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(1978). Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics, 19(1), 25–52.","status":"active","vx_hash":"33539ecfbd55e389a7bcaabd5e1b27d4ee99aae1b1a756a5eedd171e750b0e12","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"8b8101c0213f68712e681c7341b98d0147c178e2aaed1a751d96e9098aa7f7db","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"63f22b35c618f281226c1f7a891b113f411816934afb509a85f9efb6e05f2e03","detail":{"divided_from":"body","block":6,"kind":"p"},"prev":"genesis","hash":"8b8101c0213f68712e681c7341b98d0147c178e2aaed1a751d96e9098aa7f7db"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d6"},{"id":"d7","kind":"p","type":null,"order":7,"text":"Key passage from the abstract: \"A large class of recursion relations x_{n+1} = λ f(x_n) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The functions considered all have a unique differentiable maximum z. With f(z) - f(x) ~ |x - z|^z (for |x - z| sufficiently small), z > 1, the universal details depend only upon z. In particular, the local structure of high-order stability sets is shown to approach universality, rescaling in successive bifurcations, asymptotically by the ratio α (α = 2.5029078750957... for z = 2).\"","status":"active","vx_hash":"a580d84c2bec0781c62c34ea5d7121c701e1957505b1ae97b0a655e2b3ced929","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"a5c5cb1b19885b897416c0fa73f7fa5db937896c4ea52e0590af73d01245c726","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"56000ad73e41d52d389dafecf898d2019274c5e98a9e6bdcd868745942820d05","detail":{"divided_from":"body","block":7,"kind":"p"},"prev":"genesis","hash":"a5c5cb1b19885b897416c0fa73f7fa5db937896c4ea52e0590af73d01245c726"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d7"},{"id":"d8","kind":"p","type":null,"order":8,"text":"Another passage: \"Then b_{n+1} - b_n / b_{n+2} - b_{n+1} → δ as n → ∞ is universal, with δ = 4.6692016091029...\"","status":"active","vx_hash":"a37036fbdfec62226af9c1f3cd45017ca1cb020d1d74e83dd87773dd464edaf6","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"53bfda0c2cc8842390b40f5734d6561920881d7a360a9c036cb076e99e60e242","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"bd8c8dc647ad24460e3d8818b1554afc44de7fbb984e36d396652210e33bf9d1","detail":{"divided_from":"body","block":8,"kind":"p"},"prev":"genesis","hash":"53bfda0c2cc8842390b40f5734d6561920881d7a360a9c036cb076e99e60e242"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d8"},{"id":"d9","kind":"p","type":null,"order":9,"text":"The introduction explains the setup with population models and recursions, showing that qualitative dynamics (sequence of period doublings to a bounded accumulation) are independent of exact f, and that quantitative scaling emerges universally.","status":"active","vx_hash":"1eb18db129b0c7c697e0ea08c04ef7f0b629e095e182079a30c709a95a25c0f0","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"887dea3295c956a3720dacc4e71a9782d5f4acad359dab025434c585fff4894e","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"1f844eb1fab70dcb1e56d09d5cd4321d98e2c9f1361726c8a04bf02b4a0888a3","detail":{"divided_from":"body","block":9,"kind":"p"},"prev":"genesis","hash":"887dea3295c956a3720dacc4e71a9782d5f4acad359dab025434c585fff4894e"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d9"},{"id":"d10","kind":"h","type":null,"order":10,"text":"## Convergence patterns evidenced","status":"active","vx_hash":"01c0d10a129b0d0dafb4380239e74913a123096fd4fd08e04cbdeffe8b94e512","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"2dfa9ee2a941243160512729a4664dc3ba8165d5f16a72d13ccb0a3895ee96f4","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"5cea5b4905f0b73bed78ae0db5b231684e1b95b64714d4536cb14fc71cb3d750","detail":{"divided_from":"body","block":10,"kind":"h"},"prev":"genesis","hash":"2dfa9ee2a941243160512729a4664dc3ba8165d5f16a72d13ccb0a3895ee96f4"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d10"},{"id":"d11","kind":"p","type":null,"order":11,"text":"The work directly evidences scale invariance through the constant α and bounded chaos through the infinite period-doubling cascade that terminates at a finite λ_∞. These match two members of the GRAIN family of energy-flow patterns: scale invariance and bounded chaos. The renormalization mechanism that produces g* links difference (parameter variation) to flow (iteration) to structure (universal attractor) to memory (persistent scaling ratios).","status":"active","vx_hash":"eb890199388d89761880ee5e647d997739c9e7f22ceba58b45a3be0892a8e7ef","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"cf8a08083ac82376fe17c22295da0610886bdba0e5c787103018b7d6baf06107","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"75c2f122459fdde046dba2cd4293f21904d1404f5fdeb170fbd4c9ccb5cacc53","detail":{"divided_from":"body","block":11,"kind":"p"},"prev":"genesis","hash":"cf8a08083ac82376fe17c22295da0610886bdba0e5c787103018b7d6baf06107"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d11"},{"id":"d12","kind":"p","type":null,"order":12,"text":"The Ladder appears here in abstract form: parameter difference drives iterated flow, which self-organizes into stable structures whose memory is encoded in universal ratios independent of microscopic details.","status":"active","vx_hash":"ccce2722f8675520a07f9fdfb76c703b88e2bb237c0b27a2c27f74cbab6d9fdf","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"9080b59bec4b0be99689339aeb13fe345cf0e0f68daee75132ef517429d73f4a","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"557a72825458313322b678c2f5518ec64dddea150c99225e19112dc892d0c217","detail":{"divided_from":"body","block":12,"kind":"p"},"prev":"genesis","hash":"9080b59bec4b0be99689339aeb13fe345cf0e0f68daee75132ef517429d73f4a"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d12"},{"id":"d13","kind":"h","type":null,"order":13,"text":"## Distance from the full OIP/GRAIN synthesis","status":"active","vx_hash":"49389d5eb4296514cce06c88691d9b044303b968dcb840447f057a6b6efac02f","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"c03faa25c536838c29ad3a98e18efbc4b222f7d62ba73ac6f1da06fa9dd7c583","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"78b248fc09f46aa233e0ee4e7ccd8141328a45bdb7ef545daf36dd82e28468ad","detail":{"divided_from":"body","block":13,"kind":"h"},"prev":"genesis","hash":"c03faa25c536838c29ad3a98e18efbc4b222f7d62ba73ac6f1da06fa9dd7c583"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d13"},{"id":"d14","kind":"p","type":null,"order":14,"text":"The paper supplies a precise mechanistic instance of scale invariance and bounded chaos inside iterated maps. It stops short of claiming these patterns appear across physical energy flows at all scales; that extension is interpretive. It supplies no statement on the Mirror Layer or the reader being inside the system. The work is therefore close on the pattern side and distant on the philosophical framing.","status":"active","vx_hash":"1b516304f8fa699eb72ae3bf495fd4a9c2d779abcfd6bb528b5c66e1c9375d15","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"e12f48327ef5c2744ef77459e9b8ea328e60d75cea1ae0a02887f04001302301","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"5914039f6541a3a13ca88060622b5ca04ab17244e2b660264b8b48d8db610937","detail":{"divided_from":"body","block":14,"kind":"p"},"prev":"genesis","hash":"e12f48327ef5c2744ef77459e9b8ea328e60d75cea1ae0a02887f04001302301"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d14"},{"id":"d15","kind":"h","type":null,"order":15,"text":"## Honest limits and disconfirming edges","status":"active","vx_hash":"827088ddd7099723528a201a7db77b233f52eee09eb3eee4361598c03ff45512","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"d901143379804f7c617829176c6456867eab3dc5c127e8b748fee1b99a0e3ef8","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"a4ff4542a1095d01f9181ea0192500270e1b6d9e89daf971127e5a818cfedd26","detail":{"divided_from":"body","block":15,"kind":"h"},"prev":"genesis","hash":"d901143379804f7c617829176c6456867eab3dc5c127e8b748fee1b99a0e3ef8"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d15"},{"id":"d16","kind":"p","type":null,"order":16,"text":"The treatment is heuristic; the paper states that an exact theory is deferred to a sequel. The universality holds only for maps with a single differentiable maximum of order z and for the local structure near the accumulation point. Global dynamics and higher-dimensional systems lie outside the result. Reductionist objections note that the constants are properties of the functional iteration, not direct predictions for every physical system; experimental confirmation in fluids and other media came later and requires additional modeling steps.","status":"active","vx_hash":"c2568d4b510b4167afc529188676ede1b76940faf637ee7578469858286856b2","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"4b80067c60012e01fd70df32e1054621af97d823879d1b39569c240809ed3378","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"a09c1935284566a277384f857a0d29a722a5f1d9d090107676af7e0dab945860","detail":{"divided_from":"body","block":16,"kind":"p"},"prev":"genesis","hash":"4b80067c60012e01fd70df32e1054621af97d823879d1b39569c240809ed3378"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d16"},{"id":"d17","kind":"h","type":null,"order":17,"text":"## Sibling 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progression and /a/oip-the-mirror-layer for the observer-inside-system framing.","status":"active","vx_hash":"3ca26cf6dad2e477e694d2d545155a966b58acb40b01ca1716eaaa78289fa36a","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"4f127e2949ff62af7d3bf832615e5dc2238e1500389e07d7acda8e7fcfb7400c","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"60f74630dfaf6c7a0774c9cbdd5c4bfad619d7d6542fafda16f07d5ac5c06e06","detail":{"divided_from":"body","block":18,"kind":"p"},"prev":"genesis","hash":"4f127e2949ff62af7d3bf832615e5dc2238e1500389e07d7acda8e7fcfb7400c"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d18"},{"id":"d19","kind":"p","type":null,"order":19,"text":"The paper supplies concrete numbers and a renormalization route that later work can test against broader energy-flow claims.","status":"active","vx_hash":"0a743670b8cfbb6f9432b860a8ac81b3bcd100efbef84411bdd8fbb1c17b0f43","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"a963184215e7125e4a49ae9f2b14a149bc72e261fa9c9ed1b67a1141a747a6c7","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:37:09.840Z","actor":"owner","text_sha":"5696fc89c7c3503124ab1cb1dc17327da6705aa6a6fd5fdc0a2bb49c2e04a78f","detail":{"divided_from":"body","block":19,"kind":"p"},"prev":"genesis","hash":"a963184215e7125e4a49ae9f2b14a149bc72e261fa9c9ed1b67a1141a747a6c7"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:37:09.840Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/paper-feigenbaum-m-j-1978-quantitative-universality-for-a-class-of-nonlinear-transform/d19"}],"voxels":[{"id":"c1","div_id":"claim:c1","kind":"claim","text":"Feigenbaum 1978 proves that period-doubling 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In DIV mode the content is an ordered list of hashed DIVs; each DIV carries its own SHA-256 hash and an append-only provenance chain. Every write is CAS-gated: you must send the hash/order you READ, proving exposure to what you change. Every successful write returns a clickable human permalink.","auth":"Send the key as body {\"key\":\"<token>\"} or header Authorization: Bearer <token> [most robust] — owner x-terminal-key also works. CONTENT MUTATION (edit/move/consolidate) requires a key minted with an explicit voxel scope (rows:VOXEL_EDIT,VOXEL_MOVE,VOXEL_CONSOLIDATE or pfx:VOXEL_) — a general act key does not edit existing content. Filing a challenge or attestation needs no key at all.","web_runtime":"WEB CHATGPT: open https://miscsubjects.com/api/model-lane first. Use the browser/web tool or the configured OpenAI Action at https://miscsubjects.com/api/openai/actions.json. Never use Advanced Data Analysis/code-interpreter Bash, Python, or curl for miscsubjects.com. 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