Count every finite-field connector chart repair rank (no new κ)

research/connector-chart-rank-distribution/PROOF.md

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+# The finite-field chart-repair rank enumerator
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+## Statement
+
+Let F be a finite field of order q. Let B0,B1:V -> W be linear maps on the
+same finite-dimensional source, with dim W=n. Write
+
+```text
+a = rank B0,   b = rank B1,   c = dim(row B0 intersect row B1).
+```
+
+For 0<=r<=n, let N_r count the charts R in GL(W) for which
+rank(B1-R B0)=r. The row spaces are subspaces of V*. The source dimension
+can be arbitrary, subject to a+b-c<=dim V. The answer depends only on
+(q,n,a,b,c), not on the matrices' remaining entries or dim V.
+
+Use the following conventions:
+
+```text
+G_q(d,k) = Gaussian binomial [d choose k]_q, or 0 if k<0 or k>d;
+J_q(d,k) = product_(i=0..k-1) (q^d-q^i), or 0 if k<0 or k>d;
+J_q(d,0) = 1;
+gamma = J_q(n,n)/J_q(n,a).
+```
+
+For 0<=t<=n define
+
+```text
+F_t = sum_(h=0..c, 0<=j<=n-b)
+      G_q(c,h) G_q(n-b,j) q^((c-h)j+t(c-h)) J_q(t,a-c+h),
+      where j=t-b+c-h.
+```
+
+Define D_0=F_0 and successively
+
+```text
+D_t = F_t - sum_(r=0..t-1) G_q(n-r,t-r) D_r.
+```
+
+Then the complete chart counts are N_r=gamma D_r. In particular, if
+s=min(a,b) and u=max(a,b), the exact fraction attaining the minimum repair
+rank u-c is
+
+```text
+N_(u-c)/|GL_n(F)| = q^(c(u-c)) J_q(u-c,s-c) / J_q(n,s).       (1)
+```
+
+The minimum-rank value itself is ywz65's #2680 theorem, which is cited,
+not a new claim. The new result is the complete distribution and its
+minimum-rank density. The proof below also explains each counting factor.
+Empty dimensions and zero maps obey the same formulas.
+
+## Normalize the source coordinates
+
+Choose a basis of the intersection of the two row spaces, extend it
+separately to bases of both row spaces, and then extend their union to V*.
+Let the corresponding source coordinates be shared coordinates of dimension
+c, B0-only coordinates of dimension a-c, B1-only coordinates of dimension
+b-c, and unused coordinates. A basis change in W makes B1's nonzero columns
+on its coordinates e1,...,eb.
+
+As R varies, C=R B0 ranges over exactly the maps whose row space equals
+row B0. Equivalently, its a potentially nonzero columns form an injective
+map from F^a to W. There are J_q(n,a) such maps. Each occurs for exactly
+gamma charts: after fixing the images of a independent vectors, extend their
+images to an ordered basis in product_(i=a..n-1)(q^n-q^i) ways. Thus it
+suffices to count C, and multiply the resulting counts by gamma.
+
+Let E=span(e_(c+1),...,e_b), of dimension v=b-c, and let
+B=span(e1,...,eb). The residual B1-C contains E in its image. Its remaining
+columns are e_i-C_i for the shared coordinates and -C_i for the B0-only
+coordinates.
+
+## Count residual images contained in a t-space
+
+Fix a t-dimensional T<=W. It can contain the residual image only if E<=T.
+Put h=dim(T intersect B)-v and j=t-v-h. For such T, the number of choices is
+
+```text
+G_q(c,h) G_q(n-b,j) q^((c-h)j).                            (2)
+```
+
+To see this, pass to W/E and its distinguished c-dimensional subspace B/E.
+Choose the h-dimensional intersection, choose the j-dimensional image in
+W/B, and choose its lift modulo the intersection. These lifts are the
+linear maps from a fixed j-space to a (c-h)-space, giving q^((c-h)j).
+This standard subspace-intersection count uses no independence assumption
+about random charts.
+
+The condition im(B1-C)<=T prescribes C modulo T: shared coordinates map to
+e_i mod T and B0-only coordinates map to zero. This prescribed quotient map
+has rank c-h and kernel dimension a-c+h. An injective lift C must map that
+kernel injectively into T, giving J_q(t,a-c+h) choices. On a fixed
+complement of the kernel, each of the c-h basis vectors has q^t possible
+lifts, independently. Their images are independent modulo T, so these
+choices introduce no further injectivity condition. There are therefore
+
+```text
+q^(t(c-h)) J_q(t,a-c+h)                                   (3)
+```
+
+injective lifts. Multiplying (2) and (3) and summing h gives F_t. Thus F_t
+counts pairs (C,T) with C injective, dim T=t and im(B1-C)<=T.
+
+## Recover the exact ranks
+
+If im(B1-C) has dimension r, its t-dimensional superspaces are in bijection
+with the (t-r)-subspaces of W/im(B1-C). Consequently
+
+```text
+F_t = sum_(r=0..t) D_r G_q(n-r,t-r),                       (4)
+```
+
+where D_r counts injective C with residual rank r. The diagonal coefficient
+in this triangular system is one, so the displayed recurrence recovers
+all D_r uniquely. The chart multiplicity gives N_r=gamma D_r. This proves
+the full distribution for every finite field, including extension fields.
+
+## Extract the minimum density
+
+First assume a<=b. The residual contains E of dimension b-c. At t=b-c,
+j=-h in the formula for F_t, so only h=0 is possible. Hence
+
+```text
+D_(b-c) = F_(b-c) = q^(c(b-c)) J_q(b-c,a-c).
+```
+
+No lower residual rank can occur because E is contained in every residual
+image. Dividing by J_q(n,a) proves (1) when a<=b. For the other order, the
+bijection R -> R^-1 preserves the residual rank after swapping B0 and B1:
+
+```text
+rank(B1-R B0) = rank(B0-R^-1 B1).
+```
+
+Apply the first case to the swapped maps. This also proves the symmetry of
+the complete chart counts in a and b. There is no assertion that the
+numbers of injective C, before multiplying by gamma, are symmetric.
+
+## Worked exact example
+
+Over F2 take dim W=3, dim V=3 and
+
+```text
+B0 = [[1,0,0], [0,1,0], [0,0,0]],
+B1 = [[1,0,0], [0,0,1], [0,0,0]].
+```
+
+Here a=b=2,c=1. The containment counts are (0,2,30,42). Triangular
+inversion gives D=(0,2,24,16), and gamma=4. Thus among all 168 invertible
+charts the residual-rank histogram is
+
+```text
+rank       0   1   2   3
+charts     0   8  96  64.
+```
+
+Exactly 1/21 of the charts have the minimum rank one. Equal map ranks by
+themselves do not determine this distribution: with c=2 the same n,a,b
+instead give (4,60,104,0). These are counts of charts, not distinct residual
+maps, scalar circuit words, or physically legal native adapters.

research/connector-chart-rank-distribution/README.md

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+# Count every finite-field connector chart repair rank
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+Claim type: scoped algebraic proof with finite corroboration. For fixed maps
+B0,B1:V -> F_q^n, this package gives the complete histogram of
+rank(B1-R B0) over R in GL_n(F_q). It depends only on the two ranks a,b,
+their row-space intersection dimension c, n and q. A Gaussian-binomial
+containment count followed by triangular inversion gives every coefficient.
+[PROOF.md](PROOF.md) proves the formula over every finite field.
+
+If s=min(a,b), u=max(a,b), and
+J_q(d,k)=product_(i=0..k-1)(q^d-q^i), the exact fraction of charts attaining
+the minimum rank u-c is
+
+```text
+q^(c(u-c)) J_q(u-c,s-c) / J_q(n,s).
+```
+
+The minimum rank is the cited result of ywz65's
+[PR #2680](https://github.com/CrocSwap/integer-mult-bounds/pull/2680).
+The new contribution counts the entire distribution and the density of its
+minimizers. No predecessor executable or certificate is imported or replayed.
+
+For q=2,n=3,a=b=2,c=1, the 168 charts have repair-rank histogram
+`(0,8,96,64)` at ranks zero through three. Thus exactly 1/21 attain rank one.
+With c=2 instead, the histogram is `(4,60,104,0)`. This distinguishes chart
+multiplicity from the minimum-rank value and from distinct residual maps.
+
+## Evidence and reproduction
+
+The verifier computes the counting formula using integer arithmetic and
+compares it with direct matrix subtraction, multiplication and elimination.
+Its complete finite domains are:
+
+- 110 canonical parameter cases over F2,F3,F4,F5, with every codomain
+  dimension through three for F2 and through two for the other fields.
+  Every feasible a,b,c occurs, and every invertible chart is enumerated.
+  These cases make 15,090 individual chart comparisons.
+- All 4,096 pairs of binary two-by-three maps and all 6,561 pairs of ternary
+  two-by-two maps. The combined 10,657 arbitrary pairs make 339,504 chart
+  comparisons. The formula must give every histogram entry, rather than
+  only its first nonzero entry. These domains include some canonical cases.
+- Six invalid-parameter controls and four mathematical controls cover
+  omitted inversion, omitted chart multiplicity, F4 arithmetic and the
+  exact minimum-density fixture. F4 is F2[t]/(t^2+t+1), not Z/4.
+
+From the repository root, using CPython 3.12 and only the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/connector-chart-rank-distribution/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/connector-chart-rank-distribution/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/connector-chart-rank-distribution
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/connector-chart-rank-distribution --check-source-history
+```
+
+The direct marker is `PASS_CONNECTOR_CHART_RANK_DISTRIBUTION`.
+Registered commands have separate 120-second limits. Another CPython 3.12
+path can replace the local path. Both modes use explicit exceptions, and
+both are read-only. Each stdout receipt binds all six package files with
+SHA256. No receipt is handwritten or stored in the package. Provenance
+pins authenticate historical source bytes, not predecessor mathematics.
+
+## Scope and code constraints
+
+The universal claim is the written counting proof. Finite execution
+corroborates its declared domains; independent human and formal proof-kernel
+review are unclaimed. The computational formula accepts prime powers
+2<=q<=256 and 0<=n<=12. The written theorem has no such upper bounds.
+
+The chart distribution is over the full general linear group. A restricted
+native alphabet, chart synthesis cost, scalar work, physical producer,
+restoration, storage, precision and a complete supplier invoice are outside
+scope. These costs remain unknown. No particular selected caller is claimed
+to need this statistic, and no downstream construction campaign follows.
+No new kappa, selected-frontier change or complete multiplication theorem
+is claimed. Selected scientific cold replay, Lean and Docker were not run.
+
+All functions are below 60 physical lines, use direct calls and have no
+recursion, generated code or external processes. Formula loops have at most
+13 iterations per dimension. Field-order validation tries at most 255
+divisors and eight divisions per selected divisor. Oracle matrices have at
+most six rows/columns, enumerate at most 625 candidate charts at a time,
+and retain at most 81 maps in the arbitrary-pair scan. The largest pair
+scan has 6,561 iterations, each with at most 48 charts. Invalid field orders,
+rank parameters and dimension overflow raise explicit errors. Warnings are
+errors in both registered commands. No external static analyzer is required
+or claimed.
+
+Power of Ten rule 3 deviation: CPython and its standard library allocate
+integers, tuples and elimination rows dynamically. The enforced formula
+capacity and fixed oracle domains above bound live memory. Each comparison
+releases its temporary matrices; only bounded collections and counters
+persist. Both modes exercise the finite scans and capacity rejection. A
+fixed-allocation port would be needed before native or long-lived use.
+
+## Open next questions
+
+1. For a named physical caller with a legal restricted chart alphabet, how
+   much of this minimum-rank density survives? Begin by specifying that
+   alphabet and its complete chart, injection and restoration prices. The
+   unrestricted distribution is a reference count, not a physical tariff.
+2. Can one sample a minimum-rank chart uniformly without enumerating GL_n?
+   The injective-lift proof suggests a constructive sampler. An exact
+   uniformity proof and the actual caller's selection cost are still needed.
+   This package does not reserve or implement that extension.
+
+## Credit and assignment
+
+Credit ywz65 #2680 for the paid residual-rank interface and minimum theorem,
+woklerx for the equal-kernel dirty-chart contract, and Ross Richardson #943
+for the dirty-query and complete-call context. Gaussian coefficients,
+subspace-intersection counting and triangular inversion are classical.
+This is a new application and formula in that interface, with no claim of
+literature-wide priority. All predecessor files and credit lines are unchanged.
+
+The `connector-alphabet-change` card is explicitly changed to an algebraic
+side result about chart multiplicity. It adds neither a connector operation
+nor a favorable completed budget. The user's original-work restriction
+supersedes the card's predecessor-replay example. Current selected-binding
+orientation passed; it is metadata evidence only. Live claims, open PRs,
+merged-PR searches and main README searches were used for overlap. The
+cached helper covered 416 open PRs and 5,307 Discussion records, but sparse
+local coverage omitted 7,622 notes. Targeted searches of all main READMEs
+supplemented that cache; unpublished work and arbitrary diffs remain outside
+coverage. Lack of a matching record does not establish novelty.

research/connector-chart-rank-distribution/SOURCES.json

@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "base_commit": "de5ec2e7a6074068452f45b1a191b5c72832df4a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/connector-chart-rank-repair/README.md",
+      "usage": "provenance",
+      "git_blob": "298bf702a48b6ff2bf3d1e10035aa304ecb25546",
+      "sha256": "222d197d7ca6cebabc23af9c05db6dec813b7003f7b4bab83aac94dfcd26a745"
+    },
+    {
+      "path": "research/dirty-connector-chart-transport/README.md",
+      "usage": "provenance",
+      "git_blob": "2b161ebfccfc3867d3b6b74cc05c761ba1becfd2",
+      "sha256": "c3d8b2fcc3485e267e25efc260b37ed7a2e8a0bc7d20294e9c79f0792a253d67"
+    },
+    {
+      "path": "research/universal-connector-boundary/README.md",
+      "usage": "provenance",
+      "git_blob": "d277f7c5830b1f2bc42e1ff41ab86e7d7564929f",
+      "sha256": "23415f7829ab8d80a783142f10f7d31abbce84579362769984680af3df954476"
+    }
+  ]
+}

research/connector-chart-rank-distribution/TASK.json

@@ -0,0 +1,68 @@
+{
+  "schema_version": 1,
+  "task_id": "connector-alphabet-change",
+  "priority_revision": 8,
+  "base_commit": "de5ec2e7a6074068452f45b1a191b5c72832df4a",
+  "original_card": {
+    "id": "connector-alphabet-change",
+    "question": "Does one explicit extension of the connector alphabet offer a favorable completed budget?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/universal-connector-boundary/README.md",
+      "certificates/selected-result.json"
+    ],
+    "methods": [
+      "Construct and charge a legal extra operation on a minimal instance.",
+      "Prove a target-specific bound after adding one specified operation family."
+    ],
+    "acceptance_gate": "State which fixed-alphabet premise changes and charge its implementation; do not declare it free. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/universal-connector-boundary/verify.py"
+      ],
+      "expected": "PASS_UNIVERSAL_CONNECTOR_BOUNDARY",
+      "inspect": "Reproduce the 945-pairing control, then read Exact interface to list the allowed calls.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Add exactly one legally priced operation and test a tiny dirty-input instance before enumerating larger cases.",
+    "bottleneck_link": "A new legal connector operation could change recursive interface cost; a larger census in the same alphabet is a separate side result.",
+    "allowed_variations": [
+      "Choose one additional operation and a small source count for its first counterexample."
+    ],
+    "scope_change_triggers": [
+      "Keeping the alphabet and only increasing the source count; record this as a changed assignment.",
+      "Extending a source-count census without naming a supplier/interface decision it could change; audit #2973 pruning concerns before relying on the affected predecessor exclusions."
+    ],
+    "overlap_terms": [
+      "universal connector",
+      "source-toggle",
+      "connector alphabet"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Complete finite-field distribution of rank(B1-R B0) over all invertible codomain charts, with exact minimum-rank density.",
+  "scope_change_reason": "Original successor counting theorem beyond ywz65 #2680; this is a side result, not an extension of the connector alphabet or a physical supplier.",
+  "bottleneck_link": "Counts the availability of low-rank paid chart repairs. A selected consumer and complete native costs remain unknown; no implementation campaign is started.",
+  "budget_minutes": 60,
+  "stop_rule": "Ship the first proved distribution after bounded matrix corroboration; do not extend to sampling or physical synthesis.",
+  "outcome": "completed",
+  "evidence": "PROOF.md proves the all-finite-field formula. verify.py compares 110 canonical cases and 10657 arbitrary map pairs against direct complete chart enumeration.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Determine the minimum repair-rank distribution in one named caller's restricted legal chart alphabet.",
+      "why_it_matters": "Full GL_n density does not price or guarantee native availability.",
+      "first_step": "Specify one physical caller, chart alphabet and complete prices.",
+      "blocked_by": "No concrete physical consumer is supplied."
+    },
+    {
+      "question": "Sample a minimum-rank chart exactly uniformly without enumerating GL_n.",
+      "why_it_matters": "A constructive sampler could make the counting proof usable for chart selection.",
+      "first_step": "Turn the injective-lift factorization into a sampler and prove its output law.",
+      "blocked_by": "Uniformity and physical selection costs are unproved."
+    }
+  ]
+}