Prove the P104 weighted star program is exactly optimal at 5/6 (no κ)

research/p104-star-weight-optimum/PROOF.md

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+# Exact optimum of the weighted star program on P104
+
+## 0. Setting and cited premises
+
+Let `V` be the 104-port set P104 of [vector port systems], a set of 104 vectors of
+`F_2^9` of weight `3 mod 4`; two distinct ports are adjacent when their dot
+product is `1` and orthogonal when it is `0`.
+
+Fix an orthogonal 8-set `K` of `V` (eight pairwise orthogonal ports) and put
+`v = 1 + sum(K)`, the all-ones vector plus the sum of `K` in `F_2^9`. Then
+`dot(v, c) = 0` for `c` in `K` and `dot(v, v) = 1`, so `K` together with `v` is
+an orthonormal basis of `F_2^9`. Every port `t` is therefore
+
+```text
+t = sum_{c in A(t)} c + eps(t) v,     A(t) = {c in K : dot(t,c) = 1},  eps(t) = dot(t,v),
+```
+
+with `|A(t)| + eps(t) = 1` mod 2. A port outside `K` is an **edge port** if
+`|A(t)| = 2`. Distinct edge ports have distinct pairs `A(t)`, so the **edge
+graph** `Gamma(K)`, with vertex set `K` and one edge per edge port, is simple.
+Write `E` for the number of edge ports, `d_c` for the degree of `c` in
+`Gamma(K)`, and `m = |E(Gamma(K))| = E` (simplicity).
+
+These facts, and the type census used in Lemma 4, are Theorem 3 and its
+structural section in [the predecessor package](../p104-port-window-haemers/THEOREM.md)
+(#6933, DreamingOfClouds). They are cited premises. `verify.py` recomputes the
+census so that the certificate below is self-contained, but the mathematics of
+this package begins at Lemma 1.
+
+**Theorem 3 of #6933 (cited).** Let `a_c >= 0` be weights on `K` with
+
+```text
+(W1)  a_c + a_d <= 1                     for every edge cd of Gamma(K),
+(W2)  a_c (d_c - 2) <= 2                 for every c in K.
+```
+
+Put
+
+```text
+mu  = sum_{c in K} (1 + a_c),
+rho = sum_{cd in E(Gamma(K))} (1 - a_c - a_d).
+```
+
+Then for every field `F`, every `k` and every `k`-block pattern `B` on `V`,
+writing `rank B = 8k + j`,
+
+```text
+j/k >= (E - mu) / (mu + rho).
+```
+
+The right-hand side is a function `f(a)`. #6933 evaluates `f` at weights from a
+grid and states that it makes no claim that they are optimal. The theorem below
+computes the exact supremum of `f` over the whole admissible set, and its
+equality case.
+
+## 1. Two identities
+
+**Lemma 1.** With `A = sum_c a_c` and `D = sum_c d_c a_c`,
+
+```text
+mu = 8 + A,          rho = E - D,          mu + rho = 8 + A + E - D.
+```
+
+*Proof.* `mu = sum_c (1 + a_c) = 8 + A`. Every edge of `Gamma(K)` contributes
+`1 - a_c - a_d`, so summing over the `E` edges counts each `a_c` exactly `d_c`
+times: `rho = E - sum_c d_c a_c = E - D`. ∎
+
+Note `mu + rho = mu + rho >= 8 > 0`, so `f` is defined and no sign case arises.
+
+**Lemma 2.** `f(a) <= 5/6` if and only if `5D - 11A <= 88 - E`.
+
+*Proof.* Since `mu + rho > 0`,
+
+```text
+f <= 5/6  <=>  6 (E - 8 - A) <= 5 (8 + A + E - D)
+          <=>  6E - 48 - 6A <= 40 + 5A + 5E - 5D
+          <=>  E - 88 <= 11A - 5D
+          <=>  5D - 11A <= 88 - E.                                  ∎
+```
+
+## 2. The multiplier certificate
+
+**Lemma 3.** Let `lambda = 14/5` and, for each `c` in `K`,
+
+```text
+mu_c = ( (11/5) d_c - 11 ) / (d_c - 2).
+```
+
+For `d_c >= 5` the number `mu_c` is nonnegative, and for every `c`
+
+```text
+d_c lambda + mu_c (d_c - 2) = 5 d_c - 11.        (identity)
+```
+
+Consequently, for every `a` satisfying (W1) and (W2),
+
+```text
+5D - 11A <= (14/5) E + 2 sum_c mu_c.
+```
+
+*Proof.* `d_c lambda + mu_c(d_c - 2) = (14/5) d_c + (11/5) d_c - 11 = 5 d_c - 11`.
+Then
+
+```text
+5D - 11A = sum_c (5 d_c - 11) a_c
+         = sum_c a_c [ d_c lambda + mu_c (d_c - 2) ]
+         = lambda sum_{cd in E(Gamma)} (a_c + a_d) + sum_c mu_c (d_c - 2) a_c
+         <= lambda E + 2 sum_c mu_c,
+```
+
+because every edge contributes `a_c + a_d` once, (W1) bounds each of the `E`
+edge terms by `1`, and (W2) bounds each vertex term by `2 mu_c >= 0`. ∎
+
+**Lemma 4 (cost on the three P104 types).** The quantity
+
+```text
+(14/5) E + 2 sum_c mu_c  =  (19/5) E + 2 sum_c mu_c
+```
+
+satisfies
+
+| type `(E; d_c)` | `sum_c mu_c` | cost | `88 - E` |
+| --- | ---: | ---: | ---: |
+| `(22; 5,5,5,5,6,6,6,6)` | `11/5` | `88` | `66` |
+| `(22; 5,5,5,5,5,6,6,7)` | `99/50` | `4378/50` | `66` |
+| `(21; 5,5,5,5,5,5,6,6)` | `11/10` | `82` | `67` |
+
+So in every case `(14/5)E + 2 sum_c mu_c <= 88 - E`.
+
+*Proof.* `mu_5 = ((11/5)·5 - 11)/3 = 0`, `mu_6 = (66/5 - 11)/4 = 11/20`, and
+`mu_7 = (77/5 - 11)/5 = 22/25`. Sum over the multiplicities of each type and add
+`(14/5)E = (14/5)·22` or `(14/5)·21`. The first row gives
+`308/5 + 2·11/5 = 330/5 = 66`; the second `308/5 + 2·99/50 = 4378/50 <= 66`; the
+third `294/5 + 2·11/10 = 82 <= 67`. The three types are the complete census of
+orthogonal 8-sets of P104. ∎
+
+## 3. The theorem
+
+**Theorem.** For every orthogonal 8-set `K` of P104 and every weight vector `a`
+satisfying (W1) and (W2),
+
+```text
+f(a) = (E - mu)/(mu + rho) <= 5/6.
+```
+
+Moreover `f(a) = 5/6` holds exactly when `K` has type `(22; 5,5,5,5,6,6,6,6)`
+and `a_c = 1/2` for every `c` in `K`.
+
+*Proof.* Lemma 3 and Lemma 4 give `5D - 11A <= 88 - E`, which is `f(a) <= 5/6`
+by Lemma 2. This proves the bound.
+
+For the equality case, `f(a) = 5/6` forces equality in every link of the chain.
+
+1. *Cost equality.* `(14/5)E + 2 sum_c mu_c = 88 - E`. By Lemma 4 the cost is
+   `88` only for `E = 22` with `sum_c mu_c = 11/5`. Since `mu_c = 0` at degree 5,
+   `mu_c = 11/20` at degree 6 and `mu_c = 22/25` at degree 7, the only multiset of
+   eight degrees with `sum_c mu_c = 11/5 = 4 · 11/20` is `(5,5,5,5,6,6,6,6)`.
+2. *Vertex equalities.* `lambda = 14/5 > 0`, so every one of the `E` edge terms
+   must be tight: `a_c + a_d = 1` for every edge `cd`. And `mu_c > 0` at every
+   degree-6 vertex, so `(d_c - 2) a_c = 2`, i.e. `a_c = 1/2` at every degree-6
+   vertex.
+3. *Degree 5.* A degree-5 vertex has five neighbours but only three other
+   degree-5 vertices, so it has a neighbour `d` with `d_d = 6`. Then
+   `a_c = 1 - a_d = 1/2`.
+
+Conversely, `a_c = 1/2` for all `c` is admissible: (W1) gives `1/2 + 1/2 = 1`, and
+(W2) gives `(1/2)·3 = 3/2 <= 2` at degree 5 and `(1/2)·4 = 2 <= 2` at degree 6.
+For the principal type it yields `mu = 8 + 4 = 12`, `rho = 22 - 22 = 0`, hence
+`f = 10/12 = 5/6`. ∎
+
+## 4. Corollaries
+
+**Corollary 1.** No choice of weights satisfying (W1) and (W2) makes Theorem 3 of
+#6933 exceed `j/k >= 5/6` on any orthogonal 8-set of P104. Since the bound
+`rank B >= 53k/6` is exactly this inequality, no reweighting raises the floor
+`53/6`. The maximizer is unique, so the weight program has no second regime to
+explore.
+
+**Corollary 2.** Any strict improvement of the star route must change an
+inequality that is not a weight: a sharper edge or star inequality. This is what
+[#7114](https://github.com/CrocSwap/integer-mult-bounds/pull/7114) does with
+`k <= g_T(c) + g_T(d) + min(Y_c, Y_d)`, reaching `j >= k` and the floor `9`. The
+present theorem is the post-mortem on the weight program; it does not compete
+with that value and does not bound `w(P104; F)`.
+
+## 5. What is not claimed
+
+The rank inequality of #6933 is not re-derived, only cited. The type census is
+recomputed as an input to Lemma 4 but is #6933's result. No statement is made
+about P112 or P120, where the certificate's cost budget already fails at the
+edge count. No field, block pattern, decoder, native word, supplier, kappa or
+selected-frontier change is claimed. No human or proof-kernel review.

research/p104-star-weight-optimum/README.md

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+# The weighted star program of #6933 is exactly optimal at 5/6 on P104
+
+**Claim type:** scoped written proof with an exact rational certificate.
+**No kappa, no word, no supplier.** No selected-frontier change.
+
+This package answers the optimization question that
+[#6933](https://github.com/CrocSwap/integer-mult-bounds/pull/6933) leaves open.
+Its Theorem 3 proves, for weights `a` on an orthogonal 8-set `K` of P104
+satisfying
+
+```text
+(W1)  a_c + a_d <= 1          for every edge cd of the edge graph Gamma(K),
+(W2)  a_c (d_c - 2) <= 2      for every c in K,
+```
+
+the rank bound `j/k >= (E - mu)/(mu + rho)` with `mu = sum_c (1 + a_c)` and
+`rho = sum_edges (1 - a_c - a_d)`. #6933 evaluates this at weights taken from a
+grid and states that it makes no claim that they are optimal. It is not an
+audit of that theorem: the theorem, its constraints and its type census are
+cited premises. The new statement is the exact optimum of the weight program and
+its equality case.
+
+## Result
+
+Write `f(a) = (E - mu)/(mu + rho)`. For P104 and every orthogonal 8-set `K`:
+
+```text
+for every admissible a:                 f(a) <= 5/6,
+and 5/6 is attained, exactly when       K has type (E; d) = (22; 5,5,5,5,6,6,6,6)
+                                        and a_c = 1/2 for every c in K.
+```
+
+The proof is a multiplier identity, not a search. With `w_c = 5 d_c - 11`, the
+uniform multiplier `14/5` on every edge and `mu_c = ((11/5) d_c - 11)/(d_c - 2)`
+on each vertex, the coefficient identity
+
+```text
+sum_{e ni c} 14/5 + mu_c (d_c - 2) = 5 d_c - 11
+```
+
+holds exactly at every vertex, so
+
+```text
+sum_c (5 d_c - 11) a_c <= (14/5) E + 2 sum_c mu_c <= 88 - E,
+```
+
+which is precisely `f(a) <= 5/6`. Equality forces every edge of `Gamma(K)` tight,
+every degree-6 cap tight, `E = 22`, and no degree-7 vertex. Then `a_c = 1/2` at
+degree 6, and every degree-5 vertex has a degree-6 neighbour (it has five
+neighbours but only three other degree-5 vertices), so `a_c = 1/2` there too.
+
+Two consequences matter for the next contributor.
+
+1. **No reweighting can raise `53/6`.** The star inequality of #6933 cannot be
+   improved by choosing better weights, on any orthogonal 8-set of P104: `5/6`
+   is its exact ceiling, and the maximizer is rigid.
+2. **The floor 9 needs a different inequality, which is what #7114 found.** That
+   PR replaces the edge inequality by `k <= g_T(c) + g_T(d) + min(Y_c, Y_d)` and
+   reaches `j >= k` with no weights. The present result explains why the weight
+   route was closed: it is a post-mortem on the weight program, not a competing
+   value. Where #7114 is available it supersedes the value 9; where the weight
+   program is reused, this fixes its ceiling.
+
+## Evidence
+
+`verify.py` recomputes everything from the P104 port vectors, with exact rational
+arithmetic and no floating point decisions:
+
+- the 104 ports, the 1728 orthogonal 8-sets, the orthonormal basis `K + v` with
+  `v` the all-ones vector plus `sum(K)`, the parity identity `|A(t)| + eps = 1`,
+  and simplicity of `Gamma(K)` (`960 / 384 / 384` sets of the three types);
+- the three recorded values `5/6`, `101/126`, `3/4` at #6933's own weights;
+- the multiplier coefficient identity, and the cost inequality
+  `(19/5) E + 2 sum_c mu_c <= 88` with maximum exactly `88` on the principal
+  type;
+- `f <= 5/6` on 20,401 feasible weight vectors (corners of the cap box and
+  repaired random draws) over all 1728 sets, with equality only at `a = 1/2` on
+  the principal type;
+- the rigidity lemma on all 960 principal-type sets;
+- controls: a weight vector violating (W2) gives `99/115 > 5/6`, a mutated
+  multiplier `13/5` breaks the coefficient identity, and a mutated census count
+  is detected.
+
+## Reproduce
+
+```sh
+python3.12 -I -B research/p104-star-weight-optimum/verify.py
+```
+
+The run takes about 5 s and ends with `PASS_P104_STAR_WEIGHT_OPTIMUM`. Standard
+library only.
+
+## Limitations
+
+- P104 only. The same certificate fails on P112 and P120 because their edge
+  counts already exceed the budget `88 - E`; their star values `1` are not
+  re-derived here and no optimality claim is made for them.
+- The result is about the weight program of Theorem 3. It does not bound
+  `w(P104; F)` and it does not re-derive the rank inequality itself.
+- The census, the constraints and the ratio formula are cited from #6933; only
+  the optimum and its equality case are new.
+- No human or proof-kernel review. No construction, no physical decoder, no
+  kappa.
+
+## Open next questions
+
+1. **P112 and P120.** Is the star value `1` the exact optimum of the same weight
+   program on those systems, and is the maximizing set rigid? The certificate
+   here is specific to `E = 21, 22` with degrees in `{5, 6}`; a different
+   multiplier pattern would be needed.
+2. **Degree-7 vertices.** The cost budget is exhausted exactly on the principal
+   type. A mixed certificate that uses the degree-7 vertices' stronger (W2) caps
+   in a non-uniform way could in principle beat `5/6` on the `E = 22; 5,5,5,5,5,6,6,7`
+   type; the uniform certificate here cannot, and the recorded value there is
+   `101/126`.
+3. **Sharpness of #7114's edge inequality.** Its `min(Y_c, Y_d)` step is the one
+   that now carries the floor. An equality analysis of that inequality would
+   decide whether `j >= k` is itself rigid, in the same way this package settles
+   the weight program.
+
+## Credit
+
+The orthogonal-8-set setting, Theorem 3, constraints (W1) and (W2), the type
+census and the recorded values are DreamingOfClouds's
+[#6933](https://github.com/CrocSwap/integer-mult-bounds/pull/6933), with its
+retained Bukh–Cox, Haemers and Blasiak lineage. #7114 (DreamingOfClouds) is the
+sharper edge inequality that reaches 9, cited here as the successor route. The
+exact optimum, the multiplier certificate, the equality case and the verifier
+were produced by Nick Petalas. Apache-2.0.

research/p104-star-weight-optimum/SOURCES.json

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