From 6ad0e646bb24e820d4b5483b71d2f7aa4167e4d3 Mon Sep 17 00:00:00 2001 From: ellipsi2 Date: Tue, 13 Oct 2026 14:23:30 +0900 Subject: [PATCH 1/2] feat(research): certify uniform proper-star rank thresholds --- research/paired-star-rank/CERTIFICATE.json | 16324 ++++++++++++++++ research/paired-star-rank/CLAIM.json | 25 + research/paired-star-rank/NOTICE.md | 30 + research/paired-star-rank/RANK-PROOF.md | 236 + research/paired-star-rank/README.md | 152 + research/paired-star-rank/RESULTS.json | 504 + research/paired-star-rank/SOURCES.json | 47 + research/paired-star-rank/VALIDATION.md | 63 + .../paired-star-rank/build_certificate.py | 74 + research/paired-star-rank/test_verify.py | 91 + research/paired-star-rank/verify.py | 247 + 11 files changed, 17793 insertions(+) create mode 100644 research/paired-star-rank/CERTIFICATE.json create mode 100644 research/paired-star-rank/CLAIM.json create mode 100644 research/paired-star-rank/NOTICE.md create mode 100644 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"66849e1475c930fd7732b1a0067ffee7e1e95f60", + "started_utc": "2026-10-13T05:02:33Z", + "claimed_utc": "2026-10-13T05:07:00Z", + "expected_finish_utc": "2026-10-13T05:32:33Z", + "verification_reserved_from_utc": "2026-10-13T05:25:03Z", + "coordination": "GitHub pinnedIssues returned an empty list; no open sprint issue. Local claim, no reservation issue opened.", + "candidates": [ + "Tighten the asymmetric Gaussian E8 weighted-clique copy floor: deferred because PR738 has just covered that mechanism.", + "Derive a uniform rank formula for global proper stars on signed k-subset ports, with an exact simultaneous-all-target puncture certificate at (p,k)=(8,4). Selected.", + "Classify minimum punctures at (p,k)=(7,4): deferred because PR747 and PR732 already cover closely related one-target structure." + ], + "object": "The dimension of the scalar global proper-star code, including the optional empty star, and the maximum number of simultaneously decodable retained ports for power-of-two k.", + "hypothesis": "Decomposition by sign monomials reduces signed-star rank to unsigned subset-incidence rank; this gives a uniform rank formula and a sharp algebraic all-target deletion threshold.", + "method": "Prove the decomposition and unsigned rank obligation; independently construct full incidence matrices and literal information-set decoders. Seed 46 chooses p=6+(46 mod 4)=8 for k=4.", + "bounded_domain": "All 1120 ports and all proper stars for (8,4), plus small signed slices for controls and predecessor crosschecks. No heuristic search or new supplier claimed.", + "stopping_rule": "Complete the explicit matrix certificate and proof, or scope the result to the certified finite domain if the uniform rank lemma cannot be justified by verification reserve.", + "expected_deliverable": "One additive research package: proof, independent exact certificate/replay, failure controls, source pins and attribution; no new kappa.", + "overlap_checked": [669, 703, 715, 717, 725, 726, 732, 738, 747, 748, 749, 750], + "distinct_contribution": "PR747 supplies the 240-port (6,4) all-target threshold. This claim targets a uniform rank mechanism and a separate 1120-port certificate, explicitly crediting that precursor.", + "status": "in progress" +} diff --git a/research/paired-star-rank/NOTICE.md b/research/paired-star-rank/NOTICE.md new file mode 100644 index 0000000000..9194d3b513 --- /dev/null +++ b/research/paired-star-rank/NOTICE.md @@ -0,0 +1,30 @@ +# Attribution and license + +Prepared for ellipsi2 (@ellipsi2), 2026, with substantial OpenAI Codex +assistance in research, mathematical derivation, implementation, verification +and writing. A separate Codex worker investigated and reviewed the rank proof; +that is AI-assisted review, not independent human peer review. + +New files are under the repository's [Apache-2.0 license](../../LICENSE). +No predecessor executable code or literature proof text is imported. + +Credit for the unsigned integral inclusion basis belongs to E. Ghorbani, +G. B. Khosrovshahi, Ch. Maysoori and M. Mohammad-Noori, +[*Inclusion Matrices and Chains*, arXiv:0709.3144v1](https://arxiv.org/pdf/0709.3144v1), +Theorem 1 and equation (8). Their Remark 4 credits Wilson and Bier for the +stacked-degree basis. The literature retains its own rights. + +The paired-cube mechanism and its star/overlap model follow icekylinx and +eumemic. Joel Pulikkan / GamingPuzzled's #231 supplies the symmetric parity +obstruction. Sam Sung / the-missionary-company's #669 studies arbitrary global +stars and local punctures. Luke Mansoor / lukemansoor's #732 supplies the +one-target distance theorem and overlap-correction absorption. Alishan Ladhani +/ aladh's #747 supplies the finite `(6,4)` all-target threshold and attaining +information set; those results are credited corroboration here. Exact source +heads and inspected README hashes are in [SOURCES.json](SOURCES.json). + +The selected-result context retains Chafik Boukhalfa's #626, Jeremy Kothe's +#622/#623, eumemic's #543 bit supplier, DreamingOfClouds' cold-helper work, +Rohan Arun's recurrence correction and the cumulative community framework. +This research does not supersede their constructions or change the selected +conditional bound. diff --git a/research/paired-star-rank/RANK-PROOF.md b/research/paired-star-rank/RANK-PROOF.md new file mode 100644 index 0000000000..557436ed4f --- /dev/null +++ b/research/paired-star-rank/RANK-PROOF.md @@ -0,0 +1,236 @@ +# Uniform rank of global proper stars and the empty-star boundary + +This note proves the dimension of the scalar global proper-star code and its +sharp algebraic all-target puncture threshold. It distinguishes the stipulated +nonempty-star code from the stronger code obtained by granting the empty star. +At `(p,k)=(8,4)` their dimensions are **447 and 448**, respectively; the minimum +numbers of deletions for simultaneous scalar decoding are **673 and 672**. +There is no circuit, paid supplier, or new kappa claim. + +Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. The experiment's track, seed +46, domain and stopping rule are recorded in [CLAIM.json](CLAIM.json). + +## 1. Definitions and statement + +Let `p >= k >= 2`. A signed port is `(A,epsilon)`, where `A` is a k-subset +of `[p]` and `epsilon: A -> {0,1}`. There are + +```text +N(p,k) = 2^k binom(p,k) +``` + +ports. A partial signed port `(L,tau)` defines the incidence function +`m_(L,tau)(A,epsilon) = 1[L subset A and epsilon|L = tau]`. +Over F2, let `C+_(p,k)` span all such functions with `0 <= |L| < k`, and let +`C-_(p,k)` span those with `1 <= |L| < k`. Thus `C+ = C- + <1>`. + +**Theorem.** Put + +```text +rho(p,k) = sum_(j=0)^(k-1) binom(p,j) binom(p-j, min(k-1-j,p-k)), +delta(p,k) = 1 if k is a power of two and p >= 2k-1, and 0 otherwise. +``` + +Then + +```text +dim C+_(p,k) = rho(p,k), +dim C-_(p,k) = rho(p,k) - delta(p,k). +``` + +For power-of-two k, allowing uniform target-overlap corrections of orders +`1,...,k-1` does not change either code. The exact all-target deletion +thresholds are therefore `N-rho+delta` for nonempty stars and `N-rho` when +the empty star is granted. + +In the range `p >= 2k-1`, every minimum in the sum is `k-1-j`. The identity +`binom(p,j) binom(p-j,k-1-j) = binom(p,k-1) binom(k-1,j)` gives the simpler +formula + +```text +rho(p,k) = 2^(k-1) binom(p,k-1). (0) +``` + +For power-of-two k in this range, the maximum number of simultaneously +decodable retained ports using nonempty stars is therefore +`2^(k-1) binom(p,k-1) - 1`. + +## 2. The unsigned slice lemma, including small characteristic + +For an arbitrary field K, let `U(n,q,d)` be the space spanned by the functions +`z_T(Q)=1[T subset Q]`, `|T| <= d`, on all q-subsets Q of `[n]`. Then + +```text +dim_K U(n,q,d) = binom(n, min(d,q,n-q)). (1) +``` + +Here is the precise external input. A subset `B={b_1<...= 2l` for every l; the empty set is ballot. Let `B_(t,q)` have rows +indexed by ballot subsets of sizes at most t, columns by q-subsets, and +entry `1[B subset Q]`. When `0 <= t <= min(q,n-q)`, this matrix has +`binom(n,t)` rows and a square minor of determinant `+1` or `-1`. +This is Theorem 1 of Ghorbani, Khosrovshahi, Maysoori and Mohammad-Noori, +[*Inclusion Matrices and Chains*, arXiv:0709.3144v1](https://arxiv.org/pdf/0709.3144v1), +printed pp. 8-9; their Remark 4, p. 11, credits Wilson and Bier for the +stacked-degree basis. This integral statement applies over every field. + +To spell out why it gives (1), let `W_(t,q)` contain *all* t-subset incidence +rows. Counting the intermediate t-subsets gives the integer identity + +```text +B_(t,t) W_(t,q) = D_(t,q) B_(t,q), +D_(t,q)[B,B] = binom(q-|B|,t-|B|). (2) +``` + +The square matrix `B_(t,t)` is unimodular. Hence every row of `W_(t,q)` +lies in the span of `B_(t,q)` over K, even if some entries of D vanish in K. +For `d <= min(q,n-q)`, stacking this conclusion for `t=0,...,d` gives the +upper bound `binom(n,d)`. The rows of `B_(d,q)` themselves occur among these +stacked rows and are independent, giving equality. No binomial coefficient +is divided out. In particular, a rank formula for one fixed-degree +inclusion matrix, which can depend on the characteristic, would not suffice. + +For `d >= min(q,n-q)`, every q-subset Q can be isolated either by `z_Q` +(degree q) or by `prod_(i not in Q)(1-z_i)` (degree n-q). The evaluation +space is therefore full and has dimension `binom(n,q)`. This proves (1), +including the saturation cases needed below. + +## 3. Separate the sign monomials + +On a signed port define + +```text +z_i = 1[i in A], +x_i = 1[i in A and epsilon_i=1]. +``` + +The two signed-coordinate indicators are `x_i` and `z_i+x_i`. On each +fixed partial support, this invertible change of variables shows that the +star span is exactly the span of + +```text +x_S z_T, S intersect T = empty, |S|+|T| <= k-1. (3) +``` + +Deleting the empty star deletes just the constant monomial in this generating +description: all the coordinate changes preserve positive total degree. + +For a fixed S of size j, the functions in its sector are + +```text +1[S subset A] (prod_(i in S) epsilon_i) h(A\S), +h in U(p-j,k-j,k-1-j). +``` + +Different S sectors form a direct sum. Indeed, on any fixed support A, the +sign monomials `prod_(i in S) epsilon_i`, `S subset A`, are an independent +basis of functions on `{0,1}^A`. A zero linear combination thus has zero +coefficient for every S and every A containing S. Conversely, all +`(k-j)`-subsets of `[p]\S` arise as `A\S`, so this identifies each sector +faithfully with the stated unsigned space. This argument uses ordinary +Boolean monomials, not characteristic-zero Fourier characters. + +By (1), the sector dimension is +`binom(p-j,min(k-1-j,p-k))`. There are `binom(p,j)` choices of S. Summing +over `j=0,...,k-1` proves the formula for `dim C+`. + +## 4. Exactly when the empty star adds a dimension + +Only the `S=empty` sector can produce the constant function, by the direct +sum just proved. The question is therefore whether 1 is in the span of +the *positive-degree* unsigned monomials `z_T`, `1 <= |T| <= k-1`, on the +k-slice of `[p]`. + +If `p <= 2k-2`, choose a fixed B of size `p-k+1 <= k-1`. Every k-subset +meets B, so `prod_(i in B)(1-z_i)=0` on the slice. Expanding expresses 1 +as a sum of nonempty monomials of permitted degree. + +If k is not a power of two, there is a `t` with `1 <= t < k` and +`binom(k,t)` odd. For completeness, in F2[X] the identity +`(1+X)^k = prod_(r: binary digit r of k is 1)(1+X^(2^r))` proves this: +choose the least set bit of k. On the slice, + +```text +sum_(|T|=t) z_T = binom(k,t) = 1, +``` + +so the empty star is again redundant. + +It remains to take k a power of two and `p >= 2k-1`. Now the ballot rows +through degree `k-1` are independent by Section 2. In their basis the row +for the empty set is 1. For each `1 <= t < k`, the empty-index diagonal +entry of (2) is `binom(k,t)=0` in F2. Consequently every degree-t row +has zero coordinate in that empty-row direction: multiplication by the +invertible `B_(t,t)` cannot create it. All positive-degree rows lie in +the span of the nonempty ballot rows. They cannot span 1. Adding the empty +star therefore increases the dimension by exactly one. The three cases +prove the formula for `dim C-`. + +## 5. From dimension to all-target punctures + +For a fixed target T and source port V put `s=|T intersect V|`, with the +intersection taken in signed coordinates. For `1 <= l < k` the sum of +all l-stars contained in T has value `binom(s,l) mod 2`. The matrix of +these values for `s,l=1,...,k-1` is triangular with diagonal 1. At s=0 +all values vanish; at s=k they also vanish when k is a power of two. +Thus each permitted correction `1[|T intersect V|=s]`, `1 <= s < k`, +belongs to **C-**, before and after any puncturing. This is the correction +absorption mechanism proved in [Luke Mansoor's PR #732](https://github.com/CrocSwap/integer-mult-bounds/blob/80431a74c31e807581f5c2258c87abe2084caacd/research/global-star-puncture/README.md#absorbing-overlap-corrections-when-k-is-a-power-of-two). + +For either code C and surviving port set F, every target can be decoded +exactly when `C|F = F2^F`: its restricted span must contain all unit vectors. +Thus `|F| <= dim C`. A generator matrix of rank `dim C` has that many +independent columns, so an information set F of exactly that size exists +and attains equality. This proves the stated exact algebraic thresholds. +It does not bound the number or cost of operations implementing the decoders. + +| (p,k) | Ports | Nonempty-star dimension | With empty star | Minimum deletions, nonempty / with empty | +| --- | ---: | ---: | ---: | ---: | +| (5,4) | 80 | 65 | 65 | 15 / 15 | +| (6,4) | 240 | 155 | 155 | 85 / 85 | +| (7,4) | 560 | 279 | 280 | 281 / 280 | +| (8,4) | 1120 | 447 | 448 | 673 / 672 | + +For the selected `(8,4)` case, the four sign-sector contributions are +`56 + 168 + 168 + 56 = 448`. The missing empty-star direction subtracts one. +The `(6,4)` row and its all-target conclusion already appear in +[Alishan Ladhani's PR #747](https://github.com/CrocSwap/integer-mult-bounds/blob/aeaf0899179a14644b412e838ede14db967c765e/research/global-star-punctures/README.md#all-target-corollary); +they are a predecessor crosscheck, not a new result of this note. + +## 6. Scope, provenance and verification obligations + +The mathematical dependency is the integral ballot-basis theorem pinned above. +The remaining steps are the explicit variable change, the unsigned counting +identity, and linear algebra supplied here. No finite replay replaces that +argument, and no proof-assistant verification is claimed. The package's exact +replay independently checks the finite dimensions and literal information-set +decoders; its command and completed coverage belong in the package README. + +The no-empty distinction concerns the full port family. Both codes are then +projected to the surviving family, and all available stars are recomputed on +that family. Coefficients are unrestricted F2 scalars. Source-dependent +overlap coefficients, degree-k stars, nonlinear decoders and other geometries +are outside the decoder theorem. The dimension formula itself does not need +k to be a power of two; the correction-absorption corollary does. + +No costs for helper preparation, source access, frames, copies, restoration, +recursive children, or hosting are priced. The thresholds are necessary and +sufficient for scalar identities with unrestricted star use; they do not +establish a profitable complete supplier or discharge the multiplication +framework's compiler, precision, tape, recovery, or analytic assumptions. + +Predecessors: Sam Sung's [PR #669](https://github.com/CrocSwap/integer-mult-bounds/pull/669) +at `308f1e6ec3bb0b6ec0584cf7bfe219addc219b1e` posed the global puncture +question, following Joel Pulikkan's #231 and the icekylinx/eumemic paired-cube +lineage. Luke Mansoor's #732 is pinned at +`80431a74c31e807581f5c2258c87abe2084caacd`; its one-target distance theorem +is not used to infer these all-target thresholds. Alishan Ladhani's #747 is +pinned at `aeaf0899179a14644b412e838ede14db967c765e`; its all-target +information-set observation is generalized here. Their implementation code +is not imported. Literature credit is to Ghorbani, Khosrovshahi, Maysoori, +Mohammad-Noori, Wilson and Bier as specified above. + +This proof was developed with substantial OpenAI Codex assistance as part of +the seed-46 contribution. New prose and derivations are under the repository's +[Apache-2.0 license](../../LICENSE). Cited literature retains its own rights; +no literature proof or predecessor implementation is vendored here. diff --git a/research/paired-star-rank/README.md b/research/paired-star-rank/README.md new file mode 100644 index 0000000000..0ef5a5879c --- /dev/null +++ b/research/paired-star-rank/README.md @@ -0,0 +1,152 @@ +# How many signed ports can proper stars decode simultaneously? + +**Research theorem and scoped negative result; no new kappa.** This package +gives the uniform rank of global proper stars on signed k-subset ports and +the resulting sharp algebraic all-target puncture threshold. It also identifies +exactly when granting the global sum changes that threshold. + +On the seed-selected `(p,k)=(8,4)` geometry there are **1,120 ports**: + +| Allowed shared totals | Maximum retained ports, all decodable | Minimum deletions | +| --- | ---: | ---: | +| Nonempty proper stars, with arbitrary off-target coefficients | **447** | **673** | +| Same, also granting the empty star (global sum) | **448** | **672** | + +Uniform overlap-class corrections do not change these numbers. Every maximum +is attained by a literal information set and one exact decoder per retained +port in [CERTIFICATE.json](CERTIFICATE.json). + +The distinction from [Alishan Ladhani's PR #747](https://github.com/CrocSwap/integer-mult-bounds/pull/747) +is substantive: that contribution establishes the `(6,4)` all-target threshold +85, where the empty star is already redundant. Here the result holds uniformly +in p and k, and the empty-star distinction first appears at `(7,4)`. +The earlier `(6,4)` result is credited and independently corroborated. + +## Uniform statement + +For `p >= k >= 2`, there are `N = 2^k binom(p,k)` signed k-ports. A proper +star for a partial signed port L is the incidence vector `1[L subset S]` +over all source ports S, where `|L| < k`. All arithmetic is over F2. +Set + +```text +rho(p,k) = sum_(j=0)^(k-1) binom(p,j) binom(p-j,min(k-1-j,p-k)). +delta(p,k) = 1 if k is a power of two and p >= 2k-1, otherwise 0. +``` + +The star code has dimension **rho** when the empty star is included, and +**rho-delta** otherwise. For power-of-two k, the permitted target-dependent +overlap corrections add no dimension, so the sharp all-target deletion +thresholds are `N-rho` and `N-rho+delta` respectively. + +In the large-p range `p >= 2k-1`, `rho = 2^(k-1) binom(p,k-1)`. +In particular the nonempty-star class with power-of-two k can retain at most +`2^(k-1) binom(p,k-1)-1` ports. This is an exact scalar obstruction even if +all implementation costs are optimistically zero. + +[RANK-PROOF.md](RANK-PROOF.md) proves the sign-monomial decomposition, invokes +the integral unsigned inclusion basis of Ghorbani, Khosrovshahi, Maysoori and +Mohammad-Noori, and proves the empty-star criterion. It distinguishes stacked +degrees from one fixed-degree inclusion matrix, whose rank can depend on the +characteristic. The all-target linear-algebra observation follows #747; +the overlap absorption is credited to [Luke Mansoor's #732](https://github.com/CrocSwap/integer-mult-bounds/pull/732). +This is a written proof with a named literature dependency, not a formal +proof-assistant development or an inference from the finite tests alone. + +## Reproduce + +Python 3.10 or newer, standard library only, from the repository root: + +```sh +python3 -I -B research/paired-star-rank/verify.py +``` + +Expected: + +```text +PASS_PAIRED_STAR_RANK: 36 cases; (8,4) ranks 447/448; all-target deletions 673/672; 8 negative controls; kappa unchanged +``` + +The verifier reads committed certificates and recomputes the entire expected +[RESULTS.json](RESULTS.json). It makes no network requests and imports neither +the builder nor predecessor code. All checks remain active under `-O`. + +```sh +python3 -I -O -B research/paired-star-rank/test_verify.py +python3 -I -B research/paired-star-rank/build_certificate.py --output /tmp/paired-star-certificate.json +``` + +The first command runs four end-to-end test groups. The second independently +regenerates the certificate; its bytes must equal `CERTIFICATE.json`. JSON +output uses UTF-8 and LF on every platform. On Windows use `python` if +`python3` is not installed, and select an existing output directory. + +## What the certificate proves + +The builder enumerates compatible coordinate subsets using combinations and +sign tuples, then performs packed binary elimination. The verifier instead +enumerates ternary words and checks literal identities without elimination: + +1. Each listed decoder is an XOR of allowed stars. On the information set + its coefficients equal one distinct unit vector, proving independence + and simultaneously providing every retained target's scalar decoder. +2. Every allowed star is reconstructed from those decoders on **every + original port**, including deleted ports. This proves the upper rank bound. + Checking only the retained coordinates would not suffice. +3. The attained rank is compared with the uniform formula. Complete port, + star and case inventories are checked before accepting any identities. + +Each case records its coordinate masks, information set and decoder coefficient +masks. Bit i of a decoder's hexadecimal coefficient mask selects entry i of +that case's `stars` list. Coordinate bit `2q+e` denotes sign e in pair q. + +Completed finite coverage is all `k=2,3,4`, `p=k,...,8`, with and without the +empty star: **36 cases, 2,706 literal unit decoders and 3,714 full-family +star-spanning identities**. All permitted corrections for every target are +checked for k=2 and k=4. The k=3 cases are positive escape controls: permitted +corrections can isolate every target without puncturing, so the all-target +star-plus-correction theorem must retain its power-of-two hypothesis. + +Eight built-in negative controls cover a decoder coefficient, duplicate +retained port, forbidden singleton star, illicit global sum, altered source, +missing decoder, false upper rank and missing case. The false-rank control +retains valid remaining unit decoders and must specifically fail the +full-family spanning check. The separate harness checks corrupted receipts +and source pins, preserves previous output on failure, and replays from an +unrelated directory with only the declared input files. + +## Scope and provenance + +The experiment changes the algebraic requirement from decoding one chosen +target to decoding **every surviving target simultaneously**. It does not +reopen the closed one-target search with more computation. The threshold +allows arbitrarily many proper stars and unrestricted target-specific binary +coefficients. Stars and overlap corrections are recomputed on the retained +family. Higher-degree stars, source-dependent overlap coefficients, nonlinear +decoders and other geometries are outside scope. + +With more than 447 retained `(8,4)` ports, even free helpers, source access, +frame changes, copies, restoration and recursive children cannot fix the +nonempty-star identity obstruction. At or below that threshold, the attaining +identities are **not** a paid construction. Preparation, dirty restoration, +legal frames, recursive costs and assembly constraints remain unpriced; no +new supplier, stage schedule or exponent follows. + +Exact base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. +[SOURCES.json](SOURCES.json) pins predecessor heads and inspected source bytes. +The accepted conditional saving stays +`51110230636731/31250000000000000 = 0.001635527380375392`, from +#626/#622/#623 with #543. Inherited compiler, common-label/mutable-basis, +hosting, precision, tape, recovery and analytic obligations remain. + +[CLAIM.json](CLAIM.json) records OS-random assignment seed **46**, three +candidate tasks and the bounded stopping rule. The seed selected +`p=6+(46 mod 4)=8`, k=4. No pinned coordination issue existed, so the claim +was local. This package is additive; selected construction and `upstream/` +are unchanged. [VALIDATION.md](VALIDATION.md) records actual checks. + +Prepared by **ellipsi2 (@ellipsi2)** with substantial **OpenAI Codex assistance** +in research, proof development, implementation, verification and writing. +One separate AI worker reviewed the uniform proof within the claimed scope. +No independent human review is claimed. See [NOTICE.md](NOTICE.md) for +predecessor and literature credit. New work is Apache-2.0. diff --git a/research/paired-star-rank/RESULTS.json b/research/paired-star-rank/RESULTS.json new file mode 100644 index 0000000000..b6fc256c32 --- /dev/null +++ b/research/paired-star-rank/RESULTS.json @@ -0,0 +1,504 @@ +{ + "status": "PASS_PAIRED_STAR_RANK", + "cases": [ + { + "p": 2, + "k": 2, + "include_empty": false, + "ports": 4, + "stars": 4, + "rank": 3, + "all_star_identities": 4, + "unit_decoders": 3, + "minimum_deletions_for_star_only_all_target": 1 + }, + { + "p": 2, + "k": 2, + "include_empty": true, + "ports": 4, + "stars": 5, + "rank": 3, + "all_star_identities": 5, + "unit_decoders": 3, + "minimum_deletions_for_star_only_all_target": 1 + }, + { + "p": 3, + "k": 2, + "include_empty": false, + "ports": 12, + "stars": 6, + "rank": 5, + "all_star_identities": 6, + "unit_decoders": 5, + "minimum_deletions_for_star_only_all_target": 7 + }, + { + "p": 3, + "k": 2, + "include_empty": true, + "ports": 12, + "stars": 7, + "rank": 6, + "all_star_identities": 7, + "unit_decoders": 6, + "minimum_deletions_for_star_only_all_target": 6 + }, + { + "p": 4, + "k": 2, + "include_empty": false, + "ports": 24, + "stars": 8, + "rank": 7, + "all_star_identities": 8, + "unit_decoders": 7, + "minimum_deletions_for_star_only_all_target": 17 + }, + { + "p": 4, + "k": 2, + "include_empty": true, + "ports": 24, + "stars": 9, + "rank": 8, + "all_star_identities": 9, + "unit_decoders": 8, + "minimum_deletions_for_star_only_all_target": 16 + }, + { + "p": 5, + "k": 2, + "include_empty": false, + "ports": 40, + "stars": 10, + "rank": 9, + "all_star_identities": 10, + "unit_decoders": 9, + "minimum_deletions_for_star_only_all_target": 31 + }, + { + "p": 5, + "k": 2, + "include_empty": true, + "ports": 40, + "stars": 11, + "rank": 10, + "all_star_identities": 11, + "unit_decoders": 10, + "minimum_deletions_for_star_only_all_target": 30 + }, + { + "p": 6, + "k": 2, + "include_empty": false, + "ports": 60, + "stars": 12, + "rank": 11, + "all_star_identities": 12, + "unit_decoders": 11, + "minimum_deletions_for_star_only_all_target": 49 + }, + { + "p": 6, + "k": 2, + "include_empty": true, + "ports": 60, + "stars": 13, + "rank": 12, + "all_star_identities": 13, + "unit_decoders": 12, + "minimum_deletions_for_star_only_all_target": 48 + }, + { + "p": 7, + "k": 2, + "include_empty": false, + "ports": 84, + "stars": 14, + "rank": 13, + "all_star_identities": 14, + "unit_decoders": 13, + "minimum_deletions_for_star_only_all_target": 71 + }, + { + "p": 7, + "k": 2, + "include_empty": true, + "ports": 84, + "stars": 15, + "rank": 14, + "all_star_identities": 15, + "unit_decoders": 14, + "minimum_deletions_for_star_only_all_target": 70 + }, + { + "p": 8, + "k": 2, + "include_empty": false, + "ports": 112, + "stars": 16, + "rank": 15, + "all_star_identities": 16, + "unit_decoders": 15, + "minimum_deletions_for_star_only_all_target": 97 + }, + { + "p": 8, + "k": 2, + "include_empty": true, + "ports": 112, + "stars": 17, + "rank": 16, + "all_star_identities": 17, + "unit_decoders": 16, + "minimum_deletions_for_star_only_all_target": 96 + }, + { + "p": 3, + "k": 3, + "include_empty": false, + "ports": 8, + "stars": 18, + "rank": 7, + "all_star_identities": 18, + "unit_decoders": 7, + "minimum_deletions_for_star_only_all_target": 1 + }, + { + "p": 3, + "k": 3, + "include_empty": true, + "ports": 8, + "stars": 19, + "rank": 7, + "all_star_identities": 19, + "unit_decoders": 7, + "minimum_deletions_for_star_only_all_target": 1 + }, + { + "p": 4, + "k": 3, + "include_empty": false, + "ports": 32, + "stars": 32, + "rank": 22, + "all_star_identities": 32, + "unit_decoders": 22, + "minimum_deletions_for_star_only_all_target": 10 + }, + { + "p": 4, + "k": 3, + "include_empty": true, + "ports": 32, + "stars": 33, + "rank": 22, + "all_star_identities": 33, + "unit_decoders": 22, + "minimum_deletions_for_star_only_all_target": 10 + }, + { + "p": 5, + "k": 3, + "include_empty": false, + "ports": 80, + "stars": 50, + "rank": 40, + "all_star_identities": 50, + "unit_decoders": 40, + "minimum_deletions_for_star_only_all_target": 40 + }, + { + "p": 5, + "k": 3, + "include_empty": true, + "ports": 80, + "stars": 51, + "rank": 40, + "all_star_identities": 51, + "unit_decoders": 40, + "minimum_deletions_for_star_only_all_target": 40 + }, + { + "p": 6, + "k": 3, + "include_empty": false, + "ports": 160, + "stars": 72, + "rank": 60, + "all_star_identities": 72, + "unit_decoders": 60, + "minimum_deletions_for_star_only_all_target": 100 + }, + { + "p": 6, + "k": 3, + "include_empty": true, + "ports": 160, + "stars": 73, + "rank": 60, + "all_star_identities": 73, + "unit_decoders": 60, + "minimum_deletions_for_star_only_all_target": 100 + }, + { + "p": 7, + "k": 3, + "include_empty": false, + "ports": 280, + "stars": 98, + "rank": 84, + "all_star_identities": 98, + "unit_decoders": 84, + "minimum_deletions_for_star_only_all_target": 196 + }, + { + "p": 7, + "k": 3, + "include_empty": true, + "ports": 280, + "stars": 99, + "rank": 84, + "all_star_identities": 99, + "unit_decoders": 84, + "minimum_deletions_for_star_only_all_target": 196 + }, + { + "p": 8, + "k": 3, + "include_empty": false, + "ports": 448, + "stars": 128, + "rank": 112, + "all_star_identities": 128, + "unit_decoders": 112, + "minimum_deletions_for_star_only_all_target": 336 + }, + { + "p": 8, + "k": 3, + "include_empty": true, + "ports": 448, + "stars": 129, + "rank": 112, + "all_star_identities": 129, + "unit_decoders": 112, + "minimum_deletions_for_star_only_all_target": 336 + }, + { + "p": 4, + "k": 4, + "include_empty": false, + "ports": 16, + "stars": 64, + "rank": 15, + "all_star_identities": 64, + "unit_decoders": 15, + "minimum_deletions_for_star_only_all_target": 1 + }, + { + "p": 4, + "k": 4, + "include_empty": true, + "ports": 16, + "stars": 65, + "rank": 15, + "all_star_identities": 65, + "unit_decoders": 15, + "minimum_deletions_for_star_only_all_target": 1 + }, + { + "p": 5, + "k": 4, + "include_empty": false, + "ports": 80, + "stars": 130, + "rank": 65, + "all_star_identities": 130, + "unit_decoders": 65, + "minimum_deletions_for_star_only_all_target": 15 + }, + { + "p": 5, + "k": 4, + "include_empty": true, + "ports": 80, + "stars": 131, + "rank": 65, + "all_star_identities": 131, + "unit_decoders": 65, + "minimum_deletions_for_star_only_all_target": 15 + }, + { + "p": 6, + "k": 4, + "include_empty": false, + "ports": 240, + "stars": 232, + "rank": 155, + "all_star_identities": 232, + "unit_decoders": 155, + "minimum_deletions_for_star_only_all_target": 85 + }, + { + "p": 6, + "k": 4, + "include_empty": true, + "ports": 240, + "stars": 233, + "rank": 155, + "all_star_identities": 233, + "unit_decoders": 155, + "minimum_deletions_for_star_only_all_target": 85 + }, + { + "p": 7, + "k": 4, + "include_empty": false, + "ports": 560, + "stars": 378, + "rank": 279, + "all_star_identities": 378, + "unit_decoders": 279, + "minimum_deletions_for_star_only_all_target": 281 + }, + { + "p": 7, + "k": 4, + "include_empty": true, + "ports": 560, + "stars": 379, + "rank": 280, + "all_star_identities": 379, + "unit_decoders": 280, + "minimum_deletions_for_star_only_all_target": 280 + }, + { + "p": 8, + "k": 4, + "include_empty": false, + "ports": 1120, + "stars": 576, + "rank": 447, + "all_star_identities": 576, + "unit_decoders": 447, + "minimum_deletions_for_star_only_all_target": 673 + }, + { + "p": 8, + "k": 4, + "include_empty": true, + "ports": 1120, + "stars": 577, + "rank": 448, + "all_star_identities": 577, + "unit_decoders": 448, + "minimum_deletions_for_star_only_all_target": 672 + } + ], + "corrections_and_escape_controls": [ + { + "p": 2, + "k": 2, + "target_correction_checks": 4 + }, + { + "p": 3, + "k": 2, + "target_correction_checks": 12 + }, + { + "p": 4, + "k": 2, + "target_correction_checks": 24 + }, + { + "p": 5, + "k": 2, + "target_correction_checks": 40 + }, + { + "p": 6, + "k": 2, + "target_correction_checks": 60 + }, + { + "p": 7, + "k": 2, + "target_correction_checks": 84 + }, + { + "p": 8, + "k": 2, + "target_correction_checks": 112 + }, + { + "p": 3, + "k": 3, + "target_correction_checks": 8 + }, + { + "p": 4, + "k": 3, + "target_correction_checks": 32 + }, + { + "p": 5, + "k": 3, + "target_correction_checks": 80 + }, + { + "p": 6, + "k": 3, + "target_correction_checks": 160 + }, + { + "p": 7, + "k": 3, + "target_correction_checks": 280 + }, + { + "p": 8, + "k": 3, + "target_correction_checks": 448 + }, + { + "p": 4, + "k": 4, + "target_correction_checks": 48 + }, + { + "p": 5, + "k": 4, + "target_correction_checks": 240 + }, + { + "p": 6, + "k": 4, + "target_correction_checks": 720 + }, + { + "p": 7, + "k": 4, + "target_correction_checks": 1680 + }, + { + "p": 8, + "k": 4, + "target_correction_checks": 3360 + } + ], + "negative_controls": [ + "wrong decoder coefficient", + "duplicate retained port", + "degree-k singleton supplied as proper star", + "unpaid global sum in nonempty class", + "changed source coordinate", + "missing decoder", + "false rank with valid remaining decoders", + "missing case" + ], + "kappa": "unchanged" +} diff --git a/research/paired-star-rank/SOURCES.json b/research/paired-star-rank/SOURCES.json new file mode 100644 index 0000000000..af80d6862e --- /dev/null +++ b/research/paired-star-rank/SOURCES.json @@ -0,0 +1,47 @@ +{ + "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60", + "dependency_policy": "No predecessor executable code imported; primary proof theorem and contextual research credited.", + "local_context_sha256": { + "certificates/selected-result.json": "b3bf402f31a69bde0629936a9b569bed48281568d7b16d5fba120b5311ea1f53", + "docs/research/community-k492-review.md": "3b731d30e80010b15408a289b0b32192700885474c8b36d48ee3320a7a8c5c47", + "research/hypercube-decoder-parity/README.md": "2dc89289d1d979aec6709b8c55a2af7cd39d97297bc20fb94adc7d4cdf867cd1" + }, + "predecessors": [ + { + "pr": 231, + "author": "Joel Pulikkan / GamingPuzzled", + "head": "3c11e696a2fb2c454585d72710de27cd6bd3be68", + "role": "Symmetric paired-cube star/overlap obstruction; retained source on base." + }, + { + "pr": 669, + "author": "Sam Sung / the-missionary-company", + "head": "308f1e6ec3bb0b6ec0584cf7bfe219addc219b1e", + "readme_path": "research/asymmetric-hypercube-decoder/README.md", + "readme_sha256": "69b624112d6294f519682fdf762d7f670063ea3f2b05ec0f25ce26e93c481c17", + "role": "Global unpunctured and target-local puncture obstruction" + }, + { + "pr": 732, + "author": "Luke Mansoor / lukemansoor", + "head": "80431a74c31e807581f5c2258c87abe2084caacd", + "readme_path": "research/global-star-puncture/README.md", + "readme_sha256": "87825a4dd666d8c32df025fe838aca4cdc23a5d3965f79c9aca4913567dba6d4", + "role": "General one-target distance and correction absorption; not claimed as new" + }, + { + "pr": 747, + "author": "Alishan Ladhani / aladh", + "head": "aeaf0899179a14644b412e838ede14db967c765e", + "readme_path": "research/global-star-punctures/README.md", + "readme_sha256": "ab6cf6e7885f31d7601b6d686e1a543ceaf887e29207c3af08955e457a9656da", + "role": "Finite (6,4) rank155 and all-target threshold85; corroborated, not claimed as new" + } + ], + "mathematical_source": { + "url": "https://arxiv.org/pdf/0709.3144v1", + "version": "v1", + "used": "Theorem 1 and equation (8), unsigned ballot inclusion basis" + }, + "accepted_kappa": "51110230636731/31250000000000000" +} diff --git a/research/paired-star-rank/VALIDATION.md b/research/paired-star-rank/VALIDATION.md new file mode 100644 index 0000000000..6e74ed83dd --- /dev/null +++ b/research/paired-star-rank/VALIDATION.md @@ -0,0 +1,63 @@ +# Validation receipt + +Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. +Prepared 2026-10-13 by ellipsi2 with substantial OpenAI Codex assistance. + +The completed result is a written uniform rank theorem plus a finite exact +certificate. It does not certify a new paid supplier or the multiplication +theorem. The literature dependency is identified in RANK-PROOF.md. + +## Executed checks + +| Environment / check | Result | +| --- | --- | +| Windows, CPython 3.11.15, `python -I -B research/paired-star-rank/verify.py` | PASS, complete receipt matched | +| Windows, CPython 3.11.15, `python -I -O -B research/paired-star-rank/verify.py` | PASS, all assertions-independent checks active | +| Windows, CPython 3.11.15, `python -I -O -B research/paired-star-rank/test_verify.py` | Four test groups passed | +| Ubuntu under WSL, CPython 3.12.3, isolated `-I -O -B` verifier | PASS, same committed receipt | +| Ubuntu under WSL, CPython 3.12.3, isolated `-I -O -B` harness | Four test groups passed | +| Deterministic certificate regeneration in both harness runs | Byte-identical UTF-8/LF certificate | +| Relocated replay with only declared package/input files | PASS, no repository or working-directory imports | +| `python -B scripts/verify_selected_result.py` | PASS selected record, exact kappa, original source inventory and proof supplement | +| `git diff --check` and staged whitespace check | PASS | + +The Windows-mounted drives were unavailable to WSL in this session. The Linux +check therefore used an in-memory archive of the same declared files, streamed +to a fresh Linux `/tmp` directory; no Windows-side imported modules or network +dependencies were available there. This was local Linux execution, not GitHub CI. + +The verifier checks all 36 declared cases (`k=2,3,4`, `p=k,...,8`, both +empty-star policies), 2,706 unit-vector decoder identities and 3,714 spanning +identities on the complete original port families. Every target's permitted +overlap corrections is checked in the power-of-two cases; k=3 supplies the +positive escape control. Every invocation runs all eight named rejection +controls and compares the entire deterministic receipt. + +The four harness groups additionally cover byte-for-byte regeneration, +relocated optimized replay, a corrupted certificate, an incomplete receipt, +modified contextual source bytes and modified source pins. Failed runs must +leave a pre-existing export file unchanged and cannot print the success line. +The false-rank internal control must reach the full-family spanning failure; +a mismatch against the theoretical rank formula alone does not count. + +A separate AI worker reviewed the unsigned-basis argument, the sign-sector +decomposition and the no-empty criterion, and inspected the identity verifier. +This is disclosed AI-assisted review, not independent human review or formal +proof-assistant checking. + +## Scope of verification + +The existing supplier generators, selected certificate and `upstream/` were +not edited. The expensive repository-wide `make verify` and selected cold +replay `make frontier626-verify` were not repeated for this additive research +package. No repository-wide or hosted CI pass is claimed. The selected-source +integrity command above checks bindings; it does not cold-replay the suppliers. + +The finite checks establish the explicit certificates. The uniform theorem +uses the written argument and cited integral inclusion-basis theorem; it is +not established by extrapolating from the tested domain. + +The pre-submission overlap refresh covered **586 open PRs through #767**. +The relevant #669, #732 and #747 heads still matched SOURCES.json. #747's +`(6,4)` all-target result is credited; the new contribution is the uniform +rank formula, exact empty-star boundary and literal `(8,4)` information sets. diff --git a/research/paired-star-rank/build_certificate.py b/research/paired-star-rank/build_certificate.py new file mode 100644 index 0000000000..6adb6ab080 --- /dev/null +++ b/research/paired-star-rank/build_certificate.py @@ -0,0 +1,74 @@ +#!/usr/bin/env python3 +"""Build literal signed-star information-set certificates (Apache-2.0). + +Prepared for ellipsi2 with substantial OpenAI Codex assistance. +The separate verifier does not import this module or its elimination routine. +""" + +import argparse +import itertools +import json +from pathlib import Path + + +def signed_sets(p, degree): + for pairs in itertools.combinations(range(p), degree): + for signs in itertools.product((0, 1), repeat=degree): + yield sum(1 << (2 * pair + sign) for pair, sign in zip(pairs, signs)) + + +def build(p, k, include_empty): + ports = list(signed_sets(p, k)) + stars = [s for d in range(0 if include_empty else 1, k) + for s in signed_sets(p, d)] + basis = {} + for index, star in enumerate(stars): + row = sum(1 << col for col, port in enumerate(ports) if port & star == star) + coefficients = 1 << index + while row: + pivot = (row & -row).bit_length() - 1 + if pivot not in basis: + basis[pivot] = (row, coefficients) + break + other, combination = basis[pivot] + row ^= other + coefficients ^= combination + pivots = sorted(basis) + for pivot in reversed(pivots): + row, coefficients = basis[pivot] + for earlier in pivots: + if earlier >= pivot: + break + previous, combination = basis[earlier] + if previous >> pivot & 1: + basis[earlier] = (previous ^ row, combination ^ coefficients) + return { + "p": p, + "k": k, + "include_empty": include_empty, + "ports": ports, + "stars": stars, + "information_set": [ports[pivot] for pivot in pivots], + "decoders_hex": [hex(basis[pivot][1]) for pivot in pivots], + } + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--output", type=Path, required=True) + args = parser.parse_args() + domains = [(p, k) for k in (2, 3, 4) for p in range(k, 9)] + payload = { + "format": "signed-star-information-sets-v1", + "assignment_seed": 46, + "principal_domain": [8, 4], + "cases": [build(p, k, empty) for p, k in domains for empty in (False, True)], + } + args.output.write_bytes((json.dumps(payload, indent=2) + "\n").encode("utf-8")) + for case in payload["cases"]: + print(f"p={case['p']} k={case['k']} empty={case['include_empty']} " + f"ports={len(case['ports'])} rank={len(case['information_set'])}") + + +if __name__ == "__main__": + main() diff --git a/research/paired-star-rank/test_verify.py b/research/paired-star-rank/test_verify.py new file mode 100644 index 0000000000..04d6c5caf0 --- /dev/null +++ b/research/paired-star-rank/test_verify.py @@ -0,0 +1,91 @@ +#!/usr/bin/env python3 +"""End-to-end failure detection and portable replay checks (Apache-2.0).""" + +import json +from pathlib import Path +import shutil +import subprocess +import sys +import tempfile +import unittest + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] + + +class ReplayTests(unittest.TestCase): + def setUp(self): + self.workspace = tempfile.TemporaryDirectory(prefix="paired-star-rank-") + self.addCleanup(self.workspace.cleanup) + self.root = Path(self.workspace.name) + self.package = self.root / "research" / "paired-star-rank" + self.package.mkdir(parents=True) + for name in ("verify.py", "build_certificate.py", "SOURCES.json", + "CERTIFICATE.json", "RESULTS.json"): + shutil.copyfile(HERE / name, self.package / name) + self.sources = json.loads((HERE / "SOURCES.json").read_text(encoding="utf-8")) + for name in self.sources["local_context_sha256"]: + dest = self.root / name + dest.parent.mkdir(parents=True, exist_ok=True) + shutil.copyfile(ROOT / name, dest) + + def run_check(self, *args): + return subprocess.run( + [sys.executable, "-I", "-O", "-B", str(self.package / "verify.py"), *args], + cwd=self.workspace.name, capture_output=True, text=True, check=False, + ) + + def assert_rejected(self, result, output): + self.assertNotEqual(result.returncode, 0) + self.assertNotIn("PASS_PAIRED_STAR_RANK:", result.stdout) + self.assertEqual(output.read_bytes(), b"previous evidence\n") + + def export_target(self): + output = self.root / "export.json" + output.write_bytes(b"previous evidence\n") + return output + + def test_relocated_optimized_replay_and_regeneration(self): + exported = self.root / "export.json" + result = self.run_check("--write", str(exported)) + self.assertEqual(result.returncode, 0, result.stderr) + self.assertEqual(exported.read_bytes(), (HERE / "RESULTS.json").read_bytes()) + regenerated = self.root / "rebuilt.json" + build = subprocess.run( + [sys.executable, "-I", "-B", str(self.package / "build_certificate.py"), + "--output", str(regenerated)], capture_output=True, text=True, check=False, + ) + self.assertEqual(build.returncode, 0, build.stderr) + self.assertEqual(regenerated.read_bytes(), (HERE / "CERTIFICATE.json").read_bytes()) + + def test_corrupted_certificate_preserves_output(self): + certificate = self.package / "CERTIFICATE.json" + data = json.loads(certificate.read_text(encoding="utf-8")) + data["cases"][-2]["decoders_hex"][0] = "0x0" + certificate.write_text(json.dumps(data), encoding="utf-8") + output = self.export_target() + self.assert_rejected(self.run_check("--write", str(output)), output) + + def test_incomplete_receipt_preserves_output(self): + receipt = self.package / "RESULTS.json" + data = json.loads(receipt.read_text(encoding="utf-8")) + data["cases"].pop() + receipt.write_text(json.dumps(data), encoding="utf-8") + output = self.export_target() + self.assert_rejected(self.run_check("--write", str(output)), output) + + def test_context_source_and_pin_changes_are_rejected(self): + name = next(iter(self.sources["local_context_sha256"])) + source = self.root / name + original = source.read_bytes() + source.write_bytes(original + b"\nchanged\n") + output = self.export_target() + self.assert_rejected(self.run_check("--write", str(output)), output) + source.write_bytes(original) + self.sources["local_context_sha256"][name] = "0" * 64 + (self.package / "SOURCES.json").write_text(json.dumps(self.sources), encoding="utf-8") + self.assert_rejected(self.run_check("--write", str(output)), output) + + +if __name__ == "__main__": + unittest.main(verbosity=2) diff --git a/research/paired-star-rank/verify.py b/research/paired-star-rank/verify.py new file mode 100644 index 0000000000..dca1af4651 --- /dev/null +++ b/research/paired-star-rank/verify.py @@ -0,0 +1,247 @@ +#!/usr/bin/env python3 +"""Check literal star decoders and spanning identities, without elimination. + +Python 3.10+, standard library only. Apache-2.0. +Prepared for ellipsi2 with substantial OpenAI Codex assistance. +""" + +import argparse +import copy +import hashlib +import itertools +import json +from math import comb +from pathlib import Path + +HERE = Path(__file__).resolve().parent +DOMAINS = [(p, k) for k in (2, 3, 4) for p in range(k, 9)] + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def geometry(p, k, include_empty): + """Independent ternary-word enumeration; builder uses combinations/signs.""" + ports, stars = set(), set() + for word in itertools.product((0, 1, 2), repeat=p): + degree = sum(symbol != 0 for symbol in word) + if degree > k: + continue + mask = sum(1 << (2 * i + symbol - 1) + for i, symbol in enumerate(word) if symbol) + if degree == k: + ports.add(mask) + elif degree or include_empty: + stars.add(mask) + return ports, stars + + +def xor_selected(values, selector): + result = 0 + while selector: + bit = selector & -selector + result ^= values[bit.bit_length() - 1] + selector -= bit + return result + + +def rank_formula(p, k, include_empty): + rho = sum(comb(p, j) * comb(p - j, min(k - 1 - j, p - k)) + for j in range(k)) + constant_missing = (k & (k - 1) == 0) and p >= 2 * k - 1 + return rho - int(constant_missing and not include_empty) + + +def check_case(case): + p, k, empty = case["p"], case["k"], case["include_empty"] + require(type(p) is int and type(k) is int and (p, k) in DOMAINS, + "outside declared domain") + require(type(empty) is bool, "empty-star flag must be Boolean") + actual_ports, actual_stars = geometry(p, k, empty) + ports, stars, info = case["ports"], case["stars"], case["information_set"] + for values, expected, name in ((ports, actual_ports, "ports"), + (stars, actual_stars, "stars")): + require(all(type(value) is int for value in values), name + " not integers") + require(len(values) == len(expected) and set(values) == expected, + name + " incomplete, duplicated or invalid") + require(all(type(value) is int for value in info), "invalid information set") + require(len(info) == len(set(info)) and set(info) <= actual_ports, + "information set duplicated or invalid") + encoded = case["decoders_hex"] + require(len(encoded) == len(info), "decoder count differs") + coefficients = [] + for value in encoded: + require(type(value) is str, "decoder encoding must be a string") + decoded = int(value, 16) + require(decoded >= 0 and decoded < 1 << len(stars) and hex(decoded) == value, + "noncanonical or out-of-range decoder") + coefficients.append(decoded) + + position = {port: i for i, port in enumerate(ports)} + information_positions = [position[port] for port in info] + information_mask = sum(1 << i for i in information_positions) + rows = [sum(1 << col for col, port in enumerate(ports) if port & star == star) + for star in stars] + decoders = [xor_selected(rows, c) for c in coefficients] + + # Every decoder is an explicit available-star combination; its restriction + # to the information set is one particular unit vector. This proves the + # lower rank bound and gives all attaining scalar decoders. + for col, row in zip(information_positions, decoders): + require(row & information_mask == 1 << col, + "decoder is not its claimed unit vector") + + # The upper rank bound is a literal identity on ALL original ports for + # EVERY allowed generator, not just a reported Gaussian-elimination rank. + for star, row in zip(stars, rows): + expansion = 0 + for port, decoder in zip(info, decoders): + if port & star == star: + expansion ^= decoder + require(expansion == row, "star not spanned on the full port family") + + rank = len(info) + require(rank == rank_formula(p, k, empty), "uniform rank formula differs") + return { + "p": p, "k": k, "include_empty": empty, + "ports": len(ports), "stars": len(stars), "rank": rank, + "all_star_identities": len(stars), "unit_decoders": rank, + "minimum_deletions_for_star_only_all_target": len(ports) - rank, + } + + +def check_corrections(p, k): + ports, _ = geometry(p, k, False) + ports = sorted(ports) + count = 0 + for target in ports: + # Binomial polynomials of positive degree, indexed by overlap r. + tables = [[comb(r, degree) & 1 for r in range(k + 1)] + for degree in range(1, k)] + if k & (k - 1) == 0: + for overlap in range(1, k): + remaining = [int(r == overlap) for r in range(k + 1)] + chosen = [] + for degree in range(1, k): + if remaining[degree]: + chosen.append(degree) + remaining = [a ^ b for a, b in zip(remaining, tables[degree - 1])] + require(not any(remaining), "overlap correction has forbidden endpoint") + # Verify every actual source, including disjoint and target. + for source in ports: + r = (source & target).bit_count() + require(sum(comb(r, d) for d in chosen) % 2 == (r == overlap), + "correction absorption failed") + count += 1 + elif k == 3: + # A positive escape control: corrections CAN add target singletons. + for source in ports: + r = (source & target).bit_count() + require(((r & 1) ^ int(r == 1)) == int(source == target), + "non-power-of-two escape failed") + count += 1 + return count + + +def check_document(document): + require(document["format"] == "signed-star-information-sets-v1", "format differs") + require(document["assignment_seed"] == 46 and document["principal_domain"] == [8, 4], + "assignment differs") + cases = document["cases"] + require([(c["p"], c["k"], c["include_empty"]) for c in cases] + == [(p, k, empty) for p, k in DOMAINS for empty in (False, True)], + "case coverage differs") + return [check_case(case) for case in cases] + + +def negative_controls(document): + principal = next(c for c in document["cases"] + if (c["p"], c["k"], c["include_empty"]) == (8, 4, False)) + mutants = [] + case = copy.deepcopy(principal) + case["decoders_hex"][0] = hex(int(case["decoders_hex"][0], 16) ^ 1) + mutants.append(("wrong decoder coefficient", case)) + case = copy.deepcopy(principal) + case["information_set"][0] = case["information_set"][1] + mutants.append(("duplicate retained port", case)) + case = copy.deepcopy(principal) + case["stars"][0] = case["ports"][0] + mutants.append(("degree-k singleton supplied as proper star", case)) + case = copy.deepcopy(principal) + case["stars"][0] = 0 + mutants.append(("unpaid global sum in nonempty class", case)) + case = copy.deepcopy(principal) + case["ports"][0] ^= 1 + mutants.append(("changed source coordinate", case)) + case = copy.deepcopy(principal) + case["decoders_hex"].pop() + mutants.append(("missing decoder", case)) + # Removing an information coordinate AND its decoder preserves every + # remaining unit-vector equation. Only the full-family spanning obligation + # detects this false upper-rank certificate. + case = copy.deepcopy(principal) + case["information_set"].pop() + case["decoders_hex"].pop() + mutants.append(("false rank with valid remaining decoders", case)) + for name, case in mutants: + try: + check_case(case) + except ValueError as error: + if name == "false rank with valid remaining decoders": + require(str(error) == "star not spanned on the full port family", + "false-rank control did not reach the spanning obligation") + continue + raise ValueError("negative control accepted: " + name) + incomplete = copy.deepcopy(document) + incomplete["cases"].pop() + try: + check_document(incomplete) + except ValueError: + pass + else: + raise ValueError("negative control accepted: missing case") + return [name for name, _ in mutants] + ["missing case"] + + +def check_sources(): + sources = json.loads((HERE / "SOURCES.json").read_text(encoding="utf-8")) + root = HERE.parents[1] + for name, expected in sources["local_context_sha256"].items(): + candidate = (root / name).resolve() + require(candidate.is_relative_to(root.resolve()), "source escapes root") + require(hashlib.sha256(candidate.read_bytes()).hexdigest() == expected, + "context source changed: " + name) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--certificate", type=Path, default=HERE / "CERTIFICATE.json") + parser.add_argument("--write", type=Path, help="export only after all checks pass") + args = parser.parse_args() + check_sources() + document = json.loads(args.certificate.read_text(encoding="utf-8")) + cases = check_document(document) + controls = negative_controls(document) + correction_checks = [ + {"p": p, "k": k, "target_correction_checks": check_corrections(p, k)} + for p, k in DOMAINS + ] + result = { + "status": "PASS_PAIRED_STAR_RANK", + "cases": cases, + "corrections_and_escape_controls": correction_checks, + "negative_controls": controls, + "kappa": "unchanged", + } + expected = json.loads((HERE / "RESULTS.json").read_text(encoding="utf-8")) + require(result == expected, "complete deterministic receipt differs") + if args.write: + args.write.write_bytes((json.dumps(result, indent=2) + "\n").encode("utf-8")) + print("PASS_PAIRED_STAR_RANK: 36 cases; (8,4) ranks 447/448; " + "all-target deletions 673/672; 8 negative controls; kappa unchanged") + + +if __name__ == "__main__": + main() From 02b59e553649c333072b5dd4dfb97e43942efa14 Mon Sep 17 00:00:00 2001 From: ellipsi2 Date: Tue, 13 Oct 2026 14:25:44 +0900 Subject: [PATCH 2/2] docs(research): record rank result and sprint handoff --- research/paired-star-rank/CLAIM.json | 10 +++++++++- 1 file changed, 9 insertions(+), 1 deletion(-) diff --git a/research/paired-star-rank/CLAIM.json b/research/paired-star-rank/CLAIM.json index d5809d0194..9b4de617b8 100644 --- a/research/paired-star-rank/CLAIM.json +++ b/research/paired-star-rank/CLAIM.json @@ -21,5 +21,13 @@ "expected_deliverable": "One additive research package: proof, independent exact certificate/replay, failure controls, source pins and attribution; no new kappa.", "overlap_checked": [669, 703, 715, 717, 725, 726, 732, 738, 747, 748, 749, 750], "distinct_contribution": "PR747 supplies the 240-port (6,4) all-target threshold. This claim targets a uniform rank mechanism and a separate 1120-port certificate, explicitly crediting that precursor.", - "status": "in progress" + "status": "complete", + "result_updated_utc": "2026-10-13T05:25:34Z", + "result": "Uniform signed proper-star rank formula and exact empty-star criterion; for (8,4), nonempty/empty ranks 447/448 and sharp algebraic all-target deletion thresholds 673/672. No new kappa or paid supplier.", + "pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/772", + "scientific_result_commit": "6ad0e646bb24e820d4b5483b71d2f7aa4167e4d3", + "completed_coverage": "36 cases: k=2,3,4 and p=k,...,8, both empty-star policies; 2706 literal unit decoders, 3714 full-family spanning identities, all target overlap controls, eight negative controls and four harness groups.", + "verification": "Windows CPython 3.11.15 and local Linux CPython 3.12.3 pass, including isolated optimized replay, byte-identical regeneration and relocated minimal inputs. Selected-record integrity passes. All 11 committed package files matched the verified working bytes before this claim-only update.", + "final_overlap": "Full pre-submission scan: 586 open PRs through #767. Immediate pre-submission refresh through #769 and publication refresh through #772 found no duplicate. #770 concerns Gaussian E8 and #771 finite invoice guards. #669/#732/#747 heads still match the credited pins. Main remains the exact base.", + "remaining_obligations": "No paid preparation, dirty restoration, frame, hosting or recursive construction supplied. Inherited compiler, precision, tape/recovery and analytic assumptions remain. No formal multiplication proof or hosted CI pass claimed." }