Research: uniform proper-star ranks and all-target puncture thresholds

research/paired-star-rank/CLAIM.json

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+{
+  "track": "Structural decoder research",
+  "assignment_seed": 46,
+  "randomness": "System.Security.Cryptography.RandomNumberGenerator.GetInt32(100); recorded before task selection",
+  "base_commit": "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": "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."
+}

research/paired-star-rank/NOTICE.md

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+# 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.

research/paired-star-rank/RANK-PROOF.md

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+# 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<...<b_i}` is ballot if
+`b_l >= 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.