A number progression is the visible trace of a hidden generator. The solver works backwards from trace to generator by testing a fixed hierarchy of families, from the cheapest diagnostic to the most specialised. The hierarchy matters because several generators can mimic one another over short stretches, and the cheapest discriminating observation is applied first.

2.1.1 Difference hierarchies

The first level of analysis studies successive gaps rather than terms themselves. Let the observed terms be denoted by $a_1, a_2, a_3, \ldots$ and the first differences by $d_i = a_{i+1} - a_i$. When the difference row itself advances by a constant step, the underlying generator is quadratic in the index, and extension proceeds by continuing the difference row rather than by guessing at the terms. A constant difference row corresponds to linear growth in the original progression, while a difference row with constant second gap corresponds to parabolic growth. The recognition logic is therefore mechanical: form the row of gaps, inspect the gap row for constancy or for linear drift, and only when the gap row shows neither pattern is the difference hypothesis abandoned. Monotonic gap rows that widen steadily point toward this family, whereas gap rows that oscillate in sign point away from it and toward alternation structures described later.

Quick example — Series continuation: Terms run 5, 9, 14, 20, 27 with first gaps $9 - 5 = 4$, $14 - 9 = 5$, $20 - 14 = 6$, $27 - 20 = 7$.

Second gaps are all $1$, so the generator is quadratic and the gap row simply extends by one.

The next gap is $7 + 1 = 8$, giving continuation $27 + 8 = 35$.

The discipline is to extend the gap row, never to guess at the terms directly.

2.1.2 Multiplicative chains with corrections

The second family generates each term from its predecessor by scaling followed by a small adjustment, in symbols $a_{i+1} = k \cdot a_i + c_i$ for a fixed multiplier $k$ and a correction sequence $c_i$. The pure case carries a constant multiplier with zero correction, visible as near-constant ratios between neighbours. The corrected case carries a multiplier of two or three together with a drifting correction such as a steadily incrementing addition. Recognition logic rests on ratios: when each term is roughly double or treble its predecessor, the multiplicative hypothesis is active even when the ratios are not exact, because the residual after dividing out the multiplier forms the low-magnitude correction row. The correction row is then inspected exactly as a difference row was inspected, and linear drift in the correction confirms the family. Rapid growth that outpaces any plausible difference drift is the characteristic signature that promotes this family above the difference family in diagnostic priority once ratios look stable.

2.1.3 Polynomial offsets around squares and cubes

The third family places each term near a perfect square or a perfect cube, in symbols $a_i = n^2 + c$ or $a_i = n^3 + c$ for a small fixed offset $c$ that may be positive or negative. Recognition proceeds by proximity comparison: each observed term is set beside the nearest square or cube, and the offsets are inspected for constancy. A stable offset across successive positions confirms the family, while drifting offsets reject it. Growth rate discriminates between the square and cube variants, since cubic proximity grows far faster than quadratic proximity. The offset itself carries meaning: an offset of positive one above squares behaves very differently under differencing from an offset of negative one below cubes, yet both are recognised by the same constancy-of-residual test. This family is considered only after additive and multiplicative readings have failed, because proximity judgments cost more attention than gap or ratio inspection.

2.1.4 Interleaving and alternation structures

The fourth family is not a single generator but a shuffle of two independent generators occupying alternate positions. Its signature is irregularity under every single-generator test: gaps that swing between small and large values, ratios that refuse to settle, and residuals that will not stabilise. The recognition move is positional splitting: terms in odd positions are read as one subsequence and terms in even positions as another, after which each subsequence is subjected independently to the difference and multiplicative diagnostics. When both subsequences resolve into clean linear or scaling patterns, the interleaving hypothesis is confirmed. A common sub-variant advances one subsequence by a fixed step and the other by a different fixed step, which explains why the merged gap row alternates between two values. The structural moral is that apparent disorder at the merged level is compatible with perfect order at the split level.

2.1.5 Prime skeletons

The fifth family uses the prime sequence as scaffolding, either as the terms themselves or as the gap row between terms. Its signature is a gap row composed entirely of prime magnitudes advancing in order, or a term row that coincides with successive primes. Recognition logic depends on familiarity with the early prime order up to modest magnitudes, stated here in words: two, three, five, seven, eleven, thirteen, seventeen, nineteen, and so on. When observed gaps follow this order, the generator is identified as prime-driven, and continuation follows the next prime in sequence rather than any arithmetic extrapolation. This family is tested last because prime recognition demands retrieval of a memorised sequence, whereas the earlier families demand only subtraction, division, or proximity comparison.

2.1.6 Recognition order as diagnostic discipline

The five families are examined in a fixed order: differences first, then multiplicative chains, then polynomial offsets, then interleaving, and finally prime scaffolding. The ordering reflects rising cognitive cost, so that the cheapest discriminating observation is always spent before costlier ones. Each stage is given only a brief inspection, and the first family that fits cleanly governs continuation. Persistent failure across all five families indicates a composite or disguised generator, and the disciplined response is elimination of whatever readings are refuted followed by disciplined skipping rather than prolonged staring.

Alphabet progressions are arithmetic progressions in disguise, with letters serving as numerals under a fixed positional mapping. All analysis begins by converting letters to positions, after which the full apparatus of gap and ratio reasoning applies unchanged.

2.2.1 Positional numbering and the complement constant

Under the standard mapping, the first letter of the alphabet carries position one and the last carries position twenty-six, with every intermediate letter carrying its natural rank and the five vowels occupying positions one, five, nine, fifteen, and twenty-one. The central structural fact is that a letter and its alphabetic opposite always sum to twenty-seven, in symbols $p + q = 27$ where $p$ and $q$ are the two positions. The opposite of any letter numbered $n$ is therefore the letter numbered $27 - n$. This complement relation explains alternations in which one subsequence climbs from the start of the alphabet while the interleaved subsequence descends from the end, since each descending term is the complement of a steadily advancing hidden counter. Recognition of complement structure proceeds by summing candidate pairs and watching for the constant twenty-seven.

2.2.2 Shift, skip, and reverse families

Constant-step progressions advance every term by the same signed displacement, corresponding to uniform gaps such as plus three or minus two in positional value. Expanding-gap progressions advance by displacements that themselves form a simple arithmetic row, such as increments of two, then three, then four, then five, so that the step row rather than the term row carries the regularity. Skip progressions step through distinguished subsets such as vowels, where a positional advance of four lands repeatedly on vowel positions. Reverse progressions descend rather than climb, and their recognition is identical to ascent recognition with negated gaps. Complement alternations interleave a rising subsequence with its falling mirror, and the diagnostic is pairwise summation to the complement constant across the interleave boundary.

2.2.3 Alphanumeric coupling

Displays that pair each letter with a numeral carry two synchronised generators, one alphabetic and one numeric. The standard coupling advances the letter by a constant positional step while the numeral follows an independent regularity such as uniform increment or successive squares with offsets of the form $n^2$. Decoupling is the recognition move: the letter track and the numeral track are analysed separately, each by its own family logic, and the two confirmed continuations are recombined. Apparent complexity at the combined level thus dissolves into two elementary regularities once the tracks are separated.

Coding displays are input-output pairs generated by a fixed transformation applied uniformly to every element. Decoding is therefore the inverse task of identifying the transformation class from its observable invariants rather than memorising isolated pairings.

2.3.1 Additive codes

The additive class maps a word to the sum of the positional values of its letters, in symbols $C = \sum p_i$ where $p_i$ ranges over the positions of the letters. Its invariant is numeric output: whenever the coded form is a numeral while the source is alphabetic, the additive hypothesis is strongly favoured, since shift and complement transformations preserve alphabetic type. Confirmation proceeds by summing the positions of a fully known pairing and checking equality with the displayed numeral. A second known pairing is then summed independently, and agreement on both pairings establishes the class before the transformation is applied to the target word.

2.3.2 Shift codes

The shift class displaces every letter by a fixed signed amount modulo the alphabet length, in symbols $q_i = p_i + k$ with wraparound at the alphabet boundary and a constant integer $k$ that may be positive for forward displacement or negative for backward displacement. Its invariant is preservation of inter-letter gaps: the positional gaps between consecutive letters are identical before and after transformation. Recognition therefore compares the first letters of source and coded forms to extract the candidate displacement, then verifies constancy of that displacement across every remaining letter position. Uniform displacement confirms the class, while any positional drift refutes it.

Quick example — Shift coding: Source CAT maps to coded FDW under a uniform forward shift.

Position gaps are $6 - 3 = 3$, $4 - 1 = 3$, $23 - 20 = 3$, so $k = 3$ with preserved inter-letter gaps.

Applying the same shift to DOG gives $4 + 3 = 7$, $15 + 3 = 18$, $7 + 3 = 10$, hence GRJ.

The invariant is constant displacement at every position, confirmed before touching the target.

2.3.3 Complement and reversal codes

The complement class replaces each letter by its alphabetic opposite under the relation $q_i = 27 - p_i$. Its invariant is pairwise summation to twenty-seven at every aligned position. Recognition proceeds by adding each source letter to its aligned coded letter and watching for the complement constant throughout. The reversal class instead preserves the multiset of letters while inverting their order, sometimes combined with a shift or complement applied after reversal. Its invariant is multiset identity between source and coded forms. The two classes are distinguished by type preservation with gap inversion in the reversal case versus positional complementation in the complement case.

2.3.4 Diagnosis by output type and invariance

The three classes are tested in an order dictated by observational cost. Numeric output selects the additive class immediately. Alphabetic output with preserved gaps selects the shift class. Alphabetic output with complement sums selects the complement class. Alphabetic output with preserved letter multiset but inverted order selects reversal, possibly compounded. Each test rules its class in or out decisively, so that diagnosis terminates at the first confirmed invariant rather than accumulating partial resemblances.

Kinship descriptions encode a small family graph in words. The solver externalises the graph on paper under strict notational discipline, because nearly all errors in this area arise from carrying gender or generation information in memory rather than in the diagram.

2.4.1 Notation and construction discipline

Generations are laid out from top to bottom, with each horizontal layer representing one generation. A square node denotes a male individual and a circular node denotes a female individual, while an individual of unspecified gender is left as an unmarked placeholder until further information arrives. A horizontal double segment between two nodes denotes a marital union, and a vertical segment from a union or an individual down to a lower node denotes parentage. Siblinghood is represented by placement on a shared generational layer beneath a common parent node. The governing discipline is that gender is inscribed on every node before any relational inference is drawn, and no node is assigned a gender that the wording does not supply. Relational vocabulary such as father, mother, brother, sister, uncle, aunt, mother-in-law, and sibling is then read directly off the finished graph as a path description between nodes rather than reasoned out in prose.

2.4.2 Inside-out reading of pointing descriptions

Descriptions in which a speaker points to a depicted person and characterises that person through a chain of relations resolve from the innermost clause outward. The innermost possessive phrase is converted first into a small subgraph, and each enclosing phrase then extends that subgraph one edge at a time until the full chain connects the speaker to the depicted person. Pronoun anchoring is the delicate step: possessives attached to the depicted person extend from that end of the graph, while possessives attached to the speaker extend from the speaker end, and the two growing fragments meet in the middle. Where wording leaves the speaker gender open, both completions are retained as parallel readings until the surviving display options eliminate one of them.

2.4.3 Kinship vocabulary

The following table fixes the path meaning of each standard term. All compound relations are compositions of these elementary paths.

Term	Path meaning
Father	Male parent of the reference person
Mother	Female parent of the reference person
Son	Male child of the reference person
Daughter	Female child of the reference person
Brother	Male child of a parent of the reference person
Sister	Female child of a parent of the reference person
Husband	Male marital partner of the reference person
Wife	Female marital partner of the reference person
Father-in-law	Father of the marital partner of the reference person
Mother-in-law	Mother of the marital partner of the reference person
Brother-in-law	Brother of the marital partner, or husband of a sibling of the reference person
Sister-in-law	Sister of the marital partner, or wife of a sibling of the reference person
Uncle	Brother of a parent of the reference person, or husband of a sibling of a parent
Aunt	Sister of a parent of the reference person, or wife of a sibling of a parent
Nephew	Son of a sibling of the reference person
Niece	Daughter of a sibling of the reference person
Cousin	Child of a sibling of a parent of the reference person
Grandfather	Father of a parent of the reference person
Grandmother	Mother of a parent of the reference person
Grandson	Son of a child of the reference person
Granddaughter	Daughter of a child of the reference person

Direction displays combine a fixed compass frame with a sequence of translations and rotations. All reasoning is performed against an explicitly drawn compass so that left and right rotations are never evaluated mentally.

2.5.1 Bearings and turn geometry

The compass frame places north at the top, south at the bottom, east to the right, and west to the left, with the four diagonal bearings bisecting the quadrants. A right turn rotates the heading clockwise through the compass order, while a left turn rotates it counter-clockwise, each turn being measured in multiples of a right angle unless an explicit degree magnitude is stated. Facing reversals and successive orthogonal legs are tracked by updating the heading after every rotation before advancing along the next translation. The discipline of redrawing the heading after each turn prevents the characteristic error of applying a later left-right instruction to an outdated heading.

2.5.2 Shadow astronomy

Shadow direction is determined by solar position. Near sunrise the sun stands in the east, so every vertical object casts its shadow toward the west. Near sunset the sun stands in the west, so every shadow points toward the east. Around midday shadows fall very short and their directional information is unreliable, which is why timed displays always specify morning or evening. The hand rule converts shadow information into heading information: the solver imagines facing each candidate heading and checks on which hand the known shadow side would fall. Agreement between the imagined hand side and the stated shadow side identifies the heading. The astronomy behind the rule is that shadow direction is a function of time alone, while the side on which a walker observes that shadow is a function of heading, so the pairing of the two determines orientation uniquely.

2.5.3 Orthogonal returns and cancellation

When two consecutive legs of a walk meet at a right angle with lengths $a$ and $b$, the shortest return path from the endpoint to the start is the hypotenuse of the right triangle, of length $\sqrt{a^2 + b^2}$, lying along the diagonal bearing that combines the two leg directions. The classic three-four-five configuration is the most frequently encountered instance of this relation. When a walk contains two equal and opposite legs along the same axis, those legs cancel exactly, and the net displacement reduces to whatever transverse legs remain. Longer walks are therefore analysed by resolving every leg into north-south and east-west components, cancelling opposite components algebraically, and applying the hypotenuse relation once to the surviving pair. Total path length and net displacement are distinct quantities throughout: the former sums magnitudes while the latter sums signed vectors.

Syllogistic items present two or more stated premises and invite judgment about which further statements are forced to be true. The semantics is set-theoretic: each categorical statement constrains the overlap pattern of two classes, and a conclusion is valid exactly when it holds in every overlap pattern compatible with the premises.

2.6.1 Venn semantics of the four categorical forms

Each of the four categorical forms carries a canonical diagrammatic meaning. A universal affirmative statement of the form all of one class lie in another is rendered as complete containment of the first disc inside the second. A universal negative statement of the form no member of one class lies in another is rendered as two disjoint discs. A particular affirmative statement of the form some members overlap is rendered as two intersecting discs with the lens region marked as occupied. A particular negative statement of the form some members of the first class lie outside the second is rendered with a marked region of the first disc exterior to the second. Validity is then a universal quantification over diagrams: the conclusion follows exactly when no diagram satisfying all premises can simultaneously falsify the conclusion. A single compatible diagram that falsifies the conclusion refutes it permanently.

2.6.2 Distribution of terms

A term is distributed in a statement when the statement makes a claim about every member of the corresponding class. The universal affirmative distributes its subject but not its predicate: a claim that every member of the first class lies in the second says nothing about members of the second class outside the first. The universal negative distributes both terms, since disjointness is symmetric information about both classes. The particular affirmative distributes neither term, since it asserts only the existence of shared members. The particular negative distributes its predicate but not its subject. These facts explain the characteristic invalid moves: any inference that treats an undistributed term as though it were distributed overreaches the premises. The undistributed middle, in which two classes each lie inside a third without any forced relation between themselves, and the illicit leap from a partial overlap to a universal claim, are both instances of this single error.

2.6.3 The exclusive translation and why reversal fails

A statement of the form only members of one class occupy another is logically equivalent to the universal affirmative with subject and predicate exchanged: every member of the second class lies in the first, in symbols the exclusive form with classes $A$ and $B$ translates to all $B$ are $A$. Diagrammatically the second disc sits entirely inside the first, leaving a crescent of the first disc that may or may not be occupied. The un-reversed reading, which places the first disc inside the second, therefore asserts strictly more than the premises supply, and the crescent region furnishes an immediate falsifying diagram whenever that stronger claim is drawn. The trap display exploits exactly this asymmetry by offering the un-reversed universal as though it were a paraphrase.

Quick example — Only reversal: The exclusive premise reads only citizens can vote, rendered as all voters are citizens.

In symbols, with $V$ for voters and $C$ for citizens, the premise is all $V$ are $C$, with disc $V$ inside disc $C$.

The valid limited conversion is some $C$ are $V$, carried by the occupied inner region.

The trap is the unreversed universal all $C$ are $V$, refuted by the possibly empty crescent of $C$ outside $V$.

2.6.4 The converse error and affirming the consequent

From premises of the form all members of one class lie in another, together with the information that a particular individual lies in the second class, nothing follows about membership in the first. Diagrammatically the individual may sit in the crescent of the outer disc that lies outside the inner disc, a placement compatible with every premise. The converse inference that the individual belongs to the inner class affirms the consequent of the universal conditional and is therefore fallacious in every instance. The same diagram refutes the symmetric universal that all members of the outer class lie in the inner class. Both fallacies share one root: containment is directional, and information about the container never locates an individual within the contained region.

2.6.5 The existential commitment

Classical syllogistic reading as used in these displays carries existential import for the subject classes of universal statements: a universal premise is understood to concern a non-empty subject class, so that containment of one disc inside another guarantees occupancy of the inner region. Under this convention, conversion by limitation is legitimate: from the containment of the first disc inside the second follows the particular claim that some members of the second class lie in the first, because the inner region is both occupied and shared. The convention is stated explicitly here because modern predicate logic withholds this commitment and would withhold the corresponding inference. Within the examination convention, the commitment stands, and the limited conversion is valid while the full reversal remains invalid.

An assumption is an unstated premise without which the stated position collapses. Three cumulative filters separate genuine assumptions from distractors. The unstated filter rejects any candidate already asserted by the wording itself, since restatement is not assumption. The necessity filter applies negation: the candidate is denied, and the consequences for the original statement are inspected. When denial destroys the coherence or support of the statement, the candidate is load-bearing and therefore genuinely assumed; when denial leaves the statement intact, the candidate is decorative rather than structural. The moderation filter rejects candidates containing absolute quantifiers or evaluative superlatives and candidates importing specialised background knowledge that the statement never invokes. Survivors of all three filters are exactly the presuppositions the speaker relies upon without articulating.

Seating displays present a set of individuals, a fixed geometry of positions, facing conditions, and relational constraints between occupants. The structure is a constraint-satisfaction task, and the solving method is propagation from a fixed anchor rather than parallel maintenance of alternative diagrams.

2.8.1 Fixing the anchor

The most constrained individual, meaning the one named in the greatest number of conditions or occupying a structurally distinguished position such as an endpoint of a row or a diametrically opposed seat in a circle, is placed first. In circular geometries one placement fixes the rotational symmetry of the entire display, so the anchor may be assigned to any convenient position, conventionally the top, with all further placements read relative to it. In linear geometries the anchor is the individual tied to an endpoint or to the centre, since such ties eliminate translational freedom. Correct anchor choice converts most remaining conditions into forced neighbour placements.

2.8.2 Propagation by neighbourhood filling

Each subsequent individual is placed adjacent to an already placed individual through the first condition that connects them. Branching conditions are resolved eagerly: when a condition admits two completions, each completion is tested immediately against the remaining conditions, and the completion that contradicts any of them is discarded at once. This eager refutation keeps at most one live diagram on paper at any moment and prevents the exponential accumulation of parallel hypotheses. The process terminates when every individual occupies a position, at which point each original condition is rechecked against the finished display as a final consistency pass.

Quick example — Seating fill: Six seats numbered $0$ to $5$ clockwise, all facing the centre, with A anchored at $0$.

B sits immediate left of A, and left for centre-facing means clockwise, so B takes $0 + 1 = 1$.

C sits opposite A, hence at $0 + 3 = 3$, and D sits immediate right of C, hence counter-clockwise at $3 - 1 = 2$.

The method is anchor first, then fill only through neighbours of placed occupants, rechecking every condition on the finished ring.

2.8.3 Left-right orientation in circular displays

Orientation vocabulary depends on facing direction. For occupants facing the centre of a circle, the left-hand side corresponds to the clockwise neighbour and the right-hand side to the counter-clockwise neighbour, verifiable by imagining an occupant at the twelve o clock position facing inward: the left hand then points toward the three o clock position, which is the clockwise direction of travel. For occupants facing away from the centre the correspondence reverses. Fixing this correspondence on paper before any placement prevents the systematic mirror error that otherwise corrupts every lateral condition.

Non-verbal displays test recognition of rigid motions and their invariants rather than vocabulary or arithmetic. Each transformation is characterised by what it preserves and what it exchanges.

2.9.1 Mirror versus water geometry

A vertical mirror placed beside a figure implements left-right reversal: every point is mapped across the vertical axis, so the top-bottom order is preserved while the left-right order is inverted. A horizontal water surface below the figure implements top-bottom reversal: every point is mapped across the horizontal axis, so the left-right order is preserved while the top-bottom order is inverted and the vertical sequence of stacked elements reverses. The symmetry inventories follow directly. Glyphs symmetric about a vertical axis are invariant under mirror reflection, while glyphs symmetric about a horizontal axis are invariant under water reflection. Asymmetric features such as shaded halves, tails, and arrowheads decide between competing displays, because a single asymmetric corner flips in opposite senses under the two transformations and therefore eliminates every display that applies the wrong reversal.

Quick example — Mirror pick: Original glyph R carries its bowl at the top and its leg at the bottom right.

A vertical mirror maps $(x, y)$ to $(-x, y)$, so the mirror form keeps the bowl at the top with the leg flipped to the bottom left.

A water surface maps $(x, y)$ to $(x, -y)$, so the water form keeps left-right layout with the bowl dropped to the bottom.

The fast pick is the bowl height, since top bowl selects mirror and bottom bowl selects water.

2.9.2 Dice opposite-face logic

The faces of a cube form three opposite pairs, and each face is adjacent to exactly four others. Two views of one die that share a visible face jointly exhibit up to four distinct neighbours of that face, and any face that never appears beside a given face across the available views is its opposite. The neighbour-exclusion principle is the working form of this logic: adjacency observed in any single view permanently rules out opposition between the two faces concerned. Rotational comparison of the shared face across views then pairs off the remaining side faces. The standard-die convention that opposite faces sum to seven is a special manufacturing rule and is invoked only when the wording explicitly designates the die as standard; otherwise opposition is derived purely from observed adjacency.

2.9.3 Paper fold symmetry

Folding superposes layers, and every perforation or cut made through the folded packet replicates once per layer upon unfolding. Reconstruction proceeds in reverse chronological order: the most recent fold is unfolded first, with each replicated mark mirrored across the corresponding fold line, and the process repeats until the sheet is flat. A single perforation through a twice-folded packet therefore yields four symmetrically placed marks arranged in a rectangle, with each unfolding doubling the pattern across one axis. Cuts that remove edge or corner regions of the folded packet produce correspondingly symmetric apertures, each mirrored across every fold line that covered its position. The invariant throughout is mirror symmetry of the final pattern with respect to every fold axis.

Number progressions resolve into five generative families ordered by diagnostic cost: difference hierarchies recognised through gap-row regularity, multiplicative chains recognised through near-stable ratios with drifting corrections, polynomial offsets recognised through constancy of residuals against squares and cubes of the form $n^2 + c$ and $n^3 + c$, interleavings recognised through positional splitting into independently regular subsequences, and prime scaffolding recognised through ordered prime gaps. Alphabet progressions are positional arithmetic under the mapping that sends the first letter to one and the last to twenty-six, with the complement constant $27$ governing opposite-pair structure and with shift, skip, and reverse families distinguished by gap constancy, gap drift, and directional sign. Coding displays realise one of three transformation classes: additive summation of the form $C = \sum p_i$ betrayed by numeric output, uniform displacement of the form $q_i = p_i + k$ betrayed by preserved gaps, and complementation of the form $q_i = 27 - p_i$ betrayed by constant pairwise sums, with reversal distinguished by preserved multisets under inverted order. Kinship descriptions externalise into layered graphs with square, circular, and placeholder nodes joined by marital and parental segments, resolved inside-out for pointing displays and read through a fixed vocabulary of elementary paths. Direction displays operate against an explicit compass in which right turns advance clockwise and left turns advance counter-clockwise, with morning shadows pointing west and evening shadows pointing east, and with orthogonal legs resolving through the hypotenuse relation $\sqrt{a^2 + b^2}$ after cancellation of opposed components. Syllogistic validity is truth in every Venn diagram compatible with the premises, with containment, overlap, and disjointness rendering the universal affirmative, the particular affirmative, and the universal negative respectively; distribution theory explains why undistributed terms cannot support universal inferences; the exclusive form translates directionally rather than symmetrically; the converse inference from container membership to contained membership is fallacious; and the examination convention retains existential import for subject classes. Assumption displays filter candidates through the unstated, necessity-by-negation, and moderation requirements. Seating displays anchor the most constrained occupant and propagate placements by neighbourhood adjacency with eager refutation of branches, under the orientation rule that clockwise is left for centre-facing occupants. Non-verbal displays distinguish left-right mirror reversal from top-bottom water reversal, derive dice opposition by neighbour exclusion across shared-face views, and reconstruct fold patterns by reverse-chronological mirroring across each fold axis.