Price the p3 H-branch boundary at kappa 0.00302015 (322/397 calls) and close four repair routes (no new κ)

research/h-boundary-budget/README.md

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+# The p3 H branch: exact boundary budget and three closed decoder routes
+
+**Negative result, no new κ.** zxv's H branch of
+[PR4359](https://github.com/CrocSwap/integer-mult-bounds/pull/4359) replaces the
+local response by an involution `H = I + BC` (`CB = -2I`). Its raw rank surplus
+is far above the current frontier, but the noncommuting p3 controller leaves three
+interval endpoints of the form `Q K Q`. Their direct correction costs 672
+rank-one requests in complex and 666 in BIT, more than the whole surplus.
+[PR4431](https://github.com/CrocSwap/integer-mult-bounds/pull/4431) closed
+completion-only fresh inverses. This package asks how much a corrected boundary
+may cost and which decoders cannot meet that price.
+
+## 1. Exact budget at the selected conditional κ
+
+Each rank-one call has weight exactly 1 in the moment at every saving `s`, so
+the largest affordable boundary bill is the certified moment of the uncorrected
+profile. The profiles are the retained ones: complex
+`controller_tariff.geometry(3, 43)` with kernel H, surplus 616, and BIT with
+the three-row minimum completion, surplus 645 = 603 + 42.
+
+| Saving | complex budget | BIT budget |
+|---|---:|---:|
+| 0.0023 | 392 | 456 |
+| **0.00302015** (#4470) | **322** | **397** |
+| 0.004 | 228 | 317 |
+
+Saving roots for each boundary bill (certified brackets in `RESULT.json`):
+
+| Boundary bill | complex | BIT |
+|---|---:|---:|
+| free | 0.0063799 | 0.0079333 |
+| one shared head conjugation (224 / 222) | **0.0040422** | 0.0051761 |
+| two (448 / 444) | 0.0017248 | 0.0024471 |
+| direct, three (672 / 666) | negative at s = 0 | negative at s = 0 |
+
+So a corrected H branch beats the frontier only if all three endpoints are
+repaired for at most 322 complex requests, roughly 1.44 head conjugations.
+These are raw rank screens. Outer rows, tails, compiler gates and the scalar and
+movement costs of a new decoder are not included, so they are ceilings.
+
+## 2. Whole-bank Q events cannot fit
+
+One Q event on a complete bank costs 112 complex requests, so the budget admits
+at most two (`224 <= 322 < 336`). PR4359's `universal_two_query_screen` already
+shows that two complete-bank events give a coefficient span of at most 1 against
+the required 3. Even three events would cost 336 and give a complex root near
+0.0029, below the frontier.
+
+## 3. Port subsets cannot be skipped
+
+For the native H112, `rank(H - I) = 8` exactly, but all 112 rows and all 112
+columns of `H - I` are nonzero. Every interval product `K` therefore moves data
+on every port, so a correction touching only part of a bank leaves a nonzero
+`Q`-dependent block.
+
+## 4. Gather decoders through the rank-8 bottleneck cost at least the direct bill
+
+The low rank suggests compressing: since `Q^2 = I`,
+
+```
+Q K Q - K = Q M Q - M,   M = K - I,
+```
+
+and `M` factors through 8 fields per stage. This fails for an elementary reason.
+On port `p`, `Q` is one rank-one request along that port's own line `q_p`. Two
+ports' requests can be merged by linearity only when their lines agree, because a
+request along `q_p` and a request along `q_r` are different transforms. The
+stage routes (`rho`, `Gamma_s`) fix each port's line, and the commutation
+`[H_s, Q b_s] = 0` only provides the full call `Q b_s` on the compressed side.
+
+The data leave no room:
+
+| | complex | BIT |
+|---|---:|---:|
+| ports | 112 | 111 |
+| distinct query lines | **112** | **111** |
+| ports touched by `H - I` | 112 | (not computed) |
+
+Computing `M (Q w)` for an arbitrary input therefore needs one request on every
+port that `M` reads: 112. Writing `Q (M ...)` back needs one on every port that
+`M` writes: another 112. The three bad roles have independent inputs, and a
+request on a sum cannot be separated afterwards. Any decoder of this kind,
+meaning free scalars and routes plus rank-one requests applied to linear
+combinations, costs at least `3 * 2 * 112 = 672`, which is exactly the direct
+bill. The direct correction is optimal within this class.
+
+**Scope.** This is a lower bound for linear gather decoders at fixed H112, fixed
+owners and the fixed p3 controller. It is not a theorem about every possible
+algorithm. A construction that changes the local family, sends several ports'
+lines to a shared line before the request, or changes the controller is outside
+it.
+
+## 5. No controller of this family has fewer Q-conjugated roles
+
+The verifier enumerates the free noncommuting controller of
+`bit/pr589_h_global_probe.py` for p = 1..4 and every idle keep mask. It keeps
+those that diagonalize exactly with parity-preserving (Q-free) boundaries:
+2, 3, 8 and 20 controllers. **Every one has exactly p roles ending as `Q K Q`
+with nonempty K.** Each local call stamps its stage letter on both streams of
+its pair, so the odd (Q-carrying) stream always has a nonempty word. The
+p3 keep-43 controller reproduces the probe's parity `[0,1,0,1,1,0]`.
+
+The H surplus of `mixed_forest_search.verify` is `224p - 8(2p+1)` (616 for
+p = 3). A direct repair costs 224 per bad role, so after it
+
+```
+224p - 8(2p+1) - 224p = -8(2p+1) < 0      for every p.
+```
+
+Each pair brings exactly as much rank credit as its own endpoint repair costs.
+Changing p or the idle mask cannot fund the direct boundary. Section 6 shows
+that shared query lines do not help either.
+
+## 6. Shared query lines do not escape either bound
+
+Sections 4 and 5 left one escape: ports that share a query line. It closes as well.
+
+The local call realizes the identity `I = K + BA`. The pair reads `K` act only on
+pairs of ports whose F2 labels are orthogonal, and `BA` (rank `h`) is decoded
+through `h` centres (compare the source construction in
+`research/e8-direct-source/PROOF.md`). The verifier checks this on the native H112:
+all 112 labels `q_p` are odd, and `J` has zero diagonal and vanishes on all
+6,608 ordered non-orthogonal label pairs. A mutated even label is rejected.
+
+Since `q.q = 1`, two ports with the same label are non-orthogonal. So `K` is zero
+on any class of `m` copies, and `BA` restricted to that class is `I_m`. A free
+rotation `X_p +- X_p'` inside the class keeps it `I_m`. Two consequences follow.
+
+* `h` is at least the total multiplicity of any clique of non-orthogonal labels,
+  so `m` copies of E8 lines need `h >= 8m`. That is exactly the rank of `m`
+  independent copies of the unit.
+* On a copy class, `H - I` equals `-I_m` up to the `h` completion rows. So a gather
+  decoder needs at least `V - h` requests on input and `V - h` on output. Exactly:
+  P112 needs 112 + 112, and the doubled unit (224 ports, 112 lines) needs 224 + 224,
+  which is 2 per port in both cases. Merging all ports into one class would need
+  only 8 + 8 (control), so the bound really comes from the labels.
+
+For any unit with `V` ports, rank `h` and `p` pairs, the H surplus is
+`2Vp - h(2p+1)` and the tails cost at least `2p(V - h)`:
+
+```
+2Vp - h(2p+1) - 2p(V - h) = -h < 0.
+```
+
+The H branch is unfunded whatever the line layout, shared or not, for every
+local call of the form `I = K + BA`.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/h-boundary-budget --check-only
+python3.12 -B scripts/research_package.py --package research/h-boundary-budget
+```
+
+`verify.py` uses only the standard library and imports the pinned
+`research/kappa-rebuild` modules. Those modules expect `research/kappa-next-regime`
+at the repository root; on main that tree is vendored under
+`research/kappa-0023-productive-jobs/dependencies/kappa-next-regime`, and the
+verifier adds that path. It runs in about 3.5 minutes on Windows / CPython 3.12.
+Controls: a budget one above the certified value is certified negative, a
+perturbed `H - I` is rejected, a duplicated query line is detected, the
+p3 keep-43 controller reproduces the probe parity, an even label is rejected,
+and a single merged class gives only rank 8.
+
+## Open next questions
+
+The structured versions are in `TASK.json`.
+
+1. **A local call outside `I = K + BA`.** Section 6 closes shared query lines
+   for every unit of this form. An escape needs a local identity whose pair
+   reads are not tied to label orthogonality.
+2. **A controller not built from Q-gauged pairs.** Section 5 closes controllers
+   with fewer Q-conjugated roles for p <= 4, since each pair's repair costs
+   exactly its credit. A different pair structure would be a new complex
+   supplier, not an H repair.
+
+## Credit
+
+The H construction, the endpoint obstruction, the retained profiles and
+receipts are zxv's (PR4359, PR4431) together with their cited predecessors.
+This package adds the budget, the root table, the native rank/support and
+query-line counts, the gather argument, the controller parity census and the shared-line closure. Prepared by Matarener (Mark) with
+substantial Claude Code assistance. Apache-2.0. No independent human review.

research/h-boundary-budget/RESULT.json

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+{
+  "bit": {
+    "budget_at_record": 397,
+    "direct_correction": 666,
+    "one_shared_head": 222,
+    "raw_surplus": 645,
+    "roots": {
+      "0": {
+        "negative_at": "0.0079333535",
+        "positive_at": "0.0079333532"
+      },
+      "222": {
+        "negative_at": "0.0051761597",
+        "positive_at": "0.0051761594"
+      },
+      "444": {
+        "negative_at": "0.0024471107",
+        "positive_at": "0.0024471104"
+      }
+    }
+  },
+  "complex": {
+    "budget_at_record": 322,
+    "direct_correction": 672,
+    "one_shared_head": 224,
+    "raw_surplus": 616,
+    "roots": {
+      "0": {
+        "negative_at": "0.0063799044",
+        "positive_at": "0.0063799042"
+      },
+      "224": {
+        "negative_at": "0.0040422294",
+        "positive_at": "0.0040422291"
+      },
+      "448": {
+        "negative_at": "0.0017248836",
+        "positive_at": "0.0017248833"
+      }
+    }
+  },
+  "controller_parity": {
+    "1": {
+      "after_direct": -24,
+      "bad_roles": 1,
+      "gauge_compatible_controllers": 2,
+      "h_surplus": 200
+    },
+    "2": {
+      "after_direct": -40,
+      "bad_roles": 2,
+      "gauge_compatible_controllers": 3,
+      "h_surplus": 408
+    },
+    "3": {
+      "after_direct": -56,
+      "bad_roles": 3,
+      "gauge_compatible_controllers": 8,
+      "h_surplus": 616
+    },
+    "4": {
+      "after_direct": -72,
+      "bad_roles": 4,
+      "gauge_compatible_controllers": 20,
+      "h_surplus": 824
+    }
+  },
+  "native_H112": {
+    "nonzero_cols": 112,
+    "nonzero_rows": 112,
+    "ports": 112,
+    "rank_H_minus_I": 8
+  },
+  "new_kappa": false,
+  "query_lines": {
+    "bit_distinct_lines": 111,
+    "bit_ports": 111,
+    "complex_distinct_q": 112,
+    "complex_ports": 112
+  },
+  "record_kappa": "60403/20000000",
+  "schema": "h-boundary-budget/1",
+  "shared_lines": {
+    "net_after_tail_floor": "2Vp - h(2p+1) - 2p(V-h) = -h",
+    "nonorthogonal_label_pairs": 6608,
+    "odd_labels": true,
+    "tail_gather_P112": [
+      112,
+      112
+    ],
+    "tail_gather_doubled": [
+      224,
+      224
+    ]
+  },
+  "status": "PASS_H_BOUNDARY_BUDGET_AND_SCOPED_DECODER_CLOSURES"
+}

research/h-boundary-budget/SOURCES.json

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