Prove six calls for the open three-value shear case (no new κ)

research/three-spectrum-shear/PROOF.md

@@ -0,0 +1,143 @@
+# Three character values already force six calls
+
+Let C have character spectrum {1,i,-1}. For any finite complex role space,
+let N be square-zero of rank three. Require the complete endpoint
+
+    S(z) = I + (z-1)N,  z in {1,i,-1}.
+
+Every input coordinate, including every helper, is arbitrary and dirty.
+Allowed free gates are constant invertible complex role matrices. Each paid
+gate applies C or C^-1 to one role. The constants are shared across characters.
+The minimum number of paid calls is six.
+
+This settles Open next question 1 in Richard Hoekstra's PR1318. Its normalization,
+direction-span split and target-image loop method are inherited and credited.
+The new observation is that the independent-direction exclusion needs only the
+two active values i and -1. No fourth character is required.
+
+## Normalize and split
+
+At z=1 the product of constant gates must be identity. Absorb the constants into
+conjugations to write a k-call word as
+
+    F(z) = R_k(z)...R_1(z),
+    R_j(z) = I + (z^epsilon_j-1)a_j b_j,
+    b_j a_j=1, epsilon_j in {+1,-1}.
+
+These rank-one idempotents need not be self-adjoint. This normalization is the
+one in PR1318 and allows arbitrarily many restored dirty helpers. Rank(F-I)<=k
+and det F(i)=i^(sum epsilon_j)=1. Thus k>=3 and k is even. Any word with fewer
+than six calls must have exactly four, with all signs equal or two of each.
+
+Put Y=im N and V=span(a_1,...,a_4). Then Y is contained in V, dim Y=3,
+and N vanishes on Y. Hence dim V is three or four. If dim V=4, the restriction
+N|V factors through V/Y, so rank(N|V)<=1.
+
+## Independent directions: the new two-value exclusion
+
+Choose a_j=e_j for j=1,...,4, extending to a basis of the entire role space.
+Later factors change only their own rows. Row one of the endpoint forces
+
+    (z^epsilon_1-1)b_1 = (z-1)N_row1.
+
+If epsilon_1=-1, dividing by z-1 gives N_row1=-z^-1 b_1 at both z=i and -1.
+The two scalars differ, so b_1=0, contradicting b_1 a_1=1. Thus epsilon_1=+1.
+Apply the same argument to F^-1=S^-1=I-(z-1)N, with the directions in reverse
+order. It forces epsilon_4=-1. In particular both all-equal sign patterns are
+excluded. The only remaining patterns are ++-- and +-+-.
+
+For ++-- put g=z-1 and d=b_2 a_1. The first two endpoint rows give
+
+    N_row1=b_1,
+    N_row2=b_2+g d b_1.
+
+The values g(i) and g(-1) differ. Hence d b_1=0, so d=0 and N_row2=b_2.
+In the first two columns these two rows have determinant
+b_1 a_1 * b_2 a_2 - b_1 a_2 * b_2 a_1 = 1.
+Thus rank(N|V)>=2, a contradiction.
+
+For +-+- row two instead gives
+
+    N_row2 = -d b_1 + z^-1(d b_1-b_2).
+
+Again the two active values force b_2=d b_1. Idempotence gives
+1=b_2 a_2=d b_1 a_2, so d is nonzero. Direct multiplication, now as an identity
+for every nonzero z, yields
+
+    R_2 R_1 = I+(z-1)M_1,
+    M_1=(a_1-d a_2)b_1,  M_1^2=0, rank M_1=1.
+
+The inverse word has the same alternating sign pattern. Its first pair
+therefore gives R_3^-1 R_4^-1=I+(z-1)M_tilde. Inverting this square-zero
+shear gives R_4 R_3=I+(z-1)M_2, where M_2=-M_tilde has rank one and square zero.
+Consequently
+
+    F(z)=I+(z-1)(M_1+M_2)+(z-1)^2 M_2 M_1.
+
+At z=i and -1, divide F(z)-S(z) by z-1 and subtract. This gives M_2 M_1=0
+and then N=M_1+M_2. Its rank is at most two, the final contradiction.
+This is the new exclusion that closes the three-value question.
+
+## Directions spanning Y: the inherited loop method at two active values
+
+For completeness, the PR1318 target-image argument is written with its actual
+sample points. Restrict all factors to Y, where their product is identity.
+Every a_j lies in the span of the other three. Otherwise a covector annihilating
+the other three detects a nonzero jth rank-one update after an invertible prefix,
+contradicting identity on Y. The unique dependency therefore uses all four.
+Write a_1=e_1,a_2=e_2,a_3=e_3 and a_4=(c_1,c_2,c_3), with all c_j nonzero.
+Let f_j=b_j|Y. From R_3 R_2 R_1=R_4^-1, row one says
+
+    (z^epsilon_1-1)f_1=(z^-epsilon_4-1)c_1 f_4.
+
+Evaluation at i and -1 forces epsilon_4=-epsilon_1 and f_1=c_1 f_4.
+Row two excludes epsilon_2=epsilon_1, since it would give
+f_2+(z^epsilon_1-1)(f_2 a_1)f_1=(c_2/c_1)f_1 at two distinct scalars,
+forcing both f_2 a_1=0 and f_2 a_1=c_2/c_1.
+Thus epsilon_2=-epsilon_1 and f_2=-(c_2/c_1)f_1.
+The determinant on Y forces epsilon_3=epsilon_1. The first pair is a shear;
+substituting it in R_3 R_2 R_1=R_4^-1 and comparing the two active values gives
+f_3=(c_3/c_1)f_1. Rescale the directions and use them as coordinates to obtain
+
+    a_1=e_1, a_2=e_2, a_3=e_3, a_4=e_1-e_2+e_3,
+    f_1=f_2=f_3=f_4=(1,1,1).
+
+For each outside input column the remaining covector entries are arbitrary
+complex beta_1,...,beta_4. Write B(z) for its outside-to-Y transfer and
+H(z)=B(z)/(z-1), t=z^-1. Direct multiplication gives the following inherited
+canonical coefficient formulas.
+
+For +-+-:
+
+    H_1=t beta_1+(1-t)beta_2-(1-t)beta_3-t beta_4,
+    H_2=-beta_2+(1-t)beta_3+t beta_4,
+    H_3=t(beta_3-beta_4).
+
+Equality H(i)=H(-1) forces beta_1=beta_2 and beta_3=beta_4. Every constant
+transfer column lies in span(e_1-e_2), which has dimension one.
+
+For -+-+:
+
+    H_1=-beta_1+(1-t)beta_2-(1-t)beta_3+beta_4,
+    H_2=t beta_2+(1-t)beta_3-beta_4,
+    H_3=-beta_3+beta_4.
+
+Equality forces beta_2=beta_3. Every constant transfer column lies in
+span(e_1,e_2-e_3), which has dimension two. Neither pattern can have rank three.
+The argument applies to any finite number of outside and helper roles because
+the same image bound holds separately for every outside column.
+
+## Attainment and scope
+
+For one source x and target y use C on x, then y+=x, then C^-1 on x,
+then y-=x. The endpoint is x restored and y+=(C-I)x for arbitrary dirty x,y.
+Apply this independently to three pairs: six calls attain the bound. Every
+rank-three square-zero N has this canonical shear form after a constant basis
+change. The six-call construction therefore covers every N in the theorem.
+
+This is a scoped written proof. The checker corroborates its two-value
+coefficient constraints, canonical image bounds and literal six-call endpoint
+using exact Q(i) arithmetic. It is not exhaustive enumeration of arbitrary
+complex projectors, a proof-assistant formalization, or a multiplication theorem.
+Other phase functions, extra character-dependent gates, clean-input contracts,
+physical tariffs and supplier admission are outside the theorem.

research/three-spectrum-shear/README.md

@@ -0,0 +1,104 @@
+# Three values already force six same-phase calls
+
+This contribution is produced by Nick Petalas. New work is Apache-2.0.
+Independent human and formal proof-kernel review are unclaimed.
+
+For a phase operator C with spectrum {1,i,-1}, three independent transfers
+`y_j += (C-I)x_j` require exactly **six C/C^-1 calls**, even with arbitrary
+constant invertible complex role mixing and any finite number of restored
+dirty helpers. Equivalently, the complete map `I+(z-1)N` for any rank-three
+square-zero N has minimum six in this alphabet.
+
+This is a scoped algebraic proof with exact finite corroboration. It answers
+Open next question 1 in Richard Hoekstra's [PR1318](https://github.com/CrocSwap/integer-mult-bounds/pull/1318),
+which proves the four-value floor and leaves the three-value case open.
+The new first/last-row argument excludes four independent phase directions
+using only i and -1. The inherited target-image loop method needs those same
+two active values. [PROOF.md](PROOF.md) gives the complete argument and credits
+the inherited normalization, span split and canonical loops. No predecessor
+verifier is imported or replayed. No literature-priority claim is made.
+
+Deleting the -i character therefore does not create the proposed four-call
+shortcut. Six direct calls attain the bound with all source, target and helper
+columns handled as arbitrary dirty inputs. This does not change a selected
+supplier, physical tariff or kappa. Extra character-dependent gates, different
+phase functions and clean-input contracts are outside the theorem.
+
+## Reproduce
+
+From the repository root, using CPython 3.12 and its standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/three-spectrum-shear/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/three-spectrum-shear/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/three-spectrum-shear
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/three-spectrum-shear --check-source-history
+```
+
+Other machines can substitute their CPython 3.12 interpreter. Expected direct
+output is `PASS_THREE_SPECTRUM_SHEAR minimum=6 loops=2 collapse_cases=81 upper_samples=6 controls=9 no_new_kappa`.
+
+The checker uses exact Q(i) arithmetic. It constructs both four-call canonical
+loops on all basis inputs, imposes equality at i and -1 by exact elimination,
+and obtains constant-transfer image ranks one and two. It tests the new pair
+collapse on 81 explicit rational cases. Six literal eight-role upper endpoints
+include two restored dirty spectator roles. The nine controls cover six omitted
+calls, the single-active-value boundary, a nonidempotent factor and zero phase.
+These are finite checks of the written proof's core, not enumeration of all
+complex projectors or all circuits. The all-dimension lower bound is the proof.
+
+`RESULT.json` is generated by the verifier and binds its mathematical fields to
+the package and inherited-input hashes. To regenerate without overwriting it:
+
+```sh
+python3.12 -W error -B research/three-spectrum-shear/verify.py --write /tmp/three-spectrum-result.json
+python3.12 -W error -B research/three-spectrum-shear/verify.py --certificate /tmp/three-spectrum-result.json
+```
+
+Source pin availability, metadata validation and passing CI are distinct from
+scientific approval. No native phase/compiler, finite-precision, fixed-tape,
+physical hosting, supplier-admission, outer theorem or Lean replay is run.
+
+## Provenance and overlap
+
+Base canonical main is `624de9dc66858af0945e5f8a0942488d62280d43`, priority revision 7.
+`SOURCES.json` pins the current card, selected record and PR1318 proof/README
+bytes. The PR1318 head is `5cc2496bbdae91e1f5d96d2ae340b42d2d776f24`.
+The `complex-new-helper-inventory` card is explicitly changed to this open
+algebraic interface question; its historical packing experiment is not the
+question being answered. The current selected source-binding control passed.
+
+The [advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19134000)
+follows live open/merged PR and body searches, distinctive all-main README
+searches, active Discussions, recent account submissions and a local question
+reservation. Sparse local notes, arbitrary diffs and unpublished work are
+coverage limits. Search absence is not a novelty certificate. No matching open
+PR was identified by the targeted PR1318/object searches at claim time.
+
+Credit Richard Hoekstra for PR1318's four-value theorem, canonical loop method,
+normalization, upper construction and explicit open question. Retain the
+Swapnil Jain/Douglas Colkitt register/restoration lineage and the distinct
+context contributions named in PR1318's unchanged NOTICE: Jeremy Diviney
+PR836/PR800, David Govea PR720, Sagar Thakrar PR990 and Richard Hoekstra PR1235.
+No predecessor source code is copied into this package.
+
+## Verification implementation limits
+
+Power-of-Ten rules 3 and 9 have narrow Python-runtime deviations in `exact.py`
+and `verify.py`. Fraction objects, matrix entries, JSON parsing and Python
+arithmetic dispatch allocate or dispatch dynamically. Matrix order is capped
+at eight, loop alphabets and products have fixed finite bounds, and no search
+queue or recursion is used. Receipt inputs are trusted committed package files,
+not a service input. Inspectable direct arithmetic and exact endpoint checks
+compensate for these runtime constraints. A fixed-capacity native arithmetic
+engine would be a separate implementation if runtime guarantees were needed.
+All functions fit within 60 physical lines. Warning-strict normal/optimized
+replays are the available diagnostics; no optional linter is required.
+
+## Open next questions
+
+Can a separately paid additional character-dependent gate reduce the complete
+three-source transfer bill? First specify the extra phase gate and its tariff,
+then exhibit a complete dirty-restoring word. No such extra gate, physical
+invoice or supplier is provided here. Extending a search in the same alphabet
+cannot beat the proved call floor.