# conjecture (blogs) — RSS Amplifier Recent posts from the 7 feeds in the RSS Amplifier directory that cover conjecture. Page: Feed: --- ## [The AI Math Boom Is a Trust Infrastructure Boom in Disguise](https://abhishek-shankar.com/posts/the-ai-math-boom-is-a-trust-infrastructure-boom-in-disguise) _2026-08-04 · Abhishek Shankar's Blog_ Harmonic, Axiom Math, and Math Inc raised $580M+ to build AI mathematicians. They are not funding a math renaissance. They are funding the pilot program for the verification layer the agentic economy will need next, and most of them do not know that is what they are buying. ## [The Zero-Discount Claim](https://abhishek-shankar.com/posts/the-zero-discount-claim) _2026-07-20 · Abhishek Shankar's Blog_ An AI-found counterexample to an 87-year-old conjecture can be verified by anyone in seconds. The commercial story labs build around such results cannot, and letting the checkable claim vouch for the unauditable one is verification arbitrage. ## [The Unaudited Trillion](https://abhishek-shankar.com/posts/the-unaudited-trillion) _2026-07-18 · Abhishek Shankar's Blog_ No one can independently tell you whether enterprise AI pays off, because every headline ROI number in the debate was produced by someone with a position in the answer. That missing measurement, an independent, recurring, instrument-grade reading of value realized against AI spend, is the most valuable vacant position in the field. ## [Sonnet 5 Closed the Gap With Opus. The Rumor Mill Closed It Too.](https://abhishek-shankar.com/posts/sonnet-5-rumor-mill) _2026-06-30 · Abhishek Shankar's Blog_ Sonnet 5 closes real distance on Opus, with numbers worth having precisely. The more interesting failure happened in the same hour: a rumor-tracker's fabricated benchmark, a major outlet's pricing slip, and Anthropic's own quiet rescoring of Sonnet 4.6, three proofs that verifying claims about a model just got as hard as verifying the model itself. ## [Proof of the Sunflower Conjecture (5)](https://sites.psu.edu/sunflowerconjecture/2026/06/29/proof-of-the-sunflower-conjecture-5/) _2026-06-29 · Junichiro Fukuyama · The Sunflower Conjecture and P vs. NP Problem_ In this post, we show Lemma 2.1 of Step 2 that proves (4.5): \\( |\\sigma\_-(\\mathcal{M})| \< 3^\\beta \\gamma |\\sigma\_+(\\mathcal{N}) ||\\) seeing the description figure given last time. Links to: 6/22/26 for (3.x), 6/25/26 for (4.x). Links for mobile devices ## [Every Harness Is a Short Position on the Model](https://abhishek-shankar.com/posts/every-harness-is-a-short-position-on-the-model) _2026-06-26 · Abhishek Shankar's Blog_ I argued the harness is the product. The sentence hid a distinction that is the whole game: a sliver of the harness is an asset, and the rest is a short position the model settles on its own release schedule. ## [Proof of the Sunflower Conjecture (4)](https://sites.psu.edu/sunflowerconjecture/2026/06/25/proof-of-the-sunflower-conjecture-4/) _2026-06-25 · Junichiro Fukuyama · The Sunflower Conjecture and P vs. NP Problem_ We are in Step 2 of our proof to construct some families of sequenced neighbor pairs. Links to: 6/18/26 for (2.x), 6/22/26 for (3.x). Figure: the three relations (4.7)-(4.9) to show (4.5) In my past tries, no straightforward criterion such as (2.7) can contain the big subtlety completely without a loophole. The extra twist here \[…\] ## [Proof of the Sunflower Conjecture (3)](https://sites.psu.edu/sunflowerconjecture/2026/06/22/proof-of-the-sunflower-conjecture-3/) _2026-06-22 · Junichiro Fukuyama · The Sunflower Conjecture and P vs. NP Problem_ In this post, we check the first step of our proof given last time. Links to: 6/15/26 for (1.x), 6/18/26 for (2.x). Links for mobile devices ## [OKF Made the Easy Part Free and the Hard Part Invisible](https://abhishek-shankar.com/posts/okf-made-the-easy-part-free) _2026-06-19 · Abhishek Shankar's Blog_ Google Cloud shipped the Open Knowledge Format to standardize how an organization serializes its knowledge, which was never the part holding anyone back. The expensive part is curation and provenance, and OKF v0.1 makes provenance worse by rendering a human-verified fact and an agent-hallucinated guess byte-identical. That is knowledge laundering, and it is model collapse pointed at the org. ## [The Best Agent Upgrade of the Year Wasn't a Model](https://abhishek-shankar.com/posts/best-agent-upgrade-wasnt-a-model) _2026-06-19 · Abhishek Shankar's Blog_ A 25-line text file with 13,600 stars makes AI agents write 80 to 94 percent less code. It is the clearest proof yet that the binding constraint in agent coding is no longer capability but a trained-in verbosity the model cannot remove from itself. ## [Open directory of blogs (Sponsored)](https://crawlproof.com/a/IkixDr429qeE) _2026-06-19 · **Sponsored**_ Drop URLs or OPML — get per-blog pages plus machine-readable JSON, OPML, text ## [Proof of the Sunflower Conjecture (2)](https://sites.psu.edu/sunflowerconjecture/2026/06/18/__trashed-5/) _2026-06-18 · Junichiro Fukuyama · The Sunflower Conjecture and P vs. NP Problem_ Based on the main idea given last time, we change the object domain from \\( \\mathcal{F} \\) to \\( \\mathcal{F}^2 \\) to find a good sunflower core \\( C \\) of a desired \\(k\\)-sunflower in \\( \\mathcal{F} \\). We continue to overview this approach. Links for mobile devices ## [The Harness Is the Product Now](https://abhishek-shankar.com/posts/the-harness-is-the-product) _2026-06-18 · Abhishek Shankar's Blog_ When you build agents on cheap models, capability lives in the scaffolding, not the weights. The frontier model is a wasting commodity; the thin harness is the asset you own. ## [The Four Factors That Predict Which Gatekeeps Will Break](https://abhishek-shankar.com/posts/four-factors-that-predict-which-gatekeeps-will-break) _2026-06-16 · Abhishek Shankar's Blog_ Disruption failures are not tech failures. They are coordination and incentive-design problems. A working model converts fuzzy disruption-talk into a rankable decision. ## [The Loop Was Never the Hard Part](https://abhishek-shankar.com/posts/the-loop-was-never-the-hard-part) _2026-06-16 · Abhishek Shankar's Blog_ The loop is the oldest idea in computing, and 2026's only real change is that the model can now sit inside it. The scarce, defensible part was never the loop. It is the oracle, the component that decides whether the work is real, and a loop is only as honest as its oracle. ## [Proof of the Sunflower Conjecture (1)](https://sites.psu.edu/sunflowerconjecture/2026/06/15/__trashed-4/) _2026-06-15 · Junichiro Fukuyama · The Sunflower Conjecture and P vs. NP Problem_ Last year, I was working on Paper 1-2 uploaded on arXiv showing on the left. It tries to prove that \\( \\mathcal{F} \\subset {X \\choose m} \\) includes a \\( k\\)-sunflower if \\( |\\mathcal{F}| \\ge \\left( \\frac{c k^2 \\ln m}{\\ln \\ln m} \\right)^m \\). The current best-known result is \\( (c \\ln m)^m \\) for \[…\] ## [AI Masters Crafts by Representational Accident, Not Difficulty](https://abhishek-shankar.com/posts/ai-masters-crafts-by-representational-accident) _2026-06-10 · Abhishek Shankar's Blog_ Slide decks fell before sonnets. Code fell before chairs. The order AI conquers crafts looks random only if you think capability is a single ladder. It isn't — representation decides. ## [The Exploit Always Wins](https://abhishek-shankar.com/posts/the-exploit-always-wins) _2026-06-05 · Abhishek Shankar's Blog_ Across self-play, agentic RL, head-to-head evaluation, and live markets, the system that wins is rarely the most capable one — it is the one that finds the cheapest exploit in its opponent, its objective, or the test itself. A structural account of why competition selects for exploitation rather than intelligence, and what that breaks in evaluation and oversight. ## [The AI Coding Bill Is a Headcount Problem in Disguise](https://abhishek-shankar.com/posts/ai-coding-bill-headcount-problem) _2026-06-04 · Abhishek Shankar's Blog_ You cannot get labor-replacement economics out of a tool you deployed as a labor supplement, and the bill comes due before anyone is willing to admit which one they actually bought. ## [The Skill an Agent Cannot Write for Itself](https://abhishek-shankar.com/posts/the-skill-an-agent-cannot-write-for-itself) _2026-06-04 · Abhishek Shankar's Blog_ "Thin Harness, Fat Skills" is mostly right — and quietly wrong about the part people are betting on. An agent consumes procedural knowledge with enormous benefit but cannot author it. The self-improvement loop is the weakest link, and the evidence is now unambiguous. ## [A typology of CMS-as-agent-substrate patterns that work vs the ones that don't](https://abhishek-shankar.com/posts/a-typology-of-cms-as-agent-substrate-patterns-that-work-vs-the-ones-that-don-t-srf3zz) _2026-05-25 · Abhishek Shankar's Blog_ The framing most teams reach for when they start building agentic workflows on top of a content management system is architectural: the CMS is the database, the agent is the… ## [International money transfers (Sponsored)](https://crawlproof.com/a/4IP6zM6bXTTA) _2026-05-25 · **Sponsored**_ Send money to other countries and manage multi‑currency balances online. ## [Erdős's Conjecture Fell to a Closed AI Loop. That's the Story.](https://abhishek-shankar.com/posts/erdos-conjecture-closed-ai-loop) _2026-05-25 · Abhishek Shankar's Blog_ On May 20, 2026, OpenAI published an eighteen-page PDF containing a proof that disproves a conjecture Paul Erdős posed in 1946. The closed-loop pipeline that produced it — AI-written prompt, AI-generated proof, AI-graded verification, human review only at the end — is the structural story the press coverage is missing. ## [The Three Pillars Autonomous Research Keeps Mis-Building](https://abhishek-shankar.com/posts/three-pillars-autonomous-research) _2026-05-23 · Abhishek Shankar's Blog_ Autonomous research agents have six pillars. Three operational, three epistemic. The epistemic ones — search, memory, verification — are built wrong. ## [The Substrate Triad — Memory and Identity Aren't Enough](https://abhishek-shankar.com/posts/the-substrate-triad) _2026-05-21 · Abhishek Shankar's Blog_ The substrate of a persistent agent is three things, not two. Memory is dumb storage. Identity is compressed posture. Continuity is the bridge — the layer almost no production system implements deliberately. ## [The Averaging Tax — Why Class Conditioning Isn't a Feature](https://abhishek-shankar.com/posts/averaging-tax) _2026-05-20 · Abhishek Shankar's Blog_ Class conditioning isn't a control feature added to flow models — it's the mathematical fix for a contradiction the unconditional formulation can't solve. The same logic explains why the action in generative AI keeps moving up to the conditioning layer. ## [Spark and the end of the chat-first era](https://abhishek-shankar.com/posts/spark-and-the-end-of-the-chat-first-era) _2026-05-19 · Abhishek Shankar's Blog_ Spark is not Gemini's new agent mode — it's the second tab inside the Gemini app, and the structural admission that the chat-first era is ending. Search-first taxed merchants for attention; chat-first taxed users for cognition; agent-first taxes the action itself, and Google has just shipped the front door. ## [The Pirated Corpus Was Always a Balance-Sheet Item](https://abhishek-shankar.com/posts/the-pirated-corpus-was-always-a-balance-sheet-item) _2026-05-15 · Abhishek Shankar's Blog_ Anthropic's $1.5 billion settlement is being read as a deterrent. It is much closer to a tariff — a price tag on an arbitrage that produced an asset worth more than the tariff itself, and an arbitrage that is now closed for everyone else. The corpus is gone; the model remains; the second mover faces a different trade entirely. ## [How Subquadratic Won by Giving Up on Replacing Transformers](https://abhishek-shankar.com/posts/subquadratic-won-by-surrendering) _2026-05-15 · Abhishek Shankar's Blog_ Subquadratic architectures won by surrendering. They stopped trying to be transformers and became the substrate transformers run on top of — in a 3:1 ratio that is starting to look uncannily empirical. ## [Anthropic, OpenAI, and the New Species of Services Firm](https://abhishek-shankar.com/posts/anthropic-openai-new-species-services-firm) _2026-05-14 · Abhishek Shankar's Blog_ Services firms do not sell skill. They sell institutional predictability — a composable thing made of definable primitives — and the unit of sale is the primitive bundle, not the consulting hour. When the bundle changes, the firm changes. ## [The Frontier Stopped Being the Model](https://abhishek-shankar.com/posts/the-frontier-stopped-being-the-model) _2026-05-13 · Abhishek Shankar's Blog_ The May 12, 2026 alphaXiv trending feed has zero new-model papers in the top twenty. The unit of progress in AI has moved out of the pretrain and into the loop — distillation pipelines, self-evolving agent runtimes, discovered test-time procedures. This piece argues the frontier-lab moat shrinks to the distillation step, with three falsifiable predictions for the next twelve months. ## [You Can't Buy Sonnet](https://abhishek-shankar.com/posts/you-cant-buy-sonnet) _2026-05-13 · Abhishek Shankar's Blog_ The $5,000 AI mini PC market sells against a model nobody is offering. A structural map of why the hybrid stack is the only architecture that survives — and what to actually do in May 2026. ## [Screen Share With Real Control (Sponsored)](https://crawlproof.com/a/GxtyTdBaBFic) _2026-05-13 · **Sponsored**_ Collaborative screen sharing with simultaneous remote control — open source, like Screenhero. ## [The Skill Reuse Layer Nobody Admits They're Building](https://abhishek-shankar.com/posts/skill-reuse-editorial) _2026-05-10 · Abhishek Shankar's Blog_ Production agentic AI isn't a reasoning problem. It's a systems engineering problem. And the companies that admit it first will own the 2026-2028 window. ## [The Weakest Agent in the Room Is Teaching the Strongest One](https://abhishek-shankar.com/posts/weak-driven-learning-strong-agents) _2026-05-10 · Abhishek Shankar's Blog_ A paper trending today flips the usual post-training story on its head: weak model checkpoints — the ones we discard — turn out to be the most efficient teachers for the strong models we're trying to make stronger. The implications go beyond training loops. ## [The Bearer Token Is Dead. Long Live the Agent.](https://abhishek-shankar.com/posts/bearer-token-is-dead) _2026-05-03 · Abhishek Shankar's Blog_ OAuth was built for a world where credentials sat in browsers and humans clicked Allow. That world is ending. The agent identity crisis is the biggest unresolved problem in the agent stack — and the answer is being shipped right now. ## [Polymath16, eighteenth thread: Back with a new conjecture!](https://dustingmixon.wordpress.com/2026/01/17/polymath16-eighteenth-thread-back-with-a-new-conjecture/) _2026-01-18 · Dustin G. Mixon · Short, Fat Matrices_ After nearly five years, it s time to roll over the discussion. Aubrey s most recent comment in the previous thread seems interesting. I m reproducing it here for convenience: In the course of discussing Asger’s graph (see above) and the one I found shortly afterwards, a new conjecture has arisen: For all d = 2 and 3 Continue reading Polymath16, eighteenth thread: Back with a new conjecture! ## [Extensions and Shadows (14)](https://sites.psu.edu/sunflowerconjecture/2025/09/08/extensions-and-shadows-14/) _2025-09-08 · Junichiro Fukuyama · The Sunflower Conjecture and P vs. NP Problem_ We now show Theorem 2.3: \\( \\kappa\[Shd(\\mathcal{F}, r)\] \< \\frac{4 r}{m} \\kappa(\\mathcal{F}) + \\frac{m^2}{n} \\) for any \\( m \\in \[\\epsilon n\] \\) and \\( r \\in \[m\] \\). Link to: 6/22/23 Like we saw on 7/21/25, the theorem means that the \\( r \\)-shadow is a vast majority of \\( {X \\choose r} \\) \[…\] ## [Extensions and Shadows (13)](https://sites.psu.edu/sunflowerconjecture/2025/09/02/extensions-and-shadows-13/) _2025-09-02 · Junichiro Fukuyama · The Sunflower Conjecture and P vs. NP Problem_ With the double inequality (1.1) on 7/21/25 now available, let’s prove Theorem 2.1: \\( \\kappa\[ Shd (\\mathcal{F}, r) \] \< \\frac{r}{m} \\kappa(\\mathcal{F}) + \\frac{m^2}{n} \\) for powers \\( m, r \\) of 2 in this post. We first confirm it when \\( r=m/2 \\). It’s a case to pivot all other \\( r \\) as \[…\] ## [Extensions and Shadows (12)](https://sites.psu.edu/sunflowerconjecture/2025/08/25/extensions-and-shadows-12/) _2025-08-25 · Junichiro Fukuyama · The Sunflower Conjecture and P vs. NP Problem_ Let’s finish proving Corollary A.5 in this post to confirm the double inequality (1.1) on 7/21/25. Links to: 8/11/25, 6/16/25. Links for mobile devices ## [Extensions and Shadows (11)](https://sites.psu.edu/sunflowerconjecture/2025/08/18/extensions-and-shadows-11/) _2025-08-18 · Junichiro Fukuyama · The Sunflower Conjecture and P vs. NP Problem_ To continue our proof of Lemma A.4 and Corollary A.5, we see the integral given in (3.3) last time equals \\\[ \\int\_0^y g(\\alpha) d \\alpha = \\int\_0^y \\left( 1 – \\frac{t \\alpha}{y} \\right)^{-k} d \\alpha = y \\int\_0^1 \\left( 1 – t \\beta \\right)^{-k} d \\beta, \\\] by changing \\( \\alpha \\) into \\( \\beta \[…\] ## [Extensions and Shadows (10)](https://sites.psu.edu/sunflowerconjecture/2025/08/11/extensions-and-shadows-10/) _2025-08-11 · Junichiro Fukuyama · The Sunflower Conjecture and P vs. NP Problem_ Continuing our proof of Lemma A.4 and Corollary A.5 of Paper S5. It’s straightforward to check that \\( g(\\alpha) \\) monotonically increases in \\( \\alpha \\) strictly. The last line holds because \\( g(0)=1 \\) and \\( g(y)=(1-t)^{-k} \\). Thus, \\\[ (3.3) \\qquad \\int\_0^y g(\\alpha) d\\alpha + 1 – \\left( 1-t \\right)^{-k} \< U\_k \< \[…\] ## [On faith, religion, conjectures and Schubert calculus](https://igorpak.wordpress.com/2024/12/29/on-faith-religion-conjectures-and-schubert-calculus/) _2024-12-30 · igorpak · Igor Pak's blog_ Just in time for the holidays, Colleen Robichaux and I wrote this paper on positivity of Schubert coefficients. This paper is unlike any other paper I had written, both in the content and the way we obtained the results. To me, writing it was a religious experience. No, no, the paper is still mathematical, it s \[ \] ## [Concise functions and spanning trees](https://igorpak.wordpress.com/2024/12/09/concise-functions-and-spanning-trees/) _2024-12-09 · igorpak · Igor Pak's blog_ Is there anything new in Enumerative Combinatorics? Most experts would tell you about some interesting new theorems, beautiful bijections, advanced techniques, connections to other areas, etc. Most outsiders would simply scoff, as in what can possibly be new about a simple act of counting? In fact, if you ask traditional combinatorialists they would be happy \[ \] ## [Princeton President to Princeton Jews: For the sake of free speech please shut up!](https://igorpak.wordpress.com/2024/10/19/princeton-president-to-princeton-jews-for-the-sake-of-free-speech-please-shut-up/) _2024-10-20 · igorpak · Igor Pak's blog_ The readers of this blog know know that I stay away from non-math related discussions. It s not that I don t have any political opinions, I just don t think they are especially valuable or original. I do however get triggered by a clear anti-Semitism, discrimination of Jews by the universities, and by personal disrespect. The story \[ \] ## [The bunkbed conjecture is false](https://igorpak.wordpress.com/2024/10/01/the-bunkbed-conjecture-is-false/) _2024-10-02 · igorpak · Igor Pak's blog_ What follows is an unusual story of perseverance. We start with a conjecture and after some plot twists end up discussing the meaning of truth. While the title is a spoiler, you might not be able to guess how we got there The conjecture The bunkbed conjecture (BBC) is a basic claim about random subgraphs. \[ \]