A dependency-free browser demonstration of two toy models: a damped attractor and a five-observer comparison. The equations, target values, controls, and export are visible in index.html. This is exploratory visualization, not a validated theory of memory, identity, consciousness, safety, or physical dynamics.
Open the live interactive lab · Read the source
The Discovery Archive presents exact counterexamples, finite search boundaries, and a printable research report. Its candidate search records exact evidence in a reproducible JSON archive. Run python3 tools/verify_discovery_archive.py examples/discovery-scan-v1.json to replay its finite domain independently; archive queries and the page generator validate evidence before returning or displaying it. A separate symbolically checked proof record documents two established identities valid for all real inputs in this model, while the scan retains its finite-grid labels. This is a small research instrument for the five-reading model, not an automated theorem prover.
The sample flip certificate extends exact single-reading reversal analysis to 3–21 rational readings, including even samples with a central-pair median. A deterministic CLI partitions every possible real change at all median transitions and factor roots; an independent checker verifies the complete partition, tie threshold, and strict-reversal witness. Its portable research report shows each route and embeds the checked JSON for offline reuse. Version 2 accepts signed odd-sample inputs and explicit movement bounds; version 3 covers even samples with explicit bounds, including the default [0,100] interval; the signed constrained report shows how limiting movement changes the nearest route. The checked even-sample report demonstrates the central-pair rule, an exact tie at 130/3, and an opposing witness. It reports the infimum of movement, which is generally a tie boundary rather than an attained strict minimum. This research tool is separate from the five-reading browser exercises.
The Sample Reversal Research Lab accepts a visitor's 3–21 readings, odd or even, stipulated target, and inclusive movement bounds entirely in the browser. Signed values and fractions are supported within documented input limits. Exact BigInt fractions produce the same versioned JSON certificate as the Python generator; seeded parity tests compare both implementations, and the downloaded certificate can be checked independently with python3 tools/verify_odd_sample_flip.py odd-sample-flip-certificate.json. No inputs are put into a URL or sent to an application server. The browser chart is an approximate visual map of exact intervals; the fraction labels and exported record carry the mathematical claim. The printable view is a conditional research report, not authentication of supplied values.
The Sample Reversal Certificate Inspector also checks these exports directly in a browser. It does not import the certificate constructor: it independently derives closed-form roots, compares squared errors at every knot and interval, checks the tie threshold and opposing witness, rejects duplicate JSON keys and altered fields, and fingerprints the exact checked bytes with SHA-256. The separate Python verifier remains available for offline audits. A passing result is conditional on the supplied values and does not authenticate the data or its author.
The interactive Conjecture Lab lets a visitor select a finite rational grid, search four precisely stated five-reading claims with exact arithmetic, inspect counterexamples, and download a reproducible experiment. It also includes a conditional claim builder: choose 3 through 7 readings (with a central-pair average median for even samples), a target condition, and an error-order prediction, then obtain an exact counterexample, a finite unrefuted status, or a separately checkable general algebraic proof for the four cases implied by the squared-distance identity. Proof-bearing v2–v6 receipts name the corresponding Lean theorem at a pinned source revision; both the browser and Python checkers independently validate that mapping. This is traceability to a separately checked real-number result, not a Lean check of arbitrary JSON. Independent experiment and conditional-claim Python checkers replay the selected grid and reject altered claims or proof records. The browser Claim Inspector separately checks the conditional claim on a phone or computer without installing Python, then saves verified JSON in the device-local Research Archive. V3–v6 receipts classify all 20 supported condition and prediction pairs for their stated sample size: four established implications or an independently checked exact counterexample. V6 constructs witnesses from two endpoint sample families that work for every sample size at least three. A separate Lean theorem classifies the full registered 20-pair language using those families for every integer sample size at least three; it does not check arbitrary JSON or prove the browser and Python routing code matches the formal encoding. CI compares the 20 Lean endpoint routes with both executable implementations at every supported sample size to catch drift. The archive rechecks exported claims and keeps Target Box certificates and older v1–v5 claims. A grid with no counterexample supplies bounded evidence, never a theorem by itself. See the method and limits.
The Target Box research lab lets the chosen target vary across an interval while any one of 3–21 readings moves through a signed interval. Odd samples use the middle reading; even samples use the central-pair average. Exact rational factor roots give every possible Mean, Median, and Tie outcome for each route, with witness pairs and a strict route guarantee only when the winner holds throughout its entire rectangle. Export versioned inward-odd-target-box-v1 or inward-even-target-box-v2 JSON and check it on a phone with the independent browser inspector or with the separate Python verifier; read the proof and reproducibility guide. Checked certificates can be kept, searched, and exported from the device's Local Research Archive, which rechecks each record on opening and does not upload it. A portable research collection backs up the checked archive, supports verified import on another device, and has a separate Python checker. The chart shows approximate locations of exact witnesses, while the complete sign partition proves the conditional result. It does not authenticate the supplied target or data.
Quantify exact gap extends a Target Box experiment to a inward-target-box-gap-v3 certificate. It computes the attained minimum and maximum of the signed difference between the mean's and median's squared errors across every one-reading route and target in the chosen interval. The proof checks exact quadratic vertices inside each median cell as well as boundaries; browser and Python verifiers recompute the extrema and witnesses separately. A strictly positive global minimum certifies a Median margin, while a strictly negative global maximum certifies a Mean margin, in squared input units. This measures a specified hypothetical rectangle, not uncertainty probabilities or measurement validity. Method and example.
Find first failure produces a inward-target-box-stability-v4 certificate with exact left and right movement thresholds from every original reading, and an attained tie witness at each excluded boundary. It reports the nearest failure over all one-reading routes or confirms that no failure exists within the stipulated movement range. The worked example proves a strict Median result for all permitted moves smaller than 8 from the original five readings, with a tie at exactly 8. Both independent verifiers and the local research archive accept the versioned record. This stability budget depends on the supplied target interval and one-reading assumption; it is not an empirical uncertainty estimate. Proof and commands.
Map exact regions produces a inward-target-box-atlas-v5 certificate and a colored two-dimensional outcome map for every one-reading route. Each strip records an exact linear tie target and the Mean/Median winner on either side; all-target vertical ties and fixed-target one-dimensional bands are handled explicitly. The browser chart is a finite-resolution rendering of exact rules available as text and JSON, checked independently in JavaScript and Python. Its colored area does not represent probability. Method and example.
Solve target horizon produces a inward-target-horizon-v6 certificate: with the movement bounds fixed, it finds exact boundaries in the target line. When no movement makes mean equal median, one estimator is strictly closer under every permitted one-reading move below the lower boundary, the other is strictly closer above the upper boundary, and a tie can be constructed for each target between them. If a movement makes both estimates equal, the certificate gives an exact all-target tie witness and correctly rules out any strict global target guarantee. Independent browser and Python verifiers check the bounds, side winners, and a constructive tie probe at the chosen target midpoint. The worked example has boundaries 17/3 and 95/12 with a tie at target 6, moved reading 4. Its portable verified research report maps all six exact route spans, embeds the checked receipt, and identifies the scope of the Lean theorem. A receipt-derived Lean specialization proves that this example's six routes and reported bounds agree with the formal model; CI regenerates and checks the Lean file. Regenerate the report with python3 tools/render_target_horizon_report.py or render another v6 receipt with python3 tools/render_target_horizon_report.py receipt.json --output report.html. Algebraic argument and scope.
Exact target margin adds a sharp, rational safety radius to a strict v6 target query. Inside that open radius the same estimator wins under every permitted one-reading move; at the nearest boundary an exact movement gives a tie. The browser inspector and Python checker both recompute the radius and its boundary witness. A target that already ties has no strict margin. Examples and scope.
Joint Reading Box asks whether Mean or Median remains strictly closer when 3–5 readings vary independently through signed rational intervals and the target is fixed or varies independently through a closed interval. An exact finite ordering-cell calculation supplies the full Mean-minus-Median and estimator-midpoint ranges, and either attained rational readings and target for a tie or a uniform strict winner. V3 gives the exact attained minimum squared-error lead; the v4 target sweep maps the entire tie interval and every rational breakpoint where the worst-case lead changes formula or witness. The standalone verified report turns that receipt into an offline, printable target map with an exact target inspector, embedded independent verifier, and original JSON download. Regenerate it with python3 tools/render_joint_reading_sweep_report.py --check or render a different v4 receipt with python3 tools/render_joint_reading_sweep_report.py receipt.json --output report.html. A separate Python v4 auditor checks each claimed line against all finite candidates at its endpoints without rebuilding the JavaScript verifier's envelope; run python3 tools/verify_joint_reading_sweep.py receipt.json. Previous v1/v2 receipts with conservative bounds remain independently verifiable. Box corners alone can overstate the minimum: an interior ordering-cell witness attains python3 tools/joint_reading_box.py --intervals=48:49,49:50,50:51,51:52,85:86 --target=0 --target-high=80 --target-sweep > joint.json followed by node tools/verify-joint-reading-box.cjs joint.json. This extends the five-reading Target Lens uniform-drift case to heterogeneous bounds, signed rationals, and even medians. It is a conditional calculation, not a measurement or probability model. Proof and limits.
Inverse margin query: Choose a required positive squared-error lead and find every target where the winner is guaranteed that lead for all readings in the stipulated intervals. The interactive browser lab runs this query on a user's own independently checked v4 sweep and exports a JSON answer bound to the exact source bytes. It can also download a self-contained HTML report for that custom sweep: the original JSON, independent verifier, exact target inspector, threshold query, and boundary witnesses travel together offline. The report rechecks its SHA-256 and mathematical claim when opened. The standalone example report demonstrates the same analysis on the included sweep. For the worked threshold of 100, Median is guaranteed on python3 tools/joint_reading_threshold.py examples/joint-reading-box-sweep-v4.json --threshold=100 > margin-regions.json and check it with node tools/verify-joint-reading-threshold.cjs examples/joint-reading-box-sweep-v4.json margin-regions.json. The input sweep is independently verified before calculating the inverse result. Exact argument.
An exact tie at your chosen target: When a v4 sweep contains a tie interval, enter a supported exact target inside it in the browser lab or an offline report. The tie witness constructor finds rational readings inside each stipulated bound that make Mean and Median equally distant from that specific target, then replays the witness through the independent verifier. Download the retargeted v4 JSON receipt to let another checker reproduce the tie. It handles all-target ties, single-point ties, and even-sample medians. Constructive argument.
Exact target tolerance: From a chosen target and required squared-error lead, the Joint Reading Box lab and offline reports now calculate how far the target can move left and right while the same strict winner retains that lead for every permitted reading. They give the largest symmetric radius within the verified sweep and exact cutoff readings where the lead becomes tight. The exported worked answer is tied to the source receipt's SHA-256 and can be independently replayed with python3 tools/joint_reading_tolerance.py examples/joint-reading-box-sweep-v4.json --verify examples/joint-reading-box-tolerance-40-v1.json. At target 40 with required lead 100, the exact symmetric radius is
Challenge a required lead: At any target where the stipulated margin fails, the lab and both offline reports now show exact permitted readings and the Mean and Median squared errors that disprove the margin. At target 46 with required lead 100, readings [48,49,51,51,85] attain only python3 tools/joint_reading_counterexample.py examples/joint-reading-box-sweep-v4.json --verify examples/joint-reading-box-counterexample-46-v1.json. Tie targets, including even-sample and all-target ties, produce target-specific zero-lead readings. Exact argument.
Formal gap and envelope core: The Lean AffineProduct, FixedMedianCell, and AffineEnvelope theorems prove that a product of nonnegative affine factors cannot dip below both endpoint products on a segment, carry a checked margin through finite convex blends of explicit reading-vector formulas, connect those factors to the mean and central-pair median squared-error difference, identify the central positions of a stipulated sorted reading order, preserve that order through convex blends, and show that checking an affine candidate at both ends of a target piece establishes its inequality at every real target in that piece. With a listed attaining candidate, these prove exact minima over their respective stipulated candidate sets; separate lemmas prove the sharp side of a threshold cutoff. GitHub Actions builds and axiom-audits the proof. Supplying every ordering cell and proving vertex completeness for arbitrary reading intervals, plus connecting JSON receipts to the theorems, remain outside Lean and are checked by the exact browser and Python programs. Scope.
The homepage offers three clear starting routes: make a blind prediction, explore the exact two-axis Decision Field, or analyze your own numbers locally. It also previews the local day's One Move puzzle with its five readings, fixed target, and current closer estimate. A starting-field plot places all five readings, the target, mean, and median on the same 0–100 scale. Short bands show each estimate’s exact distance from the target; choosing a reading highlights its dot without exposing a reversal. Choose one of the five reading tiles to carry that starting choice into the dated puzzle; the canonical share invitation stays blind and does not include the choice. After a strict flip, the preview shows the best move saved on that device and invites a smaller move when one exists. If the exact minimum is already saved, it celebrates that result and offers a replay. A compact seven-day edition trail shows saved progress on this device, distinguishes exact minima in gold, and opens a missed date directly from the homepage. After a successful daily flip, the next action opens the most recent unsolved edition in that trail before continuing to the course. A 1200 × 630 challenge PNG shows today’s starting readings, target, and exact dated puzzle address while hiding the closer estimate and winning move. The image is made locally; visitors can download it, copy its exact dated link, or share the PNG directly where the browser supports file sharing. The blind invitation remains a separate action. The preview and trail refresh at local midnight and when the page regains focus. The live fifth-reading instrument and three guided cases remain on the homepage, with deeper links to Flip Point, Target Lens, the Outcome Atlas, and teaching materials.
The homepage hero links directly to today's Daily Flip preview, showing the local date and saved progress on this device. An exact minimum takes precedence in the hero when a newer exploratory draft also exists; the preview still shows that draft.
Share the five-minute mean vs median lesson — two opposing worked examples, revealable answers, a short activity, and links into the lab. It is a static page that can be shared without an account.
Print the three-case worksheet — a student sheet with predictions and calculation space, plus a worked teacher answer key on a separate printed page. Download the two-page PDF for predictable Letter-size printing. Cases A, B, and C yield median, mean, and equal-error outcomes. On screen, each case opens an exact blind challenge that keeps its answer hidden until the student chooses. The exercises use invented values and stated targets; the targets are not independently verified. Print page 1 for students and page 2 for the answer key. Regenerate the PDF with python tools/render_worksheet_pdf.py (requires ReportLab and DejaVu Sans).
Move the reference in Target Lens — hold five readings fixed and slide a hypothetical target from 0 to 100. A decision strip and error curves show where the mean or median has smaller absolute error. For distinct estimates, the exact tie is halfway between their values; when the estimates coincide, every target ties. Three cases match the worksheet, and a shareable view can preserve five invented or public readings and the chosen target. The workbench offers a Target Lens link when its five readings and supplied target fit the URL limits. This is a what-if reference, not proof that any target is true.
Target Lens now includes a constructive target witness for every fixed set of five readings. If mean and median differ, the target equal to the mean gives a strict mean win, the target equal to the median gives a strict median win, and their midpoint ties. The three exact target buttons display both errors and update the live map; coincident estimates instead prove every target ties. The squared-error identity (mean − median)(mean + median − 2×target) explains the verdict, and all comparisons use integer units at the page's six-decimal target precision. This establishes that neither estimator wins for every freely chosen target on a fixed distinct-estimate case. It is a mathematical what-if result, not evidence that an arbitrary target is true.
The whole-range certificate extends that result to a closed interval of possible targets. Enter exact lower and upper bounds or span the two estimates. It distinguishes a strict winner at every real target in the interval, a winner plus a tie when an endpoint touches the crossover, both strict winners plus a tie when the interval crosses it, an exact tie at a singleton, and a permanent tie when estimates coincide. The nearest endpoint's distance to the crossover quantifies the margin only for a strict whole-range win. The proof follows from the single zero of the squared-error difference; it makes no probability claim and does not verify a target. Exact views preserve both bounds as well as the selected point. Older links without bounds open with a two-unit range around their target.
The exact joint-uncertainty certificate asks a second question: suppose each of the five readings can change independently by at most ε within 0–100 while the target lies anywhere in a closed interval. The squared-error difference factors into (mean − median)(mean + median − 2×target). The second factor is monotone in each reading and in the target, so all-low readings with the highest target minimize it, and all-high readings with the lowest target maximize it. The exact extrema of mean − median come from 30 choices of a median and two readings below it. Within each ordering, pushing the other readings to their allowed limits makes this factor decrease as the median rises, so only the limiting feasible median values need checking. Integer millionth units preserve all input endpoints; factor readouts have up to seven decimals because a mean can have fifth-millionth steps. If neither factor's range includes zero, their signs prove a strict winner for every allowed reading configuration and every real target in the interval. If a factor spans zero, continuity proves that a tie is attainable. An endpoint tie displays explicit readings and target; an interior tie displays two allowed configurations and an exact rational fraction along their connecting path. Each reading and target at that fraction is a rational number, even if its decimal representation does not terminate on the input grid. The 2ε absolute-error bound remains available as a simpler sufficient shortcut. For readings 48, 49, 50, 51, 85, target range 0–1, and ε=4, it is inconclusive while the exact certificate proves the median throughout. At range 48–52 and ε=1.3, the page constructs a tie at target 52. Exact links preserve eps (default 1 for older links). These are hypothetical limits, not observed measurement uncertainty or a verified target.
Portable proof receipt. The Proof Inspector opens three worked cases—strict median guarantee, rational tie, and permanent tie—and can print a checked one-page sheet with a SHA-256 fingerprint of the exact JSON text checked. Target Lens can download a versioned JSON receipt with the five readings, target interval, drift, exact factor ranges, tie-target band, and a rational tie witness when one exists. Open the browser proof inspector to check a local receipt without installing anything, or run node tools/verify-target-lens.cjs path/to/inward-target-lens-proof.json offline. The browser and Node share an integer-only verifier; an independent Python checker scans all feasible median-order breakpoints. Both recompute the factor extrema and verdict, check any witness against the allowed box and exact tie identity, and print the same fingerprint for the same receipt text. See the receipt format and proof. Each checker exits nonzero for an altered or invalid claim. The selected target is included for context; the certificate quantifies over the entire specified range. This is a reproducible mathematical check under the stated model, not authentication of the inputs or evidence that a chosen target is true.
The cross-language quality gate runs seeded valid and corrupted receipts through both command-line verifiers in CI, with regression cases for JSON key order, integral decimal encodings, malformed JSON, and exact byte fingerprints. The commercial validation brief names the current product boundary and the buyer tests needed before offering licenses or an API.
The reachable tie-target band maps the current reading drift onto possible reference targets. The midpoint (mean + median)/2 is continuous and monotone in every reading, so across independent clipped reading intervals its image is exactly the closed interval from the all-low midpoint to the all-high midpoint. If an allowed arrangement has mean = median, that one arrangement ties at every target from 0 to 100, including targets outside the midpoint band. The page shows this exact tie set alongside the selected target interval on a 0–100 ruler. For case A at ε=1.3, midpoint targets 52–54.6 are reachable, touching the selected 48–52 range at 52. At ε=4 with target range 0–1, the midpoint band 49.3–57.3 is disjoint and the estimates cannot coincide, so the median remains strictly closer throughout. At ε=5.333334, the estimates can meet and every target can tie. Band endpoints may need seven decimals even though input settings have six; the certificate covers every real target between the selected endpoints.
Expand the target × drift map below that ruler to inspect the whole reachability geometry. Before the two estimates can coincide, the tie targets at each ε lie between the all-low and all-high midpoint curves. Both curves are affine between the finitely many drift values where a reading clips at 0 or 100; the map joins exactly those breakpoints. At the exact rational ε where mean and median can first meet, the tie set expands to all targets, shown in gold. A rose segment places the selected target interval on the current ε line. The SVG places the exact formulas on a finite pixel grid for inspection; numeric certificates and witnesses retain the exact values. Its vertical scale expands to include the selected drift and the all-target boundary, up to ε=100.
The tie depth atlas below the map extends this geometry to budgets of zero through five movable readings at the current ε. For any fixed subset, the midpoint image of its clipped reading box is a closed interval through the original midpoint. Every subset shares that original point, so the union over subsets of size at most k has no gaps. Its exact endpoints are the smallest all-low and largest all-high midpoint among those subsets. A subset that can make mean equal median extends the possible tie targets to all of 0–100. These attainable sets are nested as k grows. The first row touching the selected target range therefore gives the exact minimum number of readings that must be allowed to move; it is checked against the independent minimum-support certificate. Rose shows the selected range, mint a midpoint tie interval, and gold an all-target tie possibility. For case A at ε=1, the bands grow from the fixed 53.3 point to 52.3–54.3 with five movable readings, while range 48–52 stays outside every row. At ε=4 that range first reaches the two-reading row; at ε=10 the three-reading row can produce equal estimates and every target can tie. This is a combinatorial statement about hypothetical allowed changes, not an identification of faulty readings.
The map's rose dot answers the inverse question for the single selected target: what is the least continuous reading drift that permits its first tie? The exact factor certificate supplies a rational threshold and an admissible five-reading witness; its adjacent six-decimal selectable setting is shown separately. The inspection button sets the uncertain target range to this one target and reveals its full stability certificate. A range is an existential choice of any target inside it, so it can tie sooner than a specific target: in case A, the range 48–52 first ties at ε=13/10, while target 50 first ties at ε=33/10. A target already at the fixed-reading midpoint ties at ε=0. This comparison does not make claims about measurement accuracy or likelihood.
The stability edge gives both the exact continuous drift threshold and its neighboring six-decimal input settings. For the monotone reference factor, the extremal readings are all at their clipped lower or upper limits. For mean − median, the 30 median-order assignments give the exact extrema at their limiting feasible medians. As ε grows, each formula is affine between a finite set of landmarks: a reading clipping at 0 or 100, or half the gap between two original readings. Scanning every assignment on every segment and solving the first zero of each factor gives two exact rational routes to a tie. The smaller root is the strict guarantee's threshold. The page highlights the first route and gives a separate rational reading/target tie witness for each factor, including the later route after the guarantee has already ended. The expanded witness lists all five exact signed reading moves, the resulting mean and median, and their equal absolute errors. Visitors can copy a plain-text certificate with assumptions, both routes and their witnesses, threshold, neighboring input settings, and a reproducible URL. Nested reading boxes then prove the original strict winner persists at every smaller real ε, and tie possibility persists at every larger ε. Integer-grid bisection independently finds the adjacent settings visitors can enter and share. For case A at target range 48–52, the target can reach the mean–median midpoint at 13/10, while the two estimates cannot meet until 16/3; the median is guaranteed through six-decimal ε=1.299999 and a tie appears at 1.3. At target range 0–1, the estimate meeting at 16/3 is first, while the target-midpoint route occurs at 75; the input edge lies between 5.333333 and 5.333334. At ε=16/3, readings [128/3, 131/3, 166/3, 166/3, 239/3] with target 0 have mean = median = 166/3. The later route at ε=75 also has an exact tie witness: readings [0, 0, 0, 0, 10] and target 1 give mean 2, median 0, and equal errors of 1. At the first selectable tie setting the page also constructs an interior tie by interpolating extremal readings at fraction 1/14333335. If the unperturbed target range already permits a tie, there is no strict baseline. These are exact results for stipulated continuous uncertainty boxes, not evidence about measurement accuracy or which estimate should be used in a population.
An independent Python stability-edge audit now recomputes both continuous routes and exact rational witnesses from a validated proof receipt. Run python3 tools/audit_stability_edge.py examples/stability-case-a.json on the included worked receipt, or pass your downloaded receipt; use --mask 5 to allow only readings one and three to move, or --json for machine-readable fractions. Its finite-landmark scan is cross-checked against the browser implementation across varied reading boxes and masks. It ignores the receipt's selected drift when finding the first possible tie.
The portable stability edge certificate packages the original Target Lens receipt, selected movable-reading mask, both exact threshold routes, and rational tie witnesses. Download it from the exact stability panel in Target Lens, or generate it with node tools/create-stability-certificate.cjs examples/stability-case-a.json > edge.json, then check it independently with python3 tools/verify_stability_certificate.py edge.json. The worked JSON example is checked into the repository. The verifier recomputes both routes and checks each supplied witness; its SHA-256 output identifies the exact file bytes without authenticating the assumed inputs.
The minimum-support certificate asks a complementary question at the selected ε: how few readings must be allowed to move before a tie is possible? Each of the 32 subsets freezes its unselected readings and gives the selected readings independent, continuous ε boxes clipped to 0–100. The exact two-factor certificate checks the entire box and target interval. Subsets are visited in increasing size, so every smaller subset is explicitly excluded before the first tie witness. The card highlights actual changed readings, displays an exact rational target, mean, median, and equal errors, and copies the assumptions and witness. For case A with target interval 48–52, ε=1.3 requires all five readings; ε=4 needs two (readings 1 and 3 in the displayed witness); ε=10 needs one (reading 3). At ε=1, all 32 subsets remain strictly median-favoring. This is a combinatorial minimum for the stated per-reading boxes and target interval; it is not a claim about which real observations are unreliable or likely to change.
Expand the reading–drift frontier to reverse that question: for a budget of at most one through five movable readings, what is the smallest selectable ε that permits any tie? For each budget, the browser checks every subset of that size and bisects the six-decimal ε input grid using the exact continuous-box certificate. A tie at the displayed setting plus strict certificates for every subset one grid step below prove the minimum on that grid; nested boxes exclude every smaller selectable setting. For case A and target interval 48–52, the five thresholds are ε=8, 1.857143, 1.625, 1.444445, and 1.3, respectively. The rows identify one successful subset and jump to its setting; the minimum-support witness then shows the actual rational tie. These grid thresholds are distinct from the exact continuous rational threshold shown for the all-five route. If a baseline tie already exists, zero movement suffices. Some smaller-support budgets cannot tie even at ε=100.
The frontier also proves the exact continuous threshold for each support budget. For each subset, its clipped reading bounds are affine between finitely many landmarks: a movable reading reaches 0 or 100, two movable boxes first overlap at half their original gap, or a movable box meets a fixed reading at their full gap. Within each segment, all 30 median-order assignments and the monotone reference factor are affine; solving the first zero of either factor gives an exact rational radius and an admissible rational reading/target witness. Compare those radii across all subsets of the same size, then compare each exact radius to the independently bisected six-decimal input setting. For case A and target interval 48–52, the exact radii for one through five movable readings are 8, 13/7, 13/8, 13/9, 13/10. At two readings, moving readings 2 and 3 to 330/7 and 337/7, while the others stay at 48, 51, and 85, gives mean 391/7, median 337/7, and target 52: both errors are 27/7. Every subset of at most two movable readings stays strict below ε=13/7. The page shows the exact fraction, neighboring selectable setting, and witness for each budget. This is a deterministic statement about specified hypothetical boxes, not a probability or a claim that particular observations should change.
The single-reading comparison breaks out every one-reading box instead of showing only the earliest one. It freezes the other four readings, proves each exact continuous boundary with a rational tie witness, and independently bisects the six-decimal settings. For case A and target interval 48–52, readings 1–5 individually need exact drift 13, 13, 8, 8, 13; at ε=8, either reading 3 or reading 4 can independently force a tie, while the other three cannot. A row that cannot tie even at ε=100 remains explicitly unreachable. This measures mathematical possibility under chosen limits, not which observed reading is faulty.
Find the Flip Point — hold four readings and a chosen target fixed, then move the fifth reading from 0 to 100. The exact piecewise solution marks every tie and separates true winner reversals from touching ties. A decision band, absolute-error curves, jump and cross controls, and an exact-value field expose how the mean and median respond. The nearest reversal is measured in reading units; reaching the boundary ties, and crossing it changes the winner. Shareable links preserve all six numbers, and the five-value workbench can open its current case in Flip Point. These are what-if values and a chosen reference, not measured truth.
Explore the Decision Field — move the reference target and the fifth reading together. An exact SVG decision map shows every mean and median winning region, the piecewise midpoint frontier where a particular target ties, and horizontal lines where both estimates coincide so every target ties. Drag the crosshair or enter precise values, jump to either kind of tie, choose among three worksheet cases, and share the six-number view. The workbench links eligible five-reading cases into this map. Its geometry is calculated in the browser from the equations, not inferred from a pixel grid; the target is still a chosen what-if reference.
Shape the Data — drag any of five readings on its own lane, or drag the chosen target. Gold and mint markers show the mean and median as both absolute-error bars respond. The comparison card measures their movements against an anchored arrangement: if exactly one reading changes, the mean moves one fifth as far, while the median follows the middle sorted value. Precise labeled controls support keyboard use and up to four decimal places. Three invented cases, an anchor/reset control, and links into the blind challenge, Decision Field, and workbench make an exact arrangement reusable. Share links contain all five readings and the author-supplied target; use invented or public numbers, never sensitive data. No data is sent while manipulating the page locally.
Rank the Flip Levers — hold a target and four of five readings fixed in turn. Five piecewise decision bands show where each movable reading makes the mean or median closer. The rack ranks the smallest change from the current input to an opposing strict winner on a four-decimal grid. White ticks are true winner reversals; blue ticks are ties without reversal. A button applies a ranked candidate, and links carry the updated five readings into Shape the Data or Decision Field. A current tie is reported without a false “flip”; a reading with no opposing winner in 0–100 is marked accordingly. This is mathematical sensitivity for a chosen reference, not evidence that a reading or person is intrinsically influential.
Play One Move — a three-round puzzle based on the same exact solver. Choose one of five readings, propose a new value without a live verdict, and commit it. Reaching a tie is not enough: the other estimate must become strictly closer to the chosen target. Two optional, progressive nudges explain how the mean and median respond, then identify a viable reading and direction without revealing the exact crossing. After each valid committed move, a side-by-side view compares the mean and median errors before and after on a shared scale, identifies the closer estimate, and explains the final lead. A second view sorts the five readings before and after, highlights the moved reading and middle position, and explains how the mean and median moved. A reversal can even make both errors grow. After a successful move, the page compares its distance with the minimum available four-decimal change and reveals a five-route atlas: the shortest strict change for each reading, including readings with no reversal within 0–100. The shortest route and the player's selected reading are marked. After a strict flip, a browser-rendered 1200 × 630 result PNG can be downloaded with the chosen reading, target, before/after errors, and move distance; the separate blind challenge link keeps the answer hidden. A custom Flip Levers arrangement with a possible strict reversal can become a shareable puzzle; invalid, tied, or unflippable cases fall back to round 1. Guesses and scores stay in the current page session and are not shared or collected. The curated targets are invented what-if references, not measured truths.
After a successful One Move reversal, Why no smaller move wins opens an independent integer-grid certificate. It covers all 5,000,005 legal final positions across the five possible reading choices without checking each one in the browser: the exact signed error gap is linear between the fixed middle values, target, and mean-zero point, so endpoint signs and the first strict grid point determine each interval. The panel gives the closest strict flip or an unreachable result for every reading, the neighboring failed grid point, and the winning reading’s interval boundaries. The certificate is checked against the visual Flip Levers solver before it claims agreement. It proves the finite puzzle minimum under the chosen target and four-decimal rule; it does not validate a target or a real-world estimator.
Return for the daily flip — the browser turns its local calendar date into a dated link. A deterministic generator uses that date to make five invented, distinct whole-number readings and a chosen target, accepting only cases with at least three viable single-reading flips and a shortest move between 0.75 and 12 units. The same dated link reopens the same puzzle on any device, independent of the viewer's date. An unfinished daily reading and proposed value stay in this device's local storage so a dated edition resumes after a reload or return visit. An explicit reading choice from the homepage takes precedence; a strict flip clears the draft and stores the best movement distance for that date. There is no account, shared leaderboard, or server score. After a strict flip, the result shows a blind duel invitation with your best move distance and the original puzzle link. A dated link reveals no readings, target, or solution in the invitation; custom puzzle URLs carry the chosen inputs needed to reproduce them. The share sheet opens where supported; elsewhere the invitation can be copied. This is a generated what-if puzzle, not an observed daily event or a verified target.
The seven-day trail shows the device's local calendar dates from six days ago through today. Each date opens its exact edition; dates completed with a strict flip show the best movement distance saved on this device. The date explorer beneath it accepts today or an earlier date from 2000 onward, reports any best move saved on this device, and opens the deterministic puzzle for that date. The homepage trail links to this control. These puzzles are generated from their dates, including dates before this feature existed; the date does not claim the puzzle was published then. Progress uses local storage only and makes no claim about other players' results.
Moving between puzzles inside One Move creates browser history entries. Back and Forward restore the dated edition, practice case, or course round from its address, including a saved daily draft when one exists.
Continue in the open run — 999,999 numbered practice cases use the same exact generator constraints as the daily editions, but a distinct seed namespace. A strict flip unlocks the next numbered case; its URL recreates the same starting arrangement for anyone. A session run ribbon shows recent cases, successful flips, and exact minimums. Completed cases can be replayed in place to improve a best move, with a return button to the newest case. The sequence wraps after case 999,999. Practice attempts and best moves remain in the current page session and do not alter the device's saved daily scores. Reloading begins a new run. A blind challenge includes the practice case number and its reproducible link, without the answer.
Send a prediction challenge — a blind mean-versus-median question with five seeded readings and a known target. A visitor chooses before seeing the exact answer, then opens the same case in the full lab. From the lab, Challenge someone with this case carries the selected seed and fifth reading into a shareable URL. The URL contains generated inputs only; it does not include a guess, score, custom workbench data, or spring history. Try another challenge chooses from a balanced set of invented examples. The static challenge preview deliberately does not reveal an answer.
Invite someone to predict opens the device share sheet when available, or copies a short invitation and exact challenge link for pasting into a message. The invitation works before or after your own answer and never includes your guess or the revealed result. Reader-supplied challenge links include all five values and the stated target; a nearby warning reminds you to share only invented or public data. The compact iframe keeps the invitation out of its fixed-height view.
Create a challenge from five readings. In the workbench, analyze exactly five values with a known target, then use Turn these values into a prediction challenge. Or expand Create a challenge with your own numbers on the challenge page. Values and target must be from 0 to 100 with at most four decimal places. The new challenge hides the answer until a prediction, can be embedded, and links back to the same values in the workbench. Try a mean-wins example. These reader-supplied values are separate from the seeded atlas. The target comes from the link author and is not verified. The URL includes all five values and the target; it can appear in browser history, hosting logs, and referrers. Use invented or public data, never sensitive values. No guess is encoded in the URL.
Explore after predicting. The full challenge reveals a counterfactual sweep after an answer. Move the fifth reading through 101 whole-number positions while the first four readings and stated target stay fixed. A color map and counts show where the mean, median, or neither has smaller absolute error; the numeric readout shows the selected position. The original prediction and receipt stay in place. Open this selection as a new blind challenge shares the exact generated seed and fifth reading, or the four supplied values, selected fifth reading, and target. If the original fifth reading is fractional, the explorer starts at its nearest whole number and states both readings. The 101 positions are a grid, not an estimated probability. The compact iframe retains its fixed-height prediction view and links to the full challenge for this exploration.
Keep the result as a picture. After answering on the full challenge page, expand Keep a picture of your result to preview and download a 1200 × 630 PNG. It records the original choice, five readings, stated target, closer estimate, and both absolute errors; moving the counterfactual slider does not alter it. The browser draws the image locally and downloads it on request. Reader-supplied cards say that their values and target have not been independently verified. The PNG exposes the answer and all five readings, so share only invented or public data. The compact iframe links to the full challenge for this feature.
Embed a case in an article or course page. Open a generated or reader-supplied challenge, expand Put this challenge on your own page, and copy the iframe code for that exact case. The compact widget asks for a prediction before revealing the result, includes the toy-model caveat, and links back to the full challenge and lab. The five-minute lesson contains a live sample. The widget has no account, tracking, or backend; answers are not collected. The publisher's own platform may still have its own analytics and iframe restrictions. The snippet uses loading="lazy" and an 800-pixel height; preview it on the target platform before publishing. The ordinary share link does not include the embed=1 display flag.
Explore the outcome atlas — a clickable map of 128 consecutive generated seeds × 101 whole-number fifth readings (12,928 deterministic cases). Gold and mint distinguish which estimator has lower absolute error against the stated target of 50; brightness shows the size of that error difference, capped at four units. Select a cell by pointer or use labeled seed-row and fifth-reading controls. The side panel exposes all five readings, both estimates and errors, column-level win counts and average errors. Buttons find a mean win or median win in the selected column; the exact cell opens in the lab or the blind challenge. Share links encode the cohort start seed, selected row, and fifth reading, all validated on load. The atlas is a teaching grid, not evidence about real observer populations or a probability of either estimator winning. On a narrow screen, the map scrolls horizontally while the numeric controls remain available.
Open the live lab above, or download the repository and open index.html in a modern browser. No build, network connection, or server is required for the downloaded version. For a quick comparison, leave seed at 42 and move the fifth observer to 50; the single-seed and 128-seed results differ in useful ways. To run the model tests, use Node.js 22 or later:
node tests/lab.test.cjsThe first screen offers three guided cases: an extreme fifth estimate at 85, a counterexample where the mean has lower error at 40, and a cohort reversal at 50. Choosing one sets seed 42 and moves to the relevant chart. Copy this exact case creates a link with validated seed and fifth parameters; opening that link restores the same witness setup. The link does not encode the spring's step history or shocks. The live reading and guided outcome update as you change the slider. The page contains a descriptive title, summary, and canonical URL for the published lab; the local file works without a network connection.
The first-screen instrument now has its own fifth-reading slider. Moving it updates the error numbers, the closer estimate, the charts, and the lower lab slider together. See a mean/median win jumps to an opposing result for the current seed when one exists; the link beside it opens the selected case as a blind challenge. Guided cases, reset, and shared URLs also update this control. The result is for these five generated values against the stated target of 50, not a general claim about the estimators.
On narrow layouts, the instrument follows the headline before the longer introduction and route links, so the first interaction appears sooner in reading and keyboard order. Its range control has a taller touch area, and its flip and challenge actions have at least 44-pixel-high targets; at phone widths those actions stack across the card. The desktop and mobile layouts use the same calculations and controls.
The checked-in social preview is generated from the same seed-42 cohort equations with python tools/render_social_card.py (Pillow required only to regenerate the image). Its chart is illustrative of this specific toy cohort; it does not report a population result.
The local workbench accepts 3–40 finite observations between −1,000,000 and 1,000,000, separated by commas, semicolons, or whitespace. An optional known target lets you compare absolute errors; without one, the workbench shows both estimates without declaring either more accurate. It plots the original observations, mean, median, and supplied target. It then removes the original observation farthest from the original median and recomputes both estimators. This leave-one-out comparison shows sensitivity; it does not establish that the removed point is invalid. An even-length dataset uses the average of its two middle sorted values. Ties for the farthest observation use the first entered value.
Use the invented outlier and mean-wins examples to learn the controls, or paste your own numbers. Save this analysis as JSON downloads the inputs, optional target, exact computed values, errors, and leave-one-out result. Analysis runs in the browser, with no application backend or analytics. The workbench does not transmit values while you edit or analyze them. If you deliberately open a challenge link, its five values and target are placed in the URL and sent in the page request. Other workbench inputs are not put into links. Display labels round to four decimal places; the saved JSON retains JavaScript's numeric values. Avoid interpreting a supplied target as independently verified ground truth.
The yellow point moves toward a fixed green target. Each discrete step applies:
velocity = 0.92 × (velocity + 0.008 × α × (target − position))
position = position + velocity
coherence proxy = exp(−distance to target / 120)
Set pull strength α between 0.5 and 3.0. Step once advances exactly one update; Run advances at a fixed 60 updates per simulated second. Add shock applies an impulse whose magnitude and direction come from an isolated seeded pseudorandom stream. The seed also generates the four fixed observer values in the second experiment. Resetting the seed and replaying shocks at the same step reproduces the same numeric trajectory. “Settled” means distance under 4 and speed under 0.5 for 45 consecutive steps; it is a display rule, not a proof of convergence under every possible input.
Four seeded observers estimate a known target of 50; a fifth estimate is adjustable from 0 to 100. The graph and telemetry show the ordinary arithmetic mean, median, and each one's absolute error against the known target. With seed 42 and the fifth observer at 85, the mean is about 55.38 (error 5.38) and the median about 50.12 (error 0.12).
The sensitivity sweep holds the four seeded values fixed and computes both absolute errors for every whole-number fifth estimate from 0 through 100. The curves, current-position marker, sample table, lower-error counts, and maximum errors make the comparison inspectable beyond one selected example. With seed 42, the median has lower error at 84 grid positions and the mean at 17; there are no ties. For example, the mean wins at 40 while the median wins at 0 and 100. These counts describe an evenly spaced grid, not the chance of either estimator winning on real data. Change the seed and reset to generate four new fixed values. The fifth estimate is a numeric perturbation; it does not imply dishonesty or establish a generally superior rule.
The cohort view repeats that grid for 128 consecutive generated seeds, starting at the selected seed and wrapping modulo 2³². Each seed generates four values with four successive draws from the same seeded linear congruential generator; the fifth estimate is shared across seeds. Its curves show average absolute error across those 128 cases. The table also shows how many individual seeds yield lower median error at each sampled fifth estimate. With start seed 42 and fifth estimate 50, the mean has lower average absolute error (about 1.43 versus 1.56) even though the median has lower error in 75 of 128 seeds. Frequency of winning and size of errors can disagree. At fifth estimate 85, the median has lower error in 126 of 128 seeds. The cohort is deterministic generated data; consecutive PRNG seeds are not an independent sample of real observers, and these numbers are not population estimates.
The lab starts paused for everyone and respects reduced motion. Set a seed, then use Step once or Run to make an observed path.
- See one step ahead previews the pull term, damped velocity, next position, and distance change without advancing. A faint ring shows the next point on the trajectory while paused. Changing pull strength or adding a shock recomputes the preview.
- Trace the path behind you freezes up to 150 recent positions. Scrub for exact coordinates and distance, or jump to the closest captured approach, next shock, or next pull-strength change. Event jumps wrap within the frozen capture and are disabled when their only event is already selected. The distance chart marks shocks in rose and pull-strength changes in lavender; the field map locates the selected point and target, rose squares for shock positions, and lavender rings for pull-change positions. Overlapping markers show both events without hiding the selected point. Copy a link to the selected captured step to replay that moment paused with its trace open, even if the live run has since changed. The link uses the event history saved when the path was captured and carries the fifth reading from that capture; it does not record earlier fifth-reading edits. Download path CSV exports every captured step with its coordinates, distance, shock count, and pull changes. A shock changes velocity at its recorded step, while position responds on the next update. Refresh to capture later steps. A close pass alone is not sustained settling.
- Compare two futures forks the current position and velocity into two 60-update, no-new-shock paths that differ only in pull strength. The cursor compares any selected update; a 60-bar ribbon counts when each path is nearer, and a jump control visits lead changes. A lead at one update is not a universal ranking. Use alternative α in live run applies that strength at the current step and records the parameter change.
- Make a visual run receipt captures trajectory, seed, step, distance, speed, pull, and shock count on a 1200×630 canvas. Refresh the frozen snapshot after changes, then save a PNG named for its captured step. This image summarizes the run; the replay link or JSON carries its exact event history.
- Export run JSON includes model constants, seed, inputs, exact position and velocity, shocks and their impulses, parameter changes, observer values and errors. The
inward-math-toy-v3record also includes the 101-position sensitivity sweep and 128-seed cohort protocol, generated records, aggregate points, and winner counts.
Copy replay link encodes seed, fifth reading, pull and shock-strength changes, shock timing, and step count. Opening it reconstructs the run and pauses for inspection. Replay links are limited to 3,600 steps and 80 changes or shocks each; they exclude workbench data. URLs can appear in browser history or hosting logs, so use invented values. The page sends no inputs to a server except the URL requested when a replay link is opened. Identical seed, changes, shock timing, and step count produce identical numeric results. Inspecting frozen traces and future previews does not advance the live model.
The original page only drew an unseeded spring animation even though its README described a second panel and telemetry. The current lab implements both panels and makes the claims and formulas inspectable.
Created by Daniel Jacob Read IV / ĀRU Intelligence Inc. with Shane Travis Horman. Development assistance included Grok (xAI). Released under the MIT License; see CITATION.cff.
Complete Inward Physics index · Daniel Jacob Read IV’s ORCID · Zenodo DOI series (record index)
These links index project-authored proposals and simulations; they do not imply independent validation of a physical model.