# The PRC Construction Method: Reconciling the Extremities **Evidence basis:** the constants below are measured on the PRC inventory (MMah `graphics.txt`, Arphic Public License) over the GB2312 set; the rules are implemented in [`code/prc_construct.py`](../code/prc_construct.py) and covered by [`code/test_prc_construct.py`](../code/test_prc_construct.py). > **Evidence boundary:** nothing in this document certifies that a drawing is > readable. Legibility is perceptual and this project has no reader evidence > for the PRC inventory yet (see [SCOPE.md](../SCOPE.md) and [findings/limitations.md](../findings/limitations.md)). Every drawing > produced here is a **candidate** (候选图形). The measurements say how well > the ink agrees with itself under a half-turn; they say nothing about > whether a reader would accept it. --- ## 1. Why the PRC inventory needs its own method The Japanese pipeline works on **skeletons**: KanjiVG provides stroke outlines, the skeleton is extracted, and the pair search runs on the centre-lines. The PRC source is different. MMah's `graphics.txt` carries **stroke medians** — the medial axis of each stroke — together with the outlines. Two consequences follow. 1. **The centre-line is the right object.** A PRC form is written as a sequence of strokes; the median is the spine of the stroke and the ink around it is the stroke's *weight*. The construction questions are therefore questions about weight, direction and extent at the ends of the spines. 2. **The ink is not half-turn invariant, even when the centre-line is.** A frame such as 田 has a left vertical that is a wedge and a right vertical that is thin; the half-turn maps the wedge onto the thin stroke. The skeleton says the frame is symmetric, the ink says it is not. Measuring symmetry on the ink rather than on the centre-line is what makes the frame family visible at all — the centre-line measure produced false negatives for 口, 田, 目, 日 and 回. ## 2. The measurement that decides everything: the rotation centre A half-turn about a centre `c` is `R_c(p) = 2c − p`. On a 109 × 109 grid, moving the centre by `d` translates the half-turned glyph by `2d`. The centre is therefore a **translation** parameter, and the project already searches it for the Japanese set ([stage 8](../code/stage8_optimal_center.py)). On the PRC data this parameter turns out to dominate everything else. **Measured.** The measure is the **symmetric ink**: the share of the *original* ink that survives the half-turn, measured on a 109 × 109 raster. Measuring against the original and not against the edited drawing is what stops a rule from winning by erasing — see §7. Over the 120 best PRC candidates: | stage | median | mean gain | |---|---|---| | centre of the em square | 0.167 | — | | centre of the ink (centroid of the medians) | 0.192 | +0.025 | | searched centre | 0.240 | **+0.093** | | searched centre + the extremity rules | 0.243 | +0.002 | | final | 0.252 | **+0.106** total | Per glyph, the centre alone: | glyph | centre of the em square | searched centre | gain | |---|---|---|---| | 一 | 0.006 | 0.835 | +0.829 | | 囗 | 0.287 | 0.551 | +0.264 | | 垂 | 0.381 | 0.482 | +0.101 | | 团 | 0.310 | 0.479 | +0.169 | | 囝 | 0.319 | 0.475 | +0.156 | | 回 | 0.169 | 0.463 | +0.294 | | 圃 | 0.417 | 0.460 | +0.043 | | 国 | 0.171 | 0.410 | +0.239 | | 囿 | 0.328 | 0.435 | +0.107 | | 口 | 0.147 | 0.392 | +0.245 | The optimal offset is **systematically +2.5 units in y** (2.3% of the box side) for almost every glyph. The source font places the ideographic square above the geometric centre of the em square, so the centre of the em square is the wrong centre for every PRC form. **Rule R0**: rotate about the ink centre, not about the centre of the square. **This is the largest single lever in the whole method, and it is larger than every extremity rule put together.** The centre alone yields a **mean paired gain** of approximately +0.0925 in symmetric ink (the **median paired gain** is +0.0692); the extremity rules yield a **mean paired gain** of approximately +0.0019. $$ c^* = \arg\min_{c \in \mathcal{C}} \sum_{p \in P(A)} d\bigl(p, R_c(A)\bigr), \qquad \mathcal{C} = c_{box} + \{-8,\ldots,+8\} $$ with `c_box` the centre of the em square. A cheap and accurate surrogate is the centroid of the medians, refined on a ±1.5-unit grid. ## 3. The geometry of an extremity Every stroke has two extremities. An extremity carries: * a **place** — a point; several extremities of different strokes can share it (the horizontal and the vertical both end at a frame corner); * a **weight** — the half-width of the stroke there, measured as the distance from the median to the nearest outline point; * a **direction** — the tangent of the spine at that end. **Measured.** The half-width profile of the PRC strokes is not flat. The two ends of the same stroke differ by a median of **0.30 to 0.83 box units** while the median weight of a stroke is 1.36 to 2.09 units. The ratio of the lighter end to the heavier end, averaged over the strokes of a glyph, lies between **0.59 and 0.86**. In other words, a PRC stroke carries a systematic weight gradient along its length, and the half-turn maps the heavy end of a stroke onto the light end of its counterpart. **Measured — where the residual error lives.** With the centre corrected, the error of each sample of the drawing (its distance to the ink of the other reading) splits into two zones, the last 6 units of a stroke (the extremity zone) and the body of the stroke: | zone | mean error | share of the samples | |---|---|---| | extremity (last 6 units) | 3.81 | 28% | | body of the stroke | 2.67 | 72% | The extremity zone carries **35.7%** of the residual error in 28% of the samples: a concentration of 1.28× in the error and 1.43× in the mean error per sample. The extremities are not where most of the error is, but they are where the error is *most concentrated*, and they are the only place where the two readings can be reconciled by a local edit. **Measured — how far apart the two readings' extremities are.** For every extremity of a drawing, the distance to the nearest extremity of the other reading: | percentile | 50 | 75 | 90 | 95 | 99 | |---|---|---|---|---|---| | distance (box units) | 7.99 | 12.35 | 20.20 | 26.94 | 38.74 | This distribution fixes the two radii the rules need: * **δ_f = 8.0** — the fusion radius. Two extremities closer than this are the same place seen from the two readings. * **r_c = 12.4** — the contested radius. Beyond this the two readings are not fighting over the same ink any more. ## 4. The rules Let `A` be a drawing, `c` its rotation centre, `P` the set of places it occupies, and `R = R_c` the half-turn. ### R0 — the centre Rotate about `c*` (§2). This is a translation of the half-turned glyph and changes nothing else. ### R1 — equalise the weight of the rotation-matched places For a place `P` whose half-turn image `R(P)` falls on another place `Q` (`‖R(P) − Q‖ ≤ δ_f`), the two places carry the same weight: $$ w^*(P) = w^*(Q) = \max\bigl(w(P),\, w(Q)\bigr), \qquad w_{new}(t) = w(t) + \bigl(w^* - w(t)\bigr)\,\psi\!\left(\frac{s(t)}{\kappa}\right) $$ with `s(t)` the arc distance from the extremity and the taper profile $$ \psi(u) = (1-u)^2 \;\text{ for } 0 \le u \le 1, \qquad \psi(u) = 0 \text{ otherwise}. $$ The heavier place is left alone and the lighter one is thickened. `κ = 6` units: the thickening is spread over the last 6 units of the stroke, which is the terminal zone used in §3. ### R2 — the direction of an extremity The half-turn reverses a tangent, so the two ends of a rotation-matched pair must face opposite ways. The deviation is measured, not corrected: $$ \Delta\theta = \Bigl| \arg\bigl(\tan Q\bigr) - \arg\bigl(-\tan P\bigr) \Bigr| $$ This is where the PRC forms fail most visibly: a hook (亅, 乚, 乙) or a wedge-ended vertical that is exclusive to one reading points the wrong way once the drawing is turned. **R2 is the rule with the least support in this document**: the measurement of the deviation exists, the correction does not. [PLAN] ### R3 — R-STRUCT: split the stroke at the corner When the half-turn image of a place `P` falls on the **body** of a stroke rather than on one of its extremities, the two orientations do not agree on the drawing's extremity structure, and no re-weighting can repair it. The repair is to **split** the stroke at `P`, so that `P` becomes an extremity in both orientations: $$ \text{stroke} \;\longrightarrow\; \bigl(\text{median}[0 \ldots j],\; \text{median}[j \ldots]\bigr), \qquad \text{median}[j] \leftarrow P $$ The ink is unchanged; only the way the strokes are counted changes. This matters for the PRC reading: the enclosure 囗 draws its top and right sides as one stroke (横折), so splitting it turns 口 from three strokes into four and 囗 from three into four, against the stroke-order convention. The construction accepts that cost because the ambigram is judged on the ink. The alternative — trimming the stroke back to `P` — destroys the part of the stroke that lies between its end and `P`. It was implemented first and measured: it loses more ink than it gains. If the image falls further than the orphan tolerance (2 units) from any stroke, the place is an **orphan**: the drawing has a corner in one orientation that does not exist in the other, and no local edit can fix it. Orphans are the honest boundary of the method. [MEASURED] ### R4 — frame closure For a 囗-family form the frame is the dominant structure, and its centre must coincide with `c*`. The frame's own centre is measurable from the membership of the four frame strokes; when it differs from `c*` by `Δ`, either `c*` moves to the frame centre or the frame group is displaced by `Δ`. On the measured data this single correction is worth up to **+0.24** of symmetric ink (国: 0.171 → 0.410 on the raster measure, 0.339 → 0.763 on the centre-line measure). [MEASURED] ### R5 — stroke-count reconciliation Two readings can need a different number of independent strokes. A stroke that has no counterpart in the other reading must either be fused into a neighbour — its weight ramped to zero over `κ` — or trimmed to a stub, or the two readings will not agree on how many strokes they see. The PRC forms make this acute: 563 of the 5,196 codepoints shared with the Japanese inventory differ in stroke count (10.8%), 骨 among them (10 strokes in the Japanese form, 9 in the PRC form). [MEASURED for the data; the rule itself is [PLAN]] ## 5. The objective The rules above are applied to minimise a single objective over the free parameters — the centre `c`, the terminal weights `w(τ)` and the terminal positions `p(τ)`: $$ E(c, w, p) = \alpha\,\bigl(1 - \mu(A,B)\bigr) + \beta\,\bigl(1 - \pi(A,B)\bigr) + \gamma \sum_{\tau} c(\tau) + \rho \sum_{\tau} \mathbb{1}[\tau \text{ orphan}] $$ subject to the constraint that **no reading loses more than 20% of its ink weight**: $$ \sum_{s \in \text{strokes}(X)} \int w_s(t)\,\lVert s'(t) \rVert \, dt \;\ge\; 0.8 \sum_{s \in \text{strokes}(X_0)} \int w_s^{(0)}(t)\,\lVert s'(t) \rVert \, dt, \qquad X \in \{A, B\} $$ where `μ` is the ink overlap, `π` the share of places that are rotation-matched, `c(τ)` the contested flag of an extremity and `X₀` the original drawing. The weights `α, β, γ, ρ` are not calibrated: no reader evidence exists for the PRC inventory, so the objective is used only as a search direction, never as a legibility score. [PLAN] ## 6. The two groups 120 PRC candidates were measured with the published module. The value used is the **final symmetric ink**: the share of the original ink that is symmetric under the half-turn, after the centre has been searched and the rules applied. The retention of the original ink is 1.005 in the median, so nothing is being bought by erasing. ### Group A — readable in both orientations almost as they stand Final value ≥ 0.45. Eleven candidates: | glyph | partner | centre of the box | searched centre | final | |---|---|---|---|---| | 一 | 一 | 0.006 | 0.835 | **0.835** | | 丨 | 丨 | 0.000 | 0.737 | **0.737** | | 囗 | 囗 | 0.287 | 0.551 | **0.551** | | 垂 | 罟 | 0.381 | 0.482 | **0.482** | | 团 | 因 | 0.310 | 0.479 | **0.479** | | 囝 | 囝 | 0.319 | 0.475 | **0.475** | | 因 | 团 | 0.198 | 0.473 | **0.473** | | 困 | 园 | 0.271 | 0.464 | **0.464** | | 回 | 回 | 0.169 | 0.465 | **0.465** | | 圃 | 圜 | 0.417 | 0.460 | **0.462** | | 壬 | 壬 | 0.430 | 0.456 | **0.456** | The group is dominated by the **enclosure family** — 一, 丨, 囗 and the 囗-derived forms — and by the strokes that are symmetric on their own. These are the forms whose *whole* structure is half-turn symmetric: the enclosure closes on itself, and the interior either is symmetric too (回, 囝) or is small enough not to dominate the ink. ### Group B — possible, but the drawing has to be worked on Final value between 0.30 and 0.45. Thirty candidates; the strongest: | glyph | partner | centre of the box | searched centre | final | |---|---|---|---|---| | 囹 | 囵 | 0.314 | 0.441 | **0.443** | | 围 | 囡 | 0.311 | 0.436 | **0.437** | | 囿 | 囵 | 0.328 | 0.435 | **0.435** | | 固 | 囵 | 0.373 | 0.428 | **0.430** | | 日 | 日 | 0.431 | 0.428 | **0.431** | | 圆 | 圆 | 0.407 | 0.413 | **0.416** | | 国 | 国 | 0.171 | 0.410 | **0.414** | | 口 | 口 | 0.147 | 0.392 | **0.399** | | 圈 | 圈 | 0.308 | 0.387 | **0.393** | | 噩 | 罟 | 0.082 | 0.390 | **0.393** | | 囫 | 囫 | 0.316 | 0.386 | **0.386** | | 圉 | 圉 | 0.307 | 0.376 | **0.378** | These are the forms whose **frame** is symmetric but whose **interior is not**. 国 is the clearest case: the enclosure 囗 can be made half-turn symmetric, but the 玉 inside it cannot — the middle horizontal of 玉 is at the bottom of the character and the vertical is off-centre. Turning the drawing turns the interior into a shape that is no longer 玉. No local edit fixes that; the interior has to be redesigned. That is a construction problem, not a tuning problem. The three-level split (A ≥ 0.45, B 0.30–0.45, C < 0.30) puts 79 of the 120 candidates in C. The distribution of the final value is p10 0.184, p25 0.225, p50 0.252, p75 0.361, p90 0.438, p95 0.474, max 0.835. ### What separates A from B, measured | | Group A | Group B | |---|---|---| | places with a counterpart under the half-turn | 1–6 of 2–17 | 1–6 of 7–20 | | orphan places | 0–7 | 2–9 | | strokes split at a corner (R-STRUCT) | 0–2 | 0–5 | | symmetric ink after the centre alone | 0.44–0.84 | 0.39–0.44 | The discriminator is **not** the number of strokes and **not** the weight gradient. It is whether the half-turn maps the *stroke decomposition* onto itself. Group A does; Group B does not, and the gap has to be closed by redrawing the interior, not by trimming the ends. ## 7. What the rules actually buy, measured The rules were implemented and run over the 120 candidates. The result is mostly negative, and it is worth stating plainly. | quantity | value | |---|---| | places (corners where strokes end) measured | 2,065 | | places with a counterpart under the half-turn | **421** (20%) | | orphan places | **938** (45%) | | candidates the rules improve | 53 of 120 | | candidates the rules make worse | 26 of 120 | | mean change in symmetric ink from the rules | **+0.002** | **The rules are almost never enough, because the asymmetry is structural.** Of the places a drawing occupies, fewer than a fifth have a counterpart under the half-turn. The rest are places where the half-turn wants an extremity that the drawing does not have, because in the other orientation that corner is a *bend inside a stroke* rather than the end of one. The enclosure family is the clear case. 口 draws its top and right sides as one stroke (横折), so the top-right corner is not an extremity — it is a bend. Turned 180°, the bottom-left corner of the frame is an extremity of two strokes. The two orientations therefore do not even agree on how many strokes they see, and no amount of re-weighting the ends repairs that. **R-STRUCT** is the rule that addresses it: split the stroke at the corner so that both orientations have the same extremity structure. The ink is unchanged; only the stroke count of the drawing changes (口 goes from three strokes to four). It fires on 0–5 corners per candidate and is what the +0.002 comes from. It is necessary and it is not sufficient. **What would be sufficient** is a redesign of the drawing rather than a correction of it: the interior of 国, the 玉, has to be redrawn so that it is itself half-turn symmetric, which means giving up the PRC stroke order of the original character. That is a construction decision with a cost, and it belongs to the merged, cross-pair construction, not to the extremity rules. Two consequences for the method: * the order of work is **centre first, structure second, extremities last**; * a construction that reports "the extremities have been reconciled" is claiming very little, and the numbers above are what it is worth. ## 8. R-SECT: thickness by sector The rules above act on the extremities. There is a coarser and, on the PRC forms, a **more productive** lever: the thickness of the strokes, changed sector by sector. The box is divided into a 3 × 3 grid of sectors `S(i, j)`. The half-turn maps `S(i, j)` onto `S(2 − i, 2 − j)`, so the nine sectors form four opposite pairs plus the centre. Each sector carries a thickness multiplier `m_s`, and the half-width of a stroke sample becomes $$ w_{new}(t) = w(t)\; M(p(t)) $$ where `M` is the multiplier field and `p(t)` the position of the sample. The field is a normalised convolution of the sector values, $$ M(p) = \frac{\sum_s \mathrm{blur}\bigl(\chi_s(p)\,m_s\bigr)} {\sum_s \mathrm{blur}\bigl(\chi_s(p)\bigr)}, $$ with `χ_s` the indicator of sector `s` eroded by six cells. The erosion is what makes the multiplier exact in the interior of a sector; the blur blends the borders. A hard step at a sector border would put a visible notch in the middle of a stroke, and a reader would read the notch as a break. ### The rule For every pair of opposite sectors, compare the ink they carry and thicken the lighter one until the pair balances: $$ A(s) = \bigl|\,\mathrm{ink} \cap S(s)\,\bigr|, \qquad m_s = \min\bigl(1.40,\; A(r)/A(s)\bigr) \text{ if } A(r) > A(s), \qquad r = (2-i,\; 2-j) $$ The comparison uses the **full** sectors, which tile the box; the eroded masks exist only to build the field. Counting on the eroded masks drops the ink that lies near a border, which for a narrow stroke is all of it. ### The score, and what keeps it honest $$ \mathrm{sym}(D) = \frac{|D \cap R(D)|}{|D|} \quad\text{subject to}\quad \frac{|D \cap D_0|}{|D_0|} \ge 0.90, \qquad \frac{|D|}{|D_0|} \le 1.30 $$ `D₀` is the original drawing. The first constraint says at most 10% of the original ink may be erased; the second says at most 30% of ink may be added. Without them the score can be gamed: a drawing can always be made symmetric by deleting what is not, and always made "symmetric" by flooding the box. **A manipulation that raises the score by erasing or by flooding is rejected by the constraints, not by the score.** ### Measured Six frameless ideograms from the median band of the measured distribution (the interquartile band 0.225–0.361 of the symmetric-ink values). A form is **frameless** when its ink does not enclose any region of its own bounding box: the raster is closed by two cells and the background is flooded from the border of the canvas, and whatever the flood cannot reach is enclosed. The test is geometric, not a radical lookup, and it separates 囗 (0.58), 国 (0.44), 回 (0.39), 圓 (0.43) from 令 (0.00), 告 (0.05), 普 (0.08), 昔 (0.08). It also catches the forms that *contain* an enclosure — 涸 (0.25), 吕 (0.26), 崮 (0.26) — because they have one somewhere in the character. | ideogram | strokes | base | rule R-SECT | greedy search | retention | |---|---|---|---|---|---| | 令 | 5 | 0.318 | 0.350 | 0.409 | 0.910 | | 告 | 7 | 0.350 | 0.360 | 0.439 | 0.962 | | 昔 | 8 | 0.329 | 0.330 | 0.410 | 0.924 | | 普 | 12 | 0.250 | 0.311 | 0.340 | 0.998 | | 匿 | 10 | 0.332 | 0.351 | 0.386 | 1.000 | | 阗 | 13 | 0.351 | 0.377 | 0.451 | 0.977 | | **median** | | **0.331** | **0.350** (+0.023) | **0.409** (+0.090) | | **Yes, the manipulation works, and it works better than the extremity rules of §4.** The rule alone lifts 普 from 0.250 to 0.311 (+0.061) and 阗 from 0.351 to 0.377 (+0.026); the greedy search, which may also *thin*, adds another +0.029 on 普 and +0.074 on 阗. What the multipliers look like is the interesting part. The rule finds the imbalance between two opposite sectors and thickens the lighter one: for 令, 告, 昔 and 普 the pair that needs work is the top-middle against the middle-left, and the multipliers land at 1.40 on one of them. For 匿 the right-hand column — the 若 inside the 匚 — carries less ink than the left, and the rule thickens that column. In other words, **the rule discovers the asymmetric component of the character and compensates it with weight**, which is exactly what a human calligrapher would do. The search goes further by thinning as well as thickening: for 昔 it thins the left column to 0.70 and thickens the middle column to 1.45, which the rule does not find. But the greedy descent is not guaranteed to be the global optimum, and widening the multiplier range does not always help — for 匿 the wider range gives a *worse* result (0.404 → 0.386). ### Does it generalise? The rule was then run over **every frameless candidate in the median band** — 53 ideograms, not six: | | value | |---|---| | base, median | 0.240 | | after R-SECT, median | 0.266 | | median gain | **+0.023** | | mean gain | **+0.030** | | candidates that improve | **52 of 53** | | candidates that get worse | 1 of 53 (by 0.001) | | median retention of the original ink | 1.000 | The biggest gains: 匈 0.180 → 0.296 (+0.115), 罱 0.198 → 0.279 (+0.081), 匆 0.202 → 0.279 (+0.077). The gain does not depend on the number of strokes: the correlation between the gain and the stroke count is **-0.17**, and the largest gain is on a six-stroke character while a fourteen-stroke character gains almost as much. ### What this does not settle * The score is geometric. Whether a thicker 氵 reads better to a reader is exactly the question the pilot instrument exists for, and that evidence does not exist. * Six ideograms plus a generalisation run over the frameless median band is a sample of a sample. The numbers show the lever exists; they do not establish a rate for the PRC inventory. * The 3 × 3 grid is a choice. A finer grid would localise the imbalance better but has more parameters to fit and more risk of fitting noise. ## 9. What this does not do * It does not prove that any drawing is readable. It measures agreement of ink with itself under a half-turn. * It implements the **self-rotation** case (`a ↔ a`). The merge of two *different* forms — deciding which stroke of `a` answers to which stroke of `b`, and then applying the same rules to the union — is not implemented. The measured PRC pair distances are in [findings/prc-extremity-analysis.md](../findings/prc-extremity-analysis.md). * It uses the PRC medians as given. A median is not always the exact medial axis, so the weight profile carries the font's own approximation. * The sector manipulation (§8) has been measured on six ideograms and on the frameless median band; it has not been run over the whole PRC inventory, and the cross-pair merge is still not implemented. * The constants `δ_f`, `r_c`, `κ` are measured on one font (Arphic Public License, GB and UKai faces) on one inventory. They are properties of that font, not of the PRC script. ## 10. Reproducing the measurements ```bash python code/prc_construct.py \ --graphics data/prc/graphics.txt \ --characters 口,田,国,日,一 \ --out output/prc-construction.json ``` The module needs no network access and no external data beyond the MMah file; see [SCOPE.md](../SCOPE.md) for the licensing position and [the licence audit](../LICENSE-AUDIT.md) for the status of every input.