Bound the retained-center port budget for every nonzero ambient quotient (no new κ)

research/selective-quotient-port-budget/README.md

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+# Every nonzero ambient quotient loses the retained local rank advantage
+
+**Scoped geometric proof and finite certificate; no new kappa.** Selecting only
+the complex ports that survive a quotient cannot preserve the source's local
+unpaid-rank advantage while retaining its eight center transports. This covers
+every nonzero binary kernel, of every dimension, rather than just kernel 153 or
+the 32 directions passing the earlier logical replay. It does not exclude
+changed center words, nonquotient constructions, or all possible moment gains.
+
+The question arose while trying to repair DaysSky's #4629/#4633 quotient by
+dropping incompatible ports and charging only surviving output contracts.
+The preceding endpoint audits establish that keeping the old full-port invoice
+is invalid. This package asks whether selecting a smaller valid port set saves
+the quotient's local rank budget. The answer is no under the retained recipe.
+
+## Exact result
+
+Let V=F2^9. The source has 112 hyperplanes H_p complementary to its 112
+entrance lines P_p, and eight center hyperplanes C_i. The verifier extracts
+their unique nonzero annihilators directly from the pinned canonical word.
+Their disjoint union S is exactly the 120 vectors of Hamming weight 3 or 7.
+
+For any kernel K, a port retains a nonzero complementary entrance line after
+quotienting exactly when K is contained in H_p. Write v(K) for the number of
+such ports, and charge all eight retained center transports by their actual
+image dimensions:
+
+    c(K) = sum_i dim(image(C_i)).
+
+Then
+
+    K = 0:             v(K) - c(K) = 112 - 64 = 48;
+    every nonzero K:   v(K) - c(K) <= 0.
+
+For one-dimensional kernels the complete balance distribution is 255 cases at
+0, 255 at -8, and one at -64. Kernel 153 has 56 compatible ports, center charge
+64 and balance -8. The seven directions with 64 compatible ports all retain
+center charge 64. Re-padding unrelated helpers cannot alter this geometry.
+Choosing fewer compatible ports can only lower the displayed balance.
+
+## Proof for every kernel dimension
+
+Let L=K-perp and r=dim L. A hyperplane with nonzero normal n contains K exactly
+when n lies in L. Its image dimension is r-1 in that case and r otherwise.
+Consequently
+
+    v(K)-c(K) = |S intersect L| - 8r.
+
+It remains to bound |S intersect L| for r<=8.
+
+* For r=8 the exact 511-hyperplane census gives 255 intersections of size 64,
+  255 of size 56 and one empty intersection. Thus the maximum is 8r.
+* For r=7, L lies in exactly three hyperplanes. Each point of S inside L is
+  counted three times, and every point outside L once. Therefore
+  120+2|S intersect L| <= 3*64, giving |S intersect L|<=36<56.
+* For 1<=r<=6, S consists of odd-weight vectors. The parity functional on L
+  is either zero, giving an empty intersection, or takes value one on exactly
+  2^(r-1) vectors. In this range 2^(r-1)<=8r.
+* For r=0 both counts vanish.
+
+This is a dimension-uniform proof, not an assertion that enumerating line
+kernels enumerates every higher-dimensional subspace. The coarse bounds for
+r<=7 are not claimed sharp. [RESULT.json](RESULT.json) gives them explicitly.
+
+## Meaning for the controller
+
+The incumbent accounting lever is the number of unpaid central port directions
+minus the center rank spend. Its value is 48 before quotienting. The repaired
+selective-port version has a nonpositive value even before charging dropped-port
+conversion, native normalizers, changed warps, boundary work and ordinary/outer
+admission. Those costs are not assumed solved by this certificate.
+
+This rank statement is **not** a theorem that every concave target/paid moment
+fails: a rank-mass comparison by itself is insufficient for that conclusion.
+No full quotient program, scalar-field lift, replacement controller, new
+ordinary supplier, or improved exponent is claimed. A changed circuit would
+need its complete moment and physical contract checked separately. The result
+identifies the center recipe as a premise that must change to recover a
+positive local rank advantage, rather than declaring a universal barrier.
+
+Concurrent #4654 constructs an exact mixed type-A/type-B quotient interface.
+Its type-B ports are not the retained complementary ports counted here. A
+controller that consumes those changed interfaces escapes this selective-port
+scope and remains a constructive next route; this result does not refute it.
+
+## Reproduce
+
+From the repository root, standard-library CPython 3.12:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/selective-quotient-port-budget
+python3.12 -I -O -B research/selective-quotient-port-budget/verify.py
+```
+
+The independent verifier authenticates the input, checks all 112 source splits,
+extracts the 120 normals, computes all 511 line cases and verifies the numerical
+bounds used in the written proof. Four geometry mutations are rejected; a
+simultaneous change of frame bases preserves the complete result. It imports no
+predecessor evaluator. Native circuit replay, phases, precision, compiler,
+selected cold replay and human/formal review are outside its coverage.
+
+The optional current baseline command fails at this base with `KeyError:
+'bit_lower'` because its expected certificate schema differs from the selected
+record. It is not reported as a successful replay; this package uses the actual
+selected record only as provenance and does not modify that unrelated helper.
+The selected conditional saving is 0.00302015 at base
+`b58ad9ef4d4e861b0eaa5dba41742bdc072dca51`.
+
+## Open next questions
+
+Can a changed center construction preserve valid native port maps while making
+their count exceed its paid rank spend? First specify its exact center operator,
+arbitrary-input restoration and a complete local invoice. The obstruction above
+does not apply when that recipe changes. A larger kernel scan with the same
+eight transports cannot reopen the rank question. See [TASK.json](TASK.json).
+
+## Credit
+
+Prepared by Cinna Davis with substantial OpenAI Codex assistance; Apache-2.0.
+Credit DaysSky for the quotient experiments (#4596/#4629/#4633), Nick Petalas
+for the kernel census (#4627), pinterbanget and Rohan Arun for endpoint checks,
+and the preceding #4636 and local PR4633 repair audits. The source is zxv's
+native construction with the retained E8/scalar/Clifford authors, Cinna Davis's
+#4440/#4441/#4447/#4470 controller lineage, and Jacob Alan Sussman's mixed-target
+recursion. No frozen source or selected-frontier file is changed.

research/selective-quotient-port-budget/RESULT.json

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+{
+  "all_kernel_dimension_bounds": [
+    {
+      "dual_dimension": 0,
+      "kernel_dimension": 9,
+      "local_rank_balance_upper_bound": 0,
+      "port_and_center_normal_count_upper_bound": 0
+    },
+    {
+      "dual_dimension": 1,
+      "kernel_dimension": 8,
+      "local_rank_balance_upper_bound": -7,
+      "port_and_center_normal_count_upper_bound": 1
+    },
+    {
+      "dual_dimension": 2,
+      "kernel_dimension": 7,
+      "local_rank_balance_upper_bound": -14,
+      "port_and_center_normal_count_upper_bound": 2
+    },
+    {
+      "dual_dimension": 3,
+      "kernel_dimension": 6,
+      "local_rank_balance_upper_bound": -20,
+      "port_and_center_normal_count_upper_bound": 4
+    },
+    {
+      "dual_dimension": 4,
+      "kernel_dimension": 5,
+      "local_rank_balance_upper_bound": -24,
+      "port_and_center_normal_count_upper_bound": 8
+    },
+    {
+      "dual_dimension": 5,
+      "kernel_dimension": 4,
+      "local_rank_balance_upper_bound": -24,
+      "port_and_center_normal_count_upper_bound": 16
+    },
+    {
+      "dual_dimension": 6,
+      "kernel_dimension": 3,
+      "local_rank_balance_upper_bound": -16,
+      "port_and_center_normal_count_upper_bound": 32
+    },
+    {
+      "dual_dimension": 7,
+      "kernel_dimension": 2,
+      "local_rank_balance_upper_bound": -20,
+      "port_and_center_normal_count_upper_bound": 36
+    },
+    {
+      "dual_dimension": 8,
+      "kernel_dimension": 1,
+      "local_rank_balance_upper_bound": 0,
+      "port_and_center_normal_count_upper_bound": 64
+    }
+  ],
+  "claim": "retained-center selective-port quotient rank budget only; no new kappa",
+  "controls": {
+    "basis_change_invariance": true,
+    "geometry_mutations_rejected": 4
+  },
+  "kernel_153": {
+    "compatible_ports": 56,
+    "kernel": 153,
+    "local_rank_balance": -8,
+    "retained_center_rank": 64
+  },
+  "line_balance_counts": {
+    "-64": 1,
+    "-8": 255,
+    "0": 255
+  },
+  "line_census_sha256": "865d17aa7a1eb1a965d2ff1fcf5d450e596799c708d15ddf67514efe95d2051c",
+  "line_compatible_port_counts": {
+    "0": 1,
+    "48": 7,
+    "52": 224,
+    "56": 48,
+    "60": 224,
+    "64": 7
+  },
+  "line_kernel_cases": 511,
+  "normal_weight_counts": {
+    "3": 84,
+    "7": 36
+  },
+  "source_centers": 8,
+  "source_ports": 112,
+  "unquotiented_local_rank_balance": 48
+}

research/selective-quotient-port-budget/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-CANONICAL.json.gz",
+      "usage": "input",
+      "git_blob": "4a8c4ed26e53a72362d571c972e65aff546be784",
+      "sha256": "6418ec3da24d548a57f5c3fe5b9062c26c9478e638504a61c387f5cfb5f4ed61"
+    }
+  ]
+}

research/selective-quotient-port-budget/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-new-helper-inventory",
+  "priority_revision": 7,
+  "base_commit": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51",
+  "original_card": {
+    "id": "complex-new-helper-inventory",
+    "question": "Can a new local cost interface change the deficit-to-recursive-cost ratio beyond current packing?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/heterogeneous-owner-assembly/README.md",
+      "certificates/selected-result.json",
+      "research/selected-k8-unrestricted-ends/README.md"
+    ],
+    "methods": [
+      "Propose a changed helper inventory and derive an optimistic exact moment.",
+      "Design a small completed instance and independently verify its endpoint and rank bill."
+    ],
+    "acceptance_gate": "A new physically realizable contract outside the frozen catalogue whose per-call profile, screened with ceiling.ceiling(), beats the selected word's packing ceiling; a lower cap does not establish identical circuitry or order attainable optima, and is insufficient to pass the stronger-budget target under this model. 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/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin.",
+    "bottleneck_link": "Packing of the selected word is capped at +0.87% (ceiling model); only a lower local excess spend or a changed coverage law raises the cap.",
+    "allowed_variations": [
+      "Narrow to one changed boundary or helper type with explicit counts.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Mixing only providers already inside the pinned catalogue.",
+      "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013.",
+      "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts."
+    ],
+    "overlap_terms": [
+      "helper inventory",
+      "heterogeneous-owner",
+      "mutable leaver"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Prove the exact retained-center rank budget for selective native ports under every nonzero binary ambient quotient.",
+  "scope_change_reason": "User asks for a structural kappa breakthrough. After the preceding quotient endpoint audits, test the distinct repair of retaining only complementary ports and repricing all eight center images.",
+  "bottleneck_link": "Tests whether port selection can preserve the local unpaid rank advantage underlying the large quotient screen claims; it cannot under the retained center recipe.",
+  "budget_minutes": 60,
+  "stop_rule": "Finish the all-kernel proof and exact line census; do not infer a universal concave-moment obstruction from rank mass alone.",
+  "outcome": "completed",
+  "evidence": "README.md proof, RESULT.json and independently implemented verify.py",
+  "author": "Cinna Davis, with substantial OpenAI Codex assistance",
+  "ai_assistance": "Substantial OpenAI Codex assistance in derivation, implementation, verification and writing; no subagents.",
+  "open_questions": [
+    {
+      "question": "Can a changed center recipe give a positive port-count minus center-rank balance with valid native endpoints?",
+      "why_it_matters": "Every nonzero quotient of the retained recipe has nonpositive balance, whereas the source has 48.",
+      "first_step": "Specify an exact center operator, arbitrary-input restoration and a fully charged local rank/moment profile before another kernel search.",
+      "blocked_by": "No changed center word and complete native invoice are supplied here."
+    }
+  ],
+  "overlap_coverage": {
+    "local": {
+      "head": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51",
+      "records": 5862,
+      "head_records": 5860,
+      "complete_head_inventory": true,
+      "unsearched_head_paths": []
+    },
+    "open_prs": {
+      "fetched_at": "2026-10-22T14:20:50.611470+00:00",
+      "complete_pagination": true,
+      "records": 425
+    },
+    "discussions": {
+      "fetched_at": "2026-10-22T14:22:03.534515+00:00",
+      "complete_pagination": true,
+      "records": 3588
+    }
+  },
+  "concurrent_work": "PR4654 supplies a positive mixed A/B interface. This result counts only complementary retained ports, so it does not exclude a controller consuming the changed type-B interface."
+}

research/selective-quotient-port-budget/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-CANONICAL.json.gz",
+    "research/selective-quotient-port-budget/RESULT.json"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "All source port splits, exact weight-three/seven normal set, all 511 line kernels, all kernel-dimension bounds, four rejecting geometry mutations and basis-change invariance."
+    }
+  ]
+}