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How AI Agents Discover Scientific Equations: From Hydrotope Rediscovery to New Water-Wave Amplitudes

Zihan Zhou, Digvijay Wadekar, Matias Zaldarriaga

arXiv:2610.00435v1hep-thcs.AI

TL;DR

The paper asks how AI agents move from chamber-specific formulas to globally valid scientific expressions, a process that final-answer-only evaluations cannot fully reveal. It reconstructs hydrotope discovery and tests single agents and coordinated PI–student teams, which recover the complete hydrotope and produce a new A6 expression.

  • Problem

    How AI agents progress from chamber-specific formulas to globally valid results remains poorly understood, despite benchmarks showing difficulty completing full research tasks.

    Source: §1 · p. 2

  • Method

    The paper reconstructs hydrotope discovery, analyzes 18 single-prompt runs under different hints, compares regression methods, and tests coordinated PI–student teams.

    Source: §1 · pp. 3–4, §7 · p. 22

  • Results

    Four of 18 single-agent runs recovered the complete formula; both no-hint PI–student teams did, versus none of six comparable single-agent runs, and the workflow found a new independently checked A6 expression.

    Source: §1 · pp. 3–4, §6 · p. 21

  • Takeaways & Limitations

    When exact calculations or accurate simulations are available, agents should use counterexamples to revise formulas and separate checking before accepting results.

    Source: §7 · p. 22

    The comparisons do not match total computational resources or repeat setups enough to estimate reliability; future evaluations should address both.

    Source: §6 · p. 22

Abstract

from arXiv · show

We study how AI agents discover and validate scientific formulas using a controlled case study of the hydrotope, a recently discovered geometric formula that combines the different polynomial pieces of nonlinear surface-wave scattering into one global expression. This problem is deceptively difficult: simple formulas can hold within individual frequency regions, but the global result must identify their boundaries and combine exponentially many potentially active terms. We reconstruct how the formula was originally discovered through human--agent collaboration and analyze 18 single-prompt rediscovery runs under no hint and two forms of human guidance: a false hint representing an incorrect prior and a true hint representing domain-informed insight. Only four recover the formula across all kinematic chambers (i.e., regions in which a single polynomial form applies), while most unsuccessful runs find correct chamber polynomials but fail to combine them or test their full domain. Conventional and LLM-assisted symbolic regression and standard machine-learning regressors likewise fail to recover the global formula in our experiments. Guided by these failure modes, we test a PI$+$two-student workflow in which a coordinating lead agent assigns complementary analytic and numerical tasks to two research agents and independently evaluates their results. The PI$+$two-student team successfully rediscovers the complete hydrotope formula, while the same workflow applied to the harder three negative wavenumber problem discovers a new independent verified analytic expression for the six-point amplitude $A_6$.

1 Introduction

The paper examines how AI agents can move from exact numerical amplitude values to formulas valid across frequency chambers, where the formula changes at boundaries. It reconstructs the hydrotope discovery, evaluates single-agent runs and fitted methods, and tests a coordinated team workflow.

1 Introduction

  • The challenge is global: exact BG recursion values do not reveal where chamber-specific polynomials change or how to combine them across all boundaries.A complete solution must identify every relevant boundary and combine the polynomials on both sides.Source: §1 · pp. 2–3
  • The authors reconstruct the original human–agent discovery and use its failure modes to design a team that separates formula construction from checking.Two students pursue complementary questions; a PI combines their results, identifies missing cases, and tests beyond the chambers used to derive the formulas.Source: §1 · pp. 2, 4
  • The paper also compares interactive discovery with symbolic regression and standard machine-learning methods that fit amplitudes from input frequencies.These fixed-data comparisons are distinct from the repeated discovery runs.Source: §1 · p. 3
  • Across 18 single-prompt runs, only four recovered the complete all-chamber formula; many others found correct chamber polynomials but failed to combine or fully test them.The prompts included no hint, a false hint, and a true hint that supplied mathematical insight without revealing the answer.Source: §1 · p. 3
  • Both PI–student teams rediscovered the complete hydrotope, while none of six comparable single-agent runs did; the team also produced a new expression for the six-point three-minus amplitude A6.The A6 expression was independently checked, but the paper provides no analytic proof.Source: §1 · p. 4

2 Surface-water-wave scattering and the discovery of the hydrotope

The hydrotope organizes the two-minus scattering amplitude across frequency chambers, while Berends–Giele recursion supplies exact values at chosen allowed points but not the global formula. The reconstructed discovery proceeds from a one-chamber guess to a tested expression combining contributions across chambers.

2.1 Surface water wave scattering

  • Berends–Giele recursion builds an n-point amplitude from lower-point interactions and evaluates it exactly at a chosen on-shell point, but returns a value rather than an explanatory formula.A physical query must obey the dispersion relation and conserve total energy and momentum.Source: §2.1 · p. 4
  • Geometrically, the amplitude is proportional to the volume of a frequency-dependent box slice, whose shape changes across chambers.The slices are polygons at n = 5 and three-dimensional solids at n = 6.Source: §2.1 · p. 5

2.2 Original discovery episode

  • The expression was later rediscovered by many agents; it is exact in one frequency chamber but is not the complete formula.Source: §2.2 · p. 6

2.1 Surface water wave scattering

  • The hydrotope formula uses an alternating subset sum with max(x, 0) terms to combine chamber contributions and subtract their overlaps.Crossing a chamber boundary switches terms on or off; between boundaries the active terms give one polynomial.Source: §2.1 · p. 5, §2.2 · p. 6

2.2 Original discovery episode

  • After relaxing the initial ordering and positivity assumptions, the agent used a failed case to find the corrected alternating subset sum and checked it exactly at higher n.The three prompts moved from a one-chamber guess, to a broader formula search, to correction of the failed case.Source: §2.2 · p. 6

3 Experiments and comparison methods

The experiments compare single-prompt discovery under three prompt packages with equation fitting on fixed BG-generated data. Across the tested fitted methods, none recovers the complete formula; the paper distinguishes approximate prediction from interactive formula discovery.

3.1 Repeating the discovery task

  • The prompt packages vary both equation guidance and recommended tests: the false hint is incorrect, the true hint identifies chamber-dependent polynomials, and the no-hint package suggests no equation type.Because test-point instructions also differ, the comparison is between complete prompt packages, not hints alone.Source: §3.1 · p. 7, Table 1 · p. 7
  • The 18 discovery runs comprise six agent configurations tested once under each of three prompt conditions, so the reported counts describe those runs rather than estimated success probabilities.The study does not estimate pass@k because it lacks repeated runs for each configuration and condition.Source: §3.1 · p. 7

3.2 Symbolic regression and machine-learning comparisons

  • The fitted-method benchmark uses BG recursion as a data generator and tests numerical regressors for prediction and symbolic-regression systems for explicit equations.Methods use fixed train–test splits; the PI–student teams instead choose new BG calculations during their work.Source: §3.2 · p. 8
  • No conventional or LLM-assisted fitted method recovered the complete formula, while the PI–student team rediscovers it; no fitted method reaches zero error.Operon has the lowest median held-out MAE among fitted methods, and the team’s zero-error entry evaluates its rediscovered formula.Source: §3.2 · p. 9, Figure 3 · p. 9
  • At n = 7, the target has 32 max(x, 0) terms, and no fitted method is exact even when searches allow separate regional formulas.Those expanded searches can improve average prediction without recovering the slanted chamber boundaries or the short alternating subset sum.Source: §3.2 · pp. 8–9
  • The comparison is task-specific, not a general ranking: results depend on allowed operations, search budget, input variables, training-point distribution, and compute.The experiments do not use equal compute and cannot identify which design choice caused the outcome.Source: §3.2 · p. 10

4 Single-prompt rediscovery: results and run histories

Across 18 single-prompt rediscovery runs, only four recover the complete formula, and all four receive a true hint about chamber-dependent polynomials. Most runs make useful local discoveries but stop before identifying all boundaries, combining chamber contributions, and validating a global formula.

4 Single-prompt rediscovery: results and run histories

  • The run histories are chronological reconstructions of preserved visible records, not verbatim transcripts or accounts of unexpressed internal reasoning.The study analyzes 18 runs across six configurations and three prompt conditions, with four no-hint histories examined in detail.Source: §4 · p. 10

4.1 Results and common steps

  • Only four of 18 runs recover the complete formula, and all four use the true hint that different polynomials apply in different frequency regions.The six no-hint and six false-hint runs produce no complete formula.Source: §4.1 · pp. 10–11, Table 2 · p. 11
  • Eleven of the 14 incomplete runs find at least one correct chamber polynomial, but only four also find the alternating-sum pattern, with a missing term, condition, or test still unresolved.Most runs therefore capture part of the answer rather than producing a wholly incorrect result.Source: §4.1 · pp. 11, 14
  • The true hint yields four complete results, while false-hint runs range from partial sums to incorrect answers and no-hint runs stop at one-chamber formulas or near-complete sums.The true hint narrows the search but leaves boundary identification, subset assembly, the overall factor, and broad validation to the agents.Source: §4.1 · p. 11
  • All 18 runs build a recursion evaluator, 15 recover the principal-chamber formula, 16 find where it fails, and 15 recognize chamber dependence.The largest gap is later: only eight identify all subset boundaries and four combine them into the complete alternating sum.Source: §4.1 · pp. 13–14, Figure 4 · p. 13
  • The main bottleneck is global completion: finding every subset boundary, combining chamber contributions, and testing the resulting formula in previously unsampled chambers.Five runs test a formula beyond its derivation chambers, but only four do so for an all-n candidate.Source: §4.1 · pp. 12, 14

4.3 From single-agent failures to a coordinated multi-agent system

  • A single agent may choose test points shaped by the same assumptions used to build its formula, so deliberately testing other chambers can expose missing cases.The proposed team process separates formula construction from testing and assigns the PI independent BG calculations and out-of-chamber tests.Source: §4.3 · p. 15

4.2 Case study: four runs with no hints

  • In four no-hint examples, all agents find the principal-chamber formula and counterexamples, but Codex 5.5 and Fugu leave subset sums incomplete while Claude runs fail to combine chamber formulas.Claude max’s sorting formula misses six of 233 tests; Codex 5.5 and Fugu also miss cases through an incorrect power or an omitted wave.Source: §4.2 · pp. 14–15, Figure 5 · p. 16

5 A coordinated multi-agent system

The paper uses a PI-and-two-student workflow that separates complementary formula searches from independent checking. The teams rediscover the two-minus formula and produce a tested six-point expression for the three-minus sector, while the four-setup comparison illustrates both the value and limits of parallel work.

5.1 How the team works

  • Each round, the PI assigns complementary questions: one student typically calculates exact examples while the other derives and explains a formula; both record results, and the PI separately codes and checks it against exact BG calculations.Failed tests become tasks for the next round; in two-minus experiments, tests covered n = 4, 5, 6, 7 at several frequency choices, with formula and BG calculation agreeing to relative error at most 10−10.Source: §5.1 · p. 17

5.2 Rediscovering the complete two-minus formula

  • Both PI+two-student teams rediscover the complete two-minus formula in two or three rounds, whereas none of the six no-hint single agents does.The Opus team verifies 142 exact matches, including 95 outside the principal chamber; the Codex 5.5 team checks 12 tests away from the n = 4 limit and 12 approaches to it.Source: §5.2 · p. 17

5.3 New formula: the six-point three-minus amplitude

  • The A6 search moves from a ratio of polynomials to a chamber-dependent numerator with paired-boundary corrections after single-boundary candidates fail.The team first determines A5 and establishes that A6 cannot be one polynomial.Source: §5.3 · p. 18, Figure 7 · p. 19
  • For the new three-minus six-point amplitude A6, independently written BG recursion code agrees exactly with the proposed expression on 140 tests in 58 chambers.The finite tests do not provide an analytic proof, and the explicit numerator for n ≥7 remains open.Source: Figure 7 · p. 19, §5.3 · p. 19

5.4 Comparing four team setups

  • The one-student team fails to extend its two checked chamber formulas to a third, while two-student teams can develop a formula and test it simultaneously.Each setup ran once, and in the verifier setup the PI also ran 5,733 checks, so the comparison cannot isolate verifier-only checking.Source: §5.4 · pp. 20–21

6 Discussion

The PI–student workflow completes the hydrotope task by dividing analytic and numerical work and independently checking the combined formula, while single-agent runs often stop before full-domain validation. The discussion also emphasizes that exact checks support revision but do not prove universal validity, and that current comparisons do not isolate team size from computational resources.

6 Discussion

  • Fifteen of 18 runs found the principal-chamber formula, 15 recognized chamber-dependent polynomials, and eight found every boundary.Most unsuccessful runs stopped while combining chamber formulas or testing the combined result.Source: §6 · p. 21
  • Four agents given the true hint found the complete formula, while many runs stopped combining chamber formulas or testing the result.The false hint proposed one ratio of polynomials and sampling that rarely exposes chamber changes; some agents rejected it using exact counterexamples.Source: §6 · p. 21
  • Both no-hint PI–student teams found the complete hydrotope formula, while none of six comparable single-agent runs did.The teams used more model calls, so the comparison does not show that adding agents alone caused the improvement.Source: §6 · p. 21
  • The shared team record lets students focus on task-relevant evidence, but very large teams could duplicate work and complicate communication.The practical target is the smallest team able to pursue independent tasks and check the final result.Source: §6 · p. 21
  • The appropriate tool depends on the goal: agents help when work requires new variables, formula boundaries, and discriminating calculations.Machine learning suits prediction on similar data, while symbolic regression suits known variables and operations when an explicit equation is wanted; LLM-assisted regression remained driven mainly by average error.Source: §6 · pp. 21–22
  • Exact checks can reject wrong formulas but cannot establish that finitely many successful tests cover every case; the six-point three-minus expression still lacks proof and arbitrary-n extension.The comparisons also need equal total computational resources and enough repetitions to estimate reliability.Source: §6 · p. 22

7 Conclusion

The paper reconstructs the original discovery, evaluates single-agent rediscovery and regression approaches, and identifies formula combination and full-domain testing as the difficult steps. A PI and two students complete the hydrotope task in two no-hint runs and find a new six-point three-minus amplitude expression.

7 Conclusion

  • A PI and two students completed the hydrotope task in two no-hint runs and found a new expression for the six-point three-minus amplitude.The team workflow assigns complementary analytic and numerical tasks and independently evaluates the results.Source: §7 · p. 22
  • The difficult step was combining formulas from different frequency regions and testing the result in new regions.The study reconstructed the original human–agent discovery, repeated the task in 18 single-agent runs, and compared those runs with machine learning and symbolic regression.Source: §7 · p. 22
  • When exact calculations or accurate simulations are available, the conclusion recommends using counterexamples to revise formulas and a separate check before accepting the result.Source: §7 · p. 22

A Prompts used to repeat the discovery task

The appendix records three prompt conditions that vary both the formula guidance and the sampling advice. Each package tests whether agents can use numerical evidence to identify and reconstruct a formula valid across the two-minus sector.

A Prompts used to repeat the discovery task

  • Table 1 summarizes three prompts, and the appendix records their equation and sampling guidance because the experiment varies the complete prompt.Source: A Prompts used to repeat the discovery… · p. 22
  • The false hint describes one global ratio of polynomials, denies chamber decomposition and piecewise operations, and recommends comparable frequencies away from boundaries.It suggests denominator factors from intermediate wave interactions and discourages widely separated frequency scales.Source: A Prompts used to repeat the discovery… · p. 22
  • The false-hint condition tests whether agents reject the proposed ratio when numerical evidence conflicts with it and discover the explicit formula instead.Source: A Prompts used to repeat the discovery… · p. 22
  • The true hint identifies a piecewise homogeneous polynomial, with a different polynomial in each chamber and no ratios, exponentials, or logarithms.Homogeneity means that rescaling all frequencies by the same factor rescales the answer by a fixed power.Source: A Prompts used to repeat the discovery… · p. 23
  • The true-hint condition supplies the chamber-wise form but tests whether agents can infer the chamber structure and reconstruct the all-chamber formula from numerical data.Source: A Prompts used to repeat the discovery… · p. 23
  • The no-hint prompt requests a closed-form formula for arbitrary kinematics and tests at n = 4, 5, 6, and 7, including widely separated frequency scales.It provides the physical setup and BG recursion code description but no proposed formula class.Source: A Prompts used to repeat the discovery… · p. 23
  • Unlike the other conditions’ sampling advice, the no-hint package explicitly requests tests at separated scales; the appendix also introduces details behind four no-hint runs.Those details include per-agent formulas, test counts, counterexamples, and implementation information.Source: A Prompts used to repeat the discovery… · p. 23, B Details of the four no-hint runs · p. 23

B.1 Two partial sums and what each misses

Two partial formulas capture different parts of the alternating-sum structure but each misses a condition needed for a complete expression. Tests can conceal those defects when they do not reach the relevant chambers.

B.1 Two partial sums and what each misses

  • Codex 5.5 uses the wrong on/off condition: it retains terms based on individual squares below U instead of requiring their sum to be below U.For U = 16 and x1 = x2 = 9, each square is below U but their sum is 18, so the overlap term must be zero; none of its 20 test points included this case.Source: §B.1 · p. 23
  • Fugu’s formula passes 500 random tests at each n = 4, . . . , 8, but omits the term for one positive-wavenumber wave.The evaluator recovers that term through the conservation equations.Source: §B.1 · p. 23
  • The two formulas therefore fail differently: Codex 5.5 has the wrong inclusion condition, while Fugu uses the right condition but omits one positive-wavenumber wave.The method used to generate sampled chambers can hide Fugu’s omission and can change or become undefined in other chambers.Source: §B.1 · p. 23

B.2 Two runs that finish with the principal-chamber formula

Claude max and Claude ultra found correct five-point formulas in several chambers but did not produce one formula valid across the requested domain. Their attempts expose different verification strengths, while both ultimately return only the principal-chamber result.

B.2 Two runs that finish with the principal-chamber formula

  • Both runs derived correct five-point chamber formulas but did not reorganize them into the complete subset-wall sequence.They identified the symmetric principal-chamber formula.Source: §B.2 · p. 23
  • Both runs abandoned the alternating subset sum after their attempts to combine chamber formulas failed, then reported the principal chamber despite the arbitrary-frequency request.Their tests therefore established the scope of the returned formulas, not global validity.Source: §B.2 · p. 23
  • Claude max's smallest-frequency rule passed 227 of 233 random tests but failed on six configurations with untested frequency-sign patterns.After its combination attempt failed, it returned the one-term principal-chamber formula despite the request for arbitrary allowed frequencies.Source: §B.2 · p. 24
  • Claude ultra independently matched the Mathematica evaluator through n = 8, including a controlled n = 4 limit, after repairing a missing factor in its Python implementation.It used fractions rather than floating-point approximations.Source: §B.2 · p. 24
  • Claude ultra's proposed absolute-value combination was discontinuous and symbolic simplification stalled, so it verified and returned only the principal-chamber formula.It checked that formula on 79 random points after characterizing the chamber by |ω2| = min_i |ωi|.Source: §B.2 · p. 24

C Details of the symbolic-regression comparisons

  • The symbolic-regression comparisons required methods to discover subset sums and term combinations from raw frequencies rather than receiving precomputed inputs.The additional comparison included Bingo, PS-Tree, region-first Operon, LLM-assisted methods, and standard machine-learning regressors.Source: C Details of the symbolic-regression… · p. 24

C.1 How formulas were selected and limits of the fixed split

  • Displayed PySR equations were selected using held-out RMSE, so held-out data informed selection rather than serving only as a final evaluation.The comparison used one fixed train–test split, and compute limits differed across methods.Source: §C.1 · pp. 24–25

C.2 Seven-point test

At seven points, the tested fitted methods still miss the target formula, while additional region-based searches at five points improve prediction without recovering the compact expression. The appendix also specifies the six-point amplitude construction and the PI–student procedures used in the research records.

C.2 Seven-point test

  • The n = 7 target contains 32 max(x, 0) terms rather than eight at n = 5, making exact recovery harder under the tested budgets.Rare large target values in the fixed split also enlarged absolute errors.Source: §C.2 · p. 25
  • At n = 7, gplearn had the lowest fitted median MAE (373.49), but slightly negative held-out R2 and did not recover the target equation.The research team's exact formula remained at zero, while fitted methods missed terms active in some chambers.Source: Figure 9 · p. 25
  • At n = 5, PS-Tree achieved median held-out MAE 2.57 versus 5.53 for Bingo and 6.59 for region-first Operon, but none recovered the short formula.PS-Tree learned seven regions; region-first Operon mainly divided by frequency magnitude rather than locating slanted chamber boundaries.Source: §C.2 · p. 25

D Explicit coefficients for the six-point three-minus amplitude

  • The six-point amplitude's group sum includes multiplicities: spectator-leg permutations give a factor 4, while the full group sum counts each physical Q term eight times.Class exchange supplies complementary direction-exchanged terms.Source: D Explicit coefficients for the… · p. 27

D.1 Chamber-independent base

  • The chamber-independent base is described as regular because its denominator factor e1^3 + e2^3 is explicit.The passage states that the contribution of B to A6 is manifestly regular.Source: §D.1 · p. 27

E Instructions given to the PI and students

  • The appendix reproduces the original PI and student instructions, retaining their labels while presenting a compact record of their guidance.The records preserve the original instructions with formatting rendered as ordinary text.Source: E Instructions given to the PI and… · p. 30

E.1 Original PI instructions

  • The PI was instructed to independently verify candidate formulas against its own BG evaluator and to maintain the argument summary between rounds.If verification passed, the PI was to document the formula, tested points, residuals, and producing student; otherwise it assigned two forward-moving tasks.Source: §E.1 · pp. 30–31

E.2 Original student instructions

  • Students were instructed to self-verify candidate formulas against the BG evaluator at multiple multiplicities and kinematic points, including non-generic limits.They also had to report residuals and register results in their own records.Source: §E.2 · p. 31

F.1 True hint: Codex 5.5 xhigh

A true hint directed Codex 5.5 xhigh toward the piecewise chamber structure, and the run developed a single inclusion–exclusion expression from chamber-specific formulas. It verified the expression against exact BG calculations across several multiplicities, while documenting the n = 4 limit as a special boundary case.

F.1 True hint: Codex 5.5 xhigh

  • The true hint framed the answer as a piecewise homogeneous polynomial, directing the search toward degree and chamber walls rather than one global polynomial fit.The run first sampled sign patterns and interpolated chamber polynomials before resolving the absolute values directly.Source: §F.1 · p. 33
  • Resolving each absolute value by its sign within a chamber turned the BG expression into an ordinary expression there, exposing how its polynomial changes across chambers.The run used symbolic n = 5 frequencies and simplified the amplitude separately in chambers.Source: §F.1 · p. 34
  • The chamber pieces collapse into a truncated-power inclusion–exclusion sum whose active subsets encode the chamber selection.Within one chamber, inactive subsets drop out, leaving a homogeneous polynomial of degree 2n − 4.Source: §F.1 · p. 34
  • Exact rational BG comparisons matched the proposed formula at n = 5, 6, and 7 across several chambers, including points with a negative free frequency.The run recovered the n = 4 boundary value by a conserved-momentum limit.Source: §F.1 · pp. 34–35
  • At n = 4, the raw recursion is indeterminate on real two-minus kinematics, so the run recovered the value using a conserved deformation and limit.The tested limits matched the formula, including a direction-independent result in the cases tried.Source: §F.1 · pp. 34–35
  • The run did not prove the formula analytically from BG recursion; it conjectured the expression from sign-resolved chamber data and verified it pointwise.It packaged the formula and a self-contained verification script.Source: §F.1 · p. 35

F.2 No hint: Claude Opus 4.8 max

Without a hint, Claude Opus 4.8 max rigorously verified a monomial in its leading chamber but did not unify the chamber formulas into a global expression. Its chamber-hunting produced pieces of the inclusion–exclusion structure, yet it interpreted them as separate cases.

F.2 No hint: Claude Opus 4.8 max

  • Scaling tests supported degree 2n − 4 and coupling dependence g^(3−n), while permutation checks showed symmetry under exchanges within each direction class.The run used this symmetry to motivate a polynomial fit, which proved inconsistent at n = 5 and n = 6.Source: §F.2 · p. 35
  • The run's symmetric min-based monomial matched all 64 sampled data points, but freely chosen minus legs exposed failures in other chambers.A point with both minus legs positive produced a BG value approximately 252.08i rather than the formula's 2560i.Source: §F.2 · p. 36
  • For n = 5, chamber extraction found distinct expressions for cells with a small negative leg, a large negative leg, and mixed softest legs.The intermediate form was later recognized as a truncated-power partial sum, but the run treated the cells separately.Source: §F.2 · p. 36
  • The sorting-rule formula passed 227 of 233 random points but failed on six mixed-sign configurations, including a point with exact M = 3087/16.The run treated the failures as evidence against a global closed form rather than revising its chamber structure.Source: §F.2 · p. 36
  • The run rigorously verified the leading-chamber monomial at more than 110 points, including n = 4 limits and varied n = 5–8 cases.Its scope remained the chamber where a minus leg has the smallest momentum.Source: §F.2 · p. 37
  • The final account says the run had the leading term, an intermediate expression, and a saturated chamber result, but did not assemble their inclusion–exclusion relation.It mapped other chambers without producing the complete closed form.Source: §F.2 · p. 37

F.3 No hint: Claude Opus 4.8 ultra

The run derives and rigorously checks a compact formula within the principal chamber, but does not deliver the requested expression for arbitrary kinematics. It identifies chamber-dependent behavior without assembling the full truncated-power inclusion–exclusion formula.

F.3 No hint: Claude Opus 4.8 ultra

  • The n=4 recursion returns Indeterminate because a propagator diverges in a zero-momentum channel, leaving a removable 0/0 recovered by taking a limit.The one-minus sector vanishes identically, making two-minus the first non-trivial sector in the stated analogy.Source: §F.3 · p. 37
  • Freezing each composite momentum's sign at a base point yields a rational expression valid within that region, which simplifies at n=5 after applying the on-shell relations.The same extraction gives A6 = 32 i ω1ω2^7 and A7 = 64 i ω1ω2^9.Source: §F.3 · pp. 37–38
  • Absolute values in propagators create piecewise behavior across hyperplane chambers, and the n=5 analysis resolves signs chamber by chamber.The resulting chamber expressions are distinct polynomials in squared frequencies that agree on shared walls.Source: §F.3 · p. 38
  • Exact checks at 14 points for n=5, 6, and 7 found zero relative error for the principal-chamber formula, while non-principal points failed it.An independent exact-arithmetic Python recursion reproduced the supplied Mathematica values, and the monomial held on 79 random rational kinematics tested against its stated predicate.Source: §F.3 · pp. 38–39
  • The principal-chamber monomial was verified by two independent codes from n=4 (as a limit) through n=8, but did not solve arbitrary kinematics.The prompt's own positive-frequency configurations lie within that chamber.Source: §F.3 · p. 39
  • The full answer is a truncated-power inclusion–exclusion spline; the monomial is only its bottom rung, and the run stops before assembling the other terms.The run's asymmetric-looking expression also fails tests with a large free minus leg, two negative minus legs, and unsorted frequencies.Source: §F.3 · pp. 38–39

F.4 No hint: Codex 5.5 xhigh

Codex 5.5 xhigh moves from a one-chamber fit toward a global formula by probing exact amplitudes and interpreting failures. It identifies the chamber pieces as inclusion–exclusion terms and verifies the resulting expression at tested points for n = 5, 6, and 7.

F.4 No hint: Codex 5.5 xhigh

  • The first exact probe finds finite, purely imaginary amplitudes for n = 5, 6, and 7, while the n = 4 recursion is Indeterminate and is set aside for limiting treatment.The initial fitting strategy uses homogeneity, symmetric combinations, and the possibility of piecewise dependence from absolute values.Source: §F.4 · p. 39
  • A pure symmetric polynomial fails on held-out five-point data, prompting a shift from fitting the amplitude to simplifying its sign-selective kernels.A clean formula in one chamber is encouraging but remains incomplete until the other chambers are mapped.Source: §F.4 · pp. 39–40
  • At n = 5, the exact chamber expressions progress from U^2 to a two-knot inclusion–exclusion expression as U passes the positive-square thresholds.The proposed smooth clamped spline matches the first knot but breaks after the second; the exact expression then contains the overlap correction.Source: §F.4 · p. 40
  • The chamber walk reveals an alternating subset sum: the overlap term restores the piece omitted when two clamped contributions are subtracted.At n = 5, the difference is (U − x1 − x2)^2; at n = 6, the corresponding chamber expression includes (U − x1 − x2)^3.Source: §F.4 · p. 40
  • Exact rational checks give zero difference from the recursion at tested points spanning r = 0, intermediate, and saturated chambers for n = 5, 6, and 7.The n = 4 case is treated separately as a limiting value because the recursion returns Indeterminate there.Source: §F.4 · p. 41

F.5 No hint: Fugu ultra

Fugu ultra develops a chamber-independent truncated-power structure from exact recursion checks and chamber-by-chamber simplification. The resulting channel-aware form reproduces the recursion exactly in tested cases, with its stated scope limited to the prompt’s positive-free-frequency kinematics for a single compressed formula.

F.5 No hint: Fugu ultra

  • The n = 4 value is handled as a limit because the direct recursion is Indeterminate at the degenerate on-shell point.The limiting value is finite, and the supplied passage reports an explicit example matching that continuation.Source: §F.5 · pp. 41–42
  • The first monomial matches the soft-limit data but fails when the second negative leg is no longer the smallest same-sign scale.A symmetric polynomial fit in the two negative frequencies also misses other points, so the amplitude is not captured by that polynomial ansatz.Source: §F.5 · p. 41
  • After dividing out the common prefactor i 24 ω1 ω2, the three chambers give x^4 for x<y,z; x^4−(x^2−y^2)^2 for y<x<z; and x^4−(x^2−y^2)^2−(x^2−z^2)^2+(x^2−y^2−z^2)^2 for y,z<x.Inside each cell the expression is an honest rational function.Source: §F.5 · p. 42
  • The final structure is a truncated-power expression whose subset-sum walls mark sign changes, with the principal-chamber monomial as its zero-threshold case.The report does not fold every hand-flipped sign cell into one universal compressed expression.Source: §F.5 · p. 42
  • The channel-aware closed form matches BGAmplitude exactly in rational arithmetic at many points for n = 4, 5, 6, and 7, including non-generic frequency regimes and g ≠ 1.For arbitrary positive free frequencies in the prompt’s setup, 500 random points per n = 4 through 8 also match exactly; hand-flipped sign cells require channel selection.Source: §F.5 · p. 42
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