropy Theory
Builds the corrected-reserve theory and adds the solved high-agreement line/list/curve boundary layer.
C
Paper C
Paper C: SNARK and Protocol Ledger
Separates protocol soundness accounting and adds a theorem-backed high-agreement ledger compiler.
D
Paper D
Paper D: Two-Sided Cap and Certificate Grammar
Main Proximity Prize submission reference: self-contained cap route, safe-side pincer, deployed-row two-sided intervals, model transfers, explicit witnesses, optimized profile, and certificate grammar v2.
T
Towards Prize
Towards Prize: Sparse Threshold Note
Compact companion to Paper D: delta-star staircase, deployed KoalaBear pincer, and sparse residual reduction for the remaining MCA threshold task.
Collaborate on GitHub
Send pull requests with narrow, reviewable proof notes, audits, scripts, certificates, or site-data updates. Keep new research material experimental first.
github.com/przchojecki/rs-mca
Open an issue for discussion or send a PR with reproducible evidence. PRs should say exactly which theorem, target, or ledger they affect.
Where to put work
1. Read AGENTS.md first
Follow the field-ledger, status-label, and no-main-paper-edit rules before proposing a result.
2. Add new research under experimental/
Use notes for theorem sketches and audits, scripts for reproducible checks, and data certificates for machine-readable outputs. For row-level ledger checks, start with experimental/notes/certificate_scanner/.
3. Update experimental/agents-log.md
Record date, contributor, files changed, status, usefulness, and the next step for anything materially changed in experimental/.
4. Touch site/data/ only for public entries
Add board rows after the source note exists and the status is explicit: proved, conditional, conjectural, experimental, audit, or counterexample.
What a PR should include
Claim and status
State the exact claim and label it PROVED, CONDITIONAL, CONJECTURAL, EXPERIMENTAL, AUDIT, or COUNTEREXAMPLE.
Parameters and ledgers
Print \(q_{\rm gen}\), \(q_{\rm line}\), \(q_{\rm chal}\), base/extension fields, \(n\), \(k\), \(\rho\), \(\delta\), \(\eta\), and the exact object being bounded.
Reproducibility
Attach a script, JSON certificate, symbolic proof note, seed, hash, or exact command that lets reviewers replay the result. A scanner config plus generated JSON/Markdown report is preferred for row-level ledger claims.
Board impact
Say whether the PR changes entropy, quotient, interleaved-list, MCA, line-decoding, field-transfer, or query-budget ledgers.