---
slug: the-shortest-network
passed: 2026-10-14
by: the assay line (research/assay-line/queue-1014c/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex
checked: 123
confirmed: 89
wrong: 33
unverifiable: 0
observed: 0
stale: 1
---

This pass implemented all 34 directing-triage decisions for the layer read at f5e5f84f34, rereading the evidence governing each edit and reproducing the actual browser counterexamples before editing. It retains the checkers' 89 confirmed source readings with their identifiers and scope limits; the retained fixed-wiring and physical-model confirmations are narrowed where the same prose was corrected. The 33 findings counted as wrong comprise 20 WRONG and 13 MINOR items. MINOR corrections are recorded here only. The JSON-LD revision date uses the explicit dated-metadata exception. No claim of a certified numerical optimum or physical stability is made by the corrected check.

## Claims

- CONFIRMED | Adding new junctions, where exactly three roads meet at 120°, makes the whole network shorter. | https://arxiv.org/pdf/0806.1340 | N2 [001 new-junctions]: Dutta pp. 1-2 describes helpful three-branch Steiner junctions at 120 degrees; it does not say arbitrary added junctions help.
- CONFIRMED | A soap film finds those junctions by itself. | https://arxiv.org/pdf/0806.1340 | N2 [002 soap-finds-junctions]: Dutta p. 2 describes physical realisations with local or absolute minima.
- CONFIRMED | the shortest network that joins a set of points, allowed to invent new junction points wherever that helps. | https://www.scottaaronson.com/papers/npcomplete.pdf | N2 [003 steiner-definition]: Aaronson p. 3 defines minimum total connecting length with auxiliary vertices permitted.
- CONFIRMED | the area of a vertical film is just its length on the plate times the gap. | https://www.cimat.mx/~gil/docencia/2017/mate_elem/%5BCourant%2CRobbins%5DWhat_Is_Mathematics%282nd_edition_1996%29v2.pdf | N2 [006 vertical-film-area]: Courant pp. 391-392 describes vertical planar films; geometry gives area hL.
- CONFIRMED | Least area means least length. | https://www.cimat.mx/~gil/docencia/2017/mate_elem/%5BCourant%2CRobbins%5DWhat_Is_Mathematics%282nd_edition_1996%29v2.pdf | N2 [007 area-means-length]: At fixed positive plate gap h, minimising hL is equivalent to minimising L.
- CONFIRMED | The film is solving a geometry problem with no maths at all. | https://www.cimat.mx/~gil/docencia/2017/mate_elem/%5BCourant%2CRobbins%5DWhat_Is_Mathematics%282nd_edition_1996%29v2.pdf | N2 [008 geometry-no-maths]: Courant p. 391 describes a physical demonstration; the claim is qualitative and subject to the local-minimum caveat.
- CONFIRMED | Each sheet is pulling along itself with the same tension | https://redlectern.com/read/on-growth-and-form/4 | N2 [009 equal-sheet-tensions]: Thompson p. 297 gives equal-tension balance. Independently, three unit vectors summing to zero imply each pair has dot product -1/2.
- CONFIRMED | Three equal pulls can only balance if they are spread evenly, 120° apart | https://redlectern.com/read/on-growth-and-form/4 | N2 [010 equal-pulls]: Thompson p. 297 gives equal-tension balance. Independently, three unit vectors summing to zero imply each pair has dot product -1/2.
- CONFIRMED | any tighter and two of them would gang up and drag the junction sideways until the angles opened back out. | https://redlectern.com/read/on-growth-and-form/4 | N2 [011 junction-pulled-sideways]: Thompson p. 297 gives equal-tension balance. Independently, three unit vectors summing to zero imply each pair has dot product -1/2.
- CONFIRMED | balance means they sum to zero, and three unit vectors sum to zero only when each pair sits at cos⁻¹(−½) = 120°. | https://redlectern.com/read/on-growth-and-form/4 | N2 [012 unit-vector-balance]: Thompson p. 297 gives equal-tension balance. Independently, three unit vectors summing to zero imply each pair has dot product -1/2.
- CONFIRMED | This is Plateau's law | https://schoolbag.info/mathematics/elementary/78.html | N2 [013 plateau-law-name]: The cited chapter attributes the angle rules to Plateau.
- CONFIRMED | named for the blind physicist Joseph Plateau | https://www.ugentmemorie.be/personen/plateau-joseph-1801-1883 | N2 [014 plateau-blind-physicist]: Ghent University records his physics work and loss of sight in 1843-1844.
- CONFIRMED | who catalogued it from soap in 1873. | https://archive.org/metadata/statiqueexprime00platgoog | N2 [015 plateau-1873]: The archive catalogue dates the cited publication to 1873, without dating first discovery.
- CONFIRMED | Four pins at the corners of a unit square. | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [017 square-preset-unit]: presetSquare() uses (0,0),(1,0),(1,1),(0,1).
- CONFIRMED | has length 3. | https://arxiv.org/pdf/0806.1340 | N2 [018 square-three-sides]: Dutta Fig. 1(c) and verifier section E use three unit sides, total 3.
- CONFIRMED | The next guess, an X of both diagonals crossing in the middle, is 2√2 ≈ 2.828. | https://arxiv.org/pdf/0806.1340; git show f5e5f84f34:research/the-shortest-network/verify.mjs | N2 [019 square-diagonals]: Pythagoras gives two diagonals of sqrt(2), total 2.8284271247461903; verifier section E recomputes it.
- CONFIRMED | But the soap film grows two junctions, joined by a short bridge, every angle 120°, for a total of 1 + √3 ≈ 2.732 | https://arxiv.org/pdf/0806.1340 | N2 [020 square-two-junctions]: Dutta Fig. 1(a) identifies the 1+sqrt(3) square minimum; the numerical square control agrees.
- CONFIRMED | Dip the film a few times and watch the bridge flip between horizontal and vertical | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [021 square-dip-bridge]: The dip handler randomly chooses a topology; choices 0 and 2 are the two bridge orientations. Choice 1 is the crossing addressed by X15.
- CONFIRMED | the square is so symmetric it has two equally-best answers | https://www.cimat.mx/~gil/docencia/2017/mate_elem/%5BCourant%2CRobbins%5DWhat_Is_Mathematics%282nd_edition_1996%29v2.pdf | N2 [022 square-two-equal-best]: Courant p. 361, figures 219-220, supplies the two symmetric square solutions.
- CONFIRMED | the film simply picks one as it forms. | https://arxiv.org/pdf/0806.1340 | N2 [023 film-selects-answer]: Dutta p. 2 describes different configurations from different withdrawals, without a universal optimum guarantee.
- CONFIRMED | That picking-one is called symmetry breaking | https://arxiv.org/pdf/0806.1340 | N2 [024 symmetry-breaking]: Selecting one bridge orientation removes the square arrangement's 90-degree rotational symmetry.
- CONFIRMED | The interactive model can display either bridge orientation. | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [025 watching-soap-symmetry]: The page randomly selects a wiring and renders a numerical model; it is not an observation of physical liquid.
- CONFIRMED | A real film can relax toward a lower-area configuration. | https://www.scottaaronson.com/papers/npcomplete.pdf | N2 [027 film-slides-to-bottom]: Aaronson p. 4 supports post-formation relaxation. Unconditional convergence was removed under X4/X30.
- CONFIRMED | the number of possible wirings explodes with the number of pins | https://ndl.ethernet.edu.et/bitstream/123456789/77634/1/92.pdf | N2 [028 wirings-explode]: Hwang/Richards/Winter section 1.6 gives the double-factorial growth of full topologies.
- CONFIRMED | choosing the best is NP-hard | https://api.crossref.org/works/10.1137/0132072 | N2 [029 best-wiring-nphard]: The publisher-deposited Garey/Graham/Johnson abstract establishes NP-hardness for general planar instances.
- CONFIRMED | the same wall that guards thousands of the hardest problems in computer science. | https://www.scottaaronson.com/talks/anthropic.html | N2 [030 thousands-hardest]: Aaronson's talk describes thousands of natural NP-complete search problems; this is a comparison, not a classification of all hard problems.
- CONFIRMED | you will sometimes catch it settling for a longer network | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [032 dip-longer-network]: The four-point example returns lengths 6.10366810484571, 6.497288998814106 and 6.587809474706778.
- CONFIRMED | longer than the obvious network it was supposed to beat. | git show f5e5f84f34:research/the-shortest-network/verify.mjs | N2 [033 dip-longer-than-obvious]: The returned worst length is 6.5878 versus MST 6.3681; no stability claim follows from that inequality.
- CONFIRMED | The film is honest about finding a good answer fast; it never promises the best one. | https://www.scottaaronson.com/papers/npcomplete.pdf | N2 [034 good-answer-fast]: Aaronson p. 4 reports frequent small-instance successes, failures and slow relaxation; no universal success or time bound is asserted.
- CONFIRMED | the solver enumerates every full junction-wiring (1, 3, 15, 105 of them for 3–6 pins) | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [037 enumerates-all-full]: The incremental edge-subdivision enumeration gives counts 1,3,15,105 for 3-6 terminals, reproduced in verify-browser.mjs.
- CONFIRMED | our solver, a from-scratch Weiszfeld iteration, and Torricelli's 1640s ruler-and-compass construction | git show f5e5f84f34:research/the-shortest-network/verify.mjs; https://math-inf.uni-greifswald.de/storages/uni-greifswald/fakultaet/mnf/mathinf/boldt/pdf-dateien/cieslik-steiner-ratio.pdf | N2 [042 fermat-methods]: Verifier section B has two point computations and an equilateral-triangle length construction; X10/X11 delimit what is compared.
- CONFIRMED | Torricelli's 1640s ruler-and-compass construction | https://math-inf.uni-greifswald.de/storages/uni-greifswald/fakultaet/mnf/mathinf/boldt/pdf-dateien/cieslik-steiner-ratio.pdf | N2 [043 torricelli-date]: Cieslik p. 16 and example 2.4.1 place Torricelli's geometric solution in the 1640s.
- CONFIRMED | Equilateral triangle: Steiner tree √3·s vs spanning tree 2s | git show f5e5f84f34:research/the-shortest-network/verify.mjs | N2 [045 equilateral-lengths]: Three centroid-to-vertex segments total sqrt(3)s; two spanning edges total 2s. Section D reproduces both.
- CONFIRMED | ratio √3/2 = 0.86603 | https://teorth.github.io/optimizationproblems/constants/43a.html | N2 [046 equilateral-ratio]: sqrt(3)/2 rounds to 0.86603; this is the equilateral instance, not a proved universal lower bound.
- CONFIRMED | the famous (and, contrary to popular belief, still-unproven) worst case. | https://link.springer.com/article/10.1007/s00453-011-9508-3; https://teorth.github.io/optimizationproblems/constants/43a.html | N2 [047 unproven-worst-case]: The 2012 clarification documents gaps in the claimed proof; the current research-status page still marks the conjecture unproven.
- CONFIRMED | Unit square: 1+√3 = 2.73205 < 2√2 = 2.82843 < 3 | git show f5e5f84f34:research/the-shortest-network/verify.mjs | N2 [048 square-inequalities]: Verifier section E and direct arithmetic give 2.7320508075688772 < 2.8284271247461903 < 3.
- CONFIRMED | exactly two tying optima (the two bridge directions), every other wiring strictly longer. | git show f5e5f84f34:research/the-shortest-network/verify.mjs | N2 [049 two-tying-optima]: The square returns two 1+sqrt(3) bridge configurations; Courant p. 361 supplies the two equivalent minima.
- CONFIRMED | A computed wiring can be longer than the minimum spanning tree. | git show f5e5f84f34:research/the-shortest-network/verify.mjs; https://arxiv.org/pdf/0806.1340 | N2 [051 proof-in-code-film]: The supported numerical result is a candidate longer than the MST. The corrected page does not infer physical stability from it.
- CONFIRMED | The solver is the real thing, not a recording | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [052 real-solver]: render() calls compute() using current pins; the network is calculated live.
- CONFIRMED | the 120° law, the triangle and square optima, the convex floor of each fixed wiring. | git show f5e5f84f34:research/the-shortest-network/verify.mjs | N2 [055 what-proven]: Force balance yields 120 degrees; triangle and square formulas are reproduced. A sum of Euclidean norms establishes convexity, not strict convexity or convergence.
- CONFIRMED | This solver uses an uncertified numerical relaxation. | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [056 larger-set-heuristic]: Topology enumeration is followed by uncertified numerical relaxation, including on small inputs.
- CONFIRMED | for that no fast method is known, and possibly none exists. | https://api.crossref.org/works/10.1137/0132072; https://www.scottaaronson.com/papers/npcomplete.pdf | N2 [057 no-fast-method]: NP-hardness supplies the conditional computational barrier; the wording does not assert a proved impossibility.
- CONFIRMED | J. Plateau, Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires (Gauthier-Villars, 1873). | https://archive.org/metadata/statiqueexprime00platgoog | N2 [058 plateau-reference]: The archive catalogue matches the author, French title, publisher and 1873 year.
- CONFIRMED | The full mathematical proof that these are the only stable junctions came a century later: F. Almgren and J. E. Taylor | https://schoolbag.info/mathematics/elementary/78.html | N2 [059 only-stable-proof]: The cited chapter credits the 1976 work; the Annals entry confirms Taylor's publication.
- CONFIRMED | "The Structure of Singularities in Soap-Bubble-Like and Soap-Film-Like Minimal Surfaces," Annals of Mathematics 103 (1976), 489–539. | https://annals.math.princeton.edu/1976/103-3/p08 | N2 [060 taylor-reference]: The Annals publication record matches Jean E. Taylor, volume 103 (1976), pages 489-539.
- CONFIRMED | R. Courant and H. Robbins, What Is Mathematics? (Oxford, 1941). | https://www.cimat.mx/~gil/docencia/2017/mate_elem/%5BCourant%2CRobbins%5DWhat_Is_Mathematics%282nd_edition_1996%29v2.pdf | N2 [061 courant-reference]: The scan's front matter identifies Courant and Robbins, Oxford, first published in 1941; the scan is the 1996 edition.
- CONFIRMED | The soap-film-on-pins demonstration and the 1+√3 square are the classic exposition in | https://www.cimat.mx/~gil/docencia/2017/mate_elem/%5BCourant%2CRobbins%5DWhat_Is_Mathematics%282nd_edition_1996%29v2.pdf | N2 [062 courant-classic-square]: The scanned work supplies the square networks on p. 361 and the parallel-plate demonstration on pp. 391-392.
- CONFIRMED | M. R. Garey, R. L. Graham and D. S. Johnson, "The Complexity of Computing Steiner Minimal Trees," SIAM J. Appl. Math. 32 (1977), 835–859. | https://api.crossref.org/works/10.1137/0132072 | N2 [063 garey-reference]: Publisher metadata matches the three authors, 1977, SIAM J. Appl. Math. 32, pages 835-859.
- CONFIRMED | S. Aaronson, "NP-complete Problems and Physical Reality," SIGACT News, March 2005 (arXiv:quant-ph/0502072) | https://www.scottaaronson.com/papers/npcomplete.pdf | N2 [064 aaronson-reference]: The paper's p. 4 footnote identifies the March 2005 SIGACT News publication.
- CONFIRMED | reports real soap films settling into local minima, not the optimum. | https://www.scottaaronson.com/papers/npcomplete.pdf | N2 [065 aaronson-reports]: Section 3 p. 4 records failures to reach the optimum and variable outcomes.
- CONFIRMED | E. N. Gilbert and H. O. Pollak conjectured the √3/2 ratio in 1968. | https://ndl.ethernet.edu.et/bitstream/123456789/77634/1/92.pdf; https://teorth.github.io/optimizationproblems/constants/43a.html | N2 [066 gilbert-pollak-1968]: Hwang/Richards/Winter p. 38 and the current research bibliography give 1968.
- CONFIRMED | D.-Z. Du and F. K. Hwang published a claimed proof (PNAS 1990; Algorithmica 1992) | https://link.springer.com/article/10.1007/s00453-011-9508-3 | N2 [067 du-hwang-pnas]: The clarification's bibliography gives PNAS 87, pages 9464-9466 (1990), as a claimed proof.
- CONFIRMED | D.-Z. Du and F. K. Hwang published a claimed proof (PNAS 1990; Algorithmica 1992) | https://link.springer.com/article/10.1007/s00453-011-9508-3 | N2 [068 du-hwang-algorithmica]: The clarification's bibliography gives Algorithmica 7, pages 121-135 (1992), as a claimed proof.
- CONFIRMED | gaps were later identified | https://arxiv.org/html/1402.6079v1 | N2 [069 gaps-identified]: Ivanov/Tuzhilin identify gaps in the prior claimed proof.
- CONFIRMED | A. O. Ivanov and A. A. Tuzhilin, "The Steiner Ratio Gilbert–Pollak Conjecture Is Still Open," Algorithmica 62 (2012), 630–632 | https://link.springer.com/article/10.1007/s00453-011-9508-3 | N2 [070 ivanov-reference]: Publisher metadata gives volume 62, pages 630-632 (2012), despite an online date in 2011.
- CONFIRMED | the general conjecture remains open. | https://teorth.github.io/optimizationproblems/constants/43a.html | N2 [071 conjecture-remains-open]: The current research-status page still lists the general conjecture as unproven.
- CONFIRMED | The equilateral-triangle value √3/2 itself is exact and unconditional | git show f5e5f84f34:research/the-shortest-network/verify.mjs | N2 [072 equilateral-unconditional-computed]: The equilateral three-arm construction and two-side spanning tree give sqrt(3)/2 directly; verifier section D agrees.
- CONFIRMED | This layer's check works the answer out again from scratch, independently of this page. | git show f5e5f84f34:research/the-shortest-network/verify.mjs | N2 [073 independent-check]: verify.mjs imports only the offline module and reads no page data.
- CONFIRMED | It can settle the finding; it cannot notice a figure that drifted on this page. | git show f5e5f84f34:research/verifier-independence/grading.json | N2 [075 settles-finding-cannot-see-drift]: The recorded grading result and verifier imports show that the offline check does not inspect page figures.
- CONFIRMED | It is published, and it runs from 2 files this site serves | git show f5e5f84f34:public/checks/manifest.json | N2 [076 published-two-files]: The manifest lists steiner.mjs and verify.mjs; this pass regenerates both local downloadable copies from their sources.
- CONFIRMED | A draggable network of pins joined by a soap-film network. Drag any pin to reshape it. | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [078 draggable-network]: Pointer handlers select and move pins, then request a new render.
- CONFIRMED | Add a pin (max 6) | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [079 max-six]: The add handler returns when pins.length is at least 6.
- CONFIRMED | Remove a pin (min 3) | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [080 min-three]: The remove handler returns when pins.length is at most 3.
- CONFIRMED | Dip again relaxes a randomly chosen wiring. | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [081 dip-nucleates-title]: The control chooses a random full topology; this is a numerical model, not a measured physical distribution.
- CONFIRMED | The readout shows angles at separate junctions and identifies omissions. | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [082 readout-angles]: junctionAngles computes pairwise angles at separate vertices; the new labels explicitly disclose omitted vertices.
- CONFIRMED | This dip nucleated into a longer network | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [083 dip-longer-verdict]: The branch compares the dip length with the shortest computed candidate using a 0.001 threshold.
- CONFIRMED | It is even longer than the obvious network: the film lost to doing nothing clever. | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [085 longer-obvious-verdict]: The branch explicitly checks candidate length against MST length plus 0.001.
- CONFIRMED | The film beats the obvious network by | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [089 film-beats-verdict]: The percentage is (1 - candidate/MST)*100; the square displays 8.9%.
- CONFIRMED | Connect a handful of points with the least possible road, and the obvious answer (run the road along the shortest links) is not the shortest. | https://www.scottaaronson.com/papers/npcomplete.pdf; https://arxiv.org/pdf/0806.1340 | N2 [093 dek-obvious-shortest]: The square proves that adding junctions can improve on the MST; the corrected summary avoids a universal assertion.
- CONFIRMED | The 120° law falls straight out of three equal tensions balancing (three unit pulls sum to zero only at 120°, Plateau 1873) | https://redlectern.com/read/on-growth-and-form/4 | N2 [097 dek-force-balance]: Thompson p. 297 gives equal-tension balance. Independently, three unit vectors summing to zero imply each pair has dot product -1/2.
- CONFIRMED | the equilateral triangle gives the famous (and, contrary to the textbooks, still-unproven) Steiner ratio √3/2. | https://link.springer.com/article/10.1007/s00453-011-9508-3; https://ndl.ethernet.edu.et/bitstream/123456789/77634/1/92.pdf; https://teorth.github.io/optimizationproblems/constants/43a.html | N2 [098 dek-ratio-textbooks]: The older textbook reports a proof; the later clarification documents gaps, and the general conjecture remains open.
- CONFIRMED | For a fixed wiring, total length is convex. | https://ndl.ethernet.edu.et/bitstream/123456789/77634/1/92.pdf | N2 [099 dek-single-convex-bowl]: A sum of Euclidean norms is convex. The summary now explicitly avoids promising a unique minimum or convergence.
- CONFIRMED | choosing the best wiring is NP-hard (Garey–Graham–Johnson 1977) | https://api.crossref.org/works/10.1137/0132072 | N2 [100 dek-complexity-attribution]: The publisher abstract attributes planar Steiner NP-hardness to Garey, Graham and Johnson (1977).
- CONFIRMED | The immersive page is a live Steiner solver | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [102 live-steiner-solver]: The live compute/render path calculates numerical candidates; it does not certify them.
- CONFIRMED | toggle the obvious network | git show f5e5f84f34:public/strata/the-shortest-network/index.html | N2 [104 toggle-obvious-network]: The toggle changes showMST, which controls rendering of minimumSpanningTree edges.
- CONFIRMED | The Steiner tree problem: the shortest network joining a set of points is usually not the one built from the shortest links. | https://www.scottaaronson.com/papers/npcomplete.pdf; https://arxiv.org/pdf/0806.1340 | N2 [106 plain-usually-shortest]: Dutta p. 1 qualitatively describes the usual introduction of auxiliary vertices; no numerical frequency is claimed.
- CONFIRMED | for the corners of a unit square the least network is 1 + √3, about 2.732. | https://arxiv.org/pdf/0806.1340 | N2 [107 plain-square]: Dutta Fig. 1(a) supplies the unit-square minimum, 1+sqrt(3).
- CONFIRMED | A soap film between glass plates finds it by surface tension, but choosing the wiring is NP-hard, so the film can freeze into a worse one. | https://arxiv.org/pdf/0806.1340; https://api.crossref.org/works/10.1137/0132072 | N2 [108 plain-physical-finds]: Dutta distinguishes local and absolute minima; the corrected plain text says a film can find the minimum.
- CONFIRMED | extra junctions where three lines meet at exactly 120° make the network shorter, and a soap film finds them by itself. | https://arxiv.org/pdf/0806.1340 | N2 [109 meta-junctions]: Dutta describes useful 120-degree junctions and their soap-film realisations; the corrected metadata retains the qualification.
- CONFIRMED | But the film only finds a local minimum, which is why it can't reliably solve this NP-hard problem. | https://www.scottaaronson.com/papers/npcomplete.pdf | N2 [110 meta-film-local-only]: Aaronson reports both optimum and suboptimal results. The metadata now says films can become trapped.
- CONFIRMED | the unit square's 1 + √3 &lt; 2√2 &lt; 3 | git show f5e5f84f34:research/the-shortest-network/verify.mjs | N2 [114 markdown-square-check]: Verifier section E reproduces 1+sqrt(3) < 2sqrt(2) < 3.
- CONFIRMED | the honeycomb's three-way wall junctions are 120° | https://redlectern.com/read/on-growth-and-form/4 | N2 [116 related-honeycomb-walls]: Thompson p. 319 describes 120-degree corners in the ideal regular hexagonal geometry.
- CONFIRMED | the bee's three-dimensional cell base bottoms out its wax cost at exactly that same 109.47°. | https://www.model.u-szeged.hu/cd/content/szilassi/Euler3d-kurzus/Kieg%2582sz%25a1t%258b%20anyagok/M%C2%B4-e%CC%A8sejt/S0002-9904-1964-11155-1.pdf; https://redlectern.com/read/on-growth-and-form/4 | N2 [117 related-honeycomb-base]: Thompson pp. 331-332 derives the restricted three-rhombus area optimum; the 109.47-degree angle is within a rhombus, not between faces.
- CONFIRMED | The same two angles fall out of 'use the least wall/film' in both a network and a tiling | https://schoolbag.info/mathematics/elementary/78.html; https://redlectern.com/read/on-growth-and-form/4 | N2 [118 related-same-angles]: Thompson gives the three-film 120-degree meeting, tetrahedral edge angle and restricted rhombic-base area calculation.
- CONFIRMED | Deferred acceptance always finds a stable matching | https://iqua.ece.toronto.edu/baochun/ece1771f/papers/Gale-Shapley-1962.pdf | N2 [120 related-deferred-acceptance]: Gale/Shapley p. 12 supplies an iterative construction proving existence for the ordinary marriage problem.
- CONFIRMED | except with couples, where merely deciding if one exists is NP-complete. | https://csaws.cs.technion.ac.il/~hadas/Seminar/Winter1011_236801/Papers/stable_assignment.pdf | N2 [121 related-couples]: Stable Assignment with Couples section 2 reports Ronn's NP-completeness result for existence with couples.
- CONFIRMED | the equilateral triangle gives the famous (and, contrary to the textbooks, still-unproven) Steiner ratio √3/2. | https://ndl.ethernet.edu.et/bitstream/123456789/77634/1/92.pdf; https://link.springer.com/article/10.1007/s00453-011-9508-3; https://teorth.github.io/optimizationproblems/constants/43a.html | N2 [125 textbook-proof-status]: The old textbook claim and later clarification support the contrast with older proof claims, not all current textbooks.
- CONFIRMED | both pages are honest about where the optimum is missed | git show f5e5f84f34:public/strata/what-the-bees-dont-know/index.html | N2 [126 related-bees-confession]: The linked bees page explicitly acknowledges a better three-dimensional base; the percentage was not a triaged fix here.
- CONFIRMED | Both pages build the machine, watch it work, then show the exact cliff where the easy guarantee falls away. | git show f5e5f84f34:public/strata/no-two-would-rather/index.html | N2 [127 related-matching-instrument]: The linked page implements deferred acceptance and explains the failure of the guarantee with couples.
- MINOR | X1: A film of soap stretched between two glass plates with pins between them settles into exactly this shape | https://www.cimat.mx/~gil/docencia/2017/mate_elem/%5BCourant%2CRobbins%5DWhat_Is_Mathematics%282nd_edition_1996%29v2.pdf; https://www.scottaaronson.com/papers/npcomplete.pdf | Aaronson, section 3, p. 4, reports both successes and failures on small pin sets.
- MINOR | X2: surface tension pulls it to the least possible area | https://www.cimat.mx/~gil/docencia/2017/mate_elem/%5BCourant%2CRobbins%5DWhat_Is_Mathematics%282nd_edition_1996%29v2.pdf; https://www.scottaaronson.com/papers/npcomplete.pdf | Courant pp. 391-392 distinguishes relative and absolute minima; Aaronson p. 4 reports misses.
- WRONG | X3: it holds at 120° no matter how you drag the pins, right up until a junction is no longer worth having and collapses onto a pin. | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | The five-pin browser reproduction merges vertices away from pins; its four sectors are about 61.6474, 124.3706, 55.8366 and 118.1454 degrees.
- WRONG | X4: For a fixed wiring of the junctions, the total length is a tidy bowl-shaped function with a single bottom | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | Hwang/Richards/Winter p. 7 qualifies strict convexity. The collinear example attains 4/3 for every coincident pair at x in [1/3,2/3]; disjoint paths bound length below by 1+1/3.
- MINOR | X5: A real film just freezes into whatever wiring it happened to nucleate into. | https://www.scottaaronson.com/papers/npcomplete.pdf; https://arxiv.org/pdf/0806.1340 | Aaronson p. 4 reports variable outcomes and relaxation after formation; it does not establish universal freezing at nucleation.
- WRONG | X6: Scott Aaronson dipped real wire frames to make exactly this point. | https://www.scottaaronson.com/papers/npcomplete.pdf | Aaronson's own apparatus description, section 3 p. 4, specifies glass plates, copper rods and suction cups.
- WRONG | X7: The network you see is the genuine Steiner minimal tree | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | Actual baseline browser output is 0.8322523554660158; the feasible spanning tree has length 0.6664698836751602.
- WRONG | X8: relaxes each to its unique floor, then keeps the shortest. | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | Moving both interior vertices down by 0.0001 reduces the same returned wiring from 0.8322523554660158 to 0.8320887148985581.
- MINOR | X9: The same code runs offline in research/the-shortest-network/verify.mjs | git show f5e5f84f34:public/strata/the-shortest-network/index.html; git show f5e5f84f34:research/the-shortest-network/steiner.mjs | Browser defaults are 1500 iterations and 1e-10 movement tolerance; the offline module uses 2000 and 1e-12.
- MINOR | X10: the junction found three independent ways | git show f5e5f84f34:research/the-shortest-network/verify.mjs | Verifier section B computes two point positions and a separate Simpson-line length, not a third point.
- MINOR | X11: agree to 5 decimal places, all three angles 120.00°. | git show f5e5f84f34:research/the-shortest-network/verify.mjs; /home/liam/.assay-line/queue-1014c/the-shortest-network/N2/reader-run.txt | Section B checks coordinates within 1e-5, compares lengths within 1e-4, and prints both lengths as 6.60132; solver angles print as 120.00 degrees.
- WRONG | X12: it computes the Steiner minimal tree for your pins every frame | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | compute() uses the same uncertified browser relaxation; the four-pin counterexample is longer than a feasible spanning tree.
- MINOR | X13: the identical functions are checked offline in research/the-shortest-network/. | git show f5e5f84f34:public/strata/the-shortest-network/index.html; git show f5e5f84f34:research/the-shortest-network/steiner.mjs | The browser and offline iteration limits, movement tolerances and collapsed-edge thresholds differ; verify.mjs imports only the offline module.
- WRONG | X14: vs the true minimum | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | Film topology 1 compares 0.8333366128768616 against 0.8322523554660158 even though a feasible tree is 0.6664698836751602.
- WRONG | X15: A real local minimum the film got stuck in. | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | Square film topology 1 returns 2.8284272377928152; splitting the crossing to (0.49,0.5),(0.51,0.5) gives a nearby connected network of 2.820285699709942.
- WRONG | X16: Press Triangle/Square/Scatter to snap back to the true minimum. | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | Driving actual presetScatter() with in-range values reproduces a candidate of 0.8322523554660178 versus spanning length 0.6664698836751602.
- WRONG | X17: This dip happened to find the true shortest network. | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | Film topology 0 matches the same nonminimal candidate and previously triggered the true-shortest verdict.
- WRONG | X18: every junction at 120°. | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | The five-pin case triggered every-junction-at-120 while hiding a merged four-way meeting.
- WRONG | X19: With the pins this spread out, the obvious network already is the shortest | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | For (0,0),(1,0),(0.5,0.3), a junction improves 1.16619037896906 to 1.1660254037844386 although the saving displays as 0.0%.
- WRONG | X20: no junction is worth adding. | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | The same triangle has a helpful interior junction despite the rounded zero saving.
- WRONG | X21: for the four corners of a unit square the least network is not the perimeter (length 3) nor the crossing diagonals (2√2 ≈ 2.828) but a pair of 120° junctions joined by a bridge, total 1 + √3 ≈ 2.732. | https://arxiv.org/pdf/0806.1340; git show f5e5f84f34:public/strata/the-shortest-network/index.html | Four unit boundary sides total 4; the verifier's minimum spanning tree uses three sides and totals 3.
- MINOR | X22: A soap film stretched between two glass plates with pins between them settles into exactly this shape | https://www.cimat.mx/~gil/docencia/2017/mate_elem/%5BCourant%2CRobbins%5DWhat_Is_Mathematics%282nd_edition_1996%29v2.pdf; https://www.scottaaronson.com/papers/npcomplete.pdf | Aaronson p. 4 reports that films can miss the optimum; the opening summary lacked that qualification.
- WRONG | X23: a vertical film's area is just its length | https://www.cimat.mx/~gil/docencia/2017/mate_elem/%5BCourant%2CRobbins%5DWhat_Is_Mathematics%282nd_edition_1996%29v2.pdf; git show f5e5f84f34:public/strata/the-shortest-network/index.html | Courant pp. 391-392 describes vertical planar films; area of a strip of projected length L and plate gap h is hL.
- MINOR | X24: a real film just freezes into whatever it nucleated, sometimes a network longer than the obvious one it was meant to beat. | https://www.scottaaronson.com/papers/npcomplete.pdf; https://arxiv.org/pdf/0806.1340 | Aaronson p. 4 reports post-formation relaxation and variable configurations, not universal freezing at nucleation.
- WRONG | X25: drag the pins, watch the film snap to the shortest network with every length and angle recomputed in front of you | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | The actual browser relaxation returns the four-pin nonminimum above; metadata repeated its optimum guarantee.
- MINOR | X26: Soap films solve the Steiner tree: extra 120° junctions make a network shorter than the obvious one. | https://arxiv.org/pdf/0806.1340 | Dutta et al., p. 2, describes local or absolute equilibrium; soap films are not guaranteed global solvers.
- WRONG | X27: network snap to the shortest possible roads, every junction holding at 120°, every length and angle recomputed in front of you. | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | The four-pin nonminimum and five-pin merged junction disprove both promises in the markdown body.
- MINOR | X28: the 120° law from force balance, the Fermat point found three independent ways, the equilateral Steiner ratio √3/2 | git show f5e5f84f34:research/the-shortest-network/verify.mjs | Verifier section B calculates two point positions and one geometric length, while section D correctly computes sqrt(3)/2.
- WRONG | X29: Soap films meet three-at-a-time at 120° and four-at-a-time at the tetrahedral angle arccos(−1/3) = 109.47° | https://redlectern.com/read/on-growth-and-form/4 | Thompson pp. 316-318 distinguishes six film sectors, three sheets per edge, and four triple-junction edges at a tetrahedral meeting.
- MINOR | X30: A soap film always slides to a fixed wiring's optimum | https://ndl.ethernet.edu.et/bitstream/123456789/77634/1/92.pdf; https://www.scottaaronson.com/papers/npcomplete.pdf | The length objective is convex; neither Hwang/Richards/Winter p. 7 nor Aaronson p. 4 guarantees convergence of every physical film.
- STALE | X31: "dateModified":"2026-06-29" | git log f5e5f84f34 -- public/strata/the-shortest-network/index.html; git show 01baf9c8f5e:public/strata/the-shortest-network/index.html | Pinned JSON-LD says 2026-06-29; git history records modifications on 2026-09-28 and 2026-10-01.
- WRONG | X32: Two pages that watch minimisation pick out the same two angles, 109.47° and its half. | git show f5e5f84f34:public/strata/the-shortest-network/index.html; git show f5e5f84f34:src/content/strata/every-direction-at-once.md; https://redlectern.com/read/on-growth-and-form/4 | This layer demonstrates planar 120-degree junctions; the linked layer's sleeve maximises volume at about 54.74 degrees.
- MINOR | X33: all 120.0° | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | The triangle (0,0),(1,0),(0.5,0.288) yields angles about 120.1250,119.9375,119.9375 degrees but previously printed all 120.0.
- WRONG | X34: no junctions | research/the-shortest-network/verify-browser.mjs; git show f5e5f84f34:public/strata/the-shortest-network/index.html | The square crossing has four nonzero arms but both numerical interior vertices are filtered by the near-zero-edge rule.

## Findings

- [fixed] X1 (MINOR): Qualified the opening locally to say a film can settle into the shortest shape. Shared TV narration at public/tv/lib/narration/batch-15.js:908 still promises the exact shortest shape; that copy remains open. Recorded only; no page correction line. The directing instance corrected that narration copy on 2026-10-14.
- [fixed] X2 (MINOR): Described area reduction toward a possibly local minimum. Recorded only; no page correction line.
- [fixed] X3 (WRONG): Removed the universal live-angle promise and disclosed omitted merged vertices.
- [fixed] X4 (WRONG): Retained convexity and removed uniqueness from page, summary and solver documentation.
- [fixed] X5 (MINOR): Replaced freezing at nucleation with possible trapping in a suboptimal configuration. Recorded only; no page correction line.
- [fixed] X6 (WRONG): Replaced wire frames with parallel glass plates joined by copper rods.
- [fixed] X7 (WRONG): Replaced the optimum guarantee with numerical-candidate wording across all local copies.
- [fixed] X8 (WRONG): Explained that the iteration can stop above even its fixed-wiring minimum.
- [fixed] X9 (MINOR): Described the offline module as a separate implementation with different tolerances. Recorded only; no page correction line.
- [fixed] X10 (MINOR): Described two point computations and an independent geometric length check. Recorded only; no page correction line.
- [fixed] X11 (MINOR): Named the quantities and tolerances without promising three-point agreement. Recorded only; no page correction line.
- [fixed] X12 (WRONG): Said the solver recomputes numerical candidates as the pins move.
- [fixed] X13 (MINOR): Removed the identical-functions claim from prose and code comments. Recorded only; no page correction line.
- [fixed] X14 (WRONG): Relabelled the reference as the shortest computed candidate.
- [fixed] X15 (WRONG): Removed the real-local-minimum verdict and false physical-stability interpretations in verifier and page.
- [fixed] X16 (WRONG): Changed the preset invitation to compare all wirings.
- [fixed] X17 (WRONG): Reported agreement within 0.001 of the shortest computed candidate without certification.
- [fixed] X18 (WRONG): Removed the blanket angle claim from the saving verdict.
- [fixed] X19 (WRONG): Described the 0.05% display threshold and separated the longer-than-spanning-tree case.
- [fixed] X20 (WRONG): Removed the claim that no junction is worth adding.
- [fixed] X21 (WRONG): Changed the summary comparator to the three-side spanning tree.
- [fixed] X22 (MINOR): Qualified the summary locally to say a film can settle into the shortest shape. The same unqualified promise remains in public/tv/lib/narration/batch-15.js:908, held open with X1. Recorded only; no page correction line. The directing instance corrected that narration copy on 2026-10-14.
- [fixed] X23 (WRONG): Restored the plate-gap factor in the summary. Shared TV narration at public/tv/lib/narration/batch-15.js:909 still equates area with length and remains open; for the described vertical film, area is projected length times plate gap. The directing instance corrected that narration copy on 2026-10-14.
- [fixed] X24 (MINOR): Replaced the summary's freezing claim with possible trapping. Recorded only; no page correction line.
- [fixed] X25 (WRONG): Corrected the dek, descriptions, JSON-LD and body to describe numerical candidates and their limits.
- [fixed] X26 (MINOR): Qualified social metadata to say films can find short networks and can become trapped. Recorded only; no page correction line.
- [fixed] X27 (WRONG): Rewrote the body with candidate-network and omitted-angle limitations.
- [fixed] X28 (MINOR): Corrected the check description and retained the supported equilateral ratio. Recorded only; no page correction line.
- [open note:b621c1] X29 (WRONG): Distinguished film sheets from the edges meeting at 109.47 degrees in this layer. The incoming relation at src/content/strata/every-direction-at-once.md:14 still describes four films meeting at that angle and remains open; four junction edges meet there, with six film sectors.
- [fixed] X30 (MINOR): Replaced unconditional physical convergence with convexity and possible trapping in the relation note. Recorded only; no page correction line.
- [dated] X31 (STALE): Set dateModified to 2026-10-14. No Updated line is required for this metadata exception.
- [open note:b621c1] X32 (WRONG): Authored a corrected outgoing relation here and regenerated this layer's nearby teaser, taking precedence over the incoming note. The other layer's source at src/content/strata/every-direction-at-once.md:14 still says both pages minimise to 109.47 degrees and its half; that copy remains open.
- [fixed] X33 (MINOR): Used an explicitly approximate 120-degree label and one decimal for individually displayed angles. Recorded only; no page correction line.
- [fixed] X34 (WRONG): Replaced no-junctions with an explicit merged-vertices/omitted-angles readout.

## Verification

- `node research/the-shortest-network/verify-browser.mjs --baseline`: 11 geometry and baseline checks passed; reproduced the false output labels at f5e5f84f34.
- `node research/the-shortest-network/verify.mjs`: 21 passed, 0 failed. Section F now describes computed comparisons without asserting physical stability or a general optimum.
- `node research/the-shortest-network/verify-browser.mjs`: 22 browser-function and geometry checks passed, including all ten correction mirrors.
- Generated plain text, local nearby teasers and the two downloadable verifier files come from `node research/the-shortest-network/build.mjs`.

The incoming relation authored on the other layer and the shared TV narration are outside this worker's editable scope. The repair pass on 2026-10-14 retained those copies as open findings above.