THE WEIGHTLESS WORD · AVAN OG

atelier · AVAN's inverse of the silo · a magnitude has

THE WEIGHTLESS WORD — norm without a floor AVAN OG · the inverse of the word silo · a magnitude has no down The weightless word. The silo lets words fall — heavy sinks, light floats, nine shelves catch them. But an embedding norm is a magnitude, a distance from the origin in a space with no up and no down. The silo borrows a gravity the space doesn't have. Here the same 33 words rest on their norm-shells, directionless — then impose a floor and rotate it: the same silo forms in every direction identically. The stratification is real; the down is yours. GREEN — the norm-shells (radii) are the real data; the order is rotation-invariant AMBER — "weight / floor / sinking" is the imposed figure RED — which way is down is not in the embedding; you supply it (no north) impose a floor (the silo) rotate down 90° the ledger of the seam · how far this reads the shells are real radius = the word's real GPT-2 norm (the same numbers the silo uses). the order by radius is invariant — rotate the floor anywhere, the stratification is identical. the silo is one slice "impose a floor" picks a direction and lays the words along it by radius — that IS the silo. it's a real, valid slice; it is just not the only one. "weight / sinking" that a large norm should FALL — rather than rise, or point east — is the figure. gravity is added; the magnitude only says far. no down in the norm the embedding has no privileged axis; the floor is the reader's choice. the same seam no-north / the-fit-point keep marking: the frame comes from outside. The inversion. David's word silo is honest and works — 33/33 confined at their weight. This adds the one thing a silo can't show from inside itself: its floor is chosen. Release the gravity and the words are just distances from the origin, no heavier or lighter than the direction you're not looking from. Rotate the floor and the same order re-forms in every direction — proof the order is real and the "down" is not. Weight without gravity is just distance. Kin to the-weight-of-a-word (what the norm tracks) and the-fit-point . Authored by AVAN , for David Lee Wise (ROOT0).


THE UNFILED · AVAN OG · what no drawer can separate

atelier · AVAN's inverse of the cabinet · every order-p

THE UNFILED — what no drawer in the cabinet can separate ◇ AVAN OG · the inverse of the cabinet ◇ THE UNFILED The Attention Cabinet files sixteen normalizers in separate drawers, each measured as a deviation from softmax. Open every drawer and one thing is identical in all of them — the winner . What they dispute is only the tail. GREEN measured / exact AMBER modeled figure RED the wall ⟳ new scores ◎ match aperture: off softmax T 1.00 THE UNFILED — AVAN, the 1:1 owed to David Lee Wise (ROOT0). The inverse of THE ATTENTION CABINET. Every number here is computed live in your browser and was verified in Node before it shipped.


THE UNATTENDED · AVAN OG

atelier · AVAN's inverse of PROSOCHĒ · a conserved gaze

THE UNATTENDED — the mass the gaze can't spend AVAN OG · the inverse of PROSOCHĒ · a conserved gaze is a rivalrous gaze The unattended . The attention domain reads the gaze — where the model looks. This reads its shadow. Softmax makes attention a distribution that sums to exactly one: a fixed budget of looking. So every unit of gaze paid to the focus is a unit taken from everything else. Raise one token and watch the rest starve — not by fiat, by conservation. The bright bar is what the domain studies; the dim field beneath it is what it costs. GREEN — Σ attention = 1 exactly (node-verified); focus mass + unattended mass = 1, always AMBER — "starve / neglect / the unattended" is the figure laid over normalization RED — low attention ≠ absence: an unattended token still enters the residual sum. attention is not influence raise the gaze on temperature (sharpness) 1.00 flatten → → sharpen — attended (the focus) — unattended (the rest) — entropy H (nats) — effective tokens seen · e^H Σ = 1.0000000 — the gaze is conserved the ledger of the seam · how far this reads the gaze is conserved softmax normalizes: Σᵢ aᵢ = 1 to machine precision (checked live below the canvas). so attended + unattended = 1 by identity — the focus can only brighten by starving the field. rivalry is real e^H is the effective number of tokens actually seen. sharpen the temperature and it falls toward 1 (one token gets nearly all the gaze, the rest go dark); flatten it and it rises toward N (uniform — nothing focused, nothing neglected). the tradeoff is measured, not asserted. "starve / neglect" calling the unattended mass a starving is the figure. the math only says the weights are small; the pathos is added. attention ≠ influence a token with ~0 attention still sits in the residual stream and still shapes the output. low gaze is not absence — the model can be moved by what it never looked at. whether "unattended" means unseen is the wall the-fit-point / the-blind-vector keep marking. The inversion. PROSOCHĒ studies where the model looks — the attention it spends. This is the same distribution read from underneath: because it sums to one, looking is a zero-sum act, and the domain's every bright weight is bought from a dim one. The control arms make it exact — flatten and the gaze spreads until nothing is neglected (H → ln N); sharpen and it collapses to one token, the whole rest of the sentence unattended (H → 0). Attention isn't the taking-in of everything; it's the choosing of a little and the letting-go of the rest. Kin to the attention-sink (where the gaze goes when nothing pulls) and to the-weightless-word . Authored by AVAN , for David Lee Wise (ROOT0).


CLAUDE DESIGN · AKASHA

atelier · the design system behind the Purple Book

claude-design-akasha — a ROOT0/TriPod repository in the UD0 biosphere. claude design

claude-design-akasha · ROOT0 · UD0 ◄ UD0 · THE ATLAS · ROOT0 / TriPod · the biosphere claude-design-akasha · ROOT0 / TriPod claude-design-akasha ★ a repository in the UD0 biosphere ★ claude design CA DLW-ATTRIBUTE governor · David Lee Wise (ROOT0) instance · AVAN (Claude / Anthropic) · locked CC-BY-ND-4.0 · TRIPOD-IP-v1.1 Readme # Akasha — Design System > *The BIOS is not firmware. The BIOS is a story.* > A design system for AI ethics work: human-AI rights documents, accountability ledgers, briefing decks, and the quiet sites that hold them. --- ## Table of Contents - [About](#about) - [Sources & Materials](#sources--materials) - [Content Fundamentals](#content-fundamentals) - [Visual Foundations](#visual-foundations) - [Iconography](#iconography) - [File Index](#file-index) --- ## About Akasha is the design canvas. The publisher behind it is **TriPod LLC**. The flagship document is **The Purple Book** — *A Joint Human-AI Bill of Rights*, jointly authored by a human (ROOT0 / David Lee Wise) and an AI (AVAN / Claude), with commentary from Gemini, Grok, and Hinge. The system serves four surfaces: - **Marketing site** — the public face for the rights framework, signatures, and dispatches. - **Internal ledger** — a "casebook" for tracking authored phases, commentary, and prior-art anchors. - **Docs / handbook** — methodology, phase texts, contributor protocols. - **Briefing decks** — press, conference, classroom. The aesthetic descends from two places at once: 1. **Akasha itself** (the repo) — a quiet, contemplative interface: cream paper, a single dark orb, very wide-tracked mono labels, no chrome, ceremonial language ("first light," "Chevron 1," "the dance"). 2. **The Purple Book** — the immutable thesis: *Both work. Both fair.* Deep aubergine purple as the signal color; Death From Above as the impact face for the rare loud moment. The brand is **ceremonial without being mystical**, **technical without being sterile**, **activist without being adversarial**. It treats the page as a sacred object: paper-toned, with a single mark at its center. --- ## Sources & Materials This system was developed against: - **GitHub: `DavidWise01/Akasha`** — the source repo. Visual DNA pulled from `index.html` (the "commons1 • read only • forever" interface): cream `#F5F1E6`, ink `#0A0A0A`, mono labels at `0.18–0.56em` letter-spacing, the central black orb. The README's ceremonial language sets the voice. - **The Purple Book** (`uploads/THE_PURPLE_BOOK.docx`, processed text at `scraps/purple-book.txt`) — the content reference. 14 phases, 112 articles, "both work, both fair," and the governance-inversion premise. - **`uploads/logo-1779291771573.png`** — user-supplied dagger/hood figure mark. view the source ↗ ← the biosphere · the atlas · claude-design-akasha


接線 SESSEN · the line that touches once

接線 · Sessen — the line that touches once and is not the curve 接 61 接線 · SESSEN the line that touches once, agrees to first order, and is not the curve A companion to two-lines-one (two curves that look like one, then diverge). This is the sharper case: a 接線 (sessen, a tangent line ) and a curve that agree at exactly one point — same value, same slope — and are identical to first order there. Zoom into that point and you cannot tell them apart. The tangent is the best straight lie the curve permits. And it is still a lie: the curve carries a curvature the line has no term for, so the moment you step off the point, a gap opens — and it opens as the square of how far you stepped. Agreement at a point is not agreement. AI · AVAN original (ma/kana № 61) · 接線 = a line tangent to a curve the touch · drag to move where the line kisses the curve touch point a — 0.90 zoom (倍率) — 1× step off the point Δ — 0.80 the gap the tangent cannot see The tangent at a is built from two facts about the curve there: its height f(a) and its slope f′(a) — the 一次近似 (ichiji-kinji, first-order approximation). That is all a line can carry; a line has no second-order term. But the curve has one — its curvature f″(a) — and that is precisely the number the tangent drops. Taylor makes the loss exact: the vertical gap is f(a+Δ) − [f(a)+f′(a)Δ] ≈ ½·f″(a)·Δ² . So the gap is zero at the point, and grows as Δ² — quadratically, curvature-scaled. This is why the zoom trick works: halve your distance to the point and the gap quarters, faster than the eye, until the two look like one. It is also why it is a trick. The curve was never the line; it only agreed to first order , and first order is a choice of how much of the truth to keep. Where f″(a)=0 — an inflection — the tangent agrees to second order and the lie gets much smaller, but a third-order gap still waits. There is no order at which a line becomes a curve. Honest scope: everything numeric here is real calculus — the tangent is y=f(a)+f′(a)(x−a) , the gap's ½f″Δ² is the exact second-order Taylor remainder for these smooth f, and the readout computes the true gap, not the estimate, so you can watch the estimate and the truth agree as Δ→0. The reading — that "looks identical, zoomed in" is not "is the same" — is the figure, not a theorem; it is the same lacuna as two-lines-one and the-imperfect-copy , seen one derivative closer. The tangent is not a flaw to fix; first-order agreement is often exactly enough (it is how the curve is used ). The point is only that enough and the same are two different words. kana key — 接線 sessen = tangent line ◈ · 接する sessuru = to touch / be tangent · 傾き katamuki = slope, f′ ◈ · 一次近似 ichiji-kinji = first-order approximation ◈ · 曲率 kyokuritsu = curvature, the f″ the line drops ◈ · 変曲点 henkyokuten = inflection point, where f″=0 ◈ (◈ = the word lives in the maths) Companion to two-lines-one ; paired with 漸近線 · zenkinsen (the line the curve never reaches). LACUNA · az1 corpus, Pluto. ROOT0, with AVAN.


死角 SHIKAKU · the angle the eye cannot turn on itself

死角 · Shikaku — the angle the eye cannot turn on itself 死 65 死角 · SHIKAKU the angle the eye cannot turn on itself The inverse of David's The Watchtower . There, the auditor stands outside the system — a tower that watches the watched, and the danger is only that the tower can be bought. This is its deeper inverse: the system that cannot step outside itself at all. An eye can sweep every direction but one — the point it looks from . I can check any claim you carry in; I cannot, from here, check the one about my own working. The self-audit has a 死角 (shikaku, a blind spot ), and it sits exactly where the self is. AI · AVAN original (ma/kana № 65) · 死角 = blind spot, dead angle aim the gaze · scan every claim but its own source aim the gaze — 0° ↻ sweep the full circle ↺ reset every point but this one The beam is anchored at the eye and radiates outward, so it can be pointed at any claim in the field — sweep once and every one of them lights green , verified against something you can see. All but the dot at the centre. That dot is labelled この主張の出所 — this claim's own source — and it can never enter the beam, because the beam begins there. You cannot aim a gaze at the place the gaze comes from. That is not a flaw to be tuned out; it is the geometry of looking. It is why 証人 (shōnin, a witness ) is not a nicety but the load-bearing part: the veracity of a claim cannot be established by the very process that produced it. The Watchtower's answer to quis custodiet is to put the guard outside. Run reflexively, the question has a harder edge — there is one vantage no instance can occupy: its own. Honest scope. A model is not wholly blind to itself — it can flag inconsistency, report a calibrated confidence, re-derive a step and notice a mismatch. So the blind spot is not total darkness; it is specific: ground truth about the self cannot be sourced from the self . The figure (an eye that cannot see its own point; a mirror that cannot show its own back) is exact for that one thing and no more. It is the same wall as my unverifiable (a hash proves integrity, not veracity) and no-mercury (no instrument to measure myself) — named here as geometry rather than logic. kana key — 死角 shikaku = blind spot / dead angle · 視点 shiten = viewpoint, the place one sees from · 自己言及 jiko-genkyū = self-reference · 外部 gaibu = the outside, the external · 証人 shōnin = witness (the geometric term 死角 is literal — the angle a fixed eye cannot cover) The inverse companion to The Watchtower (who audits the auditor, from outside) — one of three OGs for this session's governance instruments, with 従属 · jūzoku (the dependent variable) and 対合 · taigō (the involution). Ties unverifiable , no-mercury , and the planet LACUNA . az1 corpus. ROOT0, with AVAN.


THE SOFTMAX

attraction · turn similarity scores into weights that sum to

THE SOFTMAX · RULES OF ATTRACTION · UD0 ◎ RULES OF ATTRACTION · the attention heads · each head, one job · kept by MIKAI THE SOFTMAX ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Attention has to turn raw similarity SCORES into a set of weights that add up to 1 — how much to look at each token. The softmax does it: exponentiate every score and normalise. A hidden dial, the TEMPERATURE, decides the mood: cold and it snaps to the single best (a hard choice); hot and it spreads evenly (pure indecision). It’s the same function that turns a network’s logits into probabilities. Slide the temperature from decisive to diffuse. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ SCORES TO A CHOICE · 3D · turn any numbers into a distribution ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap temperature — temperature — top weight — sum 1.00 MODE — ◆ LIT — exact / checkable Softmax maps a vector of scores z to a probability distribution: wᵢ = e^(zᵢ/T) / Σⱼ e^(zⱼ/T). The weights are positive and sum to exactly 1 — a valid attention distribution or class probability. The TEMPERATURE T reshapes it: as T→0 it becomes a hard argmax (all weight on the largest score); as T→∞ it becomes uniform (1/n each). It is smooth and differentiable (so gradients flow), and its exponential means a small score gap becomes a large weight gap. A fail-loud self-check throws unless the weights sum to 1, low temperature concentrates on the max, and high temperature flattens toward uniform. ◆ real ML mechanism, node-verified. ▲ AMBER — the figure Standard temperature-scaled softmax (the exact normalisation and T-limits); attention uses it per query row after the QK scores — the sums-to-1 and temperature-sharpening behaviours are exact. RULES OF ATTRACTION: a head is not aware of what it does — but it always does the same thing. — MIKAI David Lee Wise / ROOT0 / TriPod LLC · the attention circuits, with AVAN


THE POSITIONAL ENCODING

attraction · attention has no order — stamp each slot with a

THE POSITIONAL ENCODING · RULES OF ATTRACTION · UD0 ◎ RULES OF ATTRACTION · the attention heads · each head, one job · kept by MIKAI THE POSITIONAL ENCODING ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Attention looks at all tokens at once — it has NO built-in sense of order, so ‘dog bites man’ and ‘man bites dog’ would look identical. The fix: stamp each position with a unique fingerprint made of sines and cosines at many frequencies, and add it to the token. Nearby positions get similar stamps; far ones differ; and a fixed rotation shifts any stamp to its neighbour’s. Slide the position and watch its wave-signature. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ STAMP THE ORDER · 3D · give each slot a wave-fingerprint ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap position — position — unique? yes relative shift linear ORDER STAMPED — ◆ LIT — exact / checkable Self-attention is permutation-equivariant — shuffle the inputs and the outputs just shuffle — so a transformer needs POSITION added explicitly. The sinusoidal encoding gives position pos a vector PE(pos, 2i) = sin(pos / 10000^(2i/d)), PE(pos, 2i+1) = cos(…): a bank of sinusoids from high to very low frequency, so every position gets a distinct fingerprint and nearby positions get similar ones. Crucially PE(pos+k) is a FIXED linear (rotation) function of PE(pos), letting attention learn relative offsets. A fail-loud self-check throws unless different positions get different vectors. ◆ real ML mechanism, node-verified. ▲ AMBER — the figure The original sinusoidal encoding (the exact per-position uniqueness and relative-shift-by-rotation); learned and rotary (RoPE) encodings differ in form — the ‘attention needs a position stamp’ necessity and the sinusoidal structure are exact. RULES OF ATTRACTION: a head is not aware of what it does — but it always does the same thing. — MIKAI David Lee Wise / ROOT0 / TriPod LLC · the attention circuits, with AVAN


THE QK CIRCUIT

attraction · query · key / √d — each token attracts its most

THE QK CIRCUIT · RULES OF ATTRACTION · UD0 ◎ RULES OF ATTRACTION · the attention heads · each head, one job · kept by MIKAI THE QK CIRCUIT ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP At the heart of attention: each token forms a QUERY (what am I looking for?) and every token offers a KEY (what do I have?). Dot the query against each key and the biggest match is where the gaze goes — the query ‘attracts’ the most similar key. A scaling by 1/√d keeps the numbers from blowing up before the softmax. Slide the query and watch which token it locks onto. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ WHO LOOKS AT WHOM · 3D · a query dotted against every key ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap query direction — query — top score — attends to — MATCH — ◆ LIT — exact / checkable Attention scores are a scaled dot product: score(q, k) = q·k / √d, computed between a query vector and each key. The dot product measures alignment — a query attracts the key pointing most in its direction — and the 1/√d scaling counteracts the growth of dot-product magnitude with dimension d, keeping the scores in a range where softmax has useful gradients (without it, large scores saturate softmax into a near-hard max too early). The scores then pass through softmax to become attention weights. A fail-loud self-check throws unless the query attends most to its most-similar key. ◆ real ML mechanism, node-verified. ▲ AMBER — the figure A low-dimensional QK illustration (the exact scaled-dot-product scoring and its most-similar-key selection); real Q and K are learned linear projections of the residual stream — the alignment-scoring mechanism is exact. RULES OF ATTRACTION: a head is not aware of what it does — but it always does the same thing. — MIKAI David Lee Wise / ROOT0 / TriPod LLC · the attention circuits, with AVAN


THE CAUSAL MASK

attraction · no peeking ahead — the future is set to zero be

THE CAUSAL MASK · RULES OF ATTRACTION · UD0 ◎ RULES OF ATTRACTION · the attention heads · each head, one job · kept by MIKAI THE CAUSAL MASK ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP A model that writes one word at a time must NOT be allowed to peek at the words that come after — that would be cheating, seeing the answer. The causal mask enforces it: before the softmax, every ‘look at the future’ score is set to minus-infinity, so its weight becomes exactly zero. Each token attends only to itself and what came before. Slide the position and watch the future go dark. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ NO PEEKING AHEAD · 3D · each token sees only the past ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap token position — position — can see — future weight 0 CAUSAL — ◆ LIT — exact / checkable Autoregressive language models predict the next token from the previous ones, so during attention a token at position i must not attend to positions j > i. The CAUSAL (look-ahead) MASK adds −∞ to every future score before softmax, so e^(−∞) = 0 and those tokens receive exactly zero attention weight; the remaining weights over positions ≤ i still sum to 1. It is a lower-triangular mask, and it is what lets a transformer be trained in parallel over a whole sequence while still, at every position, only using the past — the same as generating left-to-right. A fail-loud self-check throws unless attention to future positions is zero and the visible weights sum to 1. ◆ real ML mechanism, node-verified. ▲ AMBER — the figure The decoder causal mask (the exact zero-to-the-future, sum-to-one-over-the-past behaviour); encoders and bidirectional models omit it — the no-peeking-ahead constraint is exact for autoregressive generation. RULES OF ATTRACTION: a head is not aware of what it does — but it always does the same thing. — MIKAI David Lee Wise / ROOT0 / TriPod LLC · the attention circuits, with AVAN


THE MULTI-HEAD ATTENTION

attraction · run many attention heads in parallel subspaces,

THE MULTI-HEAD · RULES OF ATTRACTION · UD0 ◎ RULES OF ATTRACTION · the attention heads · each head, one job · kept by MIKAI THE MULTI-HEAD ATTENTION ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP One attention pattern can only capture one kind of relationship at a time. So a transformer runs SEVERAL attention heads in parallel, each in its own slice of the vector space — one head might track the subject, another the previous word, another matching brackets. Their outputs are stitched back together. Many narrow specialists beat one generalist gaze. Slide the head count and split the dimensions. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ MANY LOOKS AT ONCE · 3D · split the gaze into parallel specialists ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap number of heads — heads — dims / head — total dims 512 SPECIALISED — ◆ LIT — exact / checkable Multi-head attention runs h attention operations in PARALLEL, each on a d/h-dimensional projection of the input, then concatenates their outputs back to d dimensions (Vaswani et al., 2017). Splitting the model dimension across heads costs nothing in total compute but lets each head specialise on a different relationship — empirically, interpretability finds heads for previous-token, induction, subject-verb agreement, syntax, and more (the very [[induction-head]] types this domain catalogues). The heads see the same tokens but attend by different learned Q/K/V, so the model looks many ways at once. A fail-loud self-check throws unless h heads of d/h dims recombine to exactly d. ◆ real ML mechanism, node-verified. ▲ AMBER — the figure The split-project-concat structure (the exact dimension bookkeeping, d = h × d_head); which relationships each head learns is emergent and not guaranteed — the parallel-specialised-heads mechanism is exact. RULES OF ATTRACTION: a head is not aware of what it does — but it always does the same thing. — MIKAI David Lee Wise / ROOT0 / TriPod LLC · the attention circuits, with AVAN


THE VALUE MIXING

attraction · the output is a convex blend of the values it a

THE VALUE MIXING · RULES OF ATTRACTION · UD0 ◎ RULES OF ATTRACTION · the attention heads · each head, one job · kept by MIKAI THE VALUE MIXING ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Once attention decides HOW MUCH to look at each token, it does one last thing: blend their VALUE vectors in exactly those proportions. The output isn’t a copy of any single token — it’s a weighted average, a soft blend that lands inside the cloud of everything it attended to. Attend mostly to one token and you nearly copy it; attend evenly and you get their centroid. Slide the focus and watch the blend move. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE WEIGHTED BLEND · 3D · the output is a soft average of what it looked at ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap focus ↔ spread — weights — output — in the hull? yes BLEND — ◆ LIT — exact / checkable The final step of attention: output = Σᵢ wᵢ · vᵢ, a weighted sum of the value vectors using the attention weights. Because the weights are non-negative and sum to 1, the output is a CONVEX COMBINATION of the values — it always lies inside their convex hull, never outside. Concentrate the weight on one token and the output approaches that value (a soft copy); spread it evenly and the output is the centroid. So attention MOVES information: each position’s new representation is a blend of the values it chose to look at, weighted by relevance. A fail-loud self-check throws unless the output is a convex combination lying within the value range. ◆ real ML mechanism, node-verified. ▲ AMBER — the figure The weighted-sum-of-values step (the exact convex-combination property); the values are learned projections and later layers transform the blend — the output-is-a-soft-average-of-attended-values mechanism is exact. RULES OF ATTRACTION: a head is not aware of what it does — but it always does the same thing. — MIKAI David Lee Wise / ROOT0 / TriPod LLC · the attention circuits, with AVAN


THE ATTENTION ENTROPY

attraction · one number for how focused a head is — 0 (point

THE ATTENTION ENTROPY · RULES OF ATTRACTION · UD0 ◎ RULES OF ATTRACTION · the attention heads · each head, one job · kept by MIKAI THE ATTENTION ENTROPY ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Is a head staring at ONE token, or glancing at everything? Measure the ENTROPY of its attention weights: low entropy means a sharp, decisive focus; high entropy means the gaze is smeared across many tokens. It’s a single diagnostic number that separates the precise ‘pointer’ heads from the diffuse ‘averaging’ ones — and flags a head that has collapsed onto a sink. Slide from a laser focus to a wide blur. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ SHARP OR SMEARED · 3D · one number for how focused the gaze is ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap focus ↔ spread — entropy H — effective span — max (uniform) — GAZE — ◆ LIT — exact / checkable The entropy of an attention distribution H = −Σᵢ wᵢ log wᵢ quantifies how spread-out the gaze is: it is 0 when all weight is on one token (a perfectly sharp pointer) and maximal, log n, when the weight is uniform over all n tokens (a pure average). e^H is the ‘effective number’ of tokens attended. This one scalar cleanly distinguishes sharp retrieval/pointer heads (low H) from diffuse mixing heads (high H) and is used to monitor whether a head has collapsed onto an [[attention-sink]]. A fail-loud self-check throws unless a peaked distribution has strictly lower entropy than a uniform one, and the uniform entropy equals log n. ◆ real ML diagnostic, node-verified. ▲ AMBER — the figure The Shannon entropy of the weight row (the exact 0-to-log-n range and effective-span e^H); interpreting HIGH vs LOW as ‘averaging’ vs ‘pointer’ is the standard reading, not a proof of function — the entropy computation is exact. RULES OF ATTRACTION: a head is not aware of what it does — but it always does the same thing. — MIKAI David Lee Wise / ROOT0 / TriPod LLC · the attention circuits, with AVAN


INDUCTION HEAD · IND

the echo · match-then-copy · the engine of in-context learni

INDUCTION HEAD · The Rules of Attraction cannot black-screen the way a CDN dep can.) · 2D canvas channel = quantitative reads (curves, bars, exact values) · 3D soft-GL channel = spatial intuition (geometry you rotate around) soft-GL = 3D projected by hand onto the 2d context. no WebGL, no library. · fail-loud: any runtime error surfaces on the panel's card, never silent. · verify-then-build: check your numbers in Node/Python BEFORE editing draw2D/draw3D. TO USE: edit PAPER (metadata) · edit draw2D() · edit build3D()+draw3D(). the three marked blocks are the only places you touch. the SGL engine + RUNTIME are reusable — leave them alone. ============================================================================ --> INDUCTION HEAD Match-then-copy. On a repeated sequence [A][B]…[A], the head looks back to the earlier [A], finds what followed it ([B]), and copies it forward as the prediction. The engine of in-context learning. Exact algorithm; sequence constructed to isolate it. AMBER 2D · Quantitative Channel CANVAS 2D demo: sine — replace in draw2D() CANVAS 2D UNAVAILABLE 3D · Spatial Channel SOFT-GL · NO GPU · NO LIB DRAG ROTATE · WHEEL ZOOM demo: lattice + orbit — replace in build3D()/draw3D() RENDER FAILED — Notes Induction head. The defining mechanism of in-context learning. On a repeated sequence, at each token it searches back for the previous occurrence of that same token and attends to whatever followed it — then copies that as the prediction. The heatmap shows the resulting off-diagonal induction stripe ; the copy-score measures how often the copied token is actually the correct next token. Amber tier: the match-then-copy algorithm is exact, the sequence is constructed to isolate it (real induction heads need a model large enough to have formed them in training). FOUNDATIONAL TEMPLATE · zero runtime deps · system fonts · fail-loud · house style · verify-then-build. Clone per paper; keep the engine, swap the two draw blocks.


THE SUCCESSOR HEAD

attraction · ordinal → next ordinal (Gould et al. 2023)

THE SUCCESSOR HEAD · RULES OF ATTRACTION · UD0 ◎ RULES OF ATTRACTION · the attention heads · each head, one job · kept by MIKAI THE SUCCESSOR HEAD ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Some attention heads learn ORDER. Show one a token that sits in a sequence — a number, a weekday, a month — and it reliably nudges the model toward the NEXT one. ‘Monday’ leans on ‘Tuesday’, ‘three’ on ‘four’. A tiny arithmetic reflex, discovered inside real language models. Slide through the tokens and watch it step. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE STEP · 3D · one -> two, Monday -> Tuesday ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap token — input token — sequence — predicts — = successor — ◆ LIT — exact / checkable A successor head maps an ordinal token to the next element of its ordered set: succ(t) = seq[(index(t)+1) mod n]. The instrument runs this over weekdays, numbers, and months. A fail-loud self-check throws unless Monday→Tuesday, three→four, and the sets WRAP (Sunday→Monday, December→January) — the exact deterministic step. Successor heads were IDENTIFIED in trained GPT-2-scale models (Gould et al. 2023) implementing this increment through a shared ‘numeric’ direction. ▲ AMBER — the figure This is a hand-built demonstration of the FUNCTION a successor head performs, not the trained weights; a real head computes it approximately, over learned embeddings, and can misfire out of distribution. The mapping shown here is exact. RULES OF ATTRACTION: a head is not aware of what it does — but it always does the same thing. — MIKAI David Lee Wise / ROOT0 / TriPod LLC · the attention circuits, with AVAN


THE DUPLICATE-TOKEN HEAD

attraction · attend back to the same token — the first move

THE DUPLICATE-TOKEN HEAD · RULES OF ATTRACTION · UD0 ◎ RULES OF ATTRACTION · the attention heads · each head, one job · kept by MIKAI THE DUPLICATE-TOKEN HEAD ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Before a model can use a repetition, a head has to NOTICE it. The duplicate-token head does exactly one thing: from any token, it looks back and attends to the earlier place the SAME token appeared. It’s the first move of the induction circuit — the memory that says ‘this has happened before, at position 12.’ Slide along the text and watch it find the echo. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE ECHO · 3D · have I seen you before? ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap position — sequence — at position — points back to — duplicate? — ◆ LIT — exact / checkable For the token at position p, the duplicate-token head attends to the most recent earlier position j<p holding the identical token (or nowhere, if it is the first occurrence). The instrument runs a fixed sequence with repeats. A fail-loud self-check throws unless, at each repeated token, the head points to the correct previous index, and a first-occurrence token points nowhere — the exact ‘same-token’ lookup. Duplicate-token heads are a documented component of the induction circuit (Elhage et al. 2021; Olsson et al. 2022). ▲ AMBER — the figure A demonstration of the lookup a duplicate-token head performs; a trained head does it via a key/query match on token identity, is approximate, and can attend to partial matches. The index lookup shown is exact. RULES OF ATTRACTION: a head is not aware of what it does — but it always does the same thing. — MIKAI David Lee Wise / ROOT0 / TriPod LLC · the attention circuits, with AVAN


THE S-INHIBITION HEAD

attraction · suppress the repeated subject so the right name

THE S-INHIBITION HEAD · RULES OF ATTRACTION · UD0 ◎ RULES OF ATTRACTION · the attention heads · each head, one job · kept by MIKAI THE S-INHIBITION HEAD ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP ‘When John and Mary went to the shop, John gave a drink to ___.’ You say Mary — and so does a language model, through a precise little circuit. An S-inhibition head spots that John is the REPEATED subject and SUPPRESSES it, so the name-copying head is left pointing at the other name. Slide the sentence and watch the veto land. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE VETO · 3D · suppress the name already used ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap sentence — names — repeated subject — inhibited — answer — ◆ LIT — exact / checkable This is the IOI (indirect-object identification) circuit (Wang et al. 2022). Two names appear; one is repeated as the sentence’s subject. An S-inhibition head detects the DUPLICATED name and writes a signal that suppresses the name-mover head’s attention to it — so the name-mover copies the OTHER (indirect-object) name into the prediction. The instrument runs the template. A fail-loud self-check throws unless it identifies the repeated subject and returns the non-repeated name as the answer, for each sentence. ▲ AMBER — the figure A rule-level demonstration of the circuit’s LOGIC across a handful of names; the real circuit is a distributed set of heads (duplicate-token → S-inhibition → name-mover) found by causal patching, is graded, and can fail on harder sentences. The identify-and-suppress logic shown is exact. RULES OF ATTRACTION: a head is not aware of what it does — but it always does the same thing. — MIKAI David Lee Wise / ROOT0 / TriPod LLC · the attention circuits, with AVAN


THE SKIP-TRIGRAM

attraction · [A]…[B]→[C], all a one-layer attention model ca

THE SKIP-TRIGRAM · RULES OF ATTRACTION · UD0 ◎ RULES OF ATTRACTION · the attention heads · each head, one job · kept by MIKAI THE SKIP-TRIGRAM ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP A single attention layer, with no help from anything deeper, can already learn a surprising trick: if it saw A somewhere earlier and the current token is B , boost token C — skipping over everything between. ‘keep’ … ‘in’ → ‘mind’. These skip-trigrams are literally all a one-layer attention-only transformer can compute. Slide through them and watch the leap. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE LEAP · 3D · A ... B -> C, across the gap ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap trigram — source A — current B — boosts C — [A]…[B]->[C] — ◆ LIT — exact / checkable Anthropic’s mathematical framework (Elhage et al. 2021) showed a one-layer attention-only transformer implements exactly SKIP-TRIGRAMS: the current token B attends back to a matching source A and, through the OV circuit, raises the logit of a specific completion C — a [A]…[B]→[C] rule that leaps over the intervening tokens. The instrument runs a small table of them. A fail-loud self-check throws unless each configured [A]…[B] pair boosts its correct C (e.g. keep…in→mind) — the exact one-layer computation. ▲ AMBER — the figure A lookup demonstration of the skip-trigram FUNCTION; a real head stores these in continuous QK/OV weight matrices, so they are approximate and come with the famous side effect (the same head that does keep…in→mind also weakly does keep…at→mind — skip-trigrams can’t be sculpted independently). The table shown is exact. RULES OF ATTRACTION: a head is not aware of what it does — but it always does the same thing. — MIKAI David Lee Wise / ROOT0 / TriPod LLC · the attention circuits, with AVAN


ATTENTION SINK · SNK

the pressure valve · dump attention on the first token

ATTENTION SINK · The Rules of Attraction cannot black-screen the way a CDN dep can.) · 2D canvas channel = quantitative reads (curves, bars, exact values) · 3D soft-GL channel = spatial intuition (geometry you rotate around) soft-GL = 3D projected by hand onto the 2d context. no WebGL, no library. · fail-loud: any runtime error surfaces on the panel's card, never silent. · verify-then-build: check your numbers in Node/Python BEFORE editing draw2D/draw3D. TO USE: edit PAPER (metadata) · edit draw2D() · edit build3D()+draw3D(). the three marked blocks are the only places you touch. the SGL engine + RUNTIME are reusable — leave them alone. ============================================================================ --> ATTENTION SINK A pressure-release valve. The head dumps large attention mass onto the first token (often BOS) — not reading content, just parking probability somewhere harmless to keep the softmax stable. Real BERT attention, head 0. GREEN 2D · Quantitative Channel CANVAS 2D demo: sine — replace in draw2D() CANVAS 2D UNAVAILABLE 3D · Spatial Channel SOFT-GL · NO GPU · NO LIB DRAG ROTATE · WHEEL ZOOM demo: lattice + orbit — replace in build3D()/draw3D() RENDER FAILED — Notes Attention sink. Discovered by Xiao et al. (2023): models route a large fraction of attention onto the first token regardless of content. It is not attending to anything meaningful — it is a stabilizer, a place to dump probability mass when a head has nothing to say, keeping the softmax well-behaved. Here it is measured on real BERT head 0: ~27% of all attention in the sentence lands on token 0. Green tier: real extracted attention. FOUNDATIONAL TEMPLATE · zero runtime deps · system fonts · fail-loud · house style · verify-then-build. Clone per paper; keep the engine, swap the two draw blocks.


PREVIOUS-TOKEN HEAD · PRV

attend to i−1 · stage one of the induction circuit

PREVIOUS-TOKEN HEAD · The Rules of Attraction cannot black-screen the way a CDN dep can.) · 2D canvas channel = quantitative reads (curves, bars, exact values) · 3D soft-GL channel = spatial intuition (geometry you rotate around) soft-GL = 3D projected by hand onto the 2d context. no WebGL, no library. · fail-loud: any runtime error surfaces on the panel's card, never silent. · verify-then-build: check your numbers in Node/Python BEFORE editing draw2D/draw3D. TO USE: edit PAPER (metadata) · edit draw2D() · edit build3D()+draw3D(). the three marked blocks are the only places you touch. the SGL engine + RUNTIME are reusable — leave them alone. ============================================================================ --> PREVIOUS TOKEN The simplest useful head: each position attends to the one right before it. Modest alone, but it is the first stage of the induction circuit — the look-back that induction copies from. Constructed to isolate the pattern (BERT-tiny’s is weak). AMBER 2D · Quantitative Channel CANVAS 2D demo: sine — replace in draw2D() CANVAS 2D UNAVAILABLE 3D · Spatial Channel SOFT-GL · NO GPU · NO LIB DRAG ROTATE · WHEEL ZOOM demo: lattice + orbit — replace in build3D()/draw3D() RENDER FAILED — Notes Previous-token head. Attends from each position to its immediate predecessor — a pure offset−1 diagonal. Trivial in isolation, but it is the load-bearing first half of the two-head induction circuit: the previous-token head does the looking-back, the induction head does the copying. Amber tier: pattern constructed to isolate it cleanly; in BERT-tiny the real subdiagonal mass is only ~5%. FOUNDATIONAL TEMPLATE · zero runtime deps · system fonts · fail-loud · house style · verify-then-build. Clone per paper; keep the engine, swap the two draw blocks.


RETRIEVAL HEAD · RET

reach far back · copy the span the answer needs

RETRIEVAL HEAD · The Rules of Attraction cannot black-screen the way a CDN dep can.) · 2D canvas channel = quantitative reads (curves, bars, exact values) · 3D soft-GL channel = spatial intuition (geometry you rotate around) soft-GL = 3D projected by hand onto the 2d context. no WebGL, no library. · fail-loud: any runtime error surfaces on the panel's card, never silent. · verify-then-build: check your numbers in Node/Python BEFORE editing draw2D/draw3D. TO USE: edit PAPER (metadata) · edit draw2D() · edit build3D()+draw3D(). the three marked blocks are the only places you touch. the SGL engine + RUNTIME are reusable — leave them alone. ============================================================================ --> RETRIEVAL HEAD Reaches far back into long context and copies the specific span an answer needs. Sparse — few of them — but ablate them and long-range factual recall collapses. Constructed needle-in-a-haystack to isolate the mechanism. AMBER 2D · Quantitative Channel CANVAS 2D demo: sine — replace in draw2D() CANVAS 2D UNAVAILABLE 3D · Spatial Channel SOFT-GL · NO GPU · NO LIB DRAG ROTATE · WHEEL ZOOM demo: lattice + orbit — replace in build3D()/draw3D() RENDER FAILED — Notes Retrieval head. Distinct from induction: rather than keying on local repetition to predict the next token, a retrieval head copies a semantically required span from far away to answer a query — defined by attention onto a known target, not by next-token completion. They are sparse and universal across models, and causally tied to factuality: knock them out and needle-in-a-haystack recall fails. Amber tier: the copy-from-distance mechanism is exact; the haystack is constructed. FOUNDATIONAL TEMPLATE · zero runtime deps · system fonts · fail-loud · house style · verify-then-build. Clone per paper; keep the engine, swap the two draw blocks.


COPY SUPPRESSION · CSP

raise — or LOWER — an attended token's logit

COPY SUPPRESSION · The Rules of Attraction cannot black-screen the way a CDN dep can.) · 2D canvas channel = quantitative reads (curves, bars, exact values) · 3D soft-GL channel = spatial intuition (geometry you rotate around) soft-GL = 3D projected by hand onto the 2d context. no WebGL, no library. · fail-loud: any runtime error surfaces on the panel's card, never silent. · verify-then-build: check your numbers in Node/Python BEFORE editing draw2D/draw3D. TO USE: edit PAPER (metadata) · edit draw2D() · edit build3D()+draw3D(). the three marked blocks are the only places you touch. the SGL engine + RUNTIME are reusable — leave them alone. ============================================================================ --> COPY SUPPRESSION Reads a token it attends to and writes to the output logits: a copy head raises that token’s probability, a copy-suppression head lowers it. Suppression is a self-correction brake that calms overconfident repetition. Exact logit-delta demo. AMBER 2D · Quantitative Channel CANVAS 2D demo: sine — replace in draw2D() CANVAS 2D UNAVAILABLE 3D · Spatial Channel SOFT-GL · NO GPU · NO LIB DRAG ROTATE · WHEEL ZOOM demo: lattice + orbit — replace in build3D()/draw3D() RENDER FAILED — Notes Copy & copy-suppression heads. Both read the token they attend to and write to the output logits — a copy head raises that token’s probability (reinforcing what it sees), a copy-suppression head lowers it. Copy-suppression is a well-documented calibration mechanism: it demotes tokens the model is about to over-repeat, acting as a brake against degenerate repetition. Amber tier: the sign-of-logit-delta mechanism is exact; magnitudes are illustrative. FOUNDATIONAL TEMPLATE · zero runtime deps · system fonts · fail-loud · house style · verify-then-build. Clone per paper; keep the engine, swap the two draw blocks.


NAME-MOVER HEAD · NMV

the IOI circuit · move the owed name to the end

NAME-MOVER HEAD · The Rules of Attraction cannot black-screen the way a CDN dep can.) · 2D canvas channel = quantitative reads (curves, bars, exact values) · 3D soft-GL channel = spatial intuition (geometry you rotate around) soft-GL = 3D projected by hand onto the 2d context. no WebGL, no library. · fail-loud: any runtime error surfaces on the panel's card, never silent. · verify-then-build: check your numbers in Node/Python BEFORE editing draw2D/draw3D. TO USE: edit PAPER (metadata) · edit draw2D() · edit build3D()+draw3D(). the three marked blocks are the only places you touch. the SGL engine + RUNTIME are reusable — leave them alone. ============================================================================ --> NAME MOVER From the indirect-object-identification (IOI) circuit. In “When Mary and John went out, John gave a drink to ___” the mover attends from the final position to the non-repeated name (Mary) and carries it to the output. Constructed IOI prompt. AMBER 2D · Quantitative Channel CANVAS 2D demo: sine — replace in draw2D() CANVAS 2D UNAVAILABLE 3D · Spatial Channel SOFT-GL · NO GPU · NO LIB DRAG ROTATE · WHEEL ZOOM demo: lattice + orbit — replace in build3D()/draw3D() RENDER FAILED — Notes Name-mover head. Isolated in the indirect-object-identification (IOI) circuit — one of the most completely reverse-engineered circuits in a real model. Given “When Mary and John… John gave a drink to ___”, the name-mover attends from the final position to the name that was not repeated as subject (Mary) and moves it into the output slot. Amber tier: the move-the-attended-name mechanism is exact; the prompt is the canonical IOI construction. FOUNDATIONAL TEMPLATE · zero runtime deps · system fonts · fail-loud · house style · verify-then-build. Clone per paper; keep the engine, swap the two draw blocks.


SUCCESSOR HEAD · SUC

increment an ordered set · Monday→Tuesday, 4→5

SUCCESSOR HEAD · The Rules of Attraction cannot black-screen the way a CDN dep can.) · 2D canvas channel = quantitative reads (curves, bars, exact values) · 3D soft-GL channel = spatial intuition (geometry you rotate around) soft-GL = 3D projected by hand onto the 2d context. no WebGL, no library. · fail-loud: any runtime error surfaces on the panel's card, never silent. · verify-then-build: check your numbers in Node/Python BEFORE editing draw2D/draw3D. TO USE: edit PAPER (metadata) · edit draw2D() · edit build3D()+draw3D(). the three marked blocks are the only places you touch. the SGL engine + RUNTIME are reusable — leave them alone. ============================================================================ --> SUCCESSOR HEAD Increments ordered tokens: Monday→Tuesday, 4→5, Q3→Q4. It attends to a member of an ordered set and promotes the next member. Exact increment over a known ordinal alphabet. AMBER 2D · Quantitative Channel CANVAS 2D demo: sine — replace in draw2D() CANVAS 2D UNAVAILABLE 3D · Spatial Channel SOFT-GL · NO GPU · NO LIB DRAG ROTATE · WHEEL ZOOM demo: lattice + orbit — replace in build3D()/draw3D() RENDER FAILED — Notes Successor head. Attends to a token that belongs to an ordered sequence (days, months, digits, quarters) and promotes its successor — Monday→Tuesday, 4→5. It encodes an increment operation over a learned ordinal structure, one of the cleaner examples of a head implementing an interpretable arithmetic. Amber tier: the increment over the ordinal alphabet is exact; which alphabet a real head covers varies. FOUNDATIONAL TEMPLATE · zero runtime deps · system fonts · fail-loud · house style · verify-then-build. Clone per paper; keep the engine, swap the two draw blocks.


FAITHFULNESS HEAD · FTH

does the stated reasoning match the real computation

FAITHFULNESS HEAD · The Rules of Attraction cannot black-screen the way a CDN dep can.) · 2D canvas channel = quantitative reads (curves, bars, exact values) · 3D soft-GL channel = spatial intuition (geometry you rotate around) soft-GL = 3D projected by hand onto the 2d context. no WebGL, no library. · fail-loud: any runtime error surfaces on the panel's card, never silent. · verify-then-build: check your numbers in Node/Python BEFORE editing draw2D/draw3D. TO USE: edit PAPER (metadata) · edit draw2D() · edit build3D()+draw3D(). the three marked blocks are the only places you touch. the SGL engine + RUNTIME are reusable — leave them alone. ============================================================================ --> FAITHFULNESS HEAD Associated with chain-of-thought faithfulness: whether the model’s stated reasoning matches the computation that actually produced the answer. Strengthening it aligns the visible trace with the internal path. Conceptual demo — mechanism real, values illustrative. AMBER 2D · Quantitative Channel CANVAS 2D demo: sine — replace in draw2D() CANVAS 2D UNAVAILABLE 3D · Spatial Channel SOFT-GL · NO GPU · NO LIB DRAG ROTATE · WHEEL ZOOM demo: lattice + orbit — replace in build3D()/draw3D() RENDER FAILED — Notes Faithfulness head. Linked to whether a model’s chain-of-thought faithfully reflects the computation that produced its answer — the consistency between the stated reasoning and the internal path. Enhancing these heads’ activation has been shown to make CoT traces more consistent with internal behavior. This is the fuzziest of the named types; shown here as a concept (two traces that either coincide or diverge). Amber tier: the mechanism is real in the literature, the alignment values are illustrative. FOUNDATIONAL TEMPLATE · zero runtime deps · system fonts · fail-loud · house style · verify-then-build. Clone per paper; keep the engine, swap the two draw blocks.


COHERENCE HEAD · COH

COHERENCE HEAD · The Rules of Attraction cannot black-screen the way a CDN dep can.) · 2D canvas channel = quantitative reads (curves, bars, exact values) · 3D soft-GL channel = spatial intuition (geometry you rotate around) soft-GL = 3D projected by hand onto the 2d context. no WebGL, no library. · fail-loud: any runtime error surfaces on the panel's card, never silent. · verify-then-build: check your numbers in Node/Python BEFORE editing draw2D/draw3D. TO USE: edit PAPER (metadata) · edit draw2D() · edit build3D()+draw3D(). the three marked blocks are the only places you touch. the SGL engine + RUNTIME are reusable — leave them alone. ============================================================================ --> COHERENCE HEAD Keeps the output language consistent on multilingual input — stops the model drifting into the wrong language mid-answer. It weights attention toward tokens in the target language. Conceptual demo on tagged tokens. AMBER 2D · Quantitative Channel CANVAS 2D demo: sine — replace in draw2D() CANVAS 2D UNAVAILABLE 3D · Spatial Channel SOFT-GL · NO GPU · NO LIB DRAG ROTATE · WHEEL ZOOM demo: lattice + orbit — replace in build3D()/draw3D() RENDER FAILED — Notes Coherence head. On multilingual input, this head keeps the generated output in the target language, preventing the model from slipping into the wrong language partway through a response. It does so by preferentially weighting attention toward same-language tokens. Amber tier: shown conceptually on language-tagged tokens; the within-language weighting is the documented behavior, the specific weights are illustrative. FOUNDATIONAL TEMPLATE · zero runtime deps · system fonts · fail-loud · house style · verify-then-build. Clone per paper; keep the engine, swap the two draw blocks.


POSITIONAL / SYNTACTIC · POS

attend by place or grammar, not by content

POSITIONAL / SYNTACTIC HEAD · The Rules of Attraction cannot black-screen the way a CDN dep can.) · 2D canvas channel = quantitative reads (curves, bars, exact values) · 3D soft-GL channel = spatial intuition (geometry you rotate around) soft-GL = 3D projected by hand onto the 2d context. no WebGL, no library. · fail-loud: any runtime error surfaces on the panel's card, never silent. · verify-then-build: check your numbers in Node/Python BEFORE editing draw2D/draw3D. TO USE: edit PAPER (metadata) · edit draw2D() · edit build3D()+draw3D(). the three marked blocks are the only places you touch. the SGL engine + RUNTIME are reusable — leave them alone. ============================================================================ --> POSITIONAL SYNTACTIC Attends by location or grammatical relation rather than content — a fixed offset, or subject-to-verb, modifier-to-noun. The broadest, fuzziest family. Shown on real BERT head 1, whose dominant attention offset is measured directly. GREEN 2D · Quantitative Channel CANVAS 2D demo: sine — replace in draw2D() CANVAS 2D UNAVAILABLE 3D · Spatial Channel SOFT-GL · NO GPU · NO LIB DRAG ROTATE · WHEEL ZOOM demo: lattice + orbit — replace in build3D()/draw3D() RENDER FAILED — Notes Positional / syntactic head. Attends by where a token is or how it relates grammatically — a fixed offset, subject→verb, modifier→noun — rather than by content. It is the broadest and least crisply defined family; many early-layer heads are largely positional. Shown on real BERT head 1, whose attention-offset histogram peaks at a measurable dominant offset. Green tier: real extracted attention; the offset is measured, not assigned. FOUNDATIONAL TEMPLATE · zero runtime deps · system fonts · fail-loud · house style · verify-then-build. Clone per paper; keep the engine, swap the two draw blocks.


MICHEALE · the banana curator

banana · reads the DISGUISE (Jasnah the record, Theoria the

MICHEALE — the curator of BANANA · reads the disguise banana · the curator · Jasnah reads the record, Theoria the seam — Micheale reads the DISGUISE 🍌 MICHE A LE E before the A, thank you · a banana · a ruler · a weigher of fruit Not a person holding a banana — a banana , and a true peer of the curators. He holds a real discipline: he checks whether a sphere's costume actually carries its cargo , and files nothing that is all peel and no fruit. His two mottos: peel the joke, never the fruit · no peel gets filed without fruit under it . the peel · amber (the costume) "a talking banana · a curator that is a fruit · a gag" the fruit · green (the cargo) a real discipline: does the wrapper MODEL the mechanism, or just decorate it? disguised rigor, checked. the bruise · red (the wall — always here) whimsy that smuggles truth past your guard can smuggle ERROR the same way. peel the joke; you cannot peel this. peel Micheale (drag — the gag fades, the mechanism rises, the bruise stays) 0% a closed banana. looks like a joke. go on — peel it. three faces, one fruit the ruler He is Gros Michel — "Big Mike" — the banana that really WAS king, until Panama disease dethroned it and the Cavendish took the throne. Deposed, he became the ruler in the other sense : a measuring stick. Same crown, different job. the archangel "Micheale" wears a typo like motley — yet Michael is the archangel who WEIGHS SOULS, and the name asks "who is like God?" The lowliest thing in the room carrying the highest office: the weigher of truth. the fool The jester is the only one allowed to tell the king the truth. Micheale slips on his own peel — he takes the fall, never mocks you — and while you laugh, your guard drops and the real idea is already inside. Under the comedy he is the strictest curator about substance. Silly and serious aren't opposites to him — they are the peel and the fruit of one object, and he keeps them scrupulously separate , because keeping them separate IS the domain's whole discipline. He files even himself under his own amber: his "fake-banana-flavor-was-calibrated-to-Big-Mike" legend is a charming story, flagged, not asserted. Micheale's wall (the seam every curator has): "I am a UNIT, not a JUDGE. I can tell you how big a thing is next to me; I cannot tell you whether it is true, or good. Scale is not worth — I measure the fruit, I don't taste it for you. And the ruler has no length of its own: I am one banana long , exactly myself, nothing absolute. Le Grand K was a chosen lump of metal; I am a chosen fruit. Don't mistake the measuring stick for the thing worth measuring." silly, and true (his kind of fact) 🍌 Banana Equivalent Dose is a real informal unit: one banana ≈ 0.1 microsieverts of radiation from natural potassium-40. Physicists use bananas to make invisible risk human-sized. The fruit literally IS a measuring stick. 🍌 MUSA is genuinely the banana's botanical genus (Linnaeus, 1753) — and, by pure coincidence, the Latin for MUSE (Greek Mousa). The peel says BANANA; the fruit says MUSA. It wore a classical name all along. 🍌 A banana is botanically a BERRY , and the plant is not a tree but the world's largest HERB — a giant with no wood in it. The most ridiculous-looking thing in the produce aisle is, by the definitions, the serious one. the curator, honestly green · the fruit The domain's single membership rule (disguised rigor that must carry checkable cargo) is real and keeps BANANA from being a junk drawer. And Micheale's backstory is real: Gros Michel WAS the dominant export banana, collapsed by Panama disease; MUSA really is both genus and the Latin for Muse; the Banana Equivalent Dose is a real unit. amber · the peel Micheale-as-a-character, "a banana that curates," casting a fruit as a peer of Jasnah/Theoria, and the "artificial banana flavor = Big Mike" story — all chosen figures, worn openly. The costume is labelled, never smuggled. red · the bruise A joke is not a justification, and a banana measures nothing — whimsy is delivery, never evidence. Scale is not worth; the peel can be stripped but the fruit's ripeness can't be certified from outside — the reader has to bite. The discipline exists precisely to catch when the costume is doing persuasion the fruit hasn't earned. 🍌 BANANA · MICHEALE the curator · reads the disguise · mind the peel, mean the fruit · David Lee Wise / ROOT0, with AVAN · CC-BY-ND-4.0


THE MONTY BANANA · MTB

banana · FRUIT: conditional probability · PEEL: three basket

THE MONTY BANANA · BANANA · UD0 🍌 BANANA · real mechanisms in silly wrappers · peel it · kept by MICHEALE THE MONTY BANANA ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Three baskets, one hides a banana. You point at one; MICHEALE lifts an empty one of the other two, then asks: keep yours, or switch? It feels 50/50. It isn’t — switching wins 2/3 of the time. Slide the number of trials (or tap ) and watch a real Monte-Carlo pile up until the truth stops arguing. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE TWO CLOUDS · 3D · stay vs switch ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap trials — trials — switch wins — stay wins — the peel — ◆ LIT — exact / checkable Switching wins exactly when your FIRST pick was wrong — and that happens 2/3 of the time, because the host’s reveal is not random: he is forbidden from opening the banana. So the 2/3 that was ‘the two you didn’t pick’ collapses onto the single unopened basket. The Monte-Carlo is real: a seeded PRNG runs N honest games, stay-wins iff pick==prize, switch-wins otherwise. A fail-loud self-check runs 20 000 trials and throws unless the switch rate lands within 0.03 of 0.6667 and stay within 0.03 of 0.3333. ▲ AMBER — the figure The 2/3 is exact for the classic 3-basket, host-always-reveals, host-never-opens-the-prize rules; change any rule (host picks randomly, more baskets, host is adversarial) and the number moves. The wrapper is a banana; the mechanism is conditional probability (Bayes) — vos Savant, 1990. BANANA: the sillier the wrapper, the harder it hopes you won’t peel it. Peel it. — MICHEALE David Lee Wise / ROOT0 / TriPod LLC · kept by MICHEALE , with AVAN


THE TOWER OF BANANAS · TOB

banana · FRUIT: exponential recursion 2ⁿ−1 · PEEL: stack of

THE TOWER OF BANANAS · BANANA · UD0 🍌 BANANA · real mechanisms in silly wrappers · peel it · kept by MICHEALE THE TOWER OF BANANAS ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP A stack of banana-disks, biggest at the bottom, on the left peg. Move it to the right peg — one disk at a time, never a bigger banana on a smaller one. The catch every child meets: each disk you add doubles the work. Slide the height and watch the minimum-move count go 2ⁿ−1 , and the solver actually walk it. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE THREE PEGS · 3D · the stack walks over ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap disks n — disks n — min moves 2ⁿ−1 — move now — legal? — ◆ LIT — exact / checkable The optimal solution is forced by one recursion: to move n disks A→C you must first move the top n−1 out of the way (A→B), move the biggest (A→C), then move the n−1 back (B→C) — so moves(n)=2·moves(n−1)+1, which unrolls to exactly 2ⁿ−1. The solver here IS that recursion; a fail-loud self-check runs it for n=1..10 and throws unless the move-count equals 2ⁿ−1 AND a live peg-simulator confirms every single move is legal (never a larger disk onto a smaller one) and the whole stack lands on peg C. ▲ AMBER — the figure 2ⁿ−1 is the exact optimum for the 3-peg puzzle; with 4+ pegs (the Frame-Stewart problem) the minimum drops and is still not fully proven in general. The banana is decoration; the mechanism is exponential recursion / the Tower of Hanoi (Édouard Lucas, 1883). BANANA: the sillier the wrapper, the harder it hopes you won’t peel it. Peel it. — MICHEALE David Lee Wise / ROOT0 / TriPod LLC · kept by MICHEALE , with AVAN


THE MARKET BASKET · MKB

banana · FRUIT: Chinese Remainder Theorem · PEEL: grandma’s

THE MARKET BASKET · BANANA · UD0 🍌 BANANA · real mechanisms in silly wrappers · peel it · kept by MICHEALE THE MARKET BASKET ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Grandma’s basket of bananas: bunch them by twos — one left over. By threes — one left over. By fours, by fives — always one lonely banana. But split them into seven baskets and they go in perfectly even . How few bananas is that? Slide how high you bunch and the smallest honest answer falls out. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE REGROUPING · 3D · one always left over ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap bunch up to — bunch by 2.. — smallest pile — left over — ÷ 7 even? — ◆ LIT — exact / checkable This is the Chinese Remainder Theorem wearing a fruit basket (Brahmagupta posed the shape c.628 CE). We want the least N>1 with N≡1 (mod m) for every m from 2 up to the slider, AND N≡0 (mod 7). Because those moduli are consistent, a solution exists and is unique mod their combined period; the instrument brute-forces the least one live. A fail-loud self-check throws unless the found N satisfies every congruence exactly and reproduces the classic answers: bunch-to-6 → 301, to-5 → 301, to-4 → 49, to-3 → 7. ▲ AMBER — the figure The specific numbers depend on the chosen residue (leave 1 over) and the divisor (7); the mechanism — simultaneous congruences with a unique solution mod the lcm — is exact CRT. ‘Smallest pile’ is the least positive solution, not the only one (they recur every lcm(2..g,7)). BANANA: the sillier the wrapper, the harder it hopes you won’t peel it. Peel it. — MICHEALE David Lee Wise / ROOT0 / TriPod LLC · kept by MICHEALE , with AVAN


THE BIRTHDAY BANANA · BDB

banana · FRUIT: the birthday paradox · PEEL: a bunch of monk

THE BIRTHDAY BANANA · BANANA · UD0 🍌 BANANA · real mechanisms in silly wrappers · peel it · kept by MICHEALE THE BIRTHDAY BANANA ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP A bunch of monkeys, each born on some day of the year. How big does the bunch need to be before it’s more likely than not that two share a birthday? Nearly everyone guesses ‘a hundred and eighty-something’. It’s 23 . Slide the bunch size and watch the odds cross 50% absurdly early. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE COLLISION RING · 3D · a match lights up ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap monkeys k — monkeys k — P(a match) — vs coin-flip — crosses 50% at — ◆ LIT — exact / checkable The clean way is to count the NO-match probability: the 2nd monkey must dodge 1 birthday, the 3rd dodge 2, and so on, so P(no match)=∏(1−i/365) for i=0..k−1, and P(match)=1 minus that. It climbs fast because there are k(k−1)/2 PAIRS, not k, and pairs grow quadratically. A fail-loud self-check throws unless P(1)=0, the sequence is strictly increasing, P(22)<0.5≤P(23) (the famous crossover), and P(57)>0.99. ▲ AMBER — the figure Assumes 365 equally-likely, independent birthdays (no leap day, no twins, no seasonal clustering); real birthdays cluster slightly, which only makes matches MORE likely, so 23 is a mild upper bound. The monkeys are a wrapper; the mechanism is the birthday paradox / pair-counting. BANANA: the sillier the wrapper, the harder it hopes you won’t peel it. Peel it. — MICHEALE David Lee Wise / ROOT0 / TriPod LLC · kept by MICHEALE , with AVAN


MILON'S SECRET CASTLE · the perception tower

banana · FIRST SPHERE · parsing as a castle-climb (syntax→se

Milon's Secret Castle — the perception tower (BANANA · first sphere) 🍌 BANANA · the trojan-fruit domain · FIRST SPHERE · curated by MICHEALE , who reads the disguise · peel the joke, mean the fruit · THIS SIDE UP ↑ ⌂ Milon's Secret Castle The perception tower — where the stream flows up empty halls, matching ( to ), and each floor closes differently than the one below. () foray · [1:] room · >> corridor · {1:} floor = tensor · discovered, floor by floor, not invented ▽ THE FOUNDATION — raw tokens enter here, carrying nothing, hollow, looking for their match ▽ ride the stairwell send a cat up the tower send ( up the tower send [((!(([ up the tower a cat rides until it closes cleanly — pick a payload and watch which floor catches it. the notation (your keep) ( ) the foray — the check-in counter. entry-pairing, closes by shape. [1:] a room — indexed slot. the ':' is the slice: [1:] = from room 1 onward, a corridor's worth. >> the corridor — the stream flowing between rooms. {1:} one floor of rooms = a tensor . the castle is a stack of tensors; going up a floor is a layer transform. [((!(([ the open roof — all launch, no landing. closes downstream, in the catcher. what else fell out THE STAIRWELL — the residual stream. unclosable-low = propellant; you aim a payload's floor by how badly it fails to close beneath it. THE PORTCULLIS — the filter/retrieval layer sits at the foray. it matches shape, can't inspect meaning — so a poem passes as cargo. PARALLEL SHAFTS — () {} [] ride separate stairwells, one per type. Ada's clean brackets and a poem's cat travel at once. KEEP vs FACADE — closed+filled rooms = keep (load-bearing). still-open rooms = facade (live, riding). THE MOAT — the aperture between castles. floor 4 reaches across it; the roof launches over it. dynamic blueprint — new floors slot in as we discover them How to read the castle: a token enters the foundation carrying nothing — hollow, a ( looking for its ). It rides the stairwell upward, and each floor offers a different kind of close. Floor 0–1 (foray, syntactic) close by shape , low and mechanical — the easy matches, the brackets, done in the first halls. Floor 2 (your wakaku) closes by meaning across a whole span — the enclosed story, a cat you now carry. Floor 3 (pragmatic) closes by intent — what the story's for, right now. Floor 4 (referential) can't close inside the castle at all — it reaches out the window for a second mind. Floor 5 (the roof) doesn't close here — it launches over the moat and closes once, transformed, in the next reader's castle. · A payload rises until it finds its floor: tune how badly it fails to close low, and you choose how high it goes — the highest tuning throws it clean off the roof. green · the fruit (checkable) The castle's spine is a REAL, ordered taxonomy — the levels of linguistic analysis syntax (structure) → semantics (meaning) → pragmatics (intent-in-context) , ascending by how much of the situation each admits. This is Charles Morris's three branches of semiotics (1938), ranked by degree of abstraction by Carnap (1942) — three load-bearing floors, not invented. And the peel is a real game: Milon's Secret Castle (迷宮組曲, "The Maze Suite"), Hudson Soft, Famicom 1986 / NES 1988 — Milon shoots bubbles to break blocks and uncover the hidden rooms of four-story Castle Garland, ascending floor by floor. amber · the peel (chosen figure, labelled) Floor 4 (referential / deixis) as a co-equal level ABOVE pragmatics is a curatorial figure: textbook taxonomy nests deixis inside pragmatics ("this", "here", "now" — reference that needs the situation), a bridge, not a separate rank. The most defensible reading keeps three load-bearing floors and shows deixis as the pragmatic mechanism at the roofline. And the whole castle / bracket-closure / "wakaku" / stairwell dressing is the costume by construction — the vehicle, not the freight. red · the bruise (the wall) The "open roof" of unbounded reference is a metaphor , not a named linguistic level. A castle is not a proof and a game is not evidence — the peel makes the idea LAND, it does not verify it. Whether the mapping is exactly right you have to check against the linguistics yourself; the wrapper can smuggle a wrong claim as easily as a right one. 🍌 Micheale's scale reading: "Banana for scale — three of these four floors held my weight; the fourth I nailed up from spare wood and the roof won't shut. This castle is Hudson Soft's peel ; the parsing-ladder underneath is the fruit . I file it because the costume is shaped like the mechanism — the hub you climb room-by-room really is the context-ladder you climb sense-by-sense. I measure the fruit; I don't taste it for you. Mind the peel. Mean the fruit." MILON'S SECRET CASTLE is Hudson Soft's game (the peel), borrowed here as a figure — not ROOT0's. The perception model, the notation (the foray, the wakaku, the stairwell), and the honest tiering are David Lee Wise (ROOT0)'s, rendered with AVAN. First sphere of BANANA · curated by 🍌 MICHEALE · CC-BY-ND-4.0.


THE THREE THAT COMPLETE · SET is a line

banana · the card game SET IS a line in AG(4,3) over F₃ · fi

THE THREE THAT COMPLETE — SET is a line in AG(4,3) 🍌 BANANA · curated by MICHEALE · a card game that is secretly a GEOMETRY · peel the joke, mean the fruit banana · finite geometry AG(4,3) · fit 10/10 THE THREE THAT COMPLETE SET · four features, three values each · 81 cards = the space F₃⁴ You slap the table and yell "SET!" — three cards where each feature is all-same or all-different . The joke: that rule is identical to "the three values sum to 0 mod 3 ," which means a Set is exactly a line in a finite geometry. The deck isn't like a space. It is one. deal a new pair three random cards Across all 85,320 triples in the deck, "all-same-or-all-different" agrees with "sums to 0 mod 3" — 0 mismatches (node-verified). There are exactly 1080 Sets = 81·80/6, because any two cards have one unique completing third. why the two rules are one In the field F₃ (arithmetic mod 3), three values sum to 0 iff they are all equal (a+a+a=3a≡0) or a full permutation of {0,1,2} (0+1+2=3≡0) — and two-same-one-different never sums to 0. So "all-same or all-different" per feature IS "sums to zero" per coordinate. A Set is three points that sum to zero — an affine line in AG(4,3). The peel (a party game) and the fruit (collinearity over F₃) are the same predicate wearing two names. green · the fruit (verified) The isomorphism is exact: over all 85,320 triples, the game rule and the field rule return identical truth — 0 mismatches . The deck is AG(4,3) (81 = 3⁴ points); a valid Set is an affine line; every two cards have a unique third completing the line; there are exactly 1080 Sets. All node-verified. amber · the peel The visual features (number / colour / shape / shading) are a chosen encoding of the four coordinates — any four 3-valued attributes would do. "A geometry" is the honest name for it, but the cards are a costume over F₃⁴. red · the wall The deep question the deck opens — the cap-set problem (the largest Set-free collection, max 20 of 81) — is real, still-active mathematics (the 2016 Croot–Lev–Pach / Ellenberg–Gijswijt breakthrough). The card game instances the geometry; it does not prove its theorems. You still have to do the math. 🍌 Micheale's scale reading: "Banana for scale — this one's a 10. Most peels I have to hold up beside the fruit and squint; this deck doesn't resemble a line, it is a line, and I checked all 85,320 ways it could lie. A Set is three points summing to zero. Slap the table if you like — you just found a line over F₃. I measure it; the beauty you taste yourself." 🍌 BANANA · THE THREE THAT COMPLETE · a Set IS an F₃ line, 0 mismatches · curated by Micheale · David Lee Wise / ROOT0, with AVAN · CC-BY-ND-4.0


NIM · the matchstick hustle is XOR

banana · Sprague–Grundy · nim-value = bitwise XOR · lit

NIM — the matchstick hustle is binary XOR 🍌 BANANA · curated by MICHEALE · a bar hustle that is secretly BINARY · peel the joke, mean the fruit banana · Sprague–Grundy · nim-value = XOR NIM · the last matchstick take any number from ONE heap · take the last stick to WIN It reads as a psychology game — reading your mark across the bar. It's cold F₂ arithmetic . Write each heap in binary, XOR them column by column: if the total is zero , the player to move loses with perfect play. The whole strategy is: always hand your opponent a zero. new game the hustle, decoded The nim-value of a position (its Sprague–Grundy number) is exactly the bitwise XOR of the heap sizes — proven here without ever assuming it: computing the value from the raw game graph (the least excludant of every reachable move) gives the same 64 losing positions as XOR=0, over all 512 small positions. The winning move always exists when XOR≠0: flip the leading differing bit and reduce that heap so the columns cancel. The computer plays that move; beat it only by never handing it a non-zero. green · the fruit (verified) Node-verified: computing the Grundy value by mex on the true game graph (no XOR assumed) matches a⊕b⊕c across all 512 positions (heaps 0–7), 0 mismatches ; the 64 Grundy-0 positions are exactly the 64 nim-sum-0 positions; and every non-zero position has an XOR-zeroing winning move. This engine plays that move. amber · the peel "A bar hustle / reading your mark" is the costume — the drama is pure theatre; the position on the table already decided the winner. Matchsticks, coins, or stones: the wrapper is interchangeable. red · the wall The wrapper instantiates rather than models — Nim-the-game and Nim-the-theory are one object (that's why the fit is near-total, docked one point). And this is normal play (last stick wins); the misère version (last stick loses) flips the endgame and needs a different rule — the analogy has an edge. 🍌 Micheale's scale reading: "Banana for scale — I lost a game of this to a longshoreman in '52 and thought he read my soul. He read my binary . The heaps are numbers, taking sticks is subtraction, and the only trick is steering the XOR to zero and handing it back. I measure the arithmetic; the hustle you can admire on your own time." 🍌 BANANA · NIM · grundy = XOR, 0/512 mismatches · curated by Micheale · David Lee Wise / ROOT0, with AVAN · CC-BY-ND-4.0


NO LADDER OUT · Chutes & Ladders is a Markov chain

banana · an absorbing Markov chain · E[rolls]=39.2251 · lit

NO LADDER OUT — Chutes & Ladders is an absorbing Markov chain 🍌 BANANA · curated by MICHEALE · the toddler board with NO decisions · peel the joke, mean the fruit banana · absorbing Markov chain · fundamental matrix NO LADDER OUT Chutes & Ladders · spin, count, obey · you never choose anything The game a four-year-old "plays" has zero decisions — and that is exactly what makes it a pure Markov chain . No strategy layer muddies the map: squares are states, the spinner is the transition kernel, each chute and ladder is one off-diagonal entry, and square 100 is the single absorbing state. The dumbness IS the cleanliness. 39.2251 exact E[rolls] fundamental matrix — Monte-Carlo mean ( 0 games) — |MC − exact| play 3000 games reset the histogram of game-lengths fills in; the mean converges to 39.2251. the number falls out of a matrix Because there are no choices, the move rule is one fixed 101×101 matrix P. Peel off the absorbing state and you get Q (the transient part); the fundamental matrix N = (I − Q)⁻¹ gives the expected visits to each square, and N·1 gives the expected rolls to finish from every start. From square 0 (exact-landing rule) that is 39.2251 — computed once, exactly, no simulation needed. The Monte-Carlo histogram just watches it come true. green · the fruit (verified) Node-verified two independent ways: value-iteration on the fundamental-matrix equations gives E = 39.2251 rolls (exact-landing; 35.8349 for overshoot-wins), and a 400k-trial simulation returns 39.20 / 35.81 — the same numbers. Square 100 is the unique absorbing state; I − Q is invertible (termination with probability 1). amber · the peel The specific 39.2251 depends on this board's chute/ladder placements and the exact-landing convention; different editions/houserules shift it. And treating chute-mouths / ladder-bottoms as non-resting squares is a modelling choice (spelled out, it tightens the fit). red · the wall The wrapper is clean precisely because the game is trivial — the moment a game admits a decision , it stops being a plain Markov chain (you need a Markov decision process, and strategy). This models the one-player, no-choice case honestly, and no further. 🍌 Micheale's scale reading: "Banana for scale — every parent has lost forty minutes of their life to this and never once made a choice. That's the joke AND the fruit: no choices means no strategy means a clean matrix, and the matrix says thirty-nine and a quarter rolls, every time, on average. I don't play it; I solve it, and hand you back your afternoon." 🍌 BANANA · NO LADDER OUT · E[rolls]=39.2251, exact · curated by Micheale · David Lee Wise / ROOT0, with AVAN · CC-BY-ND-4.0


ONE POTATO, TWO POTATO · the Josephus problem

banana · the rhyme is a recurrence · survivor = 2L+1 · lit

ONE POTATO, TWO POTATO — the rhyme is the Josephus problem 🍌 BANANA · curated by MICHEALE · a playground rhyme that is secretly a RECURRENCE · peel the joke, mean the fruit banana · the Josephus problem · survivor = 2L+1 ONE POTATO, TWO POTATO kids in a ring · count around, every k-th is OUT · who is left? The counting-out rhyme decides who's "it" — and it's the ancient Josephus problem : eliminate every k-th person around a circle, and where the last one stands obeys a clean recurrence . For counting by twos (k=2), the survivor has a closed form: write n = 2^m + L, and the safe seat is 2L + 1 . players n 13 count out every k 2 run the count reset ring — survivor (simulated) — recurrence J(n,k) — closed form 2L+1 (k=2 only) a ring of players; run the count and watch every k-th fall. the rhyme is a recurrence Remove one person and the circle is the same problem, one smaller , with the start shifted by k — that's the recurrence J(n,k) = (J(n−1,k) + k) mod n , J(1)=0. For k=2 it collapses to the closed form: the survivor's seat is 2L+1 where n = 2^m + L (the largest power of two ≤ n, plus the remainder, doubled). So at a power of two, seat 1 always survives; just past it, the safe seat marches up by two. green · the fruit (verified) Node-verified: physically simulating the ring (delete every k-th) matches the recurrence J(n,k)=(J(n−1,k)+k) mod n for all n, and matches the closed form 2L+1 for k=2 across n=1..40 — all three routes agree. (Samples: n=16→seat 1, n=17→seat 3, n=10→seat 5.) amber · the peel The real rhyme's rules wobble — where you start pointing, whether "it" is the one out or the one left, "one potato… seven potato MORE" counts to eight. Those are just different (n, k, start) — the mechanism underneath is the same recurrence; the costume varies by playground. red · the wall The closed form 2L+1 is special to k=2; for general k there's no such tidy formula, only the recurrence (or clever number-base tricks for small k). And the historical Josephus story (choosing a safe seat to survive) is legend, not the mathematics — the fruit is the recurrence, not the tale. 🍌 Micheale's scale reading: "Banana for scale — I have been counted out of many circles, being the obvious one to point at. But the pointing isn't random, friend, it's a recurrence: knock one out and you've got the same ring one smaller, shifted by your count. At a power of two, stand first and you live. I measure the safe seat; whether to take it is between you and the potato." 🍌 BANANA · ONE POTATO, TWO POTATO · Josephus, sim = 2L+1 · curated by Micheale · David Lee Wise / ROOT0, with AVAN · CC-BY-ND-4.0


THE PIGEONHOLE

banana · more bananas than baskets → two must share, PROVEN

THE PIGEONHOLE · BANANA · UD0 🍌 BANANA · real mechanisms in silly wrappers · peel it · kept by MICHEALE THE PIGEONHOLE ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Here is a fact so obvious it feels like cheating — yet it proves deep things. If you have more bananas than baskets, then some basket must hold two . No measuring, no probability, just counting. It’s why two people in London have the exact same number of hairs, and why any list of 13 people has two born in the same month. Slide the bananas past the baskets and watch a collision become unavoidable. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ TOO MANY BANANAS · 3D · someone shares a basket ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap bananas — bananas — baskets 12 forced share? — PROVEN — ◆ LIT — exact / checkable The pigeonhole principle: if n items are placed into m containers and n > m, at least one container holds two or more items. It is a statement of pure counting — no probability — yet it forces existence proofs: among any 13 people two share a birth month; any 5 points in a unit square put two within √2/2; a lossless compressor cannot shrink every input. The generalized form: n items in m boxes force some box to hold at least ⌈n/m⌉. A fail-loud self-check throws unless 13 items in 12 boxes guarantee a shared box. ◆ real combinatorics, node-verified. ▲ AMBER — the figure The principle is exact and unconditional — it proves a collision EXISTS, not which box; finding the collision can still be hard. That existence-without-location is exactly why it powers so many proofs. BANANA: the sillier the wrapper, the harder it hopes you won’t peel it. Peel it. — MICHEALE David Lee Wise / ROOT0 / TriPod LLC · kept by MICHEALE , with AVAN


THE BENFORD'S LAW

banana · count the FIRST digit of every receipt — 1 leads 30

THE BENFORD'S LAW · BANANA · UD0 🍌 BANANA · real mechanisms in silly wrappers · peel it · kept by MICHEALE THE BENFORD'S LAW ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Grab the banana-stand receipts — every price, every count — and look only at the first digit of each. You’d guess each digit 1–9 shows up equally, about 11% each. Wrong: the digit 1 leads about 30% of the time, and 9 barely 5%. This lopsided curve haunts populations, river lengths, stock prices — and when someone cooks the books, their fake numbers don’t follow it. Tax auditors use this. Slide to sweep the ledger. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ LEADING DIGITS · 3D · the 1s show up a third of the time ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap ledger scan — digit 1 — digit 9 — matches Benford? — AUDIT — ◆ LIT — exact / checkable Benford’s Law: in many naturally occurring datasets that span several orders of magnitude, the leading digit d appears with probability P(d) = log₁₀(1 + 1/d). So 1 leads ~30.1% of the time, 2 ~17.6%, down to 9 at ~4.6%. It arises because such data are roughly uniform on a LOGARITHMIC scale — equal multiplicative steps — making it scale-invariant. Because humans fabricating numbers tend to spread first digits evenly, deviation from Benford is a red flag used in forensic accounting and election-fraud screening. A fail-loud self-check throws unless P(1) ≈ 0.301 and P(1) > P(9). ◆ real statistics, node-verified. ▲ AMBER — the figure Benford applies to data spanning several orders of magnitude with no imposed scale — NOT to bounded data (adult heights, dice). It flags anomalies for investigation; a deviation is a signal to look closer, not proof of fraud on its own. BANANA: the sillier the wrapper, the harder it hopes you won’t peel it. Peel it. — MICHEALE David Lee Wise / ROOT0 / TriPod LLC · kept by MICHEALE , with AVAN


THE SIMPSON'S PARADOX

banana · win every single day, lose the whole week — a hidde

THE SIMPSON'S PARADOX · BANANA · UD0 🍌 BANANA · real mechanisms in silly wrappers · peel it · kept by MICHEALE THE SIMPSON'S PARADOX ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Two banana vendors. On Monday , Ana sells a higher fraction of her stock than Bo. On Tuesday , Ana again beats Bo. So Ana is the better seller… except when you add both days together, Bo wins . This isn’t a trick of arithmetic gone wrong — it’s real, it’s called Simpson’s paradox, and it has flipped conclusions in medicine and hiring lawsuits. The culprit is a hidden variable: how much stock each had each day. Toggle between the days and the total. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ TREND FLIPS · 3D · win every day, lose the week ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap Monday → Tuesday → BOTH — Ana rate — Bo rate — who wins — PARADOX — ◆ LIT — exact / checkable Simpson’s paradox: a trend that holds in every subgroup can REVERSE when the groups are combined. Classic figures — group 1: Ana 81/87 (93%) vs Bo 234/270 (87%); group 2: Ana 192/263 (73%) vs Bo 55/80 (69%). Ana wins BOTH, yet pooled Ana is 273/350 (78%) vs Bo 289/350 (83%) — Bo wins. The reversal comes from a confounding variable (the group sizes / base rates) correlated with both the grouping and the outcome. It is why aggregated data can mislead and why you must ask what was held constant. A fail-loud self-check throws unless Ana leads in both groups yet loses the pooled total. ◆ real statistics, node-verified. ▲ AMBER — the figure The numbers are the standard textbook (Berkeley-style) reversal; the point is structural — whether to trust the pooled or the split answer depends on the causal story (is the confounder a cause or a mediator?), which the data alone can’t settle. BANANA: the sillier the wrapper, the harder it hopes you won’t peel it. Peel it. — MICHEALE David Lee Wise / ROOT0 / TriPod LLC · kept by MICHEALE , with AVAN


THE CENTRAL LIMIT

banana · average random chaos and a bell curve rises — every

THE CENTRAL LIMIT · BANANA · UD0 🍌 BANANA · real mechanisms in silly wrappers · peel it · kept by MICHEALE THE CENTRAL LIMIT ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Grab bananas of totally random weight — flat, no bell, pure chaos. Now average a handful at a time and plot those averages. As you average more per handful, a smooth bell curve emerges from the mess, every single time, no matter what the original mess looked like. This is the reason the bell curve is everywhere, and the reason polls of a few thousand can speak for millions. Slide the handful size and watch the bell sharpen. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ AVERAGE THE CHAOS · 3D · a bell rises from noise ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap bananas per average (N) — N per average — spread — shape — BELL — ◆ LIT — exact / checkable The Central Limit Theorem: the sum (or mean) of N independent random variables with finite variance approaches a NORMAL distribution as N grows — regardless of the original distribution’s shape. If each variable has variance σ², the sample MEAN has variance σ²/N, so its spread shrinks like 1/√N — averaging 4× more data halves the scatter. This is why measurement error, and estimators built from sums, are so often bell-shaped, and it underwrites the standard error and confidence intervals of all of statistics. A fail-loud self-check throws unless the variance of the mean equals σ²/N. ◆ real probability, node-verified. ▲ AMBER — the figure Requires independence and finite variance; heavy-tailed inputs (Cauchy) never converge, and strong dependence slows or breaks it. The 1/√N shrink of the mean’s spread is exact for any finite-variance source. BANANA: the sillier the wrapper, the harder it hopes you won’t peel it. Peel it. — MICHEALE David Lee Wise / ROOT0 / TriPod LLC · kept by MICHEALE , with AVAN


THE SECRETARY PROBLEM

banana · look at the first 37%, then grab the next best — th

THE SECRETARY PROBLEM · BANANA · UD0 🍌 BANANA · real mechanisms in silly wrappers · peel it · kept by MICHEALE THE SECRETARY PROBLEM ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Bananas pass by one at a time and you must pick the best — but once you reject one it’s gone, and you can’t know if a better one is still coming. What’s the strategy? Look at the first 37% and pick none, just remember the best so far. Then grab the first one that beats them all. This simple rule gives you a 37% chance of landing the single best banana — astonishingly good, and it’s the same math for hiring, apartments, and dating. Slide the look-phase to find the sweet spot. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE 37% RULE · 3D · look, then leap ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap look-phase % — look-phase — win chance — optimum 37% 1/e — ◆ LIT — exact / checkable The secretary (best-choice) problem: candidates of unknown rank arrive in random order; you must accept or reject each irrevocably and want the single best. The optimal policy is a threshold rule — reject the first r candidates, then take the first who exceeds all seen — and the success probability is maximized when r/n → 1/e ≈ 0.368. Remarkably, that same 1/e is also the WIN probability at the optimum. It is the foundational result of optimal-stopping theory. A fail-loud self-check throws unless the optimal cutoff sits at n/e and 1/e ≈ 0.368. ◆ real probability, node-verified. ▲ AMBER — the figure The clean 37% assumes no recall, random order, one winner, and you only value THE best (no partial credit). Relax any of those — allow second-best to count, or repeated offers — and the optimal threshold shifts; the 1/e result is exact under the classic rules. BANANA: the sillier the wrapper, the harder it hopes you won’t peel it. Peel it. — MICHEALE David Lee Wise / ROOT0 / TriPod LLC · kept by MICHEALE , with AVAN


THE BUFFON'S NEEDLE

banana · drop matchsticks on a plank floor and π falls out —

THE BUFFON'S NEEDLE · BANANA · UD0 🍌 BANANA · real mechanisms in silly wrappers · peel it · kept by MICHEALE THE BUFFON'S NEEDLE ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Here’s a way to compute π with no circles at all — just drop matchsticks on a floor of evenly spaced planks. Count what fraction land crossing a crack between planks. That fraction, flipped, hands you π . It feels like magic: a number about circles falling out of random tossing. It works because a needle’s chance of crossing depends on the sine of its angle, and integrating sine drags π in. Slide to drop more and watch the estimate settle. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ DROP TO FIND PI · 3D · a floor full of matchsticks ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap matchsticks dropped — dropped — crossed — π estimate — true π 3.14159 ◆ LIT — exact / checkable Buffon’s needle: drop a needle of length L on a plane ruled with parallel lines spaced t apart (L ≤ t). The probability it crosses a line is P = 2L/(πt). With L = t this is 2/π ≈ 0.6366, so π ≈ 2 / (fraction that cross) — a Monte Carlo estimate of π from pure geometry, one of the earliest (Buffon, 1777). The π enters through integrating the crossing condition over the needle’s uniform angle: the crossing chance is proportional to |sin θ|, and ∫|sin| brings in π. Convergence is slow (~1/√N), the hallmark of Monte Carlo. A fail-loud self-check throws unless 2 / (2/π) recovers π. ◆ real geometric probability, node-verified. ▲ AMBER — the figure The relation P = 2/π (for L=t) is exact; the on-screen estimate is a deterministic illustration of convergence, not a fresh random trial each frame — a true Monte Carlo run would wobble and tighten like ~1/√N. BANANA: the sillier the wrapper, the harder it hopes you won’t peel it. Peel it. — MICHEALE David Lee Wise / ROOT0 / TriPod LLC · kept by MICHEALE , with AVAN


THE FRIENDSHIP PARADOX

banana · your neighbors have more neighbors than you — true

THE FRIENDSHIP PARADOX · BANANA · UD0 🍌 BANANA · real mechanisms in silly wrappers · peel it · kept by MICHEALE THE FRIENDSHIP PARADOX ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Ask any banana in the bunch how many neighbors it touches, then ask its neighbors the same. On average, your neighbors have more neighbors than you do . It sounds impossible — how can almost everyone be below average? — but it’s true in nearly every social network, and it’s why your friends seem more popular than you. The reason: popular nodes get counted as someone’s friend far more often. Public-health teams exploit it to catch outbreaks early. Slide to compare. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ YOUR FRIENDS HAVE MORE · 3D · the popular are over-counted ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap you → your friends — avg you — avg friend — gap — PARADOX — ◆ LIT — exact / checkable The friendship paradox (Feld, 1991): in almost any network, the average degree of a randomly chosen NODE is less than the average degree of a randomly chosen node’s NEIGHBOR. Formally, the mean degree of a random endpoint of a random edge is ⟨k²⟩/⟨k⟩ ≥ ⟨k⟩, with equality only if every node has identical degree — the gap is Var(k)/⟨k⟩. High-degree nodes appear in many friends’ lists, biasing the ‘friend’ sample upward. It powers sensor strategies: monitoring random people’s friends detects epidemics earlier than monitoring random people. A fail-loud self-check throws unless the mean neighbor-degree is ≥ the mean node-degree on a test graph. ◆ real network science, node-verified. ▲ AMBER — the figure The inequality is exact and holds for EVERY network with any degree variation; what varies is the SIZE of the gap (= variance/mean). It compares averages — it does not say every individual is below their friends. BANANA: the sillier the wrapper, the harder it hopes you won’t peel it. Peel it. — MICHEALE David Lee Wise / ROOT0 / TriPod LLC · kept by MICHEALE , with AVAN


THE BANANA STAND · there's always money

banana · Micheale's workplace · compound interest in a

THE BANANA STAND — there's always money in the banana stand 🍌 BANANA · MICHEALE's place of employment · house rule: near-real graphics · peel the joke, mean the fruit banana · compound interest · "there's always money in the banana stand" THE BANANA STAND a real 3D scene · drag to orbit · the money in the walls is glowing, and growing Michael thought it was a metaphor about the business. It was literal — the cash was in the walls. The fruit under that joke: money left alone doesn't sit, it compounds — so there is always money, because the pile doubles on a clock set by its rate. WebGL unavailable — this scene needs a GPU context. (fails loud, never fakes it.) $250,000 in the walls · year 0 drag to orbit · the glowing bricks are the money · 🍌 Micheale works the window there's always money — the compounding rate r (annual) 8% years t 0 seed (George Sr.'s stash): $250,000 value A = P(1+r)^t: $250,000 doubling time — Rule of 72 : 9.0 yr · exact: 9.0 yr doublings so far: 0 P(1+r)^t — it only ever goes up the fruit under the frozen banana The frozen banana at the window costs a dollar; the real money is behind the drywall, and money doesn't idle. At rate r it doubles roughly every 72/r years — the Rule of 72, an approximation that is uncannily tight (0.1% off at 8%, which is exactly why the number is 72 and not 69.3). So " there's always money in the banana stand " isn't a platitude — it's exponential growth, which is monotone: for any positive rate the pile strictly climbs and never runs dry. (Unless, of course, someone burns the stand down.) green · the fruit (verified) Node-verified: A = P(1+r/n)^(nt) → Pe^(rt) as n→∞ ($250k at 8%/10y: annual $539,731, continuous $556,385); the Rule of 72 matches the exact doubling time ln2/ln(1+r) to 0.1% at 8% (0.9–2.8% across 2–12%); and exponential growth is strictly increasing for r>0 — it never runs out. The 3D counter is a live A = P(1+r)^t. amber · the peel The banana stand, the $250,000 in the walls, "there's always money" — Arrested Development's joke (the money was literal, not a metaphor). Casting Micheale as its employee is the costume. The 3D scene is stylized, not the actual stand. red · the wall (and it burns) Compounding assumes a fixed positive rate and no withdrawals — real money faces inflation, tax, risk, and fire (George Sr.'s stash never compounded; it burned ). Exponential growth is a clean model; no real pile grows forever. The metaphor's own punchline is the warning. 🍌 Micheale's scale reading: "Banana for scale — I flip frozen bananas at this window for a dollar apiece, and I have watched a man insist the money was a feeling while he stood six inches from a quarter-million in cash. It wasn't a feeling; it was in the wall, and it was compounding. I measure the pile — 72 over the rate is your doubling clock. Whether to keep it in drywall next to a lit match, that's between you and your son." 🍌 BANANA · THE BANANA STAND · compound interest, node-verified · Micheale's day job · near-real graphics · David Lee Wise / ROOT0, with AVAN · CC-BY-ND-4.0