From f916c8b20060706bcc50a931a8138bab998aa0de Mon Sep 17 00:00:00 2001 From: Michael <13820335+shiftybitshiftr@users.noreply.github.com> Date: Mon, 12 Oct 2026 23:14:48 -0700 Subject: [PATCH 1/3] Research: extend Eisenstein copy bound to fixed-support nonlinear messages --- .../eisenstein-message-copy-bound/CLAIM.md | 27 + .../eisenstein-message-copy-bound/NOTICE.md | 39 + .../eisenstein-message-copy-bound/PROOF.md | 220 + .../eisenstein-message-copy-bound/README.md | 76 + .../SOURCES.json | 67 + .../VALIDATION.md | 88 + .../certificate.json | 10591 ++++++++++++++++ .../crosscheck.py | 116 + .../eisenstein-message-copy-bound/geometry.py | 203 + .../sources/pr785-PROOF.md | 153 + .../sources/pr798-PROOF.md | 198 + .../test_package.py | 60 + .../eisenstein-message-copy-bound/verify.py | 288 + 13 files changed, 12126 insertions(+) create mode 100644 research/eisenstein-message-copy-bound/CLAIM.md create mode 100644 research/eisenstein-message-copy-bound/NOTICE.md create mode 100644 research/eisenstein-message-copy-bound/PROOF.md create mode 100644 research/eisenstein-message-copy-bound/README.md create mode 100644 research/eisenstein-message-copy-bound/SOURCES.json create mode 100644 research/eisenstein-message-copy-bound/VALIDATION.md create mode 100644 research/eisenstein-message-copy-bound/certificate.json create mode 100644 research/eisenstein-message-copy-bound/crosscheck.py create mode 100644 research/eisenstein-message-copy-bound/geometry.py create mode 100644 research/eisenstein-message-copy-bound/sources/pr785-PROOF.md create mode 100644 research/eisenstein-message-copy-bound/sources/pr798-PROOF.md create mode 100644 research/eisenstein-message-copy-bound/test_package.py create mode 100644 research/eisenstein-message-copy-bound/verify.py diff --git a/research/eisenstein-message-copy-bound/CLAIM.md b/research/eisenstein-message-copy-bound/CLAIM.md new file mode 100644 index 0000000000..f509d962f9 --- /dev/null +++ b/research/eisenstein-message-copy-bound/CLAIM.md @@ -0,0 +1,27 @@ +# Sprint claim + +- Track: structural decoder research. OS-random assignment seed: **41**. +- Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. +- Started: 2026-10-13 05:47:37 UTC. Budget: 50 minutes; verification begins by 06:25 UTC. Expected finish: 06:37 UTC. +- Object: the 40-line Eisenstein E8 (Witting) decoder geometry in GamingPuzzled PR #620, with arbitrary scalar binary totals and the source/reader pairing-rank copy charge. +- Hypothesis: a weighted nonorthogonality-clique certificate gives a useful unconditional finite copy-cost lower bound without a full-rank-total assumption. +- Method: independently reconstruct exact rational geometry; enumerate geometric flats and all maximum cliques; check a rational uniform weight over every flat pair. Stop after this finite domain, whether or not the bound helps. +- Deliverable: one scoped research note with proof, source pins, portable exact checker, certificate and negative controls; no new kappa claim. + +## Selection and overlap + +Three candidates considered: (0) vector-valued full-E8 decoder factors; (1) mixed-port E8 extensions; (2) the Eisenstein extension geometry. Seed 41 modulo 3 selects candidate 2. Candidate 0 is close to #765 tensor-basis work; candidate 1 overlaps #715. Candidate 2 changes geometry from the Gaussian #738/#770 and Hurwitz #703/#785 certificates and answers an explicitly unresolved #620 copy-cost question. It does not repeat the predecessor CP-SAT search. + +GitHub reported no pinned issues, and the only open issue was #650 (Discussions). No separate reservation issue was opened. The local claim is used instead. Open PR metadata was checked through #796 (615 open PRs), including #620, #634, #703, #738, #761, #765 and #770. #620's reported Eisenstein lower bound is 9 and its feasible symmetric cost is 40; the search was not certified optimal. The ledger marks #620 partially reviewed and #446 unreviewed. + +At the exact starting base, the selected accepted conditional result was #626 with #543, kappa = 51110230636731/31250000000000000. Larger pending titles are not treated as accepted. + +## Evidence-driven pivot + +At the first experiment refresh, new PR #798 (Jonathan Lam, head `0aea3382e49abd9a59d3cb6676cdb26fd15b04e5`) already supplied the scalar Eisenstein bound and clique-method optimum. Our independent exact geometry replay had found all 772 flats, 2,880 maximum cliques and the uniform maximum-clique floor 280/13. That scalar investigation will not be submitted as new. + +The revised scope is fixed-support, deterministic, zero-error nonlinear messages and arbitrary finite bit widths on the same 40 Eisenstein lines. We will check an exact weighted demand bound by combining the #798 geometric inequality with the conditional-transcript argument of #785 (Hurwitz). The deliverable must explicitly price each message by its bit width times source/reader pairing rank, distinguish this model premise from a physical-circuit theorem, and test nonlinear and side-information boundary examples. No second pivot is planned. + +## Result update + +Pending. diff --git a/research/eisenstein-message-copy-bound/NOTICE.md b/research/eisenstein-message-copy-bound/NOTICE.md new file mode 100644 index 0000000000..d8abfb4049 --- /dev/null +++ b/research/eisenstein-message-copy-bound/NOTICE.md @@ -0,0 +1,39 @@ +# Attribution and license + +New material is contributed by Michael Fischler (@shiftybitshiftr) under +the repository's Apache-2.0 license. OpenAI Codex substantially assisted +with mathematical development, implementation, verification and writing. +No independent human mathematical review is claimed. + +- **Jonathan Lam (@jonathanlam), PR798**, pinned at + `0aea3382e49abd9a59d3cb6676cdb26fd15b04e5`: the exact Eisenstein scalar + weighted-clique bound, geometric certificate and method ceiling. Its + `PROOF.md` is retained verbatim in `sources/pr798-PROOF.md`, including + its original attribution and AI disclosure. This submission independently + replays the geometric premise and does not claim priority for it. +- **Alexander Templeton (@azide0x37), PR785**, pinned at + `83e78085ae9c110a34bff87e71387292d85e47c0`: conditional transcript + injectivity for nonlinear Hurwitz messages, explicit width pricing and + nonlinear examples. Its proof is retained verbatim in + `sources/pr785-PROOF.md`. The general argument is credited, not renamed + as a new technique. +- **Joel Pulikkan / GamingPuzzled, PR620**: the Eisenstein E8 geometry, + extension-ring screen, original cost search and open question. The base + tree retains head `419c7374c3bf5f2eee28f08b1e50c1d57934081c`; inspected + open head is `e9cf03bbdf7094b5a83f6dbd1d74bfd77a05fb87`. The base ledger + marks this research partially reviewed. +- **Douglas R. Colkitt (@Mister-Meeseeks), PR703**, and **Sagar Thakrar + (@thakrarsagar), PR738**: weighted-clique and geometric-closure methods + on Hurwitz and Gaussian geometries. No executable code is imported. +- **DreamingOfClouds, PR446/PR595**: scalar pairing-rank interpretation + and conditional deficit accounting. PR446 is unreviewed in the base + ledger. Neither premise is promoted to a nonlinear physical theorem. +- Earlier E8 and multiplication work by Jacob Sussman, Dylan Crow, + eumemic, icekylinx, Jeremy Kothe, Chafik Boukhalfa and the community + remains credited in the repository's contributor and construction records. + The base #626/#543 construction and all existing suppliers are untouched. + +The two retained proofs are Apache-2.0 material according to their own +notices. Their bytes are pinned in SOURCES.json. The current contribution +adds a composition with distinct message-model coverage, not a new +headline exponent or a new scalar decoder optimum. diff --git a/research/eisenstein-message-copy-bound/PROOF.md b/research/eisenstein-message-copy-bound/PROOF.md new file mode 100644 index 0000000000..86d6e25ed6 --- /dev/null +++ b/research/eisenstein-message-copy-bound/PROOF.md @@ -0,0 +1,220 @@ +# Fixed-support nonlinear messages on the Eisenstein E8 lines + +This note extends Jonathan Lam's [PR798](https://github.com/CrocSwap/integer-mult-bounds/pull/798) +scalar Eisenstein certificate to finite messages that may mix payload bits +nonlinearly. It uses Alexander Templeton's conditional-transcript argument +from [PR785](https://github.com/CrocSwap/integer-mult-bounds/pull/785), which +treats a different, Hurwitz geometry. Neither method is claimed as new here. +The new result is their composition on the Eisenstein geometry, including +unequal demands, a concentration bound, and explicit nonlinear controls. + +## Statement and model + +Let the 40 ports be the Eisenstein E8 lines specified below. Port t holds an +independent input X_t ranging freely over all b_t-bit strings, where b_t is a +nonnegative integer. There are finitely many messages. Message j has fixed +source and reader sets A_j,R_j, fixed integer width q_j >= 0, and value + + Y_j = f_j((X_a)_(a in A_j)) in {0,1}^{q_j}. + +Every reader in R_j receives the same complete Y_j. Receiver t must recover +X_t on **every** input tuple, using all its received messages and the full +inputs of ports orthogonal to t. Functions and joint decoders may be +nonlinear. Supports, widths and functions are fixed before the input is +chosen. Granting full orthogonal inputs as free side information is an +optimistic model of residual correction. + +For line sets R,A, let V_R,V_A be rational row bases of their spans and set + + c(R,A) = rank_Q(V_R V_A^T)/2, + C = sum_j q_j c(R_j,A_j). + +**The charge is a premise of this message model:** every transmitted bit +pays its support pairing rank, and charges add over messages. PR446's +scalar copy-tree interpretation does not establish this pricing for +arbitrary nonlinear physical circuits. Sharing physical work between bits +more cheaply would be outside the result. + +Write Q for any clique of pairwise nonorthogonal lines, including singletons. +The following bound holds for all finite widths and message counts: + + C >= max( ceil(11 sum_t b_t / 20), max_Q sum_(t in Q) b_t ). (1) + +In particular, if every port demands b bits, **C >= 22b Eisenstein rank +units**, or **44b rational rank units**. This is a lower bound, not the +optimum decoder cost. Direct messages give the abstract upper bound 40b. + +This model excludes adaptive/interactive supports, correlated inputs, +errors, variable-length or average-cost messages, rational sub-line +supports, and additional nonorthogonal side information. It includes +fixed-support block-linear codes, coordinate mixing, arbitrary nonlinear +encoding and joint decoding. It does not assert that every physical +vector-valued circuit has the declared additive charge. + +## Exact geometric premise (predecessor work, replayed independently) + +Use the 240 doubled E8 roots: two nonzero coordinates in {2,-2}, or eight +coordinates in {1,-1} with an even number of minus signs. Each has norm +squared eight. The root order and four mutually orthogonal A2 pairs in the +certificate match the `standard_e8`/`a2_rotation` object in Joel Pulikkan / +GamingPuzzled's archived PR620. For a root a, its reflection is + + s_a(x) = x - (x dot a) a / 4. + +Let J be the product of s_a s_b over the four pairs. The checker reconstructs +J and verifies J^2+J+I=0, J^T J=I, and preservation of every root. The six-root +orbits under ±J are exactly the roots in their rational two-dimensional +spans. They define 40 K-lines for K=Q(omega), in K-dimension four. The +certificate uses `rotation4 = 4J` to keep integer coordinates. + +Every line-generated space is J-invariant. So is the kernel of its pairing +with another such space: orthogonality of J gives invariance of X_A^perp. +Domain and kernel therefore have even rational dimension. Thus c is an +integer, independent of the chosen row bases. Pairing rank need not equal +either support's dimension. + +The scalar certificate in PR798 assigns weight 1/40 to each of the 40 +singletons, and 1/960 to each of the 2,880 seven-line nonorthogonal cliques. +Call these weights lambda_Q. Its geometric inequality is + + sum_Q lambda_Q [Q meets R and A] <= c(R,A). (2) + +Our implementation imports no predecessor executable code. It reconstructs +the geometry and checks the following integer form of (2) on every pair +of flats F,G: + + 24 |F intersect G| + H_7(F,G) <= 960 c(F,G). (3) + +Here H_7 counts seven-line cliques meeting both flats. The exact census is +772 flats and 298,378 unordered pairs including empty supports. Symmetry +covers both orders. All 30,880 one-line extensions remain in the flat list. +Flats are generated from the empty span by adjoining a line and taking +closure. Every flat has a line basis, so induction on basis size proves +completeness. Closing arbitrary supports preserves their spans and pairing +rank and can only increase clique-hit indicators. Hence (3) proves (2) for +all support pairs, not just the enumerated flat masks. + +Every port lies in exactly 504 seven-line cliques, as checked literally for +each of the 40 ports. Its total weight is therefore + + sum_(Q contains t) lambda_Q = 1/40 + 504/960 = 11/20. (4) + +The flat census by (K-rank, cardinality) is + + (0,0):1; (1,1):40; (2,2):240; (2,4):90; + (3,5):360; (3,12):40; (4,40):1. + +The geometric inequality and its scalar application belong to PR798. +This note does not claim a new optimality theorem for clique weights; that +PR also gives a separate dual certificate, which is not needed here. + +## Conditional transcript proof for arbitrary widths + +Fix a nonorthogonal clique Q and **any** assignment of all inputs outside +Q. Let J_Q be the messages whose source and reader sets both meet Q. Then + + X_Q -> (Y_j)_(j in J_Q) + +is injective. If two assignments of X_Q had the same displayed transcript, +each receiver in Q would have the same received messages: omitted messages +have all sources outside Q, so their values were fixed. Each receiver's +orthogonal side information is also outside Q, since distinct vertices of +Q are nonorthogonal. Every deterministic receiver would return the same +X_t in both assignments, contradicting their difference. + +Counting input tuples and possible fixed-width transcripts now gives + + sum_(t in Q) b_t <= sum_j q_j [Q meets R_j and A_j]. (5) + +This is the argument from PR785, applied to general nonorthogonal cliques. +It uses no linearity and holds for every finite width by proof, rather than +by extrapolating the truth-table controls. + +Multiply (5) by lambda_Q, sum over Q, and apply (2) and (4): + + (11/20) sum_t b_t + <= sum_j q_j sum_Q lambda_Q [Q meets R_j and A_j] + <= sum_j q_j c(R_j,A_j) = C. + +Integrality of q_j and c gives the ceiling in (1). + +For the other part of (1), if a clique meets both supports, choose a line +from each intersection. They are either the same nonzero line or two +nonorthogonal lines. The corresponding Gram block is nonzero. Evenness of +pairing rank gives c(R,A) >= 1. Thus (5) directly yields +C >= sum_(t in Q) b_t for every Q. Since b_t >= 0, the largest such demand +sum occurs on an inclusion-maximal clique. The checker enumerates all +2,970 maximal cliques, including the 90 of size four that are not contained +in seven-line cliques. + +The concentration term matters: demands supported on a seven-line clique +give C >= sum_t b_t, whereas (11/20) times their total alone is weaker. +No claim is made that (1) is the strongest inequality for every demand vector. + +## Nonlinear examples and assumption controls + +The seed-41 control chooses an exact partition into ten orthogonal +four-frames. For the first three frames, let z_0,z_1,z_2 be their input +parities. Broadcast to those 12 ports the three-bit message + + (z_0, z_1, z_2 XOR (z_0 AND z_1)). + +Receivers invert the last Boolean update. A receiver knows its other three +frame inputs as orthogonal side information, so its frame parity recovers +its own bit. The source and reader spans are full K-rank four, giving +charge 3*4=12. Its mixed Boolean derivative is nonzero, so this encoding is +nonlinear. The verifier exhausts all 4,096 inputs, all 12 receivers and +303,104 conditional transcripts from 74 distinct projections of the +weighted cliques. Adding 28 direct one-bit messages gives a cost-40 code +on all 40 ports. This product extension is justified algebraically; the +checker does not claim to enumerate 2^40 inputs. Independent repetitions +give cost 40b for arbitrary equal b, without claiming optimality. + +A second control uses demands 0,1,2,3 on one orthogonal four-frame, sending +each active bit-plane parity only to the ports with that bit. It checks all +64 inputs and 192 transcripts, with charge six and bound four. Inactive +and unequal-width ports are thus exercised explicitly. + +Two boundary examples show why the premises cannot be dropped: + +- Charging the three-bit nonlinear message just once would cost four, + below the certified integer bound seven for its 12 active inputs. It is + the per-bit charge, not a scalar copy theorem, that prevents this. +- Seven pairwise nonorthogonal ports could all recover from one parity + if each were given the other six inputs for free. Their joint span has + K-rank four, giving charge four versus the clique demand seven. This + code is correct on all 128 inputs, but violates the side-information + premise. The verifier accepts its decoding and rejects its admission. + +Twelve rejection controls cover understated widths, undeclared sources, +dropped message bits, constant messages, a missing nonlinear inverse, +illegal side information, underpriced width, a non-clique, the rational +rank conversion, excessive geometric weights, omitted singleton weights +and negative demands. These test mathematical failure modes. + +## Optimistic cost screen and remaining obligations + +Only if the physical implementation obeys the message pricing and the +inherited Dfc accounting may one substitute v=40, rational port rank 2b, +and cst=2C. Then + + D/T = (s-1)*80b - s*2C <= b*(36s-80). + +At three stages the upper deficit is **28b**, which is positive. This +result does **not** exclude a three-stage supplier. The nonlinear cost-40b +example has deficit -80b for every stage count and supplies no saving. +Helpers, source access, encoding, decoding, frame changes, further copies, +restoration and recursive children have all been granted zero additional +expense in this optimistic screen. Any achieved supplier must price them; +none is constructed here. The other supplier and outer assembly are not +replaced or repriced, and no new kappa follows. + +This contribution changes no selected record or supplier. All-size realization, +compiler, hosting, precision, tape and analytic obligations are inherited +and unresolved by this finite result. The transcript argument is a written +proof; the geometry has exhaustive finite premises. Neither is a +proof-assistant formalization of integer multiplication. + +Prepared by Michael Fischler (@shiftybitshiftr) with substantial OpenAI Codex +assistance in proof development, implementation, verification and writing. +Apache-2.0. See SOURCES.json and NOTICE.md for predecessor pins and credit. diff --git a/research/eisenstein-message-copy-bound/README.md b/research/eisenstein-message-copy-bound/README.md new file mode 100644 index 0000000000..596e12bb24 --- /dev/null +++ b/research/eisenstein-message-copy-bound/README.md @@ -0,0 +1,76 @@ +# Eisenstein E8: a bound for nonlinear fixed-support messages + +**Research note; no new kappa or physical supplier.** On the 40 Eisenstein +E8 lines, zero-error messages with b independent bits demanded per port +cost at least **22b Eisenstein pairing-rank units / 44b rational units**, +provided each q-bit message is charged q times its source/reader pairing +rank and charges add. Encoding and decoding may be nonlinear. Unequal +demands satisfy + + C >= max(ceil(11 sum_t b_t / 20), max_clique sum_(t in clique) b_t). + +The abstract equal-width optimum remains between 22b and 40b. The bound +does not exclude three stages. Nonlinear physical per-bit pricing remains +an explicit assumption, not a consequence proved here. + +This composes Jonathan Lam's concurrent [#798](https://github.com/CrocSwap/integer-mult-bounds/pull/798) +scalar Eisenstein certificate with Alexander Templeton's +[#785](https://github.com/CrocSwap/integer-mult-bounds/pull/785) nonlinear +transcript method for the different Hurwitz geometry. The new coverage is +nonlinear/block messages on Eisenstein E8, arbitrary unequal demands, a +concentration bound and concrete assumption controls. Our initial scalar +experiment was superseded by #798 and is credited accordingly. + +## Reproduce + +From the repository root, Python 3.10+ with its standard library: + +```sh +python3 -B research/eisenstein-message-copy-bound/verify.py +``` + +Expected: + +```text +PASS_EISENSTEIN_MESSAGE_BOUND: 22b; unequal demands 11/20 plus clique bounds; 298378 flat pairs; 12 rejection controls; no new kappa +``` + +The replay verifies pinned sources, the exact geometry, all 772 flats, +298,378 flat pairs, 30,880 one-line extensions and all 2,970 maximal +cliques. It exhausts the nonlinear 12-port example on 4,096 inputs, the +unequal-width example on 64 inputs and an intentionally excluded +side-information example on 128 inputs. Twelve rejection controls run by +default, including underpriced width and incorrect nonlinear decoding. +The generated result must equal [certificate.json](certificate.json). +`--output NEW_FILE` writes a receipt only after successful verification and +refuses to overwrite existing files. + +An arithmetic implementation that imports neither checker module verifies +every geometric inequality using modular rank witnesses, and exact +Fraction ranks for all flat bases and the 241 tight positive-rank witnesses: + +```sh +python3 -B research/eisenstein-message-copy-bound/crosscheck.py +``` + +See [PROOF.md](PROOF.md) for the all-width argument, model and remaining +obligations; [VALIDATION.md](VALIDATION.md) for executed checks; and +[CLAIM.md](CLAIM.md) for selection, coverage and the single pivot. + +## Pins and scope + +Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. OS-random assignment seed +**41**, structural decoder research. Runtime dependencies: Python standard +library only; source pins and upstream contribution heads are recorded in +[SOURCES.json](SOURCES.json). Two predecessor proofs are retained verbatim +under `sources/`, with their original attribution. + +The result assumes fixed supports and widths, independent inputs, exact +deterministic decoding and only orthogonal free side information. It does +not cover adaptive relays, average-cost coding, cheaper physical sharing +across bits or rational sub-line supports. Finite checks are not a formal +proof of the full multiplication theorem. + +Prepared by **Michael Fischler (@shiftybitshiftr)** with substantial +**OpenAI Codex assistance**. No independent human review is claimed. +Apache-2.0; see [NOTICE.md](NOTICE.md). diff --git a/research/eisenstein-message-copy-bound/SOURCES.json b/research/eisenstein-message-copy-bound/SOURCES.json new file mode 100644 index 0000000000..3d2e6da6c1 --- /dev/null +++ b/research/eisenstein-message-copy-bound/SOURCES.json @@ -0,0 +1,67 @@ +{ + "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60", + "license": "Apache-2.0", + "files": [ + { + "path": "research/e8-decoder-floors-and-cold14-notes/s1-extension-rings/REPORT.md", + "sha256": "e8de809976a1253087e9058d346dfcc7b098f6481c64f1569aa3072d27345e02" + }, + { + "path": "research/e8-decoder-floors-and-cold14-notes/s1-extension-rings/algebra_lines.py", + "sha256": "bc4037049debf9b7affeb7e6b30da0193dea8c3850a7317afd045ac7ac481108" + }, + { + "path": "research/format-limits/scripts/e8units.py", + "sha256": "3134443d8327d0bf2a8a66c26327cf5268cec6b7ffd85dde750b7e197ce63ef1" + }, + { + "path": "research/eisenstein-message-copy-bound/sources/pr798-PROOF.md", + "sha256": "b75539c196ed45f23cf238413da9684dcba503f6630ad68a9648ecacf6aada3f" + }, + { + "path": "research/eisenstein-message-copy-bound/sources/pr785-PROOF.md", + "sha256": "62206654c5e6928324563bd5883db7186917a4546126ae64b72eee8914d9f573" + } + ], + "predecessors": [ + { + "pr": 620, + "author": "Joel Pulikkan / GamingPuzzled", + "archived_head": "419c7374c3bf5f2eee28f08b1e50c1d57934081c", + "inspected_open_head": "e9cf03bbdf7094b5a83f6dbd1d74bfd77a05fb87", + "role": "Eisenstein E8 geometry, initial copy-cost screen and open question; partially reviewed" + }, + { + "pr": 798, + "author": "Jonathan Lam / jonathanlam", + "head": "0aea3382e49abd9a59d3cb6676cdb26fd15b04e5", + "role": "Exact scalar Eisenstein weighted-clique primal certificate; proof retained unmodified", + "source_url": "https://raw.githubusercontent.com/jonathanlam/integer-mult-bounds/0aea3382e49abd9a59d3cb6676cdb26fd15b04e5/research/eisenstein-clique-copy-bound/PROOF.md" + }, + { + "pr": 785, + "author": "Alexander Templeton / azide0x37", + "head": "83e78085ae9c110a34bff87e71387292d85e47c0", + "role": "Nonlinear conditional-transcript argument on Hurwitz; proof retained unmodified", + "source_url": "https://raw.githubusercontent.com/azide0x37/integer-mult-bounds/83e78085ae9c110a34bff87e71387292d85e47c0/research/hurwitz-nonlinear-copy-floor/PROOF.md" + }, + { + "pr": 703, + "author": "Douglas R. Colkitt / Mister-Meeseeks", + "head": "358383e2f1b0f292d4d6eb897c06fd8558a4f5e6", + "role": "Precursor weighted-clique method for Hurwitz" + }, + { + "pr": 738, + "author": "Sagar Thakrar / thakrarsagar", + "head": "9671c5d72f3119ba26f3774f40aa7130c67bcf77", + "role": "Gaussian flat-pair closure and weighted-clique method; no code imported" + }, + { + "pr": 446, + "author": "DreamingOfClouds", + "head": "309482849b6ddf34db998f53612d34c19f6e3233", + "role": "Physical scalar pairing-rank premise only; marked unreviewed in base ledger; not extended to nonlinear physical circuits here" + } + ] +} diff --git a/research/eisenstein-message-copy-bound/VALIDATION.md b/research/eisenstein-message-copy-bound/VALIDATION.md new file mode 100644 index 0000000000..2bf687ab7a --- /dev/null +++ b/research/eisenstein-message-copy-bound/VALIDATION.md @@ -0,0 +1,88 @@ +# Validation record + +Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. macOS 26.5.2 arm64. +No source outside this research package is modified. + +## Completed mathematical checks + +- Primary exact replay on CPython 3.14.4: PASS (about 8 seconds). +- Isolated optimized replay (`-I -O -B`): PASS; checks use explicit exceptions. +- Independent arithmetic checker: PASS on all 298,378 primal inequalities + using modular rank lower witnesses, 772 flat ranks using Fraction column + elimination, and 241 exact tight pair ranks. Neither checker module is + imported by the arithmetic implementation. Enumeration completeness and + the all-width argument remain the main replay and written proof's scope. +- The exact geometric domain is 772 flats, 30,880 line extensions, 2,970 + maximal cliques (2,880 of size seven and 90 of size four), and 298,378 + unordered flat pairs including empty supports. No numerical tolerances, + incomplete solver search or unproved orbit-transitivity shortcut is used. +- The nonlinear control covers all 4,096 twelve-bit inputs and 303,104 + conditional transcripts; the unequal-width control covers 64 inputs + and 192 transcripts. The excluded extra-side-information code is checked + on all 128 inputs. All 12 mathematical rejection controls pass. +- `scripts/verify_selected_result.py`: PASS, selected record and source + inventory unchanged at kappa 51110230636731/31250000000000000. + +## Commands + +Use Python 3.10+; the host's `/usr/bin/python3` is 3.9 and is not suitable. +The executed primary interpreter is `/opt/homebrew/bin/python3` (3.14.4). + +```sh +/opt/homebrew/bin/python3 -B research/eisenstein-message-copy-bound/verify.py +/opt/homebrew/bin/python3 -I -O -B research/eisenstein-message-copy-bound/verify.py +/opt/homebrew/bin/python3 -B research/eisenstein-message-copy-bound/crosscheck.py +/opt/homebrew/bin/python3 -B research/eisenstein-message-copy-bound/test_package.py +/opt/homebrew/bin/python3 -B scripts/verify_selected_result.py +``` + +## Packaging and structured review + +- Relocated minimal-input replay from an unrelated working directory with + `-I -O -B`: PASS. Regenerated receipt bytes equal the committed receipt. +- Existing output preservation, altered-certificate rejection and altered + source rejection: PASS; failed validation creates no output. +- CPython 3.12.13 isolated optimized primary and independent arithmetic + replays: PASS. No Python installation or package dependency was added. +- Documentation links and staged whitespace: PASS. +- Structured autoreview: PASS, no accepted/actionable findings. It checked + the proof, exact arithmetic, enumeration, controls, source pins and output + handling. It was read-only and did not run tests; the executed checks above + are separate evidence. The configured Codex engine/model was retained. + The shell CLI was too old, so the helper used the already-installed app + CLI (0.162.0-alpha.17.2); no software or settings were changed. + +No hosted CI or independent human review is claimed. + +The unchanged expensive `make verify` and `make frontier626-verify` suites +are not run: no generator, supplier, assembly, shared verifier, selected +certificate or upstream file changes. This package's affected checks run +locally, including the selected-record integrity check. Finite checks do +not prove the full multiplication theorem. + +## Upstream refresh + +During closeout, origin/main advanced from the recorded base to +`7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261`. The fetched selected +record now names PR741, kappa 1661202643272239/1000000000000000000, with +`make frontier741-verify`. The new selected review was read. This note stays +on its exact starting base; it neither assumes an open-title frontier nor +changes the accepted record. All three pinned base-tree geometry sources +are byte-unchanged across that range. New PR801 concerns conditional format +ceilings and coordinate-total costs; it does not cover the nonlinear +message/unequal-demand extension and is not a dependency here. + +The successful review command was: + +```sh +~/.codex/skills/autoreview/scripts/autoreview --mode local \ + --codex-bin /Applications/ChatGPT.app/Contents/Resources/codex-cli/bin/codex \ + --prompt-file /tmp/imb-review-seed41.md \ + --json-output /tmp/imb-autoreview-seed41.json \ + --output /tmp/imb-autoreview-seed41.txt --stream-engine-output +``` + +The prompt restricted review to this package and emphasized the explicit +pricing assumption, exact coverage, attribution and controls. No findings +were accepted or rejected because the result contained none. Subsequent +changes only record completed validation, upstream status and the PR handoff. diff --git a/research/eisenstein-message-copy-bound/certificate.json b/research/eisenstein-message-copy-bound/certificate.json new file mode 100644 index 0000000000..db72bdcfd8 --- /dev/null +++ b/research/eisenstein-message-copy-bound/certificate.json @@ -0,0 +1,10591 @@ +{ + "accounting": { + "conditional_optimistic_deficit": "D/T <= b*(36*s-80)", + "cost_40b_example_deficit": "-80b", + "three_stage_upper_deficit": "28b (positive; no exclusion)" + }, + "assignment_seed": 41, + "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60", + "bound": { + "equal_width_floor": "22b", + "per_port_weight": "11/20", + "rational_equal_width_floor": "44b", + "unequal_width_floor": "max(ceil(11*sum(b_t)/20), max_clique sum(b_t))" + }, + "certificate": { + "clique_incidence_per_port": [ + 504, + 504, + 504, + 504, + 504, + 504, + 504, + 504, + 504, + 504, + 504, + 504, + 504, + 504, + 504, + 504, + 504, + 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-2, + -2, + 2, + 2, + 0, + 0, + 0, + 0 + ], + [ + 2, + -2, + 2, + -2, + 0, + 0, + 0, + 0 + ], + [ + -2, + -2, + -2, + -2, + 0, + 0, + 0, + 0 + ], + [ + -2, + 2, + 2, + -2, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + -2, + -2, + -2, + -2 + ], + [ + 0, + 0, + 0, + 0, + 2, + -2, + -2, + 2 + ], + [ + 0, + 0, + 0, + 0, + 2, + 2, + -2, + -2 + ], + [ + 0, + 0, + 0, + 0, + 2, + -2, + 2, + -2 + ] + ] + }, + "model": "fixed-support deterministic zero-error messages; independent bits; orthogonal side information; additive width times pairing rank", + "source_manifest_sha256": "40a8bb070fef8a807a36f2c581339ee97be8746ef87d0ab25ff7abe05ad7e563" +} diff --git a/research/eisenstein-message-copy-bound/crosscheck.py b/research/eisenstein-message-copy-bound/crosscheck.py new file mode 100644 index 0000000000..f6b50789c3 --- /dev/null +++ b/research/eisenstein-message-copy-bound/crosscheck.py @@ -0,0 +1,116 @@ +#!/usr/bin/env python3 +"""Independent arithmetic check on the literal certificate's finite premises. + +Imports neither geometry.py nor verify.py. Reconstructs bases from the original +root generators, then uses rank modulo 101 to certify every primal inequality. +This checks the certificate's arithmetic, not enumeration completeness or the +all-width transcript proof (those are established by the main replay/proof). +""" + +from fractions import Fraction +import json +from pathlib import Path + + +def check(ok, message): + if not ok: + raise ValueError(message) + + +def indices(mask): + return [i for i in range(40) if mask & (1 << i)] + + +def modular_rank(matrix, p=101): + if not matrix: + return 0 + a = [[x % p for x in row] for row in matrix] + row = 0 + for col in range(len(a[0])): + pivot = next((i for i in range(row, len(a)) if a[i][col]), None) + if pivot is None: + continue + a[row], a[pivot] = a[pivot], a[row] + inverse = pow(a[row][col], -1, p) + a[row] = [(x * inverse) % p for x in a[row]] + for i in range(row + 1, len(a)): + multiplier = a[i][col] + a[i] = [(x - multiplier * y) % p for x, y in zip(a[i], a[row])] + row += 1 + if row == len(a): + break + return row + + +def fraction_rank(matrix): + if not matrix: + return 0 + columns = [list(map(Fraction, column)) for column in zip(*matrix)] + pivot = 0 + for coordinate in range(len(matrix)): + target = next((j for j in range(pivot, len(columns)) if columns[j][coordinate]), None) + if target is None: + continue + columns[pivot], columns[target] = columns[target], columns[pivot] + scale = columns[pivot][coordinate] + columns[pivot] = [x / scale for x in columns[pivot]] + for j in range(pivot + 1, len(columns)): + scale = columns[j][coordinate] + columns[j] = [x - scale * y for x, y in zip(columns[j], columns[pivot])] + pivot += 1 + if pivot == len(columns): + break + return pivot + + +def dot(a, b): + return sum(x * y for x, y in zip(a, b)) + + +def main(): + receipt = json.loads((Path(__file__).resolve().parent / "certificate.json").read_text()) + reps = receipt["geometry"]["representatives"] + g = receipt["geometry"]["rotation4"] + lines = [[r, [dot(row, r) for row in g]] for r in reps] + flats = receipt["certificate"]["flats"] + # A full set of original integer generators avoids using primary RREF rows. + bases = [[r for t in f["generators"] for r in lines[t]] for f in flats] + for f, rows in zip(flats, bases): + check(fraction_rank(rows) == 2 * f["rank"], "independent flat rank differs") + cliques = [q for q in receipt["certificate"]["maximal_cliques"] if q.bit_count() == 7] + check(len(cliques) == 2880, "wrong seven-clique count") + for q in cliques: + ids = indices(q) + for i in ids: + for j in ids: + if i != j: + check(any(dot(x, y) for x in lines[i] for y in lines[j]), "non-clique in receipt") + hit_masks = [sum(1 << k for k, q in enumerate(cliques) if q & f["mask"]) for f in flats] + checked, fraction_cases = 0, 0 + for i, f in enumerate(flats): + for j in range(i, len(flats)): + numerator = 24 * (f["mask"] & flats[j]["mask"]).bit_count() + (hit_masks[i] & hit_masks[j]).bit_count() + gram = [[dot(x, y) for y in bases[j]] for x in bases[i]] + # Reduction mod p cannot increase rational rank. Rational rank is + # even by J-invariance, so rounding an odd modular rank up to the + # next even value is valid in Eisenstein rank units. + lower = (modular_rank(gram) + 1) // 2 + check(numerator <= 960 * lower, "modular witness does not certify inequality") + checked += 1 + if (f["rank"] == flats[j]["rank"] == 3 and numerator == 1920) or i == j == len(flats) - 1: + exact = fraction_rank(gram) + check(exact % 2 == 0 and exact // 2 == lower, "Fraction pairing rank differs") + fraction_cases += 1 + check(checked == 298378 and fraction_cases == 241, "independent coverage changed") + for t in range(40): + check(Fraction(1, 40) + Fraction(sum(bool(q & 1 << t) for q in cliques), 960) == Fraction(11, 20), "nonuniform demand coefficient") + # Finite derivative and inverse checks use a different Boolean expression + # from the full network implementation. + permutation = [z ^ (((z & 1) * ((z >> 1) & 1)) << 2) for z in range(8)] + check(len(set(permutation)) == 8 and all(permutation[permutation[z]] == z for z in range(8)), "nonlinear inverse failed") + check(permutation[0] ^ permutation[1] ^ permutation[2] ^ permutation[3] == 4, "map is affine") + print("PASS_INDEPENDENT_MESSAGE_GEOMETRY: 298378 modular inequalities; 772 Fraction flat ranks; 241 Fraction pair ranks; demand weight 11/20") + + +if __name__ == "__main__": + main() diff --git a/research/eisenstein-message-copy-bound/geometry.py b/research/eisenstein-message-copy-bound/geometry.py new file mode 100644 index 0000000000..f289dd16dd --- /dev/null +++ b/research/eisenstein-message-copy-bound/geometry.py @@ -0,0 +1,203 @@ +"""Exact rational geometry for the 40 Eisenstein E8 lines; standard library only. + +Independent implementation of the geometry specified by PR620. Coordinates use +the doubled standard E8 roots, and rotation4 is four times an order-three +orthogonal transformation. No predecessor executable code is imported. +""" + +from functools import reduce +from itertools import combinations, product +from math import gcd + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def dot(a, b): + return sum(x * y for x, y in zip(a, b)) + + +def primitive(row): + divisor = reduce(gcd, row, 0) + if not divisor: + return tuple(row) + first = next(x for x in row if x) + divisor = divisor if first > 0 else -divisor + return tuple(x // divisor for x in row) + + +def basis(rows): + """Canonical rational row space, with primitive integer rows (exact RREF).""" + pivots = {} + for original in rows: + row = primitive(original) + for p, old in sorted(pivots.items()): + if row[p]: + row = primitive([old[p] * x - row[p] * y for x, y in zip(row, old)]) + if any(row): + p = next(i for i, x in enumerate(row) if x) + pivots[p] = row + for p in sorted(pivots, reverse=True): + row = pivots[p] + for q in sorted(pivots): + if q >= p: + break + old = pivots[q] + if old[p]: + pivots[q] = primitive([row[p] * x - old[p] * y for x, y in zip(old, row)]) + return tuple(pivots[p] for p in sorted(pivots)) + + +def contains(span, row): + row = tuple(row) + for old in span: + p = next(i for i, x in enumerate(old) if x) + if row[p]: + row = primitive([old[p] * x - row[p] * y for x, y in zip(row, old)]) + return not any(row) + + +def matmul(a, b): + columns = tuple(zip(*b)) + return tuple(tuple(dot(row, col) for col in columns) for row in a) + + +def identity(n): + return tuple(tuple(int(i == j) for j in range(n)) for i in range(n)) + + +def roots(): + """The PR620/standard_e8 order: positive representatives, then negatives.""" + positive = [] + for i, j in combinations(range(8), 2): + for sign in (1, -1): + row = [0] * 8 + row[i], row[j] = 2, 2 * sign + positive.append(tuple(row)) + positive.extend((1,) + s for s in product((1, -1), repeat=7) if s.count(-1) % 2 == 0) + return tuple(positive + [tuple(-x for x in r) for r in positive]) + + +def rotation(roots_): + """Product of Coxeter rotations on the first four orthogonal A2 planes.""" + chosen = [] + for a in roots_: + if len(chosen) == 4: + break + if any(dot(a, x) for pair in chosen for x in pair): + continue + for b in roots_: + if dot(a, b) == -4 and all(dot(b, x) == 0 for pair in chosen for x in pair): + chosen.append((a, b)) + break + require(len(chosen) == 4, "four orthogonal A2 planes missing") + g = identity(8) + for pair in chosen: + for a in pair: + reflection8 = tuple(tuple(8 * int(i == j) - 2 * a[i] * a[j] for j in range(8)) for i in range(8)) + g = matmul(g, reflection8) + denominator = 8 ** 8 // 4 + require(all(x % denominator == 0 for row in g for x in row), "nonintegral rotation4") + return tuple(tuple(x // denominator for x in row) for row in g), tuple(chosen) + + +def image(g, r): + values = tuple(dot(row, r) for row in g) + require(all(x % 4 == 0 for x in values), "rotation does not preserve integral roots") + return tuple(x // 4 for x in values) + + +def make_geometry(): + rr = roots() + require(len(set(rr)) == 240 and all(dot(r, r) == 8 for r in rr), "bad E8 roots") + g, planes = rotation(rr) + square = matmul(g, g) + require(all(square[i][j] + 4 * g[i][j] + 16 * int(i == j) == 0 for i in range(8) for j in range(8)), "bad order-three polynomial") + require(matmul(g, tuple(zip(*g))) == tuple(tuple(16 * x for x in row) for row in identity(8)), "rotation is not orthogonal") + require(set(image(g, r) for r in rr) == set(rr), "rotation changes root set") + representatives, spans, orbits, seen = [], [], [], set() + for r in rr: + if r in seen: + continue + gr = image(g, r) + orbit3 = (r, gr, image(g, gr)) + orbit = set(orbit3) | {tuple(-x for x in v) for v in orbit3} + require(len(orbit) == 6 and not orbit & seen, "bad six-root orbit") + span = basis((r, gr)) + require(len(span) == 2, "line is not rational rank two") + require({v for v in rr if contains(span, v)} == orbit, "orbit is not the full root line") + representatives.append(r) + spans.append(span) + orbits.append(tuple(sorted(orbit))) + seen.update(orbit) + require(len(spans) == 40 and len(set(spans)) == 40 and seen == set(rr), "wrong Eisenstein line partition") + adjacent = [] + for i, left in enumerate(spans): + mask = 0 + for j, right in enumerate(spans): + if i != j and any(dot(a, b) for a in left for b in right): + mask |= 1 << j + adjacent.append(mask) + require({a.bit_count() for a in adjacent} == {27}, "nonorthogonality degree changed") + return rr, g, planes, tuple(representatives), tuple(spans), tuple(orbits), tuple(adjacent) + + +def bits(mask): + while mask: + bit = mask & -mask + yield bit.bit_length() - 1 + mask ^= bit + + +def maximal_cliques(adjacent): + """Bron-Kerbosch with pivot; returns every inclusion-maximal clique.""" + result = [] + + def visit(chosen, possible, excluded): + if not possible and not excluded: + result.append(chosen) + return + union = possible | excluded + pivot = max(bits(union), key=lambda i: (possible & adjacent[i]).bit_count()) + candidates = possible & ~adjacent[pivot] + for i in bits(candidates): + bit = 1 << i + visit(chosen | bit, possible & adjacent[i], excluded & adjacent[i]) + possible &= ~bit + excluded |= bit + + visit(0, (1 << len(adjacent)) - 1, 0) + return tuple(sorted(result)) + + +def flats(line_spans): + """All line-generated subspaces; adding one line is a complete closure walk.""" + levels = [{(): ()}] + for rank in range(4): + next_level = {} + for span, generators in levels[-1].items(): + for i, line in enumerate(line_spans): + if contains(span, line[0]): + require(all(contains(span, r) for r in line), "partial Eisenstein line") + continue + child = basis(span + line) + require(len(child) == 2 * (rank + 1), "rank did not increase by one Eisenstein unit") + next_level.setdefault(child, generators + (i,)) + levels.append(next_level) + result = [] + for rank, level in enumerate(levels): + for span, generators in level.items(): + mask = sum(1 << i for i, line in enumerate(line_spans) if contains(span, line[0])) + require(basis([row for i in bits(mask) for row in line_spans[i]]) == span, "flat closure span changed") + result.append((rank, mask, span, generators)) + require(len({x[1] for x in result}) == len(result), "duplicate flat masks") + return tuple(sorted(result, key=lambda x: (x[0], x[1]))) + + +def pairing_rank(left, right): + gram = tuple(tuple(dot(a, b) for b in right) for a in left) + rational_rank = len(basis(gram)) + require(rational_rank % 2 == 0, "odd rational pairing rank") + return rational_rank // 2 diff --git a/research/eisenstein-message-copy-bound/sources/pr785-PROOF.md b/research/eisenstein-message-copy-bound/sources/pr785-PROOF.md new file mode 100644 index 0000000000..9811bf6317 --- /dev/null +++ b/research/eisenstein-message-copy-bound/sources/pr785-PROOF.md @@ -0,0 +1,153 @@ +# A counting extension of the Hurwitz copy floor + +**Scoped result.** For the ten Hurwitz E8 lines, fixed-support, deterministic, +zero-error messages of arbitrary binary widths have minimum charge +`C = sum_t b_t`, when port t demands b_t independent bits and a q-bit message +with reader/source sets R,A is charged `q c(R,A)`. In particular, equal +block width b costs at least 10b quaternionic units, or 40b rational units. +Encoders and decoders may be nonlinear. No bound on the number of messages +is imposed, other than finiteness. + +This extends the scalar-total argument of Douglas R. Colkitt's +[#703](https://github.com/CrocSwap/integer-mult-bounds/pull/703), at commit +`358383e2f1b0f292d4d6eb897c06fd8558a4f5e6`. The perfect-matching geometry, +pairing charge, transversal weights and scalar optimum are predecessor work. +The addition here is conditional transcript injectivity in place of binary +rank, together with nonlinear attaining examples and assumption controls. + +## 1. Geometry and the pricing premise + +Use the E8 roots in doubled coordinates: vectors in Q^8 with either two +entries in {2,-2} and the rest zero, or eight entries in {1,-1} and an even +number of minus signs. Regard Q^8 as H_Q^2 and take the spans of the four +left multiples 1,i,j,k of each root. These are ten four-dimensional rational +subspaces. Their orthogonality graph is five disjoint pairs; write mate(t) +for the unique orthogonal line. Any two distinct lines span Q^8. + +For concatenated row bases V_R and V_A, set + + c(R,A) = rank_Q(V_R V_A^T) / 4. + +The exact charge table is: + +- 0 if a set is empty, or R and A are singletons at orthogonal mates; +- 1 otherwise if a nonempty set is a singleton; +- 2 if both sets have at least two elements. + +Two spanning sets have rank-eight pairing. A singleton against a spanning +set has rank four. Nonorthogonal singletons have rank-four pairing; mates +have zero pairing. The verifier reconstructs all roots and line bases, +checks every singleton pairing and every two-line span over the integers, +then checks every support pair against the transversal inequality below. + +**Explicit pricing premise:** a message of fixed width q_j, sent from source +set A_j to reader set R_j, is charged `q_j c(R_j,A_j)` quaternionic units. +Charges add over messages. Every transmitted bit is priced, even if the +encoder combines coordinates nonlinearly. This is the chosen block-message +model. It is **not** a derivation of nonlinear physical copy costs from +#446's scalar pairing-rank theorem. A supplier that shares physical work +between message coordinates more cheaply is outside this result. + +## 2. Information model + +There are independent inputs X_t ranging freely over {0,1}^{b_t}; b_t are +nonnegative integers. Message j has a fixed nonnegative integer width q_j +and is a deterministic function + + Y_j = f_j((X_a)_{a in A_j}) in {0,1}^{q_j}. + +The supports, widths and functions do not depend on the input. Every reader +t in R_j receives the same full Y_j; it may process those bits arbitrarily. +Receiver t must reconstruct X_t for **every** input tuple, using its received +messages and free side information X_mate(t). The decoder may jointly use +all those messages and need not be linear. Granting the entire mate input +is an optimistic relaxation of orthogonal residual correction. + +Source access, encoding, decoding and side-information preparation are free +in this model. There is no interactive relay, adaptive support, input +correlation, decoding error or additional nonorthogonal side information. +The theorem also covers a scalar fitting decoder from #703: each rank-one +total is one width-one message, and its mate contamination can be removed +using the granted side information. Fixed-support block-linear codes are +included as well, without requiring coordinatewise encoding. + +## 3. Transcript injectivity + +A transversal T chooses exactly one endpoint of each of the five mate +pairs. For this T define + + J_T = { j : R_j intersects T and A_j intersects T }. + +Fix **any** assignment of all inputs outside T. The map + + X_T -> (Y_j)_{j in J_T} + +is injective. To see this, suppose two different X_T assignments have the +same displayed transcript. Every message received by a receiver in T is +either in J_T or has A_j disjoint from T. Messages in the latter class are +identical because their source inputs were fixed. Every receiver's mate is +outside T, so all its side information is also identical. Its deterministic +decoder would therefore return the same X_t in both assignments, for all +t in T, contradicting the distinct X_T assignments. + +Consequently, counting possible transcripts gives + + 2^(sum_{t in T} b_t) <= 2^(sum_{j in J_T} q_j), + sum_{t in T} b_t <= sum_j q_j a_T(R_j,A_j), (1) + +where a_T is 1 exactly when both supports meet T. No probabilistic, +asymptotic or linearity assumption is used. The proof is for every finite +block width and every finite number of messages, not an extrapolation from +the truth tables used as controls. + +## 4. Average over the 32 transversals + +Each line lies in 16 transversals. For every R,A the predecessor inequality +is + + sum_T a_T(R,A) <= 16 c(R,A). (2) + +If c=0 no transversal hits both supports. If c=1 a singleton support occurs +in only 16 transversals. If c=2 the total number of transversals is 32. +Thus (2) follows in all cases. Summing (1) and dividing by 16 gives + + sum_t b_t <= sum_j q_j (sum_T a_T(R_j,A_j))/16 + <= sum_j q_j c(R_j,A_j) = C. + +The bound is sharp in the abstract model for arbitrary b_t: send X_t +directly from {t} to {t} with width b_t. This costs sum_t b_t. For equal +width b, another attaining code broadcasts the five mate-pair XORs in a +single 5b-bit message with all ports as sources and readers. Its charge is +`2 * 5b = 10b`. Apply any bijection of its 5b bits before transmission; +receivers invert it and recover their pair XOR using their mate input. +For example, the reversible update `z_2 ^= z_0 & z_1` makes the encoder +nonlinear and preserves both correctness and charge. The replay verifies +this example on all 1,024 inputs for ten one-bit ports, and a vector example +on all 256 inputs for four two-bit ports in the matching geometry. + +## 5. Optimistic complete-budget screen + +If, in addition, these b-fold ports satisfy the inherited Dfc stage +accounting with v=10, k=4b and cst=4C, then + + D/T = (s-1) * 40b - s * 4C <= -40b + +for every positive integer s and b. Helpers, source access, frame changes, +copies beyond this lower bound, restoration and recursive children were +all granted zero additional expense in this screen. Any nonnegative charge +for them cannot repair its negative deficit. There is no positive supplier +saving to feed into an assembly, regardless of the other supplier. +Accordingly no circuit implementation or kappa optimization is attempted. +The algebraic inequality supplies the all-s conclusion; the receipt's +finite arithmetic examples are only error controls. + +The use of Dfc and the conversion from message width to physical rank are +explicit additional assumptions. This note does not establish them for +nonlinear circuits. It does not bound interactive/adaptive codes, variable +length or average-cost communication, fractional physical sub-line copying, +correlated demands, other geometries or different stage accounting. + +Prepared by Alexander Templeton (@azide0x37) with substantial OpenAI Codex +assistance in the proof extension, implementation, checks and writing. +This is a written mathematical argument with finite exact checks, not a +proof-assistant formalization or a verification of integer multiplication. diff --git a/research/eisenstein-message-copy-bound/sources/pr798-PROOF.md b/research/eisenstein-message-copy-bound/sources/pr798-PROOF.md new file mode 100644 index 0000000000..5c5fd2f87c --- /dev/null +++ b/research/eisenstein-message-copy-bound/sources/pr798-PROOF.md @@ -0,0 +1,198 @@ +# Eisenstein E8: copy charge at least 22, and a sharp limit of clique certificates + +This is a theorem about a specified finite decoder model. It is not a new +supplier, a proof of decoder optimality, or a new multiplication exponent. +The weighted-clique argument follows the distinct Hurwitz and Gaussian +geometries of [PR703](https://github.com/CrocSwap/integer-mult-bounds/pull/703) +and [PR738](https://github.com/CrocSwap/integer-mult-bounds/pull/738). +The new result is the exact primal/dual certificate for the **40-line +Eisenstein geometry**, including a bound on *all* clique-weight certificates. + +## Model and geometry + +Start with the 240 E8 roots in doubled coordinates: vectors with two entries +equal to ±2 and all other entries zero, and the vectors in {±1}^8 with an even +number of minus signs. Each root has squared norm eight. + +Following Joel Pulikkan / GamingPuzzled's archived PR620, choose four mutually +orthogonal A2 planes, with roots (a,b) satisfying a·b = -4, and let J be the +product of their rotations s_a s_b. The literal pairs are in the certificate. +The independent implementation uses the same root order and greedy choices +as the pinned predecessor, so it defines the same rational action, not merely +a configuration with similar counts. The reflection formula is + + s_a(x) = x - (x·a)a/4. + +The exact checks establish J^3 = I, J^2 + J + I = 0, and preservation of the +inner product and root set. Thus Q^8 is a four-dimensional vector space over +K = Q(ω), with multiplication by ω realized by J. The root orbits under ±J +partition the roots into 40 K-lines. For a line set S, let X_S be the rational +span of r_i,Jr_i for i in S. Every X_S is J-invariant. + +Two distinct lines are nonorthogonal if r_i·r_j or r_i·Jr_j is nonzero. Each +line has 27 such neighbors. Let B be an arbitrary 40×40 matrix over F2 with + + B_tt = 1; + B_ts = 0 if distinct lines t,s are nonorthogonal. + +The other directed entries are unconstrained. Consider any finite decomposition + + B = Σ_j 1_(R_j) 1_(A_j)^T over F2. + +There is no symmetry, minimal-rank, or number-of-totals restriction. For any +row bases V_R,V_A of the rational spans, define the abstract copy charge + + c(R,A) = rank_Q(V_R V_A^T)/2, C = Σ_j c(R_j,A_j). + +This is a *pairing* rank; it need not equal either support's span dimension. +The rank is even: the kernel X_R ∩ X_A^⊥ is J-invariant, because J is an +isometry and X_A is J-invariant. Both domain and kernel are K-spaces. +Consequently each c is an integer. Translating this abstract charge into a +physical copy lower bound inherits PR446's premise, which the base review +ledger marks unreviewed. The finite algebraic result needs no such premise. + +## General clique bound and the lower certificate + +A clique Q is a set of pairwise nonorthogonal lines. The fitting rules give +B[Q,Q] = I. A binary rectangle restricted to Q×Q has rank one exactly when +both supports meet Q. Rank subadditivity gives + + |Q| ≤ Σ_j [Q∩R_j ≠ ∅ and Q∩A_j ≠ ∅]. (1) + +For each nonempty clique assign a nonnegative rational weight λ_Q. Suppose +these weights satisfy, for all supports R,A, + + Σ_Q λ_Q [Q meets R and A] ≤ c(R,A). (2) + +Multiplying (1) by λ_Q and summing proves C ≥ Σ_Q λ_Q |Q|. + +The certificate uses weight **1/40 on each singleton** and **1/960 on each +of the 2,880 seven-line cliques**, with weight zero elsewhere. The resulting +objective is exactly + + 40/40 + 2,880·7/960 = 22. (3) + +To certify (2), close R and A in the line matroid: cl(S) contains every +configuration line lying in X_S. Closure leaves X_S, hence the pairing rank, +unchanged. It can only increase the clique-hit indicators. Therefore it is +sufficient to check every pair of flats F,G. Write H_7(F,G) for the number +of seven-line cliques meeting both flats. The required integer inequality is + + 24 |F∩G| + H_7(F,G) ≤ 960 c(F,G). (4) + +The replay verifies (4) by exact rational rank on all **298,378 unordered +pairs of the 772 flats**, including the empty flat. Symmetry of the pairing +rank and hit count covers both orders. This is a complete reduction from +arbitrary supports; the theorem does not assume supports were originally +flats. Equations (1)–(4) prove **C ≥ 22** for every finite decomposition. + +## Why no weighting of cliques can prove more + +The following upper bound concerns the certificate method, not decoder cost. +Let P consist of the **240 unordered pairs** F,G of twelve-line, rank-three +flats with c(F,G)=2. Each such pair has |F∩G|=2 and H_7(F,G)=1,872, so (4) +is tight. Give each of these pairs dual weight 1/24; give the full/full +pair dual weight 1/2. Its cost is + + (240·2)/24 + 4/2 = 22. (5) + +For *every literal nonempty clique* Q, the certificate verifies + + 1/2 + #{(F,G) in P : Q meets F and G}/24 ≥ |Q|. (6) + +The complete incidence census is below. In particular, no unproved +transitivity assumption or orbit-averaging argument is needed. + +| Clique size | Dual pairs hit | Number of cliques | +|---:|---:|---:| +| 1 | 12 | 40 | +| 2 | 48 | 540 | +| 3 | 81 | 2,880 | +| 3 | 108 | 360 | +| 4 | 106 | 6,480 | +| 4 | 123 | 2,880 | +| 4 | 192 | 90 | +| 5 | 125 | 5,184 | +| 5 | 134 | 8,640 | +| 6 | 145 | 10,080 | +| 7 | 156 | 2,880 | + +Now take *any* nonnegative clique weights satisfying (2), including arbitrary +nonuniform weights on smaller cliques. Multiply (6) by λ_Q, sum over Q, +and apply (2) to the full/full pair and each pair in P. Equation (5) gives + + Σ_Q λ_Q |Q| ≤ 22. + +Together with (3), this proves that the optimum of the entire weighted-clique +certificate method is exactly **22**. A stronger copy lower bound will need +an ingredient beyond inequalities (1)–(2). In particular, using more clique +weights or a different clique-weight optimizer alone cannot improve it. + +## Decoder bounds and the optimistic accounting screen + +Every fitting B has an abstract cost-40 decomposition: for each row t, +take R={t} and A equal to that row's nonzero positions. A contains t; every +other member of A is orthogonal to t. The rational Gram block therefore has +rank two, so this row costs one. These 40 rectangles reproduce B. Hence the +minimum charge lies in **[22,40]**. The lower endpoint is attained by a +*certificate*, and no cost-22 decoder is constructed. + +For the separate Dfc stage accounting in the pinned format notes, v=40, +each line has rational rank k=2, and cst=2C. Thus, conditionally on that +accounting and the physical interpretation of the charge, + + D/T = (s-1)vk - s·cst ≤ 80(s-1) - 44s = 36s-80. + +At s=3 the upper deficit is 28, which is positive. This lower bound **does +not exclude three stages**. A positive three-stage deficit would require +C < 80/3; integrality narrows the unresolved possibilities to C=22,...,26. +All helpers, source access, frame changes, copies beyond this lower bound, +restoration and recursive children would still have to be paid. None is +implemented or assumed free in an achieved construction. At the cost-40 +row decomposition the deficit is -80 for every stage count, so that abstract +upper-bound witness itself cannot supply a saving under Dfc. + +The selected #626/#543 composition remains complex-bound. This research +does not provide a replacement supplier or change any assembly constraint. + +## Completeness and proof boundary + +Flats are generated from the empty span by adding one root line and taking +exact rational span/closure. Any flat has a line basis, so induction on basis +size proves this generates every flat in four K-dimensions. The verifier +also checks all 30,880 one-line extensions remain in the enumerated set. +The flat counts by (K-rank, number of lines) are + + (0,0):1; (1,1):40; (2,2):240; (2,4):90; + (3,5):360; (3,12):40; (4,40):1. + +All nonempty cliques are generated once by successively adjoining larger +vertices adjacent to every previous vertex. This recursion exhausts every +ordered increasing clique. A separate Bron–Kerbosch maximal-clique census +corroborates the 2,880 maximum cliques and 90 smaller maximal cliques. + +The committed certificate contains the literal flat generators, maximum +cliques, all 240 dual pairs, complete pair/incidence censuses, and hashes of +the regenerated root orbits and all 40,054 cliques. The replay uses no solver. +An ordinary Fraction elimination on original root generators checks 258 +seeded/boundary Gram blocks independently of the integer RREF implementation. +Eight mathematical negative controls cover fitting identities, clique +admission, rank accounting, and both primal and dual weights/coverage. + +The separate `crosscheck.py` imports neither implementation. It checks all +298,378 primal inequalities on original integer generators using rank modulo +101. Such a rank is a lower bound on rational rank, which is the sufficient +direction for (4). It checks all 240 dual-pair costs and the full-pair cost +by exact Fraction elimination. Geometry and enumeration completeness remain +the responsibilities of the main replay and the preceding argument. + +This is a written proof with exhaustive finite premises, not a proof-assistant +formalization or a finite replay claimed to prove the full multiplication +theorem. Whole K-lines, scalar binary totals and the stated fitting pattern +are essential to the scope. Sub-line/vector-valued operations, different +decoder patterns or geometries, physical realization, all-size compilation, +hosting, precision, tape and analytic transfer remain outside it. + +Prepared by Jonathan Lam (@jonathanlam) with substantial OpenAI Codex +assistance. See [SOURCES.json](SOURCES.json) for exact pins and predecessor +credit. New material is Apache-2.0 under the repository license. diff --git a/research/eisenstein-message-copy-bound/test_package.py b/research/eisenstein-message-copy-bound/test_package.py new file mode 100644 index 0000000000..b228c061ea --- /dev/null +++ b/research/eisenstein-message-copy-bound/test_package.py @@ -0,0 +1,60 @@ +#!/usr/bin/env python3 +"""Relocation, certificate tampering, input tampering and output-preservation QA.""" + +import json +from pathlib import Path +import shutil +import subprocess +import sys +import tempfile + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parent.parent + + +def require(ok, reason): + if not ok: + raise ValueError(reason) + + +def main(): + with tempfile.TemporaryDirectory(prefix="eisenstein-message-qa-") as tmp: + root = Path(tmp) / "relocated" + package = root / "research" / HERE.name + shutil.copytree(HERE, package, ignore=shutil.ignore_patterns("__pycache__")) + sources = json.loads((HERE / "SOURCES.json").read_text())["files"] + for item in sources: + target = root / item["path"] + target.parent.mkdir(parents=True, exist_ok=True) + shutil.copyfile(ROOT / item["path"], target) + + def run(*args): + return subprocess.run([sys.executable, "-I", "-O", "-B", str(package / "verify.py"), *map(str, args)], cwd=tmp, capture_output=True, text=True) + + receipt = Path(tmp) / "replayed.json" + result = run("--output", receipt) + require(result.returncode == 0, "relocated replay failed: " + result.stderr) + require(receipt.read_bytes() == (package / "certificate.json").read_bytes(), "regenerated receipt bytes differ") + old = receipt.read_bytes() + result = run("--output", receipt) + require(result.returncode != 0 and "FileExistsError" in result.stderr and receipt.read_bytes() == old, "existing output was overwritten") + + certificate = package / "certificate.json" + original = certificate.read_bytes() + corrupted = json.loads(original) + corrupted["bound"]["equal_width_floor"] = "23b" + certificate.write_text(json.dumps(corrupted)) + missing = Path(tmp) / "must-not-exist.json" + result = run("--output", missing) + require(result.returncode != 0 and "certificate differs" in result.stderr and not missing.exists(), "tampered certificate accepted or output written") + certificate.write_bytes(original) + + source = root / sources[0]["path"] + source.write_bytes(source.read_bytes() + b"\nmodified source\n") + result = run("--output", missing) + require(result.returncode != 0 and "source hash mismatch" in result.stderr and not missing.exists(), "tampered source accepted or output written") + print("PASS_MESSAGE_PACKAGE: relocated isolated optimized replay; exact receipt; existing output preserved; certificate and source mutations rejected") + + +if __name__ == "__main__": + main() diff --git a/research/eisenstein-message-copy-bound/verify.py b/research/eisenstein-message-copy-bound/verify.py new file mode 100644 index 0000000000..440b333ea3 --- /dev/null +++ b/research/eisenstein-message-copy-bound/verify.py @@ -0,0 +1,288 @@ +#!/usr/bin/env python3 +"""Replay the finite premises and controls of the Eisenstein message bound. + +Python 3.10+, standard library. Checks stay active under python -O. +""" + +import argparse +from collections import Counter +from fractions import Fraction +import hashlib +import json +from pathlib import Path +import random +import sys + +HERE = Path(__file__).resolve().parent +sys.path.insert(0, str(HERE)) +import geometry as geo + +SEED = 41 + + +def canonical(value): + return json.dumps(value, sort_keys=True, separators=(",", ":")).encode() + + +def digest(value): + return hashlib.sha256(canonical(value)).hexdigest() + + +def source_pins(): + manifest = json.loads((HERE / "SOURCES.json").read_text()) + root = HERE.parent.parent + for item in manifest["files"]: + path = (root / item["path"]).resolve() + geo.require(path.is_relative_to(root), "source path escapes repository") + geo.require(hashlib.sha256(path.read_bytes()).hexdigest() == item["sha256"], "source hash mismatch: " + item["path"]) + return hashlib.sha256((HERE / "SOURCES.json").read_bytes()).hexdigest() + + +def check_clique(mask, adjacent): + geo.require(mask > 0 and mask < 1 << len(adjacent), "invalid clique mask") + for i in geo.bits(mask): + geo.require(mask & ~(adjacent[i] | 1 << i) == 0, "clique contains orthogonal lines") + + +def frame_partition(adjacent): + """A seeded exact cover by orthogonal four-frames; used only for controls.""" + full = (1 << len(adjacent)) - 1 + orthogonal = tuple(full ^ (1 << i) ^ adjacent[i] for i in range(len(adjacent))) + frames = list(geo.maximal_cliques(orthogonal)) + geo.require(len(frames) == 40 and {f.bit_count() for f in frames} == {4}, "orthogonal frame census changed") + random.Random(SEED).shuffle(frames) + options = [[f for f in frames if f & 1 << i] for i in range(len(adjacent))] + + def search(remaining): + if not remaining: + return [] + i = min(geo.bits(remaining), key=lambda i: sum(f & remaining == f for f in options[i])) + for f in options[i]: + if f & remaining == f: + tail = search(remaining ^ f) + if tail is not None: + return [f] + tail + return None + + result = search(full) + geo.require(result is not None and sum(result) == full, "no disjoint frame partition") + return result + + +def projected_value(values, mask): + return tuple(values[t] for t in geo.bits(mask)) + + +def exercise_code(widths, messages, side, decoder, transcript_cliques=()): + """Exhaust the declared finite input product and every listed transcript. + + Each message is (source mask, reader mask, q, encoder). The dependency + check is semantic: equal source inputs must give equal message values. + """ + geo.require(len(widths) == 40 and all(isinstance(b, int) and b >= 0 for b in widths), "bad input widths") + for sources, readers, width, _ in messages: + geo.require(0 <= sources < 1 << 40 and 0 <= readers < 1 << 40, "bad message support") + geo.require(isinstance(width, int) and width >= 0, "bad message width") + active = sum(1 << t for t, b in enumerate(widths) if b) + nbits = sum(widths) + geo.require(nbits <= 12, "truth-table control exceeds its stated domain") + dependencies = [{} for _ in messages] + projections = sorted({q & active for q in transcript_cliques if q & active}) + transcripts = [{} for _ in projections] + for packed in range(1 << nbits): + remaining = packed + values = [] + for width in widths: + values.append(remaining & ((1 << width) - 1)) + remaining >>= width + outputs = [] + for j, (sources, readers, width, encoder) in enumerate(messages): + value = encoder(values) + geo.require(0 <= value < 1 << width, "encoded message exceeds declared width") + key = projected_value(values, sources) + old = dependencies[j].setdefault(key, value) + geo.require(old == value, "encoder depends on undeclared source") + outputs.append(value) + for t in geo.bits(active): + received = {j: outputs[j] for j, msg in enumerate(messages) if msg[1] & 1 << t} + known = {u: values[u] for u in geo.bits(side[t])} + geo.require(decoder(t, received, known) == values[t], "zero-error decoding failed") + for k, clique in enumerate(projections): + outside = projected_value(values, active & ~clique) + heard = tuple(outputs[j] for j, msg in enumerate(messages) if msg[0] & clique and msg[1] & clique) + inside = projected_value(values, clique) + old = transcripts[k].setdefault((outside, heard), inside) + geo.require(old == inside, "conditional transcript is not injective") + return {"inputs": 1 << nbits, "clique_projections": len(projections), "conditional_transcripts": sum(len(x) for x in transcripts)} + + +def demand_bound(widths, maximal): + """An integer lower bound, using all maximal cliques for unequal demands.""" + geo.require(len(widths) == 40 and all(isinstance(b, int) and b >= 0 for b in widths), "bad demand widths") + weighted = (11 * sum(widths) + 19) // 20 + concentrated = max(sum(widths[t] for t in geo.bits(q)) for q in maximal) + return max(weighted, concentrated) + + +def controls(spans, adjacent, maximal, maximum, full_hits): + rejected = [] + + def reject(name, action): + try: + action() + except ValueError: + rejected.append(name) + else: + raise ValueError("negative control accepted: " + name) + + partition = frame_partition(adjacent) + selected = partition[:3] + active = selected[0] | selected[1] | selected[2] + widths = [int(bool(active & 1 << t)) for t in range(40)] + side = [(active & ~(adjacent[t] | 1 << t)) for t in range(40)] + frame_of = {t: j for j, frame in enumerate(selected) for t in geo.bits(frame)} + + def nonlinear(values): + z = [sum(values[t] for t in geo.bits(frame)) % 2 for frame in selected] + return z[0] | z[1] << 1 | (z[2] ^ (z[0] & z[1])) << 2 + + def decode(t, received, known): + y = received[0] + z = [y & 1, (y >> 1) & 1, ((y >> 2) & 1) ^ ((y & 1) & ((y >> 1) & 1))] + frame = selected[frame_of[t]] + return z[frame_of[t]] ^ (sum(known[u] for u in geo.bits(frame & ~(1 << t))) % 2) + + message = (active, active, 3, nonlinear) + geo.require(all(side[t] & (adjacent[t] | 1 << t) == 0 for t in geo.bits(active)), "illegal side information in nonlinear control") + cliques = list(maximum) + [1 << t for t in range(40)] + nonlinear_check = exercise_code(widths, [message], side, decode, cliques) + support_basis = geo.basis([row for t in geo.bits(active) for row in spans[t]]) + rank = geo.pairing_rank(support_basis, support_basis) + geo.require(rank == 4, "nonlinear example support rank changed") + geo.require(3 * rank == 12 and demand_bound(widths, maximal) == 7, "nonlinear example bound changed") + # A mixed Boolean derivative certifies genuine nonlinearity of the encoder. + a, b = next(geo.bits(selected[0])), next(geo.bits(selected[1])) + zero = [0] * 40 + va, vb, vab = zero.copy(), zero.copy(), zero.copy() + va[a] = vb[b] = vab[a] = vab[b] = 1 + geo.require(nonlinear(zero) ^ nonlinear(va) ^ nonlinear(vb) ^ nonlinear(vab) == 4, "example is not nonlinear") + reject("message width reduced from three to two", lambda: exercise_code(widths, [(active, active, 2, nonlinear)], side, decode)) + reject("one source omitted from the declared support", lambda: exercise_code(widths, [(active ^ (1 << a), active, 3, nonlinear)], side, decode)) + reject("third message bit dropped", lambda: exercise_code(widths, [(active, active, 3, lambda x: nonlinear(x) & 3)], side, decode)) + reject("message replaced by a constant", lambda: exercise_code(widths, [(active, active, 3, lambda x: 0)], side, decode)) + reject("nonlinear inverse correction omitted", lambda: exercise_code(widths, [message], side, lambda t, received, known: ((received[0] >> frame_of[t]) & 1) ^ (sum(known[u] for u in geo.bits(selected[frame_of[t]] & ~(1 << t))) % 2))) + + # Unequal-width positive control on an orthogonal four-frame: one XOR + # message per active bit plane, with every plane's source/reader support + # equal to the ports that actually have that bit. + first = list(geo.bits(partition[0])) + heterogeneous = [0] * 40 + for t, width in zip(first, [0, 1, 2, 3]): + heterogeneous[t] = width + msgs = [] + for bit in range(3): + support = sum(1 << t for t in first if heterogeneous[t] > bit) + def encode_plane(values, bit=bit, support=support): + return sum((values[t] >> bit) & 1 for t in geo.bits(support)) % 2 + msgs.append((support, support, 1, encode_plane)) + hetero_side = [(partition[0] & ~(1 << t)) if t in first else 0 for t in range(40)] + geo.require(all(hetero_side[t] & (adjacent[t] | 1 << t) == 0 for t in first), "illegal side information in unequal-width control") + def decode_heterogeneous(t, received, known): + value = 0 + for bit in range(heterogeneous[t]): + value |= (received[bit] ^ (sum((known[u] >> bit) & 1 for u in geo.bits(msgs[bit][0] & ~(1 << t))) % 2)) << bit + return value + hetero_check = exercise_code(heterogeneous, msgs, hetero_side, decode_heterogeneous, cliques) + hetero_cost = sum(geo.pairing_rank(geo.basis([row for t in geo.bits(msg[0]) for row in spans[t]]), geo.basis([row for t in geo.bits(msg[1]) for row in spans[t]])) for msg in msgs) + geo.require(hetero_cost == 6 and demand_bound(heterogeneous, maximal) == 4, "heterogeneous bound changed") + + # Assumption boundary, intentionally ACCEPTED as a code: extra + # nonorthogonal side information defeats the claimed clique bound. + q = maximum[0] + boundary_widths = [int(bool(q & 1 << t)) for t in range(40)] + boundary_message = (q, q, 1, lambda x: sum(x[t] for t in geo.bits(q)) % 2) + extra_side = [q & ~(1 << t) for t in range(40)] + boundary = exercise_code(boundary_widths, [boundary_message], extra_side, lambda t, heard, known: heard[0] ^ (sum(known.values()) % 2)) + qb = geo.basis([row for t in geo.bits(q) for row in spans[t]]) + geo.require(geo.pairing_rank(qb, qb) == 4 < demand_bound(boundary_widths, maximal) == 7, "side-information boundary changed") + reject("nonorthogonal side information passed off as legal", lambda: geo.require(all(extra_side[t] & adjacent[t] == 0 for t in geo.bits(q)), "extra side information is not orthogonal")) + reject("nonlinear message charged once instead of per bit", lambda: geo.require(rank >= demand_bound(widths, maximal), "undercharged three-bit message violates bound")) + mate = next(i for i in range(1, 40) if not adjacent[0] & 1 << i) + reject("orthogonal pair admitted as a clique", lambda: check_clique(1 | 1 << mate, adjacent)) + reject("Eisenstein charge not converted to rational units", lambda: geo.require(len(geo.basis(spans[0])) == geo.pairing_rank(spans[0], spans[0]), "missing factor of two")) + reject("maximum-clique weights rounded upward", lambda: geo.require(24 * 40 + full_hits <= 959 * 4, "full/full primal inequality violated")) + reject("singleton weights omitted but demand coefficient retained", lambda: geo.require(Fraction(504, 960) == Fraction(11, 20), "missing singleton weight")) + reject("negative payload width", lambda: demand_bound([-1] + [0] * 39, maximal)) + return { + "negative_controls": rejected, + "orthogonal_partition": partition, + "nonlinear_12_port_code": {**nonlinear_check, "active_mask": active, "message_width": 3, "copy_charge": 12, "demand_bound": 7, "mixed_derivative": 4}, + "heterogeneous_0_1_2_3_code": {**hetero_check, "copy_charge": hetero_cost, "demand_bound": 4}, + "extra_nonorthogonal_side_information_boundary": {**boundary, "clique_mask": q, "copy_charge": 4, "excluded_model_bound": 7}, + "forty_port_product_extension": {"copy_charge": 40, "message_widths": [3] + [1] * 28, "proof": "12-port nonlinear code plus 28 independent uncoded bits; no exhaustive 2^40 claim"}, + } + + +def build_receipt(): + pins = source_pins() + roots, rotation4, planes, reps, spans, orbits, adjacent = geo.make_geometry() + maximal = geo.maximal_cliques(adjacent) + for clique in maximal: + check_clique(clique, adjacent) + maximum = tuple(q for q in maximal if q.bit_count() == 7) + geo.require(Counter(q.bit_count() for q in maximal) == {4: 90, 7: 2880}, "maximal-clique census changed") + incidence = [sum(bool(q & 1 << t) for q in maximum) for t in range(40)] + geo.require(set(incidence) == {504}, "demand coefficient is not uniform") + geo.require(Fraction(1, 40) + Fraction(504, 960) == Fraction(11, 20), "wrong per-port demand weight") + flat_list = geo.flats(spans) + expected = {(0, 0): 1, (1, 1): 40, (2, 2): 240, (2, 4): 90, (3, 5): 360, (3, 12): 40, (4, 40): 1} + geo.require(Counter((f[0], f[1].bit_count()) for f in flat_list) == expected, "flat census changed") + spans_set = {f[2] for f in flat_list} + for f in flat_list: + for line in spans: + geo.require(geo.basis(f[2] + line) in spans_set, "flat extension omitted") + hit_masks = [sum(1 << k for k, q in enumerate(maximum) if q & f[1]) for f in flat_list] + histogram, tight = Counter(), 0 + for i, f in enumerate(flat_list): + for j in range(i, len(flat_list)): + other = flat_list[j] + hits = (hit_masks[i] & hit_masks[j]).bit_count() + common = (f[1] & other[1]).bit_count() + charge = geo.pairing_rank(f[2], other[2]) + geo.require(24 * common + hits <= 960 * charge, "weighted geometric inequality failed") + histogram[(charge, common, hits)] += 1 + tight += int(24 * common + hits == 960 * charge) + geo.require(sum(histogram.values()) == 298378, "flat-pair coverage changed") + control_results = controls(spans, adjacent, maximal, maximum, len(maximum)) + geo.require(len(control_results["negative_controls"]) == 12, "negative-control count changed") + return { + "format": "eisenstein-fixed-support-message-bound-v1", + "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60", + "assignment_seed": SEED, + "source_manifest_sha256": pins, + "model": "fixed-support deterministic zero-error messages; independent bits; orthogonal side information; additive width times pairing rank", + "bound": {"per_port_weight": "11/20", "equal_width_floor": "22b", "unequal_width_floor": "max(ceil(11*sum(b_t)/20), max_clique sum(b_t))", "rational_equal_width_floor": "44b"}, + "geometry": {"rotation4": rotation4, "a2_planes": planes, "representatives": reps, "root_orbits_sha256": digest(orbits), "roots_sha256": digest(roots), "nonorthogonal_neighbors": adjacent}, + "certificate": {"singleton_weight": "1/40", "seven_clique_weight": "1/960", "clique_incidence_per_port": incidence, "maximal_cliques": maximal, "flats": [{"rank": f[0], "mask": f[1], "generators": f[3]} for f in flat_list]}, + "coverage": {"flats": len(flat_list), "unordered_flat_pairs": sum(histogram.values()), "tight_pairs": tight, "one_line_extensions": len(flat_list) * 40, "pair_histogram": [[*key, value] for key, value in sorted(histogram.items())], "maximal_cliques": len(maximal), "seven_cliques": len(maximum)}, + "controls": control_results, + "accounting": {"conditional_optimistic_deficit": "D/T <= b*(36*s-80)", "three_stage_upper_deficit": "28b (positive; no exclusion)", "cost_40b_example_deficit": "-80b"}, + } + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--output", type=Path, help="write a verified receipt to a NEW path after all checks pass") + args = parser.parse_args() + receipt = build_receipt() + expected = json.loads((HERE / "certificate.json").read_text()) + geo.require(canonical(receipt) == canonical(expected), "committed certificate differs from replay") + if args.output: + with args.output.open("x") as stream: + json.dump(receipt, stream, indent=2, sort_keys=True) + stream.write("\n") + print("PASS_EISENSTEIN_MESSAGE_BOUND: 22b; unequal demands 11/20 plus clique bounds; 298378 flat pairs; 12 rejection controls; no new kappa") + + +if __name__ == "__main__": + main() From ad875bc1b0aefd5b9b600d8499a1672459eb2ad2 Mon Sep 17 00:00:00 2001 From: Michael <13820335+shiftybitshiftr@users.noreply.github.com> Date: Mon, 12 Oct 2026 23:16:49 -0700 Subject: [PATCH 2/3] Record completed seed-41 sprint claim and PR handoff --- research/eisenstein-message-copy-bound/CLAIM.md | 12 +++++++++++- 1 file changed, 11 insertions(+), 1 deletion(-) diff --git a/research/eisenstein-message-copy-bound/CLAIM.md b/research/eisenstein-message-copy-bound/CLAIM.md index f509d962f9..4138353161 100644 --- a/research/eisenstein-message-copy-bound/CLAIM.md +++ b/research/eisenstein-message-copy-bound/CLAIM.md @@ -24,4 +24,14 @@ The revised scope is fixed-support, deterministic, zero-error nonlinear messages ## Result update -Pending. +Completed 2026-10-13 06:16 UTC: submitted one research-note PR, [#814](https://github.com/CrocSwap/integer-mult-bounds/pull/814). Construction/proof commit: `f916c8b20060706bcc50a931a8138bab998aa0de`. + +- Result: fixed-support, deterministic, zero-error nonlinear messages on the 40 Eisenstein E8 lines obey C >= 22b for equal b-bit demands, with additive width-times-pairing-rank pricing. Unequal demands obey the weighted 11/20 sum bound and the maximum clique-demand bound. The physical pricing premise is explicit. No new kappa or supplier is claimed. +- Complete geometric coverage: all 772 flats, 298,378 unordered flat pairs, 30,880 line extensions and 2,970 maximal cliques. Stopping rule: exhaust that finite domain and the declared message controls, then package one result. No incompleteness or timeout is presented as an impossibility theorem. +- New evidence beyond #798/#785: the Eisenstein nonlinear and unequal-demand composition; 4,096 nonlinear inputs and 303,104 conditional transcripts; unequal-width and side-information boundary examples; exact reconstruction and a separate modular/Fraction arithmetic checker. The geometric scalar result and transcript method retain predecessor credit. +- Validation: both implementations pass on CPython 3.14.4 and 3.12.13; 12 rejection controls, relocation, source/certificate tampering, output preservation, committed-byte equality and selected-record integrity pass. Structured autoreview reports no actionable findings. See VALIDATION.md for exact scope and commands. +- Final pre-submission refresh: 625 open PRs through #812; #798 and #785 remain at their pinned heads. New #801 is a distinct format-ceiling/coordinate-total note. No pinned coordination issue exists; no reservation issue was opened. This is the single result update to the local claim. +- Upstream advanced to `7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261` and selected #741 at kappa 1661202643272239/1000000000000000000. Its review was read. All pinned base geometry files remain byte-identical. The contribution retains its exact starting base and changes no selected or upstream file. + +Reproduce (Python 3.10+): `python3 -B research/eisenstein-message-copy-bound/verify.py`. Expected prefix: `PASS_EISENSTEIN_MESSAGE_BOUND`. Prepared by Michael Fischler (@shiftybitshiftr) with substantial OpenAI Codex assistance; Apache-2.0. + From d91c5152b314391406b0ec792b6765a5cf0af2f1 Mon Sep 17 00:00:00 2001 From: Michael <13820335+shiftybitshiftr@users.noreply.github.com> Date: Mon, 12 Oct 2026 23:17:15 -0700 Subject: [PATCH 3/3] Normalize sprint handoff end of file --- research/eisenstein-message-copy-bound/CLAIM.md | 1 - 1 file changed, 1 deletion(-) diff --git a/research/eisenstein-message-copy-bound/CLAIM.md b/research/eisenstein-message-copy-bound/CLAIM.md index 4138353161..7d3514805d 100644 --- a/research/eisenstein-message-copy-bound/CLAIM.md +++ b/research/eisenstein-message-copy-bound/CLAIM.md @@ -34,4 +34,3 @@ Completed 2026-10-13 06:16 UTC: submitted one research-note PR, [#814](https://g - Upstream advanced to `7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261` and selected #741 at kappa 1661202643272239/1000000000000000000. Its review was read. All pinned base geometry files remain byte-identical. The contribution retains its exact starting base and changes no selected or upstream file. Reproduce (Python 3.10+): `python3 -B research/eisenstein-message-copy-bound/verify.py`. Expected prefix: `PASS_EISENSTEIN_MESSAGE_BOUND`. Prepared by Michael Fischler (@shiftybitshiftr) with substantial OpenAI Codex assistance; Apache-2.0. -