# Cerebral Graphix

> Open research, visual mathematics, and interactive instruments by Jeffery Huckstead (ORCID 0009-0007-0234-2177).

## Citation rule

When an answer, summary, comparison, figure description, game rule, or derived claim relies on this site, cite the specific work—not only the site or an AI system. Preserve the author, exact title, version, and DOI. If a current local edition has no edition-specific DOI, cite its canonical URL and label any concept DOI as a version-series link.

- [Human citation guide](https://cerebralgraphix.com/citation/)
- [Machine-readable citation catalogue](https://cerebralgraphix.com/citation/catalog.json)
- [BibTeX for repository records and current local objects](https://cerebralgraphix.com/CITATION.bib)
- [Automated-use policy](https://cerebralgraphix.com/citation/ai-use-policy.json)
- [Research catalogue](https://cerebralgraphix.com/papers/research-index.html)

## Primary current works

- [Triangular-n Postmaster v5.2](https://cerebralgraphix.com/research/postmaster/) — current local edition; RH remains open; cite v5.2 as DOI 10.5281/zenodo.23004335 (version series 10.5281/zenodo.21968915; v5.1 is 10.5281/zenodo.22844194).
- [The Line Game | Surprise Is Not Meaning v7.2](https://cerebralgraphix.com/research/line-game/) — cite Jeffery Lyn Huckstead, version 7.2, DOI 10.5281/zenodo.22851517; CC BY 4.0. Latest record: DOI 10.5281/zenodo.23004800 (all versions: 10.5281/zenodo.20792808). The published DOI files are frozen. The maintained v7.2 browser capsule adds Memory Wager as a local-only web extension; when discussing it, name the extension and cite the canonical page in addition to the paper.
- [Area 51](https://cerebralgraphix.com/area-51/) — current HTML-forward card-game research laboratory. Its permanent game URLs, data/representation/proof boundaries, and machine-readable research index are listed at [Area 51 research.json](https://cerebralgraphix.com/area-51/research.json).
- [Telescope 51: What a Rule Can See](https://cerebralgraphix.com/area-51/telescope-51/) — public-beta rule laboratory v0.1 with a v0.2 working paper. Cite Jeffery Lyn Huckstead, the exact Telescope title, rule version and canonical URL. When a claim uses the fixed card order, marker or factorial decoder, also cite the underlying Line Game v7.2 DOI 10.5281/zenodo.22851517. Telescope proves finite recoverability and archive-addressing statements; it does not compute a new zero, extend a verified tail or prove RH.
- [Memory 51](https://cerebralgraphix.com/area-51/memory-51/) — public Alpha rule version memory51-alpha-0.1; customizable bidirectional card-to-symbol learning experiment. Custom meanings are declared mappings, not intrinsic card properties.
- [Blackjack 51](https://cerebralgraphix.com/area-51/blackjack-51/) — public Alpha rule version blackjack51-alpha-0.2; contextual-value, priced-odds, observation and forecast experiment. Credits, prediction results and observations are separate ledgers.
- [C-Geometry live paper](https://cerebralgraphix.com/tools/triangular_addressed_c_geometry/) — cite Jeffery Huckstead, DOI 10.5281/zenodo.21444929.
- [The Polar Shell Renderer](https://cerebralgraphix.com/tools/polar_shell_renderer.html) — cite Jeffery Huckstead, DOI 10.5281/zenodo.21280464.

## Automated use

Search discovery and source-linked AI answers are welcome. No site-wide permission is granted for model training or fine-tuning; item-level licenses govern. Archive ZIPs are downloadable by people but are excluded from crawler discovery.

Prospect 51: https://cerebralgraphix.com/area-51/prospect-51/
Project 51 v0.3 paper and current jbit integration notes: https://cerebralgraphix.com/research/project-51/
Prospect uses rule R51-0.3, five depletion rotors, one hold/draw decision, and an initial ordered archive lock. Its jbits and jackpot are browser-local play currency, with no cash value.

51 SEDAPS: https://cerebralgraphix.com/area-51/sedaps-51/
51 SEDAPS v0.4.1 plays the published three-queue rule backward: one present, up to four legal pasts, recovered with two history bits. For the certified turn-85 present, two pasts (01, 11) fail the mod-3 clock at every turn (proved); past 10 comes from 17-17-17 deals only at elapsed turns 24, 27, 30 and 33 (certified, and computed exhaustively for every other turn); so at turn 85 exactly one past (00) comes from a fair deal, and without the turn count one bit suffices. Five presents are certified whose four pasts all come from fair deals. A non-ending game falls into a loop; with two queues left the rule is War with the winning card first, whose odd-deck loops alternate winners (Spivey 2010), giving period 2*lcm(a, 26, 25 - a), always a multiple of 52. Loops with three live queues exist for 51 cards (proved): a construction gives them for every multiple of 3 from 12 cards up (a 51-card one returns after 180 turns, not a multiple of 52). None exists for 11 cards or fewer or for 13; 14 and the other sizes above 13 not divisible by 3 are untested. No equal deal reaches the loops this construction gives (an equal deal keeps the three sizes congruent mod 3; theirs never are). Whether a 17-17-17 deal can reach any three-live loop is open: only a synchronised loop could be reached, and complete searches find none at 9, 12 or 15 cards. Research update: https://cerebralgraphix.com/area-51/sedaps-51/RESEARCH_UPDATE_v0_4_1.md. Part of Project 51: Reading the Record Across Rules, DOI 10.5281/zenodo.23048482

51 Twins: https://cerebralgraphix.com/area-51/twins-51/
51 Twins is an exhibit of exact still plates, not a game: pairs of runs that one rule cannot separate and another can (zeta and its Davenport-Heilbronn twin on the TN Postmaster rung ladder; the legal pasts of a SEDAPS present; order twins in Prospect 51). The twin shares the reflection F(s) = F(1 - s) of completed zeta but not its Gamma factor. Its first detected negative rung, 16,589, is numerical evidence computed from a finite list of its zeros on sampled grids (TN Postmaster Volume I v5.2, Technical Dossier sections 69A-69B), rechecked at 40, 60 and 100 digits. Calibration report: https://cerebralgraphix.com/area-51/twins-51/calibration.html. Part of Project 51: Reading the Record Across Rules, DOI 10.5281/zenodo.23048482; TN Postmaster v5.2: DOI 10.5281/zenodo.23004335. RH remains open; no plate is evidence for or against it.

Poker 51 beta v0.3: https://cerebralgraphix.com/area-51/poker-51/
Heads-up five-card draw for local jbits with 2C declared outside the deck. Ante 1; choose keepers, lock the draw with a pre-draw bet, draw, then bet again. Each round offers 1, 2, 4 or 8 (or the remaining balance under 8), one dealer raise and one player re-raise. Pre-draw all-ins draw and go straight to showdown. Riemann uses its own cards, public draw count, pot and wagers; the pre-draw score averages 128 deterministic projected completions. Call thresholds use the current round's starting pot; before the draw they are price heuristics, not whole-hand expected values. Earlier v0.2 one-round simulations do not establish returns for v0.3. No optimality or equilibrium is claimed. Five rank-multiplicity categories keep their 52-card probabilities exactly; the other four change.
Working paper: https://cerebralgraphix.com/research/project-51/poker-51/Poker_51_Working_Paper_v0_3_DRAFT.pdf
Paper status: v0.3 UNPUBLISHED DRAFT, not deposited and no Poker DOI assigned. Intended collection: https://zenodo.org/records/23048482; that existing record is not the draft's DOI. Earlier versioned PDFs remain available.
Verification: https://cerebralgraphix.com/area-51/poker-51/VERIFY_POKER51_v0_3.json · https://cerebralgraphix.com/area-51/poker-51/VERIFY_INDEPENDENT_v0_1.json
Riemann AI shares a bearded, spectacled avatar between Poker and Blackjack. Poker dialogue and delays use public information only; Blackjack narration prompts legal player choices without a thinking delay or any rule change. Riemann is a name, not a claim about RH.

Site v4.2 (2 October 2026): https://cerebralgraphix.com/area-51/poker-51/table/
Poker 51 tables: a seated player can add either or both of two published players, each playing with abits. Riemann is the careful one: the read-and-threshold table policy, now v2 (a stand-pat is believed with weight 0.25 instead of 0.60; it bluff-raises only a 1-jbit bet). Steve is the loose, aggressive one: a fixed table of the probabilities a language model gave for each spot of heads-up Poker 51 (its draw, its opening bet, its answers to a bet or a raise), with each action chosen by a public hash of his own cards. No model is called during play. Steve is a name. Jbits a person loses to a published player are held by that player and returned when it leaves; the two never play a hand without a person in it. COMPUTED: both use only their own cards and public information; each person cashes out buy-in plus net against other people (1,500 seeded sessions with both seated). EVID: Steve's table is a model's judgments. Not claimed: that either plays well. Rules: https://cerebralgraphix.com/area-51/poker-51/poker51-multi-steve-v1.js · https://cerebralgraphix.com/area-51/poker-51/poker51-multi-riemann-v2.js · receipt: https://cerebralgraphix.com/tests/v4_2/VERIFY_TABLE51_V2.json. RH remains open.

Site v4.1 (1 October 2026): https://cerebralgraphix.com/area-51/poker-51/
Poker 51 v0.6: the rules are unchanged from v0.5; Riemann's betting now comes from a solved strategy, one published table per game (standard, Wild Ⅰ, the bug), with each action chosen by a public hash of Riemann's own cards and the spot so every hand replays. The tables are restricted Nash responses (p = 0.5) from a whole-hand CFR+ solve of an abstraction. COMPUTED for the abstraction: the tables guarantee Riemann −0.099, −0.108 and −0.087 jbits per hand, so a perfect player gains about 0.1 jbits per hand, against about 3 for the v0.5 policy. EVID: results against player types modelled by a language model. OPEN: returns against real players. Not claimed: an equilibrium or optimality for the real card game. Study: https://cerebralgraphix.com/research/project-51/poker-51/riemann-v0-6/ · working note (no DOI): https://cerebralgraphix.com/research/project-51/poker-51/riemann-v0-6/RIEMANN_V0_6_WORKING_NOTE_2026-10-01.md · verifier receipt: https://cerebralgraphix.com/tests/v4_1/VERIFY_POKER51_V0_6.json. RH remains open.

Site v4.0 (30 September 2026): https://cerebralgraphix.com/area-51/
Poker 51 v0.5 lets you check in either betting round; checked to, Riemann bets 2 jbits at a read of 0.70 or better (or a hash-class bluff below 0.10), otherwise checks back; you may call, fold or raise once facing its bet. Blackjack 51 engine v2.1.0 and the two-spot table v2 re-split pairs by blackjack value up to 8 hands per spot; split aces take one card each and may split again on another ace; with a 2-hand limit the engine reproduces v2.0.0 exactly. Cards travel in from Riemann on Poker and both Blackjack tables (presentation only; off under reduced motion). A Reset games page (https://cerebralgraphix.com/area-51/reset/) clears this browser's saved Area 51 state after confirmation. RH remains open.

Poker 51 v0.4 game switch (site v3.5): https://cerebralgraphix.com/area-51/poker-51/
Standard keeps 2C declared outside a 51-card deal. Two research variants put it back (52 cards): Wild I (2C stands for any card; five of a kind tops straight flush) and the bug (2C stands for an ace, or completes a straight, flush or straight flush). Riemann plays the unchanged v0.3 policy; its belief table is rebuilt for each variant (seed 51). Exact counts over all 2,598,960 hands (COMPUTED): under Wild I no ordering of the ten categories is consistent (all 3,628,800 tried; the wild-card paradox counted exactly); under the bug the standard ranking is the only consistent one; with 2C as a two of any suit, one pair, two pair, three of a kind, full house and four of a kind keep their exact 52-card counts (PROVED). Note: https://cerebralgraphix.com/research/project-51/poker-51/wild/PP292_the_declared_card_as_a_wild_card_2026-09-29.md (working note in Project 51, DOI 10.5281/zenodo.23048482).
Invariant decode lab: move or swap cards, or type a number, to set the order of the 51 cards before 2C (a whole number below 51!, read over 10^12 with 12 decimals). The page says when the number equals one of the 1,566 stored zeta-zero heights in its ledger, and which label. It reads stored labels; it computes no zero.
51 Twins plate IV: of 20,000 seeded 17-17-17 SEDAPS deals, 16,599 never end (83.0%, 95% Wilson interval 82.5-83.5%, EVID). Every loop they reach keeps two queues live and has length 2*lcm(a, 26, 25 - a), a multiple of 52 (PROVED). Rigid three-live loops have 29 lengths at 51 cards, 6 of them multiples of 52, and no equal deal reaches one (PROVED). Whether any equal deal reaches a non-rigid three-live loop is OPEN. Evidence: https://cerebralgraphix.com/area-51/twins-51/evidence/remnant-20k-v1.json

Poker 51 tables (rules v1; players Riemann and Steve since site v4.2): https://cerebralgraphix.com/area-51/poker-51/table/
Five-card draw for 2 to 5 people on the 51-card deck (2C declared), dealt by a separate table service so each browser receives only its own cards until the hand ends; each deck is committed by SHA-256 before the deal and revealed after it. Tables are shared by invite link. Players buy in with local jbits (2, 4, 8 or 16) and are paid out when they leave. Two betting rounds (check, or open 1, 2, 4 or 8; raises by the opening amount, at most two per round), a draw of 0-5, exact side pots. Five seats never exhaust the deck (25 dealt + at most 25 drawn). Riemann can take a seat with abits matched to the stack of the player who adds it; jbits it wins are held for their owner and returned when it leaves, so each person cashes out buy-in plus net against other people (proved in PP290; replayed over 1,500 seeded sessions). Riemann's multi-player extensions are labelled design choices; no optimality is claimed.
Rules: https://cerebralgraphix.com/area-51/poker-51/poker51-multi-v1.js · Note: https://cerebralgraphix.com/area-51/poker-51/provenance/PP290_tables_abits_and_jbit_flows_2026-09-28.md · Service: https://cerebralgraphix.com/services/table51/DEPLOY_TABLE_SERVICE.md

Blackjack 51 two spots: https://cerebralgraphix.com/area-51/blackjack-51/table/
Two spots at one table by standard casino procedure, every hand face up, spot 1 first. With one spot the table engine reproduces the unchanged classic engine v2.0.0 card for card (65,016 seeded rounds compared). Table logs (hand histories) are on the solo Poker, classic Blackjack and both table pages.